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abstract: 'We present and analyze [*Far Ultraviolet Spectroscopic Explorer*]{} (FUSE) observations of six solar analogs. These are single, main-sequence G0–5 stars selected as proxies for the Sun at several stages of its main-sequence lifetime from $\sim$130 Myr up to $\sim$9 Gyr. The emission features in the FUSE 920–1180 Å wavelength range allow for a critical probe of the hot plasma over three decades in temperature: $\sim10^4$ K for the H [i]{} Lyman series to $\sim6\cdot10^6$ K for the coronal Fe [xviii]{} $\lambda$975 line. Using the flux ratio C [iii]{} $\lambda$1176/$\lambda$977 as diagnostics, we investigate the dependence of the electron pressure of the transition region as a function of the rotation period, age and magnetic activity. The results from these solar proxies indicate that the electron pressure of the stellar $\sim$10$^5$-K plasma decreases by a factor of $\sim$70 between the young, fast-rotating ($P_{\rm rot}=2.7$ d) magnetically active star and the old, slow-rotating ($P_{\rm rot}\sim35$ d) inactive star. Also, we study the variations in the total surface flux for specific emission features that trace the hot gas in the stellar chromosphere (C [ii]{}), transition region (C [iii]{}, O [vi]{}), and corona (Fe [xviii]{}). The observations indicate that the average surface fluxes of the analyzed emission features strongly decrease with increasing stellar age and longer rotation period. The emission flux evolution with age or rotation period is well fitted by power laws, which become steeper from cooler chromospheric ($\sim10^4$ K) to hotter coronal ($\sim10^7$ K) plasma. The relationship for the integrated (920–1180 Å) FUSE flux indicates that the solar far-ultraviolet (FUV) emissions were about twice the present value 2.5 Gyr ago and about 4 times the present value 3.5 Gyr ago. Note also that the FUSE/FUV flux of the Zero-Age Main Sequence Sun could have been higher by as much as 50 times. Our analysis suggests that the strong FUV emissions of the young Sun may have played a crucial role in the developing planetary system, in particular through the photoionization, photochemical evolution and possible erosion of the planetary atmospheres. Some examples of the effects of the early Sun’s enhanced FUV irradiance on the atmospheres of Earth and Mars are also discussed.'
author:
- 'Edward F. Guinan, Ignasi Ribas, and Graham M. Harper'
title: 'Far-UV Emissions of the Sun in Time: Probing Solar Magnetic Activity and Effects on Evolution of Paleo-Planetary Atmospheres[^1]'
---
Introduction
============
The Sun’s magnetic activity is expected to have greatly decreased with time (Skumanich 1972; Simon, Boesgaard, & Herbig 1985; Dorren & Guinan 1994; Guinan, Ribas, & Harper 2002) as the solar rotation slows down because of angular momentum loss in the stellar wind and resultant reduction in magnetic dynamo-related activity. The study of the young Sun’s far-ultraviolet (FUV) fluxes using solar proxies provides important diagnostics for the state of the younger Solar System and the physics of the much more active early Sun. The comprehensive “Sun in Time” project, begun in 1988 (Dorren & Guinan 1994), focuses on the study of the long-term evolution of the outer atmosphere of an early G star, from the zero-age main sequence to the terminal-age main sequence. A crucial component of the program is a carefully-selected sample of nearby solar analogs (Güdel, Guinan, & Skinner 1997) with spectral types confined between G0-5 V and with well-determined physical properties (including temperatures, luminosities, and metal abundances). We have obtained extensive photometry ($UBVRI$) of these stars from ground-based observatories for over a decade to determine their rotation periods, investigate starspots and possible activity cycles. In addition, we have been able to estimate the stellar ages by making use of their memberships in clusters and moving groups, rotation period–age relationships, and, for the older stars, fits to stellar evolution models.
The sample of solar proxies within the “Sun in Time” program (see Guinan, Ribas & Harper 2002) contains stars that have masses close to 1 M$_{\sun}$ and cover most of the Sun’s main sequence lifetime from $\sim$130 Myr to $\sim$9 Gyr. Basically, the program stars have similar convective-zone depths and chemical abundances to those of the Sun and vary only by their age and rotation periods ($P_{\rm rot}$), and hence dynamo-generated magnetic activity, i.e., chromospheric, transition region (TR), and coronal emissions. In essence, we use these one solar mass stars with $P_{\rm rot}$ between 2.7 and $\sim$35 days as laboratories to study and test solar and stellar dynamo theories by varying essentially only one parameter: rotation period.
The “Sun in Time” is a comprehensive and multi-frequency program that addresses a variety of topics: study of short- and long-term magnetic evolution; physics and energy transfer mechanisms of the chromosphere, transition region (TR) and corona; evolution of the spectral irradiances of the Sun and their effects on paleo-planetary environments and atmospheres. To this end we utilize observational data spanning almost the entire electromagnetic spectrum, which allows us to probe the structure of stellar/solar atmospheres. The FUV has not been readily accessible until the successful launch of the [*Far Ultraviolet Spectroscopic Explorer*]{} (FUSE). FUSE allows a critical probe of the hot plasmas over nearly three decades in temperature: e.g., $\sim10^4$ K for the H [i]{} Lyman series (Ly$\beta$, Ly$\gamma$, ...) through O [vi]{} $\lambda\lambda$1032, 1038 at $\sim3\cdot10^5$ K, and the recently identified Fe [xviii]{} $\lambda$975 coronal emission feature at $\sim6\cdot10^6$ K. Several of the strongest emission features, such as C [iii]{} $\lambda\lambda$977,1176 and O [vi]{} $\lambda\lambda$1032,1038, originate in TR plasmas, and are pivotal for understanding the mechanisms of chromospheric and coronal heating. Thus, spectrophotometry with FUSE fills the energy gap between IUE/HST (1160–3200 Å) and EUVE (80–720 Å) to yield a complete and comprehensive picture of a solar-type star’s atmosphere at different ages and rotation periods.
In this paper we present and analyze FUSE observations of six bright solar analogs that span a wide range of rotation periods (and ages) from 2.7 days ($\sim$130 Myr) to $\sim$35 days ($\sim$9 Gyr). The stars, whose properties are listed in Table \[tabprop\] along with the Sun’s, are: EK Dra, $\pi^1$ UMa, $\kappa^1$ Cet, $\beta$ Com, $\beta$ Hyi, and 16 Cyg A. The present study focuses on two particular aspects of the “Sun in Time” program that can be effectively addressed with FUV observations: [*1.)*]{} The flux ratios between the C [iii]{} $\lambda$1176 and C [iii]{} $\lambda$977 lines (both within the FUSE spectral range) yield empirical measures and estimates of the electron pressure ($P_e$) of the TR; [*2.)*]{} Integrated fluxes for selected emission lines can be used to study the evolution of the solar FUV irradiances at different ages and rotation periods.
Observations {#secobs}
============
FUV observations of the six targets discussed here were obtained with FUSE. The FUSE instrument consists of four spectroscopic channels (LiF1, LiF2, SiC1, and SiC2) that cover a combined wavelength range 905–1187 Å. Detailed information on the FUSE mission and its on-orbit performance are provided by Moos et al. (2000) and Sahnow et al. (2000).
The observations of the targets were secured during FUSE Cycles 1, 2 and 3. Spectra of three stars ($\kappa^1$ Cet, $\beta$ Com, and $\beta$ Hyi) were taken as part of the Guest Investigator program A083 (P.I. Guinan), one star ($\pi^1$ UMa) was observed as part of the program B078 (P.I. Guinan), and to additional stars (EK Dra and 16 Cyg A) were acquired within program C102 (P.I. Guinan). The large aperture (LWRS) was employed in all cases to minimize photon losses due to telescope misalignments, and this caused a fairly severe contamination of day-time spectra from geocoronal emission lines and scattered solar radiation. Information on the dates and exposure times of the observations is provided in Table \[taball\].
The FUSE data were retrieved from the Multimission Archive at Space Telescope ([*MAST*]{}), and then re-calibrated with CALFUSE 2.0.5[^2]. For each star, the time-tagged data for the sub-exposures were combined into a single dataset. The spectral extraction windows were individually aligned to ensure correct extraction of the stellar spectra. We employed the CALFUSE 2.0.5 event burst screening and default background subtraction. The data were then calibrated and screened separately for orbital night and day, allowing us to estimate any contamination of the stellar spectra from “airglow” and scattered solar light. The scattered light was found to be negligible compared to the stellar source, except for the faint FUV target 16 Cyg A, which could suffer from some contamination especially in the O [vi]{} features.
The calibrated spectra are oversampled so they were re-sampled by binning over four wavelength intervals, thereby increasing the signal-to-noise ratio. A comparison of the emission line fluxes from different channels, e.g., C [iii]{} 977 Å SiC2A and SiC1B, provides an independent check on data artifacts, and whether the star was in both apertures for the same length of time.
Analysis {#secan}
========
For illustration we show the extracted spectrum of $\kappa^1$ Cet in Figure \[figsp\]. Note that we only show the night-time data that presents less contamination by terrestrial airglow and geocoronal emission. Thus, all features in Figure \[figsp\] are of stellar origin, although the H [i]{} Lyman series still has some geocoronal contamination. The FUV spectra of the other four targets are qualitatively similar, and only the spectrum of EK Dra and $\pi^1$ UMa are somewhat noisier. For these two stars the increased noise arises from the failure of the SiC2A and LiF2A observations and the consequent use of the less sensitive segment B of detector 1 (SiC1B and LiF1B) for those parts of the spectra. As can be seen in Figure \[figsp\], the continuum level is negligible and the flux is in the form of emission lines. Individual channels are represented in the figure because the FUSE effective area is heavily dominated by only one channel at each wavelength. In this case, SiC2A, LiF1A, and LiF2A spectra are shown in the upper, central, and lower panels, respectively.
Studies of FUSE observations of late-type stars, AB Dor and Capella, have been presented by Ake et al. (2000) and Young et al. (2001), respectively. The authors carried out thorough investigations of the chromospheric and TR emission features in the FUSE wavelength range and provided identification spectra. A comparison with our sample spectrum in Figure \[figsp\] indicates a striking resemblance to these spectra, except for small to moderate changes in the relative strengths of some features. We have thus been able to identify the most significant features in our spectra by using the results of Ake et al. (2000) and Young et al. (2001). Among the strongest FUV emission lines in coronal spectra of late-type stars are the C [iii]{} singlet at 977.020 Å, the C [iii]{} multiplet around 1176 Å, and the O [vi]{} doublet at 1031.925 Å and 1037.614 Å. We focus here primarily on the analysis of stellar features that are not significantly affected by geocoronal contamination and day-time spectra can be used without any loss of accuracy. We thus combined the night-time and day-time data by using a exposure-time weighting scheme.
All of our FUSE spectra show clean profiles for these lines that permit reliable integrated flux measurements, which were carried out by means of NOAO/IRAF tasks. Different methods (plain addition, Gaussian fitting) were employed to measure the integrated line fluxes, study the associated uncertainties, to check the line profiles, and also to assess the possible presence of interstellar absorption (see below for further discussion). The total integrated fluxes for each star and their intrinsic uncertainties (random errors) are provided in Table \[taball\].
The ratio of the fluxes for the C [iii]{} $\lambda$1176 and $\lambda$977 emission features is provided in Table \[tabR\]. This ratio, which yields an excellent electron pressure/density diagnostic, is the topic of discussion for §\[secne\]. The ratio of the two O [vi]{} features ($\lambda$1038/$\lambda$1032) is not provided in Table \[tabR\] but it is in all cases very close to 1:2. This is the value expected for an effectively-thin plasma where the flux ratio is proportional to the ratio of the collision strengths for the two transitions, which for O [vi]{} is very close to 1:2 (Zhang, Sampson, & Fontes 1990).
Plasma model fits to extensive X-ray observations of the three youngest stars in the sample (EK Dra, $\pi^1$ UMa, and $\kappa^1$ Cet) indicate the presence of high-temperature coronal material (Güdel et al. 1997). As discussed by Feldman & Doschek (1991), Young et al. (2001) and Redfield et al. (2003), coronal emission lines of highly-ionized ions arising from forbidden transitions are expected in the FUSE wavelength range. A coronal Fe [xviii]{} forbidden line at 974.85 Å was identified for the first time in FUSE spectra of Capella (Young et al. 2001). FUSE spectra of EK Dra, $\pi^1$ UMa and $\kappa^1$ Cet also clearly show this emission feature (see Figure \[figfe\] for an example). Very recently, Redfield et al. (2002) have reported the detection of a coronal Fe [xix]{} $\lambda$1118 emission line in the FUSE spectra of several nearby cool stars. The FUSE spectra of EK Dra, $\pi^1$ UMa and $\kappa^1$ Cet show weak Fe [xix]{} $\lambda$1118 features barely above the noise level. As pointed out by Redfield et al. (2003), this line can be corrupted by a nearby C [i]{} multiplet. Because of the poor S/N of the flux measurements for Fe [xix]{} $\lambda$1118 and concerns with nearby line contamination have focused only on the clean Fe [xviii]{} $\lambda$975 line as a tracer of the hot coronal plasma. Integrated flux measurements have been carried out using the same method described above and are reported in Table \[taball\]. No trace of this feature is detected for the older stars in the sample down to the noise level of the spectra, and flux upper limits are therefore provided in Table \[taball\].
In addition to integrating the fluxes, we also obtained line centroid positions and FWHM measurements for the selected features in Table \[taball\]. The motivation for such measurements was the search for possible line shifts or broadenings that could arise from plasma outflows or extended material. A critical point when measuring line shifts is an accurate wavelength scale. Aperture centering problems caused by misalignments may result in systematic velocity shifts among the different channels. Redfield et al. (2002) used ISM features to derive the absolute wavelength scale of their spectra. In our case ISM absorption features are not measurable and we followed a different approach. Namely, we aligned the O [vi]{} $\lambda\lambda$1032,1038 doublet, which is available in all four channels, using the LiF1 channel as the reference (this is the channel used for guiding). This procedure should ensure relative velocities accurate to 5–10 km s$^{-1}$ in all channels. The velocity corrections were generally quite relevant with values up to 40 km s$^{-1}$. With a corrected wavelength scale, we measured the line centroids and velocity shifts for several features. The results are given in Table \[tabRV\]. No significant line shifts with respect to the photospheric velocity were observed down to the precision of the measurements. The expected $\sim$10 km s$^{-1}$ redshift of TR features (e.g., Redfield et al. 2002) is only observed in the most active star of the sample, EK Dra. The rest of the stars actually exhibit slightly blueshifted TR features, albeit with very low statistical significance.
FWHM measurements for all the emission lines in Table \[taball\] yielded relatively small widths of $0.20-0.25$ Å, which are equivalent to velocities of $60-70$ km s$^{-1}$. Only EK Dra, the most active of our targets, was found to have broader line profiles. In this case, the measured FWHM was about 0.35 Å or 110 km s$^{-1}$. (The actual rotational velocity of this star is $v_{\rm rot}\approx 17$ km s$^{-1}$.) A closer inspection of the C [iii]{} $\lambda$977 and O [vi]{} $\lambda$1032 line profiles revealed extended line wings in some of the targets. This is a well-known phenomenon (see, e.g., Linksy & Wood 1994) that is usually modeled by assuming double Gaussians with components of different widths. For the stars in our sample, only EK Dra shows unambiguous evidence of a broad component, whereas $\pi^1$ UMa and $\kappa^1$ Cet show somewhat extended line wings but the two-component fits yielded inconclusive results. Two-Gaussian fits to the O [vi]{} $\lambda$1032 line of EK Dra indicate a ratio between the broad component flux and the total flux of 0.38. Thus, extended line profiles appear to correlate with stellar activity (and age), in agreement with the results of Redfield et al. (2002) and Wood, Linsky, & Ayres (1997). The presence of broad and narrow components in high-temperature lines is still not well understood. A plausible model (Wood et al. 1997) suggests that these may arise from high-velocity nonthermal motions during magnetic reconnection events (microflares).
Interstellar absorption corrections {#secism}
-----------------------------------
Before analyzing and modeling the line integrated fluxes, the possible effects of interstellar medium (ISM) absorption must be investigated. The C [iii]{} $\lambda$977 line and the H [i]{} Lyman series lines are the most sensitive to attenuation effects because of possible superimposed ISM absorption. As shown in Table \[tabprop\], the stars in the sample are nearby and lie within 7–34 pc of the Sun and have negligibly small values of $E(B-V)$ from ISM dust. However, even for the small ISM column densities at these close distances ($N_{\rm H}\sim10^{18}$ cm$^{-2}$; Redfield & Linsky 2000), one expects some absorption in the lines mentioned above because they are among the strongest ISM features in the entire spectrum. Unfortunately, both the resolution and S/N of our FUSE spectra are not sufficiently high for reliable direct estimations of the (expected weak) ISM FUV absorptions. We therefore adopted an alternative procedure and carried out indirect determinations of the flux corrections for the individual spectral features.
To estimate the ISM column densities in the line of sight of our targets we made use of the local ISM observations and model by Redfield & Linsky (2002). These authors analyzed high-resolution observations of Fe, Mg and Ca features to determine column densities of these elements toward stars within 100 pc of the Sun. One of our targets, $\kappa^1$ Cet, is included in Redfield & Linsky’s study and thus direct ISM measurements are available. For the other five targets we adopted the local ISM characteristics of neighboring stars with direct measurements: DK UMa for EK Dra and $\pi^1$ UMa, HZ 43 for $\beta$ Com, $\zeta$ Dor for $\beta$ Hyi, and $\delta$ Cyg for 16 Cyg A. The C abundance was computed from the observed column densities of Mg [ii]{} and Ca [ii]{} and local ISM log abundances (relative to H) of $-3.66$, $-5.58$ and $-7.64$ for C, Mg [ii]{}, and Ca [ii]{}, respectively, from Wood et al. (2002b) and Redfield & Linsky (2000). The C [iii]{} column density was subsequently obtained by adopting the local ISM ionization ratio of C [iii]{}/C [ii]{}$\approx$0.02 from Wood et al. (2002b). (Note that C [ii]{} is the dominant ionization species in the ISM.)
The ISM absorption equivalent widths were estimated from the C [iii]{} column densities and the transition oscillator strengths through the curve of growth method (see Spitzer 1978). For the C [iii]{} $\lambda$977 transition, despite its strength, the ISM absorption features for all our targets were found to be relatively weak (unsaturated) because of the small column densities. Then, using the ISM component velocities and the observed C [iii]{} line profile, we calculated the fraction of stellar flux absorbed by the ISM. Our results indicate that the flux corrections for the C [iii]{} $\lambda$977 line are very small (3-7%) for EK Dra, $\pi^1$ UMa, $\kappa^1$ Cet, and $\beta$ Com, and somewhat larger (11-13%) for $\beta$ Hyi and 16 Cyg A.
The H [i]{} Lyman series and total FUSE fluxes {#lyman}
----------------------------------------------
In addition to the features mentioned above, we attempted an estimation of the total integrated fluxes within the FUSE bandpass (920–1180 Å). However, even when restricting to night-time spectra, contamination of the stellar H [i]{} Lyman features by geocoronal emission is very significant. This is because FUSE observations in this program were obtained through the large aperture to ensure, where possible, that the important C [iii]{} 977 Å emission line was detected in the SiC channels. As a consequence, the Lyman geocoronal emission is much broader than for the medium aperture, reaching about $\pm100$ km s$^{-1}$ near Ly$\beta$. This emission overlies and in most cases completely dominates the stellar emission profile. The stellar signal is expected to be a self-reversed emission feature with a dark ISM absorption core.
To estimate the stellar fluxes for Ly$\beta$ we took the difference between the night and day spectra, and assumed that the shape of geocoronal emission remains constant. A fraction of this geocoronal emission was then subtracted from the night spectra so that the residual flux at the location of the ISM feature is consistent with the flux predicted by a simulation of the stellar profile attenuated by the ISM. If emission peaks remained in the subtracted spectrum we assummed that the remaining signal was stellar. Only $\kappa^1$ Cet and $\pi^1$ UMa had any significant signal to satisfy these requirements. To estimate the shape of the underlying stellar emission profile, required to estimate the total stellar flux, we used the functional form of the “universal” profile for strong partially coherent scattered lines derived by Gayley (2002). This allowed us to use the observed flux in the wings, which is relatively uncontaminated by geocoronal emission, to extrapolate towards line center and provide a total flux estimate. The model stellar profile was then attenuated by an ISM model (described above) and convolved with a PSF of $R\sim
15000-18000$ to mimic the FUSE spectral resolution. From the fits we obtained total Ly$\beta$ fluxes of 4.0$\cdot$10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ and 2.4$\cdot$10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ for $\pi^1$ UMa and $\kappa^1$ Cet, respectively. Given the uncertainty of the spectra, the adopted ISM model and the assumed stellar profile shapes, we expect that the accuracy of estimated stellar flux is no better than a factor of 1.5–2. Future observations of H Ly$\alpha$ and MDRS observations of Ly$\beta$ are warranted and required to improve the quantification of the total stellar FUV emission.
A correction to account for the rest of the H [i]{} Lyman series was derived by comparison with SOHO SUMER spectra for the Sun. We roughly estimate that the total flux contributed by the H [i]{} Lyman lines within the FUSE bandpass is about 1.6 times the Ly$\beta$ flux. Then, the total flux in the range 920–1180 Å was calculated by adding the contributions from the individual features discussed above and the integrated H [i]{} Lyman series. We obtained values of 10.5$\cdot$10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ and 8.7$\cdot$10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ for $\pi^1$ UMa and $\kappa^1$ Cet, respectively. Our integrations indicate that the H [i]{} Lyman features in this wavelength range (Ly$\beta$, Ly$\gamma$, etc) are important contributors ($\sim$40–60%) to the total flux of both $\pi^1$ UMa and $\kappa^1$ Cet. This is also presumably the case of the remaining solar-type stars in the sample but the weak stellar signal and strong geocoronal contamination of the FUSE observations prevented us from carrying out reliable measurements.
Emission line fluxes in the FUV {#secir}
===============================
One straightforward application of the measured line fluxes presented in Table \[taball\] is the analysis of their evolution along the main sequence of a one solar mass star. To compare the absolute irradiances, we scaled the observed fluxes to surface fluxes at a radius of 1 R$_{\odot}$. To estimate the actual radius of each target we made use of the stellar luminosity (computed from the Hipparcos distances and apparent magnitudes) and effective temperature (obtained from spectroscopic measurements in the literature). The radii of the targets are included in Table \[tabprop\] and using these we calculated the equivalent flux for a star of 1 R$_{\odot}$ in radius. As can be seen in Table \[tabprop\], the radii of all stars but $\beta$ Hyi and 16 Cyg A are very close to 1 R$_{\odot}$ (within 5-8%) and the correction was typically very small. The situation is somewhat different for $\beta$ Hyi because of its advanced evolutionary stage and its mass being about $\sim$1.1 M$_{\odot}$. In this case the flux correction was quite significant.
The adopted final stellar surface fluxes for the four emission features studied are provided in Table \[tabsurf\]. Also in this table are the total integrated FUV fluxes in the FUSE wavelength range (920–1180 Å) for two of the targets ($\pi^1$ UMa and $\kappa^1$ Cet). All fluxes have been corrected for ISM absorption (see §\[secism\] and \[lyman\]). Before proceeding with a numerical analysis of these data, an illustrative visual impression of the surface flux changes for the targets stars can be seen in Figure \[figOvi\]. We have plotted details of the spectra (flux corrected to the stellar surface) in a narrow 10-Å wavelength interval that contains the O [vi]{} doublet at $\lambda\lambda$1032,1038. A rough comparison of the line variations as a function of stellar age does reveal a clear and definite trend or decreasing strength with increasing age.
For a more quantitative analysis we used the integrated surface fluxes in Table \[tabsurf\], which are plotted in Figure \[figIrr\] as a function of the stellar rotation period. The features represented are C [ii]{} $\lambda$1037, C [iii]{} $\lambda$977, O [vi]{} $\lambda$1032, and Fe [xviii]{} $\lambda$975. The typical peak formation temperatures of the C [ii]{}, C [iii]{}, O [vi]{}, and Fe [xviii]{} ions are approximately $20,000$ K, $60,000$ K, $300,000$ K, and 6 MK, respectively (Arnaud & Rothenflug 1985). Thus, these lines probe a wide interval of plasma temperatures. As can be seen, the newly-detected coronal line of Fe [xviii]{} is very important to our investigation because it extends the coverage to the hot plasma component. Note the much smaller surface fluxes of $\beta$ Hyi with respect to the younger solar analogs for all the features studied. Roughly, $\beta$ Hyi’s flux levels are comparable to, or slightly smaller than, those of today’s Sun. In addition, Figure \[figIrr\] also depicts the total integrated FUV fluxes in the FUSE bandpass for $\pi^1$ UMa and $\kappa^1$ Cet.
The random measurement errors affecting the fluxes in Figure \[figIrr\] are discussed in §\[secan\]. However, additional systematic uncertainties can be caused by the intrinsic stellar activity cycle. From an analogy with the Sun, we expect peak-to-peak cycle amplitudes in the FUV of about 30–50%. Since the target stars are observed at different stages of their activity cycles, these activity variations can be a source of scatter in the irradiance plot. Fortunately, this activity-related scatter is rendered negligible by the large relative differences between the fluxes of the targets (with factors of over 30 in flux).
The fluxes of Figure \[figIrr\] can be fit to good accuracy with power laws of different slopes. These are listed in Table \[tabslopes\], which contains the power-law slopes using both rotation period ($P_{\rm rot}$) and age as independent variable. Note that only two FUSE measurements are available for the total flux within 920–1180 Å. To constrain the fit we used the flux values for the quiet sun obtained through integration of the solar reference spectrum by Heroux & Hinteregger (1978) in the FUSE wavelength window. The resulting surface flux was 2.4$\cdot$10$^4$ erg s$^{-1}$ cm$^{-2}$.
Interestingly, the slopes not only change from line to line, but also they show a clear trend. The power law becomes steeper as we move from cooler to hotter material. The most extreme trend is that of the flux for the coronal ion Fe [xviii]{}. The results indicate that the flux is reduced by a factor of over 1000 when the rotation period increases ten-fold. To serve as a reference, plasma of similar (or maybe slightly lower) temperature is probed when analyzing fluxes in the X-ray domain. Güdel et al. (1997) studied X-ray fluxes (0.1–2.4 keV) of about a dozen solar analogs covering also a wide period range. A power law fit to the data yielded a slope of $-2.64\pm0.12$, which is in very good agreement with our value for the decrease of the Fe [xviii]{} flux.
The results indicate that the emission from hotter plasma decreases more rapidly than that from cooler plasma. Analogous behavior was observed by Güdel et al. (1997) when comparing the emission measures derived from observations in hard and soft X-rays. A possible scenario to explain the steeper flux decrease for hotter plasma could be the weakening of the stellar dynamo as the stars spin down with age and the corresponding decline in the efficiency of heating mechanisms (e.g. flares) and in the strength of magnetic fields that confine the gas.
It should be noted that our program stars, except for age and rotation period, have similar properties to the Sun (mass, radius, effective temperature, metal abundance). Importantly, these stars should all have similar convection zone depths, which together with rotation, is an important parameter in most modern magnetic dynamo theories. Thus, the stars in our sample can serve as laboratories for testing the generation (and dissipation) of magnetic dynamo-related energy and heating in solar-type stars where rotation (angular velocity) is the only important variable. The observed power law dependencies on rotation of the various emission line fluxes should be important as inputs for testing and constraining stellar/solar dynamo theories.
Electron pressures in the transition region {#secne}
===========================================
The C [iii]{} emission line at 977 Å and the C [iii]{} multiplet at 1176 Å occur in the FUSE wavelength region. The ratio of the C [iii]{} $\lambda$1176/$\lambda$977 emission line fluxes has long been recognized to be a sensitive diagnostic for measuring the electron pressure ($P_e$) of hot ($\sim$60,000 K), optically-thin plasmas (e.g., Dupree, Foukal, & Jordan 1976; Keenan & Berrington 1985). FUSE offers an excellent opportunity to exploit the full potential of this important diagnostic (${\cal R}\equiv{\cal
F}^{\mbox{\tiny C{\sc iii}}}_{\mbox{\tiny $\lambda$1176}}/{\cal F}^{\mbox{\tiny
C{\sc iii}}}_{\mbox{\tiny $\lambda$977}}$) because both C [iii]{} features can be measured simultaneously. Furthermore, unlike the UV C [iii]{}\] $\lambda$1909 line, these FUV features are not contaminated by the stellar continuum or nearby line emissions. Details of the C [iii]{} $\lambda$977 and $\lambda$1176 emission features for two of the targets in the sample are shown in Figure \[figcomp\].
To compute ${\cal R}$ ratios we adopted a simple C [iii]{} model which includes the first 10 fine-structure levels. The energy levels are taken from Moore (1970), the highest level in the model being $2p^2\>^1S_0$ at $18219.88\>{\rm cm}^{-1}$. The electric dipole oscillator strengths are taken from the compilation of Allard et al. (1990), except for transitions calculated by Tachiev & Froese Fischer (1999; 2002, priv. comm.) and Fleming, Hibbert, & Stafford (1984). The electric quadrupole and magnetic dipole oscillator strengths are from Nussbaumer & Storey (1978) and Tachiev & Froese Fischer (1999) and the $2s2p\>^3P_2^o - 2p^2\>^1S_0$ and $2s^2\>^1S_0 - 2s2p\>^3P_2^o$ magnetic quadrupole strengths are from Shorer & Lin (1977) and Tachiev & Froese Fischer (1999), respectively. The electron collision strengths are taken from Berrington (1985) and Berrington et al. (1985). The proton collision rates for transitions within $2s2p\>^3P^o$ are from Ryans et al. (1998) and within $2p^2\>^3P$ are from Doyle, Kingston, & Reid (1980). We solved the statistical equilibrium equations, under the assumption that the plasma is in steady state, with no photo-excitation from external sources. The ratios presented here are computed under the assumption that all the C [iii]{} transitions are optically thin. We have also computed the ratios under optically thick conditions using escape probabilities with hydrogen columns of $10^{20}\>{\rm cm}^{-2}$ and assuming solar abundances (Grevesse & Sauval 1998). We find that ${\cal R}$ is quite insensitive to optical depth effects under these conditions, in agreement with the findings of Bhatia & Kastner (1992).
The $\lambda$1176/$\lambda$977 flux ratio of Be-like C [iii]{} is well known to be sensitive to both electron density and temperature and is therefore sensitive to the [*shape of the emission measure distribution*]{} between $\log
T_e=4.5$ and 4.9. The density sensitivity arises from collisional thermalization of the lower levels of the 1176 multiplet ($2s2p\>^3P^o$) which connect to the ground state ($2s^2\>^1S$) by an intercombination electric dipole and a weak magnetic quadrupole transition at 1908.7 Å and 1906.7 Å, respectively. The contribution functions for 1176 Å and 977 Å contain the product of the collisional excitation which increases with $T_e$, the ionization balance of C [iii]{} which peaks near $\log T_e=4.85$ (Arnaud & Rothenflug 1985), and the shape of the emission measure distribution between $\log T_e=4.5$ and 4.9 which is a strong declining function of $T_e$, e.g., Jordan et al. (1987). The peak contribution is expected to lie below $\log
T_e=4.85$. For example for the Sun, Judge et al. (1995) find a peak as low as $\log T_e=4.5$ for the 977-Å transition, while MacPherson & Jordan (1999) find $\log T_e=4.7$. For the solar analog transition regions, we consider that the C [iii]{} emission lines are formed at a constant electron pressure, and we derive the ratio ${\cal R}$ as a function of electron pressure rather than at a (less realistic) single temperature.
The emission measure distribution at C [iii]{} temperatures may change as a function of stellar magnetic activity. However, as pointed out by Jordan (2000), both the analyses of main sequence stars by Jordan et al. (1987) and the similarity of the C [iv]{}/C [ii]{} UV emission line ratios for a wide range of active coronal stars (Oranje 1986), indicate that the shape of the emission measure distribution in the C [iii]{} forming region approximately follows the shape of the inverse of total radiative loss curve between $\log
T_e=4.3$ and 5.3. If the density sensitivity of the ionization balance and abundance gradients can be neglected, then the relative electron pressures derived for our sample should be reliable.
To quantify the effects of assuming differential emission measure distributions (DEM) with different slopes, we adopt a form $$\label{eqDEM}
{\rm DEM}\left(T_e\right)= n_e n_H {dr\over{d\ln T_e}} \propto T_e^{-\alpha}$$ where $n_e$ and $n_H$ are the electron and hydrogen densities, respectively, and $r$ is the radial coordinate. Here we assume that the shape of DEM is similar to the emission measure over the C [iii]{} 977 Å formation region. Note that $\alpha \simeq 2$ is a reasonable approximation to the radiative power-loss function for this temperature range. For each pressure we integrate over temperature from $\log T_e=4.3$ to 5.3, where the fractional abundance of C [iii]{} exceeds 0.01, for each DEM to derive the total fluxes and derive ${\cal R}$. In Figure \[figPres\] we show the ratio ${\cal R}$ for $\alpha$=1.5, 2.0, and 2.5. This figure shows that the resulting ratio is not too sensitive to uncertainties in the shape of the emission measure distribution. Indeed, uncertainties in the collision strengths lead to greater uncertainties in the electron pressure.
The empirical C [iii]{} ratios ${\cal R}$ computed for the six stars in the sample are given in Table \[tabR\]. Also provided are the ${\cal R}$ values corrected for ISM absorption. From these one can estimate the electron pressures through the relationship discussed above. When doing so, we obtain $P_e$ values between $10^{14}$ and $10^{16}$ cm$^{-3}$ K, approximately. The actual electron pressures computed for each of the stars in the sample are given in Table \[tabR\]. Both the ${\cal R}$ values and the derived plasma electron pressures are plotted in in Figure \[figRobs\] as a function of the stellar rotation period. For comparison, the value for the quiet Sun was determined to be ${\cal R}$=0.29 (Dupree, Foukal, & Jordan 1976), and the active Sun (measured in solar active regions such as sunspots) has a C [iii]{} ratio of ${\cal R}$=0.44 (Noyes et al. 1985; Doyle et al. 1985). Note that the ${\cal R}$ values determined for our stars correspond to the integrated stellar disk average, thus including both active and inactive regions, weighted towards region of large areal filling factor and high $n_e$. Interestingly, the observed data for the target stars suggest a strong correlation between the activity level of the star (= age or $P_{\rm rot}$) and ${\cal R}$ or $P_e$.
The elusive density of the TR has been probed for the first time along the evolutionary path of a solar-type star, from ZAMS to TAMS. Our results show that the electron pressure of the TR decreases monotonically as the star spins down (and magnetic dynamo activity decreases). From the fit in Figure \[figRobs\] we infer a power-law relationship with a slope of $\sim-1.7$. This suggests that the electron pressure of the solar TR may have decreased by about a factor of $\sim$40 since the beginning of its main sequence evolution. A likely explanation for the greater plasma electron pressure for the young and active (rapidly rotating) stars is a stronger magnetic confinement of the emitting material.
Most notably, the power-law slope for the electron pressure and the C [iii]{} 977 Å flux are very similar, namely $\simeq -1.65$. Note, however, that the flux power-law relationship has been computed from the integrated flux and it thus represents an average over the stellar surface. If, however, the emitting TR plasma only occupies a fraction ($A$; $A\le1$) of the surface, known as filling factor, then the specific flux emitted by the active regions will be correspondingly larger. The pressure power-law does not depend on the area factor if only one atmospheric component dominates. Thus, assuming that the same fraction of C [iii]{} 977 Å is emitted over the same $d\ln{T_e}$ interval for our sample then we can adopt $${F_{A=1}\over{A}} \propto P_e^2 {dr\over{d\ln T_e}}$$ where $F_{A=1}$ represents the surface integrated flux. Using the power-law relationships derived above, one can write $$\label{eqffact}
{{d\ln T_e}\over{dr}}\propto A\;P_{\rm rot}^{-1.77}$$ as a function of the rotation period. Thus, with empirically measured filling factors for the solar proxies or with a reliable relationship between $A$ and $P_{\rm rot}$ (or Rossby number[^3]) the relationship above can yield an estimate of the dependence of the electron temperature gradient with age. For example, using the empirical fit of Montesinos & Jordan (1993) that relates the magnetic filling factor with the Rossby number, we can write for our solar proxies $$\log A \simeq -0.86\;R_{\circ} \simeq -0.068\;P_{\rm rot}$$ and thus, the temperature gradient can be expressed as $${{d\ln T_e}\over{dr}}\propto P_{\rm rot}^{-1.77} \; 10^{-0.068\;P_{\rm rot}}$$ A representation of the relative variations of the temperature gradient, the electron pressure and the specific C [iii]{} $\lambda$977 flux (i.e., flux divided by filling factor) are provided in Figure \[figPhys\]. Note that these relationships are only represented for rotation periods shorter than that of the Sun, where the filling factor is better defined. Strikingly, the specific C [iii]{} $\lambda$977 flux remains relatively constant with rotation period: as the filling factor increases, the total stellar flux increases correspondingly. However, since the electron density increases with decreasing rotation period, the temperature gradient must also increase to keep the specific flux relatively constant. It is worth noting that, given the large change in the implied temperature gradient across the sample, one or more of the several assumptions on which Eq. \[eqffact\] is based may no longer hold.
FUV irradiances and effects on paleo-planetary atmospheres
==========================================================
The relationships obtained from the solar proxies indicate that the total solar FUSE/FUV flux was about twice the present value 2.5 Gyr ago and about 4 times the present value about 3.5 Gyr ago. Note also that the FUSE/FUV flux of the Zero-Age Main Sequence Sun could have been stronger than today by as much as 50 times.
As discussed by Canuto et al. (1982, 1983), Luhmann & Bauer (1992), Ayres (1997), Guinan, Ribas & Harper (2002) and others, the strong FUV and UV emissions of the young, more active Sun could have played a major role in the early development and evolution of planetary atmospheres – especially those of the terrestrial planets. The expected strong X-ray–UV irradiance of the young Sun can strongly influence the photochemistry and photoionization (and possible erosion) of the early planetary atmospheres and even surfaces (in the case of Mercury, Moon and Mars) and also may play a role in the origin and development of life on Earth as well as possibly on Mars. For example, Canuto et al. (1982, 1983) discuss the photochemistry of O$_2$, O$_3$, CO$_2$, H$_2$O, etc, in the presumed outgassed CO$_2$-rich early atmosphere of the Earth. Ayres (1997) discusses the effect of the young Sun’s increased ionizing X-ray–UV flux, and possible accompanying enhanced solar wind, on the erosion of the early atmosphere of Mars about 3–4 Gyr ago. Also, Lammer et al. (2003) utilize irradiance data from the “Sun in Time” project and solar-wind estimates by Wood et al. (2002a) to evaluate the mechanisms for loss of water from Mars and study the implications for the oxidation of the Martian soil.
Similarly, our data can also provide insights into the so-called Faint Sun Paradox. The paradox arises from the fact that standard stellar evolutionary models show that the Zero-Age Main Sequence Sun had a luminosity of $\sim$70% of the present Sun. This should have led to a much cooler Earth in the past while geological and fossil evidence indicate otherwise. A solution to the Faint Sun Paradox proposed by Sagan & Mullen (1972) was an increase of the greenhouse effect for the early Earth. The gases that have been suggested to account for this enhanced greenhouse effect are carbon dioxide (CO$_2$), ammonia (NH$_3$) or methane (CH$_4$). However, recent results for atmospheric composition of the early Earth (Rye, Kuo, & Holland 1995) are in conflict with the high levels of CO$_2$ and H$_2$O needed to explain the stronger greenhouse effect. Moreover, ammonia is a likely candidate except that it is quickly and irreversibly photodissociated. However, Sagan & Chyba (1997) proposed that hydrocarbon-based aerosols could shield the ammonia from damaging incoming solar radiation. Another alternative explanation for the enhanced greenhouse effect is from atmospheric methane as discussed by Pavlov et al. (2000). Methane is a very strong greenhouse gas (10$^3$–10$^4$ times stronger than CO$_2$) and the necessary amount of CH$_4$ could have been provided by methanogenic bacteria in the proposed early Earth’s reducing atmosphere.
In addition to the stronger greenhouse effect, the expected higher X-ray–UV irradiance of the young Sun should heat up the upper atmosphere of the early Earth. Although the stronger high-energy solar radiation cannot by itself explain the Faint Sun Paradox, the photoionization and photodissociation reactions triggered could play a major role in what greenhouse gases are available. For example, the high levels of FUV-UV radiation of the young Sun could strongly influence the abundances of ammonia and methane in the pre-biotic and Archean planetary atmosphere some 2-4 Gyr ago. Similarly, the photochemistry and abundance of ozone (O$_3$) is of great importance to study life genesis in the Earth. Ozone is an efficient screening mechanism for the enhanced UV radiation of the young Sun, thus protecting the emerging life on the Earth’s surface.
To fully evaluate the influence of FUV radiation in the developing planetary system one must first estimate the flux contribution of the strong H [i]{} Ly$\alpha$ feature, which is not included in the FUSE wavelength range. Preliminary estimates using HST/STIS spectra indicate that Ly$\alpha$ $\lambda$1216 contributes a significant fraction (up to 90%) of the total FUV flux. Further observations and measurements in the near future will allow us to obtain this missing piece of information and complete the FUV irradiance study.
Conclusions
===========
In this study we have utilized FUV spectra acquired with the FUSE satellite to investigate the FUV emission characteristics of solar analogs. Our six targets are well-known G0–5 solar-type stars especially selected to serve as proxies for the Sun at different ages, nearly covering the entire solar main sequence lifetime from 130 Myr to $\sim$9 Gyr. Here we have focused on two aspects of the “Sun in Time” program which are the study of TR plasma electron pressures (using C [iii]{} $\lambda$1176/$\lambda$977 line ratio diagnostics) for stars that differ only in rotation period (and age) and the evolution of irradiances for specific features covering emitting plasma with temperatures from $\sim10^4$ K to $\sim10^7$ K.
To analyze the plasma density of the TR we have used a theoretical relationship between the ratio of C [iii]{} $\lambda$1176 and $\lambda$977 emission line fluxes and the electron pressure ($P_e$) of the material responsible for the emission. Fortunately, both C [iii]{} transitions are within the FUSE wavelength range so that they can be measured simultaneously. Our results indicate a power-law relationship between the electron pressure and the stellar rotation period and overall magnetic-related activity (both related to age). The slope of this relationship has been found to be $\sim$$-$1.7, which suggests that the electron pressure of $\sim$10$^5$-K material in the Sun has decreased significantly since the beginning of its main sequence evolution. The higher values of electron pressure found for the more active stars are best explained by stronger magnetic confinment of the TR plasma.
The measured fluxes for four emission features – C [ii]{} $\lambda$1037, C [iii]{} $\lambda$977, O [vi]{} $\lambda$1032, and Fe [xviii]{} $\lambda$975 – were referred to a radius of 1 $R_{\odot}$ and also corrected for ISM absorption whenever necessary. The typical formation temperatures of the studied features are $20,000$ K, $60,000$ K, $300,000$ K, and 6 MK, respectively. Our analysis indicates that the evolution of these fluxes with stellar rotation period or age can be accurately fit with power-law relationships of different slopes. Interestingly, the slopes not only change from line to line, but also they show a clear trend: The power law becomes steeper as we move from cooler to hotter plasmas, with the most extreme trend being that of the coronal ion Fe [xviii]{}. Also, the evolution total integrated flux in the FUSE wavelength range (920–1180 Å) can be fit with a power-law relationship with a slope of $\sim$$-$1.7, which indicates a factor of $\sim$50 flux decrease along the solar main sequence evolution as the magnetic dynamo activity decreases.
The high levels of FUV line emission fluxes (and related high-energy emission) of the early Sun could have played a crucial role in the photochemistry and photoionization of terrestrial and planetary atmospheres. To address this point we are completing spectral irradiance tables covering 1 Å to 3200 Å for our program stars that can serve as input data for evolution and structure models of the paleo-atmospheres of the Solar System planets. The FUSE observations fill a critical wavelength and energy gap in the “Sun in Time” program and complement observations of the same stars in the X-ray and EUV regions (corona) made with ROSAT, SAX, ASCA, XMM, Chandra, and EUVE, and in the UV (TR and chromosphere) made using IUE and HST. The only missing piece of information is the evolution of the irradiance of the strong chromospheric H [i]{} Ly$\alpha$ $\lambda$1216 FUV feature. We are currently carrying out this part of the study and we expect to complete it shortly. Full spectral irradiance tables for five solar proxies are thus nearing completion and will be made available in a forthcoming publication (Ribas & Guinan 2003, in preparation).
We anticipate that these irradiance results will be important for the study of paleo-atmospheres of the Solar System planets. In particular, preliminary analyses indicate that the high X-ray and EUV emission fluxes of the early Sun could have produced significant heating of the planetary exospheres and upper-atmospheres thus enhancing processes such as thermal escape. The early Sun’s strong FUV and UV fluxes penetrate further into the atmosphere and probably influenced the photochemistry of, e.g., methane and ammonia, which are important greenhouse gases. Thus, this study has strong implications for the evolution of the pre-biotic and Archean atmosphere of the Earth as well as for the early development of life on Earth and possibly on Mars.
We thank Seth Redfield for making available to us the data from the Colorado LISM model, which we have used to correct the measured fluxes for ISM absorption. The referee, Tom Ake (Johns Hopkins Univ.), is thanked for helpful comments and suggestions that led to significant improvements. We acknowledge with gratitude the support for the “Sun in Time” program from NASA-FUSE grants NAG 5-08985, NAG 5-10387, NAG 5-12125 and also from NSF-RUI grant AST-00-71260. G.M.H.’s research was funded by NASA grant NAG5-4808 (LTSA). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.
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[lrlrcccccl]{} EK Dra & 129333 & G0 V &33.9 &5818&1.07&0.95&2.75 &0.13 & Pleiades str.\
$\pi^1$ UMa & 72905 & G1.5 V&14.3 &5840&0.98&0.97&4.68 &0.3 & UMa str.\
$\kappa^1$ Cet& 20630 & G5 V & 9.2 &5700&1.01&0.94&9.2 &0.65 & $P_{\rm rot}$-Age rel.\
$\beta$ Com & 114710 & G0 V & 9.2 &5950&1.11&1.08&12.4 &1.6 & $P_{\rm rot}$-Age rel.\
Sun & – & G2 V & 1 AU&5777&1.00&1.00&25.4 &4.6 & Isotopic dating\
$\beta$ Hyi & 2151 & G2 IV & 7.5 &5800&1.09&1.88&$\sim$28&6.7 & Isochrones\
16 Cyg A & 186408 & G2 V &21.6 &5740&0.99&1.27&$\sim$35&$\sim$9& Isochrones\
[lclccrrrrr]{} EK Dra & C1020501 & 2002 May 14 & 24.2&11.1$\pm$0.7 & 7.6$\pm$0.4 & 7.3$\pm$0.3 & 3.4$\pm$0.2 &0.47$\pm$0.07& 0.81$\pm$0.19\
$\pi^1$ UMa & B0780101 & 2001 Dec 5 & 16.5&14.7$\pm$0.7 & 9.3$\pm$0.5 & 9.6$\pm$0.3 & 4.8$\pm$0.2 &0.74$\pm$0.09& 0.65$\pm$0.26\
$\kappa^1$ Cet& A0830301 & 2000 Sep 10 & 13.0&16.4$\pm$0.7 &10.7$\pm$0.4 &12.4$\pm$0.4 & 6.2$\pm$0.2 &1.07$\pm$0.12& 0.44$\pm$0.14\
$\beta$ Com & A0830401 & 2001 Jan 26 & 15.0&11.0$\pm$0.5 & 5.5$\pm$0.2 & 5.9$\pm$0.2 & 2.8$\pm$0.1 &0.63$\pm$0.08&$<$0.06\
$\beta$ Hyi & A0830101 & 2000 Jul 1 & 18.0&15.1$\pm$0.5 & 5.8$\pm$0.2 & 6.1$\pm$0.2 & 2.8$\pm$0.1 &1.02$\pm$0.09&$<$0.04\
16 Cyg A & C1020101 & 2002 Jul 30 & 35.0&0.66$\pm$0.14&0.20$\pm$0.09&0.43$\pm$0.05&0.20$\pm$0.03&$<$0.05&$<$0.04\
[lrrrrrrrrr]{} EK Dra &$-$30.5&$-$23 & 8 &$-$15 & 16 &$-$25 & 6 &$-$26 & 5\
$\pi^1$ UMa &$-$12.0&$-$23 &$-$11 &$-$24 &$-$12 &$-$19 & $-$7 &$-$51 &$-$39\
$\kappa^1$ Cet& 19.9& 12 & $-$8 & 16 & $-$4 & 16 & $-$4 & 1 &$-$19\
$\beta$ Com & 6.1& 3 & $-$3 & 0 & $-$6 & 4 & $-$2 & – & –\
$\beta$ Hyi & 22.7& 6 &$-$17 & 20 & $-$3 & 16 & $-$7 & – & –\
16 Cyg A &$-$25.6&$-$40&$-$14 &$-$29 & $-$3 &$-$22 & 4 & – & –\
[lllllr]{} EK Dra &1.17$\pm$0.18 &28.4$\pm$1.7 &18.2$\pm$0.8 &2.02$\pm$0.51 &–\
$\pi^1$ UMa &0.32$\pm$0.04 &6.52$\pm$0.33 &4.10$\pm$0.12 &0.28$\pm$0.11 & 45$\pm$20\
$\kappa^1$ Cet&0.20$\pm$0.02 &3.30$\pm$0.15 &2.32$\pm$0.07 &0.082$\pm$0.025&16.4$\pm$3.8\
$\beta$ Com &0.090$\pm$0.011&1.64$\pm$0.07 &0.842$\pm$0.030&$<$0.0085 &–\
$\beta$ Hyi &0.032$\pm$0.003&0.543$\pm$0.019&0.190$\pm$0.007&$<$0.0012 &–\
16 Cyg A & $<$0.03 &0.423$\pm$0.090&0.246$\pm$0.028&$<$0.02 &–\
[lcll]{} FUV (920–1180 Å) &$\sim10^4-10^5$&$-$1.74&$-$1.03\
C [ii]{} &$2\cdot10^4$ &$-$1.49&$-$0.88\
C [iii]{} &$6\cdot10^4$ &$-$1.61&$-$0.95\
O [vi]{} &$3\cdot10^5$ &$-$1.82&$-$1.07\
X-rays (0.1–2.4 keV)&$2\cdot10^6$ &$-$2.64&$\;\;\; -$\
Fe [xviii]{} &$6\cdot10^6$ &$-$3.2 &$-$1.9\
[lccl]{} EK Dra &0.68$\pm$0.05&0.67$\pm$0.05&$\ga$16.0\
$\pi^1$ UMa &0.63$\pm$0.05&0.61$\pm$0.04&$15.4^{+0.6}_{-0.2}$\
$\kappa^1$ Cet&0.65$\pm$0.04&0.60$\pm$0.04&$15.3^{+0.5}_{-0.2}$\
$\beta$ Com &0.50$\pm$0.03&0.48$\pm$0.03&$14.8^{+0.1}_{-0.1}$\
$\beta$ Hyi &0.38$\pm$0.02&0.33$\pm$0.02&$14.3^{+0.05}_{-0.05}$\
16 Cyg A &0.30$\pm$0.15&0.27$\pm$0.13&$14.1^{+0.4}_{-0.4}$\
[^1]: Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985.
[^2]: http://fuse.pha.jhu.edu/analysis/calfuse.html
[^3]: The Rossby number ($R_{\circ}$), which is proportional to the rotation period of the star and inversely proportional to the turnover time at the base of the convective zone, is commonly used as a measure of the activity level of a star. For our targets, which have very similar masses and spectral types, one can safely assume that the Rossby number is simply proportional to $P_{\rm rot}$.
|
---
abstract: 'We show that during cosmological inflation the nonsymmetric metric tensor theory of gravitation develops a spectrum which is potentially observable by cosmic microwave background observations, and may be the most sensitive probe of the scale of cosmic inflation.'
author:
- 'Tomislav Prokopec$^*$ and Wessel Valkenburg'
title: The cosmology of the nonsymmetric theory of gravitation
---
0.1cm
[*1. Introduction.*]{} The metric tensor of the general relativistic theory of gravitation is well known to be symmetric under the exchange of indices, $g_{\mu\nu}(x)=g_{\nu\mu}(x) \equiv g_{(\mu\nu)}(x)$, where $ g_{(\mu\nu)} = (1/2)(g_{\mu\nu} + g_{\nu\mu})$ denotes the symmetric part of the metric tensor. No physical principle precludes us from considering a more general theory of gravitation, in which the metric tensor, $g_{\mu\nu}\rightarrow \bar g_{\mu\nu}$, contains a small antisymmetric admixture [@Moffat:1979], $$\bar g_{\mu\nu}(x) = g_{\mu\nu}(x) + B_{\mu\nu}(x)
\,,\qquad g_{\mu\nu} = \bar g_{(\mu\nu)}
\,,$$ where $B_{\mu\nu} =
\bar g_{[\mu\nu]} = (1/2) (\bar g_{\mu\nu}-\bar g_{\nu\mu})$ denotes the antisymmetric part of the metric tensor. This type of generalisation of the general relativistic theory of gravitation was first considered by Einstein [@Einstein:1925], in an attempt to unify gravitation with electromagnetism, whereby $B_{\mu\nu}$ was interpreted as the electromagnetic field strength tensor. From the tests of the Einstein theory of gravitation [@Will:2001], we know that the antisymmetric part of the metric tensor is small, and hence the action can be well approximated by the Einstein-Hilbert action, plus a linearised antisymmetric contribution [@DamourDeserMcCarthy:1992], $$\begin{aligned}
{\cal S} &=& {\cal S}_{\rm EH} + S_{\rm NGT}
\label{action}
\\
{\cal S}_{\rm EH} &=& - \frac{1}{16\pi G_N}\int d^4x \sqrt{-g}\,
({\cal R} +2\Lambda)
\label{action:EH}
\\
{\cal S}_{NGT} &=& \int d^4x \sqrt{-g}\, \Big(\frac{1}{12}
g^{\mu\alpha}g^{\nu\beta}g^{\rho\gamma}
H_{\mu\nu\rho}H_{\alpha\beta\gamma}
\nonumber\\
&&\hskip 1.8cm -\, \frac 14 \, m_B^2 g^{\mu\alpha}g^{\nu\beta}
B_{\mu\nu}B_{\alpha\beta}\Big)
\,,\quad
\label{action:NGT}\end{aligned}$$ where ${\cal R}$ denotes the Ricci scalar, $\Lambda$ the cosmological term, $G_N$ the Newton constant, and $$H_{\mu\nu\rho} = \partial_\mu B_{\nu\rho}
+ \partial_\nu B_{\rho\mu}
+ \partial_\rho B_{\mu\nu}
\,
\label{field strength}$$ is the field strength associated with the antisymmetric tensor (Kalb-Ramond) field $B_{\mu\nu}$. We work in natural units, in which $c=1$ and $\hbar = 1$.
In Eq. (\[action:NGT\]) we added a mass term for stability reasons [@DamourDeserMcCarthy:1991+1993; @DamourDeserMcCarthy:1992]. Moffat has recently argued [@Moffat:2004] (see also [@Moffat:1994]) that the Einstein theory plus a small massive antisymmetric component provides a viable explanation for the missing matter problem of the Universe, which is standardly cured by adding a dark (nonbaryonic) matter of unknown composition and origin. The inverse mass scale, $m_B^{-1}$ gives the scale at which the effective strength of the gravitational interaction changes. At distances smaller than $m_B^{-1}$ the Newton force is equal to the observed value, at distances larger than $m_B^{-1}$ the Newton force is stronger, explaining thus the rotation curves of galaxies, as well as the gravitational lensing of light by galaxies and clusters of galaxies. Note that when the mass term in (\[action:NGT\]) is nonzero, the NGT theory (\[action:NGT\]) ceases to be equivalent to the Kalb-Ramond axion [@GasperiniVeneziano:2002].
In this Letter we consider the cosmological aspects of the nonsymmetric theory of gravitation. By canonically quantising the physical components of the nonsymmetric tensor field during inflation, we study the growth of quantum fluctuations during inflation [@MukhanovChibisov:1981], and then evolve them during radiation and matter eras. The nonsymmetric tensor field appears naturally in flux compactifications in string theory, where it is disguised as the Kalb-Ramond axion, whose cosmological relevance has been studied in detail [@GasperiniVeneziano:2002; @VernizziMelchiorriDurrer:2000].
0.1in
[*2. Conformal space-times.*]{} The metric tensor of spatially homogeneous conformal space times, which include cosmic inflation and Friedmann-Lemaïtre-Robertson-Walker (FLRW) space-times, has the form $$g_{\mu\nu} = a^2(\eta)\eta_{\mu\nu}
\,,
\label{metric tensor}$$ where $\eta_{\mu\nu} = {\rm diag}(1,-1,-1,-1)$ denotes the Minkowski metric, and $a=a(\eta)$ is the conformal (scale) factor. In conformal space-times the action of the nonsymmetric tensor theory (\[action:NGT\]) simplifies to, $$\begin{aligned}
{\cal S}_{NGT} \rightarrow
{\cal S}_{NGT}^{\rm conf} &=& \int d^4x\, \Big(\frac{1}{12}\frac{1}{a^2}
\eta^{\mu\alpha}\eta^{\nu\beta}\eta^{\rho\gamma}
H_{\mu\nu\rho}H_{\alpha\beta\gamma}
\nonumber\\
&&\hskip 0.9cm -\, \frac 14 \, m_B^2 \eta^{\mu\alpha}\eta^{\nu\beta}
B_{\mu\nu}B_{\alpha\beta}\Big)
\label{action:NGT:conformal}
.\end{aligned}$$ Unlike vector gauge fields [@ProkopecPuchweinWoodard:2003], the antisymmetric tensor field does not couple conformally to gravitation. The corresponding equation of motion is easily obtained by varying the action (\[action:NGT:conformal\]), -0.5cm $$\Big(\partial^2 + a^2 m_B^2 \Big) B_{\mu\nu}
- 2\frac{a^\prime}{a}H_{0\mu\nu} = 0
\,,
\label{eom}$$ where $\partial^2 \equiv \eta^{\mu\nu}\partial_\mu\partial_\nu$. Note that the antisymmetric tensor field $B_{\mu\nu}$ is anti-damped by the Universe’s expansion. Upon taking the divergence $\eta^{\mu\alpha}\partial_\alpha$ of (\[eom\]) divided by $a^2$, we arrive at the following consistency (gauge) condition, -0.8cm $$\eta^{\mu\nu}\partial_\mu B_{\nu\rho} = 0
\,,
\label{Lorentz condition}$$ which is analogous to the Lorentz gauge condition of electromagnetism. Equations (\[eom\]) and (\[Lorentz condition\]) fully specify the dynamics of a massive antisymmetric tensor field in conformal space-times.
We now make use of the following electric/magnetic decomposition of the antisymmetric tensor field, -0.5cm $$B_{0i} \!=\! E_i \!=\! - B_{i0}
\,,\;\, B_{ij} \!=\! -\epsilon_{ijl} B_{l}
\; (i,j,l=1,2,3)
\,,
\label{electric magnetic decomposition}$$ where $\epsilon_{ijl}$ denotes the totally antisymmetric symbol ($\epsilon_{123} = 1, \epsilon_{321} = -1$, [*etc.*]{}). With this decomposition, Eq. (\[eom\]) becomes -0.6cm $$\begin{aligned}
\Big(\partial^2 + a^2 m_B^2 \Big) \vec E &=& 0
\label{eom:E}
\\
\Big(\partial^2 + a^2 m_B^2 \Big) \vec B
- 2\frac{a^\prime}{a}\Big(\partial_\eta \vec B
+ \vec \partial \times \vec E
\Big) &=& 0
\,,\qquad
\label{eom:B}\end{aligned}$$ while the Lorentz condition (\[Lorentz condition\]) implies the following ‘constraint’ equations -0.6cm $$\begin{aligned}
\quad
\vec \partial\cdot \vec E &=& 0
\label{eom:E constraint}
\\
\vec \partial_\eta \vec E - \vec \partial \times \vec B &=& 0
\,.
\label{eom:B constraint}\end{aligned}$$ Note that these four equations are equivalent to the three vacuum Maxwell equations (\[eom:E constraint\]), (\[eom:B constraint\]), and $\partial_\eta \vec B + \vec \partial \times \vec E = 0$. An important difference with respect to the Maxwell theory is that the constraint equation, $\vec \partial \cdot \vec B = 0$, is missing. This then implies that, unlike in the Maxwell theory, the longitudinal magnetic component $\vec B^L$ of the antisymmetric tensor field is dynamical. Equations (\[eom:B\]–\[eom:B constraint\]) imply that, in the massive case, the transverse electric component, $\vec E^T$, and the longitudinal magnetic component, $\vec B^L$, comprise the three physical degrees of freedom, while $\vec E^L = 0 $ and $\vec B^T$ is specified in terms of $\vec E^T$, as given in (\[eom:B constraint\]). In the massless theory however, there is a remaining gauge freedom [@KalbRamond:1974]. Similarly to the gauge fixing, $A_0 = 0$, in electromagetism, one can choose the gauge, $\vec E^T=0$. This then implies that the longitudinal magnetic component, $\vec B^L$, is the only remaining physical degree of freedom of the massless nonsymmetric tensor theory.
0.1in
[*3. De Sitter inflation.*]{} Let us now consider de Sitter inflation, in which the scale factor is a simple function of conformal time $\eta$, -0.5cm $$a=-\frac{1}{H_I\eta}
\,\qquad (\eta\leq -1/H_I)
\,\qquad {\tt (de\; Sitter)}
\,,
\label{scale factor:de Sitter}$$ and $H_I$ is the Hubble parameter during inflation. Let us for the moment assume that the mass scale $m_B^{-1}$ is larger than any relevant physical scale in the theory, such that it can be omitted from Eqs. (\[eom:E\]–\[eom:B\]). In this limit the transverse electric component couples conformally to gravitation, such that its evolution corresponds to that of conformal vacuum, whith the identical correlations as the Minkowski vacuum of gauge fields, and hence are of no relevance for cosmology.
We therefore focus on the longitudinal magnetic component. We now perform a canonical quantisation, $$\begin{aligned}
\hat{\vec B^L} = a\!\!\int\! \frac{d^3 k}{(2\pi)^3}
{\rm e}^{i\vec k\cdot \vec x}
\vec \epsilon^{\,L}\!(\vec k)
\Big[
B^L_{\vec k}(\eta) \hat b_{\vec k}
+ {B^L_{-\vec k}}\!\!^*(\eta) \hat b^\dagger_{\!-\vec k}
\,\Big]
,\;\;
\label{canonical quantisation:BL}\end{aligned}$$ with $\big[\hat b_{\vec k}, \hat b^\dagger_{\vec k^\prime} \big]
= (2\pi)^3\delta(\vec k-\vec k^{\;\prime}\,)$. The (conformally rescaled) mode functions obey Eq. (\[eom:B\]), with $m_B=0$, $$\begin{aligned}
\Big(
\partial_\eta^2+\vec k^2 + \frac{a^{\prime\prime}}{a}
- 2\Big(\frac{a^\prime}{a}\Big)^2
\Big) B^L_{\vec k}(\eta) &=& 0
\,.
\label{eom:BL:2}\end{aligned}$$ In de Sitter inflation (\[scale factor:de Sitter\]), ${\hat{\vec B^L}}(\vec x,\eta)/a$, couples conformally, $$\big(
\partial_\eta^2+\vec k^2
\big) B^L_{\vec k}(\eta) = 0
\,\qquad {\tt (de\; Sitter\; era)}
\,,
\label{eom:BL:3}$$ implying the following amplitude of vacuum fluctuations, $
B^L_{\vec k}(\eta) = (2k)^{-1/2}\, {\rm e}^{-i k\eta}
$, where $k=\|\vec k\|$, $\vec \epsilon^{\,L}(\vec k)$ is the longitudinal polarization vector, $\vec k\times \vec \epsilon^{\,L}(\vec k) =0 $, and the Wronskian reads, ${\bf W}\big[B^L_{\vec k}(\eta),{B^L}^*_{\vec k}(\eta)\big] = i$. Hence, during de Sitter inflation, the physical field, $\vec B^{\,L}$, exhibits conformal vacuum correlations.
0.1in
[*4. Radiation and Matter Era.*]{} Let us now consider radiation and matter era, in which the scale factors read, -0.5cm $$\begin{aligned}
a &=& H_I \eta
\,\quad (\eta_{\rm e}\geq \eta \geq 1/H_I)
\,\quad {\tt (radiation\;era)}
\,
\label{scale factor:radiation}
\quad\,\,
\\
a &=& \frac{H_I}{4\eta_{\rm e}}\Big(\eta + \eta_{\rm e}\Big)^2
\,\quad (\eta \geq \eta_{\rm e})
\,\quad {\tt (matter\;era)}
\, ,
\label{scale factor:matter}\end{aligned}$$ where we assumed a sudden radiation-to-matter transition. $\eta_{\rm e}=(H_IH_{\rm e})^{-1/2}$ denotes the conformal time at the matter and radiation equality, and it is defined by, $a_0/a_{\rm e} \equiv 1+z_{\rm e}
= (\eta_0/\eta_{\rm e})^2$, where $a_0=a(\eta_0)$ denotes the scale factor today, and $z_{\rm e} = 3230\pm 200$ [@Spergeletal:2003WMAP] is the redshift at the radiation-matter equality.
In radiation era Eq. (\[eom:BL:2\]) reduces to -0.5cm $$\Big(
\partial_\eta^2+\vec k^2 - \frac{2}{\eta^2}
\Big) B^L_{\vec k}(\eta) = 0
\,.
\label{eom:BL:radiation era}$$ This is the Bessel equation with the index, $\nu = 3/2$, and whose general solution is a linear combination of Hankel functions, $H_{3/2}^{(1)}(k\eta)$ and $H_{3/2}^{(2)}(k\eta)$ (analogous to the Bunch-Davies vacuum in de Sitter space), $$B^L_{\vec k}(\eta) = \frac{1}{\sqrt{2k}}
\Big[
\alpha_{\vec k}\Big(1\!-\!\frac{i}{k\eta}\Big)
{\rm e}^{-ik\eta}
+ \beta_{\vec k}\Big(1\!+\!\frac{i}{k\eta}\Big)
{\rm e}^{ik\eta}
\Big]
\,.
\label{BL:radiation era}$$ Upon choosing the coefficients $\alpha_{\vec k}$ and $\beta_{\vec k}$ such that $|\alpha_{\vec k}|^2-|\beta_{\vec k}|^2 = 1$, the Wronskian becomes canonical, ${\bf W}\big[B^L_{\vec k}(\eta),{B^L}^*_{\vec k}(\eta)\big] = i$. Note that the mode amplitude (\[BL:radiation era\]) exhibits a $1/k$ infrared enhancement in radiation era on superhubble scales.
This enhancement leads to mode mixing at the inflation-radiation transition. Indeed, upon performing a continuous matching of $B_{\vec k}^{\,L}$ and $\partial_\eta B_{\vec k}^{\,L}$ at the inflation-radiation transition, we arrive at $$\alpha_{\!\vec k} \!=\! - \frac{1}{2}\frac{H_I^2}{k^2}
\Big[
1 - 2i \frac{k}{H_I} - 2\Big(\frac{k}{H_I}\Big)^2\,
\Big]{\rm e}^{2ik/H_I}
\!\!,\;
\beta_{\vec k} = - \frac{1}{2}\frac{H_I^2}{k^2}
,
\label{alpha-beta}$$ such that for superhubble modes at the end of inflation, $$\beta_{\vec k} \simeq \alpha_{\vec k} = -\frac{H_I^2}{2k^2}
\, \qquad (k\ll H_I)
\,.
\label{alpha-beta:superhubble}$$
0.2cm
On the other hand, in matter era (\[scale factor:matter\]), Eq. (\[eom:BL:2\]) becomes -0.6cm $$\Big(
\partial_\eta^2+\vec k^2 - \frac{6}{(\eta+\eta_{\rm e})^2}
\Big) B^L_{\vec k}(\eta) = 0
\,.
\label{eom:BL:matter era}$$ The fundamental solutions are proportional to $H_{5/2}^{(1)}\big(k(\eta+\eta_{\rm e})\big)$ and $H_{5/2}^{(2)}(k(\eta+\eta_{\rm e})\big)$. More precisely, $$\begin{aligned}
B^L_{\vec k}(\eta) \!\!&=&\!\! \frac{1}{\sqrt{2k}}
\Big[
\gamma_{\vec k}
\Big(1-\frac{3i}{k\tilde\eta}-\frac{3}{(k\tilde\eta)^2}
\Big)
{\rm e}^{-ik\tilde\eta}
\label{BL:matter era}
\\
&&\hskip 0.6cm +\, \delta_{\vec k}
\Big(1+\frac{3i}{k\tilde\eta}-\frac{3}{(k\tilde\eta)^2}
\Big)
{\rm e}^{ik\tilde\eta}
\Big]
\,\quad {\tt (matter)}
\nonumber\,,\end{aligned}$$ with $|\gamma_{\vec k}|^2-|\delta_{\vec k}|^2 = 1$ and $\tilde\eta=\eta+\eta_{\rm e}$. Note that the mode amplitude (\[BL:matter era\]) exhibits a $1/k^2$ infrared enhancement on superhubble scales.
We now continuously match $B_{\vec k}^{\,L}$ and $\partial_\eta B_{\vec k}^{\,L}$ at the radiation-matter transition, $\eta=\eta_{\rm e}$ ($\tilde\eta=2\eta_{\rm e}$), to get $$\begin{aligned}
\gamma_{\vec k}{\rm e}^{\!\!-ik\tilde\eta_{\rm e}} \!\!\!&=&\!\!
\alpha_{\vec k}
\Big(\!1
\!+\! \frac12\frac{i}{k\eta_{\rm e}}
\!-\! \frac18\frac{1}{(k\eta_{\rm e})^2}
\Big){\rm e}^{\!-ik\eta_{\rm e}}
+ \beta_{\vec k}
\frac18\frac{{\rm e}^{ik\eta_{\rm e}}}{(k\eta_{\rm e})^2}
\,,\qquad
\nonumber\\
\delta_{\vec k}{\rm e}^{ik\tilde\eta_{\rm e}} \!\!\!&=&\!\!
\alpha_{\vec k}
\frac18\frac{{\rm e}^{\!-ik\eta_{\rm e}}}{(k\eta_{\rm e})^2}
+ \beta_{\vec k}
\Big(\!
1
\!-\! \frac12\frac{i}{k\eta_{\rm e}}
\!-\! \frac18\frac{1}{(k\eta_{\rm e})^2}
\Big){\rm e}^{ik\eta_{\rm e}}
\,.
\nonumber
$$
0.1in
[*5. The spectrum.*]{} Since we are interested in how a nonsymmetric tensor field $B_{\mu\nu}$ may affect large scale structures of the Universe, the spectrum of $B_{\mu\nu}$ can be defined (in analogy to matter density and magnetic field perturbations [@ProkopecPuchwein:2004]) in terms of the corresponding stress-energy tensor, ${T_{\mu\nu}}_{\rm NGT} = [2/\sqrt{-g}]
\delta S_{NGT}/\delta g^{\mu\nu}$. When expressed in terms of the electric and magnetic components, and setting $m_B\rightarrow 0$, one finds, $${T_{0}^{\;0}}_{\rm NGT} = \frac{1}{2a^6}
\Big[
(\partial_\eta \vec B + \vec \partial\times\vec E)^2
+ (\vec \partial\cdot \vec B)^2
\Big] \equiv \rho_{\rm NGT}
\,,
\label{T00}$$ where $\rho_{\rm NGT}$ denotes the energy density of $\vec B^L$. Transforming into momentum space, we then get for the spectrum ([*cf.*]{} Ref. [@ProkopecPuchwein:2004]), $$\begin{aligned}
{\cal P}_{\rm NGT} &=& \frac{k^3}{2\pi^2}{T_{0}^{\;0}}^{\rm NGT}(\vec k,\eta)
\nonumber
\\
&=& \frac{k^3}{4\pi^2 a^4}
\Big[
\Big|\partial_\eta B_{\vec k}^L+\frac{a^\prime}{a}B_{\vec k}^L\Big|^2
+ k^{\,2}
|B_{\vec k}^L|^2
\Big]
\,.\quad
\label{spectrum}\end{aligned}$$
-0.2in
-0.1in
-0.1in \[figure one\]
To obtain the spectrum in radiation era, we insert the mode functions (\[BL:radiation era\]) into (\[spectrum\]), with the matching coefficients given in (\[alpha-beta:superhubble\]). The result is, $${\cal P}_{\rm NGT}^{\tt rad} = \frac{H_I^4}{8\pi^2 a^4}
\bigg\{
1+\frac12\frac{1}{(k\eta)^2}
-\frac12\frac{\cos(2k\eta)}{(k\eta)^2}
- \frac{\sin(2k\eta)}{k\eta}
\bigg\}
\label{spectrum:radiation era}
\,.$$ In figure \[figure one\] we plot $\log[{\cal P}_{\rm NGT}]$ as a function of $\log(k\eta)$. Note that in the infrared $(k\eta\ll 1)$ the spectrum scales as, ${\cal P}_{\rm NGT}\propto k^2$ (dashed red line), which is enhanced with respect to the inflationary spectrum, ${\cal P}_{\rm NGT}\propto k^4$. On subhubble scales $(k\eta\gg 1)$ the massless field spectrum reduces to a constant (solid red line), ${\cal P}_{\rm NGT} \simeq H_I^4/(8\pi^2a^4)$ ($m=0$). This amplitude is the same as that of gravitational wave fluctuations, and scales as a relativistic matter, plus a decaying oscillating component. The spectrum of the massive field (longitudinal magnetic component) exhibits in addition linear growth on large scales, $k/a \ll m_B$, and the spectrum for various masses is shown in figure \[figure one\] (as a function of increasing $m_BH/k^2$). This implies that at any given time there is an enhancement in power on large scales ($k/a \ll m_B$) by a factor $m_Ba/k$ when compared with the power in gravitational waves, rendering a massive nonsymmetric field potentially a more sensitive probe of inflationary scale than gravitational waves. (The transverse components in (\[eom:E\]) do not exhibit a significant amplification, and hence we do not discuss them here.) Since in a geometric theory, $m_B^2 \sim \Lambda$, an observation of any imprint in cosmic microwave background may be a signal for a genuine cosmological term.
A complete analytical expression for the spectrum (\[spectrum\]) in matter era, with the modes given by (\[BL:matter era\]), is rather complicated. To illustrate its main features, we plot the matter era spectrum for a massless nonsymmetric field in figure \[figure two\]: (a) at the time of electron-proton recombination ($z=z_{\rm rec} \simeq 1089$, solid red), (b) at the time of structure formation ($z \simeq 10$, dashed blue) and (c) today ($z=0$, dot-dashed violet). The main features of the spectrum are as follows. Up to a small oscillating correction on subhubble scales, corresponding to the momenta, $k\eta \gg [z_{\rm e}/z(\eta)]^{1/2}$ ($z_{\rm e} = 3230\pm 200$), the spectrum in matter era stays flat, ${\cal P} \simeq H_I^4/(8\pi^2a^4)$. On subhubble scales the nonsymmetric tensor field scales as nonrelativistic matter, implying an enhancement relative to the radiation era spectrum, which is given by the ratio of the scale factors at the hubble crossing and at time $\eta$, $a(\eta)/a(k^{-1})$. This results in the characteristic feature seen in figure \[figure two\], with ${\cal P}\propto 1/k^2$ ($1\ll k\eta \ll [z_{\rm e}/z(\eta)]^{1/2} $), to which decaying oscillations are superimposed.
-0.2in
-0.1in
-0.15in \[figure two\]
0.1in
[*6. Discussion.*]{} How the antisymmetric tensor field affects microwave background anisotropies, depends on the precise nature of the coupling to the photon field, and can be realised either [*via*]{} a geodesic equation, or a direct coupling to an electromagnetic field [@JanssenProkopec].
In this Letter we consider cosmological implications of the nonsymmetric tensor theory, by canonically quantising the physical component of the nonsymmetric tensor field in de Sitter inflation, and subsequently evolving it in radiation and matter era. We find that in the massless limit the relevant physical excitation (the longitudinal magnetic component), exhibits an approximately scale invariant (energy density) spectrum in radiation and matter era on subhubble scales, and an infrared-safe spectrum on superhorizon scales, ${\cal P}_{\rm NGT}\propto k^2$, in both radiation and matter eras. The spectrum of a massive theory gets amplified on large scales ($k/a \ll m_B$) such that the amplitude exceeds that of gravitational waves, rendering the nonsymmetric theory of gravitation potentially the most sensitive probe of inflationary scale.
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|
---
abstract: 'Dense, star-forming, cores of molecular clouds are observed to be significantly magnetized. A realistic magnetic field of moderate strength has been shown to suppress, through catastrophic magnetic braking, the formation of a rotationally supported disk during the protostellar accretion phase of low-mass star formation in the ideal MHD limit. We address, through 2D (axisymmetric) simulations, the question of whether realistic levels of nonideal effects, computed with a simplified chemical network including dust grains, can weaken the magnetic braking enough to enable a rotationally supported disk to form. We find that ambipolar diffusion, the dominant nonideal MHD effect over most of the density range relevant to disk formation, does not enable disk formation, at least in 2D. The reason is that ambipolar diffusion allows the magnetic flux that would be dragged into the central stellar object in the ideal MHD limit to pile up instead in a small circumstellar region, where the magnetic field strength (and thus the braking efficiency) is greatly enhanced. We also find that, on the scale of tens of AU or more, a realistic level of Ohmic dissipation does not weaken the magnetic braking enough for a rotationally supported disk to form, either by itself or in combination with ambipolar diffusion. The Hall effect, the least explored of these three nonideal MHD effects, can spin up the material close to the central object to a significant, supersonic rotation speed, even when the core is initially non-rotating, although the spun-up material remains too sub-Keplerian to form a rotationally supported disk. The problem of catastrophic magnetic braking that prevents disk formation in dense cores magnetized to realistic levels remains unresolved. Possible resolutions of this problem are discussed.'
author:
- 'Zhi-Yun Li, Ruben Krasnopolsky, Hsien Shang'
title: Nonideal MHD Effects and Magnetic Braking Catastrophe in Protostellar Disk Formation
---
Introduction {#intro}
============
The formation and early evolution of disks is a long-standing fundamental problem in star formation. Early work in the field had concentrated on the simpler problem of disk formation from the collapse of a rotating dense core in the absence of a magnetic field, as reviewed in @Bodenheimer1995 and @Boss1998. Dense star-forming cores are observed to be significantly magnetized, however. There is increasing theoretical evidence that disk formation is greatly modified, perhaps even suppressed, by a dynamically important magnetic field.
The most comprehensive measurements of the magnetic field strength in dense cores of low-mass star formation come from @TrolandCrutcher2008. They carried out an OH Zeeman survey of a sample of nearby dark cloud cores, probing densities of order $10^3$–$10^4{{\rm\,cm}}^{-3}$. The inferred mean value for the dimensionless mass-to-flux ratio, relative to the critical value $(2\pi
G^{1/2})^{-1}$ ([NakanoNakamura1978]{}, [ShuLi1997]{}), is $\lambda_{los}\approx 4.8\pm 0.4$. It was obtained from the measured line-of-sight component of the magnetic field $B_{los}$ without any geometric correction. Correcting for geometric effects statistically would lower the mass-to-flux ratio by a factor of 2–3 [@Shu_1999], bringing the mean value of the intrinsic mass-to-flux ratio to a few, i.e., $\lambda \sim$ 2–3. Such values of $\lambda$ are naturally produced in the scenario of ambipolar diffusion-regulated dense core formation in strongly magnetized (magnetically subcritical) clouds, with or without the assistance of supersonic turbulence (e.g., [LizanoShu1989]{}; [BasuMouschovias1994]{}; [NakamuraLi2005]{}; [KudohBasu2011]{}). It may also be consistent with the scenario of dense core formation from turbulence compression in more weakly magnetized background clouds, since the core is expected to be more strongly magnetized relative to its mass (i.e., lower $\lambda$) than the cloud as a whole ([TilleyPudritz2005]{}; [Dib\_2007]{}). Well ordered magnetic fields are also inferred from polarization maps of dust continuum emission, both on the core scale [@Ward-Thompson_2000] and smaller (e.g., [Girart\_2006]{}). We therefore expect the dense cores to be rather strongly magnetized based on both observational data and core formation theories.
A moderately strong magnetic field can suppress disk formation in the ideal MHD limit. This was first demonstrated in @Allen_2003, who carried out 2D (axisymmetric) simulations of the collapse of rotating cores magnetized to a level of $\lambda \leq
10$. The basic reason for disk suppression is that, in the ideal MHD limit, flux freezing allows the infalling material to drag a finite amount of magnetic flux into the central object, creating a split magnetic monopole whose (poloidal) field strength increases rapidly with decreasing radius (as $r^{-2}$, [Galli\_2006]{}; see their Fig. 1). The increased field strength close to the central object, coupled with a long magnetic lever arm from severe equatorial pinching of (poloidal) field lines, is responsible for the catastrophic magnetic braking that suppresses the formation of a rotationally supported disk. This magnetic braking catastrophe was confirmed numerically by @MellonLi2008 and @HennebelleFromang2008 using, respectively, 2D and 3D simulations (see also [PriceBate2007]{}), at least when the magnetic and rotation axes are aligned (see, however, [Machida\_2010]{} and discussion in §\[discussion\]).
When the magnetic and rotation axes are misaligned, @HennebelleCiardi2009 found in their AMR MHD simulations that magnetic braking efficiency is reduced relative to the aligned case. A potential concern is that the misalignment increases the flow complexity, which may enhance the numerical magnetic diffusion that is considerable on small scales (see a nice discussion in §5.3 of [Hennebelle\_2011]{}). A more obvious possibility for avoiding the magnetic braking catastrophe is to relax the ideal MHD approximation. Since dense cores are known to be lightly ionized [@BerginTafalla2007], non-ideal MHD effects (including ambipolar diffusion, Hall effect and Ohmic dissipation, e.g., [Nakano\_2002]{}) are to be expected. The first non-ideal MHD effect considered in this context was Ohmic dissipation. @Shu_2006 suggested that in order for Ohmic dissipation to weaken the field strength over a large enough region so that a large-scale rotationally supported disk of tens of AUs or more can potentially form, the resistivity must be at least one order of magnitude above the classical (microscopic) value. This suggestion was confirmed by @Krasnopolsky_2010, who found that large, $100{{{{\rm\,AU}}}}$ scale disks can indeed form, as long as the resistivity is enhanced by a large enough factor, to a value of order $10^{19}{{\rm\,cm}}^2{{\rm\,s}}^{-1}$ or more. @Machida_2010 carried out core collapse calculations including a distribution of resistivity with density and temperature (from a fit to the resistivities computed in [Nakano\_2002]{}) and found that, even with just the classical resistivity, a small rotationally supported disk can form at the beginning of the protostellar accretion phase (see also [DappBasu2010]{}) and grow to larger, 100-AU scales at later times. Part of the apparent discrepancy between @Machida_2010 and @Krasnopolsky_2010 may be due to different simulation setup. Another difference may be in the level of numerical magnetic diffusivity (see discussion in §\[discussion\]). Further investigation is needed to clarify the situation in the limiting case that includes only Ohmic dissipation.
Ohmic dissipation is important at high densities (above $\sim
10^{11}{{\rm\,cm}}^{-3}$, [Nakano\_2002]{}; see also [KunzMouschovias2010]{}). Before reaching such densities, dense core material must evolve through lower densities, where the Hall effect and especially ambipolar diffusion dominate over Ohmic dissipation. The effect of ambipolar diffusion on disk formation was investigated semi-analytically by @KrasnopolskyKonigl2002, and numerically by @MellonLi2009([-@MellonLi2009]; see also [DuffinPudritz2009]{} and [HoskingWhitworth2004]{}). Compared to the ideal MHD case, a new ingredient is the ambipolar diffusion-induced accretion shock, driven by the magnetic flux left behind by the material that has gone into the central object ([LiMcKee1996]{}; [CiolekKonigl1998]{}; [Contopoulos\_1998]{}; [TassisMouschovias2007]{}). @KrasnopolskyKonigl2002 demonstrated using a 1D semi-analytic model that the strong magnetic field piled up inside the ambipolar diffusion (AD) shock can in principle brake the post-shock material efficiently. @MellonLi2009 showed through 2D (axisymmetric) simulations that this is indeed the case. They started their calculations from a self-similar rotating, magnetized isothermal toroid [@Allen_2003]. A power-law dependence on the neutral density is assumed for the ion density, so that the subsequent collapse remains self-similar, even in the presence of ambipolar diffusion. The self-similarity provides a powerful check on the validity of the numerically obtained collapse solutions. It imposes, however, strong restrictions on both the initial core properties and charge densities.
One objective of the present paper is to investigate the role of ambipolar diffusion in disk formation without the restrictive simplifications made in @MellonLi2009. We do this by following both the pre-stellar evolution of the rotating, magnetized dense core and the protostellar mass accretion phase after a central stellar object has formed and by computing the charge densities self-consistently using a simplified chemical network that includes dust grains ([Nishi\_1991]{}; [Nakano\_2002]{}); they can affect the magnetic diffusivities greatly (e.g., [WardleNg1999]{}). Another objective is to extend @MellonLi2009’s ([-@MellonLi2009]) calculations by including both Ohmic dissipation and Hall effect in addition to ambipolar diffusion. The Hall effect was explored previously in the context of disk-driven outflows ([WardleKonigl1993]{}; see also [Konigl\_2010]{}) and accretion disk dynamics [@SanoStone2002]; it is only starting to be explored in the context of core collapse and disk formation ([Krasnopolsky\_2011]{}; [Braiding2011]{}).
We find that, on scales greater than $10^{14}{{\rm\,cm}}$ (or $6.7{{{{\rm\,AU}}}}$) that we are able to resolve in our non-ideal MHD calculations, no rotationally supported disks form in dense cores with a moderately strong magnetic field (with $\lambda \sim$ several), largely because of the excessive braking due to the magnetic field trapped interior to the AD shock. On these scales, Ohmic dissipation affects the flow dynamics relatively little. The Hall effect, on the other hand, can torque up the spun-down post AD-shock material to significant, supersonic, rotation speeds. The rotation speeds remain well below Keplerian and rotationally supported disks are not formed. Our non-ideal MHD calculations re-enforce the idea that disk formation is difficult in the presence of a moderately strong magnetic field, at least in 2D (assuming axisymmetry). We discuss possible ways to get around this difficulty in the discussion section (§\[discussion\]).
The rest of the paper is organized as follows. In §\[setup\], we describe the problem setup, including the initial and boundary conditions, the nonideal MHD code used in this work, and the computation of charge densities including dust grains. Disk suppression by a moderately strong magnetic field under a range of realistic conditions is demonstrated in §\[suppression\]. We discuss potential disk formation in the case of weaker magnetic fields in §\[weakfield\] and the spin up of an initially non-rotating collapsing envelope by the Hall effect in §\[PureHall\]. Our main results are summarized §\[summary\].
Problem Setup {#setup}
=============
Initial and Boundary Conditions {#initialBC}
-------------------------------
Low-mass pre-stellar cores in nearby star-forming regions are observed to have relatively simple dynamical structures [@BerginTafalla2007]. We idealize such cores as initially uniform spheres of $R=10^{17}{{\rm\,cm}}$ in mass and $1{{\rm\,M_\odot}}$ in mass (see [HennebelleFromang2008]{} for a similar setup). The initial core mass density is thus $\rho_0=4.77\times
10^{-19}{{{{\rm\,g}}}}{{\rm\,cm}}^{-3}$, corresponding to a volume density for molecular hydrogen $n_{H_2} = 10^5{{\rm\,cm}}^{-3}$ (assuming 10 hydrogen nuclei for each He), and a free fall time $t_{ff}
=3\times 10^{12}{{\rm\,s}}$. A simple isothermal equation of state is adopted below a critical density $\rho=10^{-13}{{{{\rm\,g}}}}{{\rm\,cm}}^{-3}$ (with a sound speed $a=0.2{{\rm\,km\,s^{-1}}}$) and $P\propto \rho^{7/5}$ at densities above (e.g., [MasunagaInutsuka2000]{}). The ratio of thermal to gravitational binding energy is $\alpha = 2.5 R a^2/(GM)=0.75$. On this uniform sphere, we impose a uniform magnetic field $B_0$ at the beginning of the simulation. We choose a fiducial value for $B_0$ of $10^{-5}$ in the Lorentz-Heaviside units that are convenient for the Zeus family of codes. It corresponds to $B_0=\sqrt{4\pi}\times
10^{-5}{{\,\rm G}}= 35.4{{\,\mu\rm G}}$ in Gaussian CGS units. The ratio of magnetic to gravitational binding energy is $\gamma = 0.13$. Another way to characterize the strength of the magnetic field is through the dimensionless mass-to-flux ratio $\lambda$, in units of the critical value $(2\pi G^{1/2})^{-1}$. For the core as a whole, the mass-to-flux ratio is $\lambda = 2.92$. On the central flux tube that passes through the origin, the mass-to-flux ratio is $\lambda_
c = 4.38$, higher than the global value by $50\%$. These fiducial values of mass-to-flux ratios are somewhat larger than the values of $\lambda \sim$ 2–3 obtained in models of ambipolar-diffusion driven core formation out of magnetically subcritical background clouds, with ([NakamuraLi2005]{}; [KudohBasu2011]{}) or without ([LizanoShu1989]{}; [BasuMouschovias1994]{}) turbulent compression. They are consistent with the mean value of $\lambda$ inferred by @TrolandCrutcher2008 after correcting for geometric effects (see §1).
One may argue that the adopted fiducial field strength of $B_0 =
35.4{{\,\mu\rm G}}$ at a number density $n_{H_2}=10^5{{\rm\,cm}}^{-3}$ is too high, because such a field strength is rarely measured in nearby regions of low-mass star formation (such as the Taurus clouds) directly using OH. However, the OH emission is dominated by relatively low density envelopes of dense cores [@TrolandCrutcher2008]. If the core material was condensed out of a more diffuse gas, its field strength would be lower before condensation. For example, at densities of order $n_{H_2}
\sim 10^{3.5}{{\rm\,cm}}^{-3}$ (probed by typical OH observations; [Crutcher\_2010]{}), the pre-condensation field strength of our model core would be only $\sim 3.5{{\,\mu\rm G}}$ (if the condensation occurs more or less isotropically under flux-freezing condition, as it should be since the core is substantially magnetically supercritical). It is lower than the medium value of $\sim 6{{\,\mu\rm G}}$ inferred for the [*atomic*]{} cold neutral medium [@HeilesTroland2005]. If anything, our adopted fiducial field strength may be on the low side. Nevertheless, we will consider some smaller values for $B_0$ as well, in view of the possibility that there may be a broad distribution of field strengths at a given density [@Crutcher_2010] and the fact that the majority of stars are formed in clusters where the magnetic field is less well observed.
One may also be concerned that our adopted initial density distribution is uniform, whereas the observed pre-stellar cores are centrally condensed [@BerginTafalla2007]. However, the density distribution changes with time, evolving through a series of centrally condensed configurations with different degrees of central-to-edge contrast, as seen in Fig. \[condensation\] in the absence of magnetic field and rotation. For example, the central density $n_{H_2}\sim 10^6{{\rm\,cm}}^{-3}$ at time $2.5\times 10^{12}{{\rm\,s}}$ is close to that inferred for the well studied pre-stellar core L1544 [@Ward-Thompson_1999], although the maximum infall speed at this time is somewhat higher than the observationally inferred value, which is roughly $10^{4}{{\rm\,cm\,s^{-1}}}$ [@Tafalla_1998]. The infall speed is reduced, however, by both rotation and especially magnetic fields that are included in the majority of our calculations. Indeed, the infall is approximately half-sonic for our reference model (see Table 1) when the central density $n_{H_2}\sim 10^6{{\rm\,cm}}^{-3}$, in agreement with observations.
An alternative approach would be to start with a centrally condensed static configuration that mimics the observed pre-stellar density profile, with a uniform magnetic field imposed at the beginning (e.g., [Machida\_2007]{}). Implicit in this approach is the assumption that there is strong evolution in the density profile of the core but no corresponding evolution in its magnetic field distribution. For our purpose of studying the efficiency of magnetic braking, it has the drawback that the central flux tubes are loaded with much more mass than those in the outer part. It would lead to a weaker magnetic braking in the early phase of protostellar collapse that, in our view, is hard to justify physically.
The rotating profile of dense cores is not well constrained by observations. For simplicity, we adopt for our initially uniform core a solid body rotation, with a fiducial angular speed $\Omega = 10^{-13}{{\rm\,s}}^{-1}$. It corresponds to a ratio of rotational to gravitational binding energy $\beta = 0.025$, which is typical of the values inferred for NH$_3$ cores based on the velocity gradient observed across the cores [@Goodman_1993]; the inferred $\beta$ has a considerable range, which motivates us to consider other values of $\beta$ as well.
As in @MellonLi2008 [@MellonLi2009], we adopt a spherical polar coordinate system $(r, \theta, \phi)$ for our axisymmetric simulation. The inner boundary is set at $10^{14}{{\rm\,cm}}$ (or $6.7{{{{\rm\,AU}}}}$) and the outer boundary at $10^{17}{{\rm\,cm}}$. Even though our inner radius is relatively large compared with other collapse studies, such as Machida et al. (2010) who set the sink particle size to $1{{{{\rm\,AU}}}}$, our smallest cell size is only $0.2{{{{\rm\,AU}}}}$, which ensures that the flow dynamics near the inner boundary is well resolved. The high resolution is particularly important for minimizing the numerical magnetic diffusion that may affect the trapping of magnetic flux at small radii, which lies at the heart of efficient magnetic braking and disk suppression. The standard outflow boundary conditions are enforced at both the inner and outer boundaries. Matter that crosses the inner boundary is collected at the origin. The central point mass interacts with the matter in the computation domain through gravity. We use a computational grid of $120\times 90$ that is non-uniform in the $r$-direction, with a spacing $\Delta r=0.2{{{{\rm\,AU}}}}$ next to the inner boundary. The spacing increases outward by a constant factor $\sim 1.0647$ from one cell to the next. The grid is uniform in the $\theta$-direction.
Induction Equation and Non-Ideal MHD Code
-----------------------------------------
At the heart of our non-ideal MHD core collapse problem lies the induction equation $${\partial {\bf B}\over \partial t }= \nabla \times ({\bf v}\times {\bf B})
- \nabla \times [\eta_O (\nabla \times {\bf B})]
- \nabla \times \left\{ \eta_H [(\nabla\times {\bf B})\times {{\bf
B}\over B}]\right\}
- \nabla \times \left\{ \eta_A {{\bf B}\over B} \times [(\nabla \times
{\bf B})\times {{\bf B} \over B}] \right\}$$ where the Ohmic, Hall and ambipolar diffusivities are related to the electric conductivity parallel to the field line $\sigma_\parallel$, Pedersen and Hall conductivities $\sigma_P$ and $\sigma_H$ through [@Nakano_2002] $$\eta_O = {c^2\over 4\pi \sigma_\parallel}; \ \ \ \ \eta_H= {c^2 \over
4\pi} {\sigma_H \over \sigma_P^2+\sigma_H^2 };
\ \ \ \ \eta_A= {c^2\over 4\pi}
\left({\sigma_P\over \sigma_P^2 +\sigma_H^2} -{1\over
\sigma_\parallel}\right).$$ These diffusivities will be discussed further in the next subsection. Here, we describe briefly our numerical treatment of the three non-ideal MHD terms.
Our code, dubbed “ZeusTW,” was derived from the ideal MHD code Zeus3D [@Clarke_1994]. It treats the Ohmic term in the induction equation (1) using an algorithm based on @Fleming_2000 ([-@Fleming_2000]; see also [Krasnopolsky\_2010]{}). Ambipolar diffusion was treated using the explicit method described in @MacLow_1995, as in @MellonLi2009. The magnetic field is evolved using a velocity that is the sum of the bulk neutral velocity ${\bf v}$ and a “drift” velocity defined as $$\Delta {\bf v}_{AD} = {\eta_A \over B^2} (\nabla \times {\bf B})
\times {\bf B}.$$ In the widely discussed limit where ambipolar diffusion is the dominant non-ideal effect and ions are well tied to the magnetic field, the velocity $\Delta {\bf v}_{AD}$ can be interpreted as the ion-neutral drift velocity. In the more general case that we are studying, this is not necessarily the case. Nevertheless, we will define an “effective ion velocity” $${\bf v}_{i,{\rm{eff}}} \equiv \Delta {\bf v}_{AD} + {\bf v},$$ to make contact with previous work. It provides a useful measure of the effect of ambipolar diffusion even in the presence of other non-ideal MHD effects. Lastly, our treatment of the Hall term was based on @SanoStone2002 and @Huba2003 ([-@Huba2003], see [Krasnopolsky\_2011]{} for detail). The time steps for evolving the ambipolar diffusion and Hall terms in the induction equation are particularly stringent near the polar axes, where the magnetic field is strong but the density is often low, due to either gravitational collapse along the field lines or outflow. In some cases, a floor is imposed on the time step, by decreasing the ambipolar or Hall diffusivity in a small volume or increasing the density in an evacuated region to limit the Alfv[é]{}n speed. We have verified that the floor has little effect on the flow dynamics.
Charge Densities and Magnetic Diffusivities {#s:charge}
-------------------------------------------
The magnetic diffusivities depend on the densities of charged particles, including molecular and atomic ions, electrons, and charged dust grains. We will follow Nakano and collaborators in computing the charge densities (e.g., [Nakano\_2002]{}; [Nishi\_1991]{}), using a simplified chemical network and simple prescriptions for grain size distribution. The network includes neutral species H$_2$, He, CO, O, O$_2$ and other heavy metals (denoted collectively by “M”) and charged species $e^-$, H$^+$, He$^+$, C$^+$, M$^+$, H$_3^+$, and other molecular ions (denoted collectively by “$m^+$”), as well as neutral and positively and negatively charged dust grains (see [Nishi\_1991]{} for detail). The grain size distribution in dense cores of molecular clouds is relatively unconstrained observationally. For illustration, we will consider the standard MRN distribution $(dn/da)\propto a^{-3.5}$ with the grain size $a$ between $5{{\rm\,nm}}< a < 250{{\rm\,nm}}$ [@Mathis_1977] and a grain mass that is $1\%$ of the total. It is likely, however, for the grains to grow substantially in dense cores of molecular clouds; the MRN distribution, which is more appropriate for diffuse clouds, may be regarded as the starting point for grain growth in dense cores. Direct evidence for grain growth comes from the Spitzer detection of the so-called “coreshine” [@Pagani_2010], which indicates that at least some grains have grown to micron-size. To illustrate the effects of grain growth, we will also consider an opposite limit where the grains have a single, large, size $a=1{{\,\mu\rm m}}$ (denoted LG distribution). The MRN and LG distributions should bracket the real situation where some grain growth is expected.
Fig. \[f:charge\] plots the fractional abundances (relative to the number density of hydrogen nuclei $n_H$) of the charged atomic and molecular species and dust grains as a function of $n_H$ (related to mass density through $\rho = 2.34\times 10^{-24}
n_H$), with the cosmic rate ionization rate $\zeta$ normalized to the standard value $10^{-17}{{\rm\,s}}^{-1}$; the normalized rate is denoted by $\zeta_{17} = \zeta/(10^{-17}{{\rm\,s}}^{-1})$. In the (simpler) large grain (LG) case, the dominant charges are metal ions ($M^+$) and electrons ($e^-$) over the whole density range of interest to us. In the MRN case, the charge densities are lower compared to the LG case, because of a large amount of small grains, which provide a large total surface area for ions and electrons to recombine on. The metal ions and electrons remain the most abundant charges at low densities (below $\sim 10^6{{\rm\,cm}}^{-3}$). At higher densities, negatively and positively charged (small) grains become more dominant. The MRN case was computed with 10 size bins equally spaced logarithmically, following @Nishi_1991. When 20 bins are used, the charge densities change by less than $3\%$ over the density range that spans more than 10 orders of magnitude.
To obtain the magnetic diffusivities in the induction equation, we need not only charge densities, but also magnetic field strength. The field strength will be computed self-consistently in our MHD simulations. The rough magnitude of the diffusivities can be illustrated using e.g. the field strength-density relation $$B= 1.43 \times 10^{-7} n_H^{1/2}$$ assumed in @Nakano_2002, relevant for a magnetically supported sheet in hydrostatic equilibrium along the field lines. The computed diffusivities are shown in Fig. \[Diffusivities\]. As expected, the ambipolar diffusivity dominates at relatively low densities for both MRN and LG cases. For the MRN case, which has a large amount of small grains, the Hall diffusivity becomes comparable to the ambipolar diffusivity over a wide range of density ($\sim 10^8$ – $10^{12}{{\rm\,cm}}^{-3}$). It has a negative value, dominated by the contribution from negatively charged grains, although the contribution from positively charged grains approaches (from below) that from the negative grains at densities greater than $\sim 10^{10}{{\rm\,cm}}^{-3}$. At densities greater than $\sim 10^{12}{{\rm\,cm}}^{-3}$ (which is generally not reached in our computational domain), Ohmic diffusivity dominates. The large Ohmic diffusivity $\eta_O$ is the reason for the ambipolar diffusivity $\eta_A$ to become negative above a density $\sim 10^{12}{{\rm\,cm}}^{-3}$, since it contributes negatively to $\eta_A$ (see eq. \[2\]). In the opposite extreme of large grain case (LG), where all small grains are removed, the Hall diffusivity is positive, dominated by the contribution from metal ions. It exceeds the ambipolar diffusivity above a density $n_H\sim 10^{12}{{\rm\,cm}}^{-3}$. In this case, the level of ionization is high enough that the Ohmic diffusivity remains unimportant throughout the plotted density range (up to $\sim 10^{15}{{\rm\,cm}}^{-3}$). Overall, the elimination of small grains in the LG case makes the magnetic field better coupled to the bulk neutral material compared to the MRN case, as expected.
Disk Suppression by a Moderately Strong Magnetic Field {#suppression}
======================================================
In this section, we will consider a magnetic field of moderate strength $B_0=35.4{{\,\mu\rm G}}$, corresponding to a dimensionless mass-to-flux ratio of $\lambda=2.92$ for the dense ($n_{H_2}=
10^5{{\rm\,cm}}^{-3}$) core as a whole and $4.38$ for the central flux tube. As we argued in §\[intro\], this field strength is likely on the conservative side. Since the ambipolar diffusivity dominates over the Ohmic and Hall diffusivities in most of the density range encountered in our calculations (see Fig. \[Diffusivities\]), we will first concentrate on its effect on core collapse and disk formation in §\[PureAD\] and §\[ADvariation\]. Similar AD-only calculations were performed by @MellonLi2009, except that we now include the prestellar phase of core evolution leading up to the formation of a central object (in addition to the protostellar phase that they studied) and a more detailed calculation of the charge densities, including charged grains. The additional effects of the Ohmic and Hall diffusivities are considered in §\[Ohmic\] and §\[Hall\], respectively.
Reference Model with Ambipolar Diffusion Only {#PureAD}
---------------------------------------------
For the reference model, we adopt the fiducial values for the initial field strength $B_0=35.4{{\,\mu\rm G}}$, cosmic ray ionization rate $\zeta=
10^{-17}{{\rm\,s}}^{-1}$, and initial core rotation rate $\Omega_0=
10^{-13}{{\rm\,s}}^{-1}$, as well as an MRN grain size distribution. Since grain growth in dense cores tends to make the magnetic field better coupled to the neutral matter (see Fig. \[Diffusivities\]) and magnetic braking more efficient, our adoption of the MRN distribution is also on the conservative side. The parameters for this (Model REF) and other models are listed in Table 1. The results are shown in Figs. \[ReferenceModel\], \[pressures\] and \[transition\].
[llllll]{} REF & MRN & 35.4 & 1 & 1 & no\
LG & LG & 35.4 & 1 & 1 & no\
LoCR & MRN & 35.4 & 0.1 & 1 & no\
HiCR & MRN & 35.4 & 3 & 1 & no\
LoROT & MRN & 35.4 & 1 & 0.5 & no\
HiROT & MRN & 35.4 & 1 & 2 & no\
REF$_{AO}$ & MRN & 35.4 & 1 & 1 & no\
REF$_{O}$ & MRN & 35.4 & 1 & 1 & no\
REF$_{AHO}^-$ & MRN & 35.4 & 1 & 1 & no\
REF$_{AHO}^+$ & MRN & 35.4 & 1 & 1 & no\
WREF & MRN & 10.6 & 1 & 1 & no\
WLG & LG & 10.6 & 1 & 1 & no\
WHiCR & MRN & 10.6 & 3 & 1 & no\
WLoROT & MRN & 10.6 & 1 & 0.5 & no\
WLoCR & MRN & 10.6 & 0.5 & 1 & yes?\
WHiROT & MRN & 10.6 & 1 & 2 & yes?\
VWREF & MRN & 3.54 & 1 & 1 & yes?\
NoROT$_{AHO}^-$ & MRN & 35.4 & 1 & 0 & no\
NoROT$_{AHO}^+$ & MRN & 35.4 & 1 & 0 & no\
Fig. \[ReferenceModel\] displays the density distribution and velocity field on the meridian and equatorial planes for the inner part ($10^3$-AU scale) of the computation domain, at a representative time $t=6\times 10^{12}{{\rm\,s}}$ (or twice the initial free-fall time), when $0.57{{{{\rm\,M_\odot}}}}$ (or $57\%$ of the initial core mass) has fallen to the center. From the left panel, it is clear that the density distribution on the meridian plane is highly flattened, especially at high densities. The dense, flattened, equatorial structure is [*not*]{} a rotationally supported disk, however. Direct evidence against such a disk comes from the right panel, which shows a transition from an outer region of rapid rotating-infall to an inner region that is neither collapsing nor rotating rapidly. The transition is shown more quantitatively in the left panel of Fig. \[pressures\], where the infall and rotation speeds on the equator are plotted. The equatorial infall is initially slowed down near a relatively large radius $r=5\times 10^{16}{{\rm\,cm}}$. It corresponds to the edge of the magnetic bubble inflated by magnetic braking (not shown in Fig. \[ReferenceModel\]), where a magnetic barrier forces the collapsing material over a large solid angle into a narrow equatorial channel (see Fig. 2 of [MellonLi2008]{} and associated discussion). Upon passing through the barrier, the material resumes rapid radial infall, spinning up as it collapses, until a second barrier is encountered at $r\sim 8\times 10^{15}{{\rm\,cm}}$. This second barrier is induced by ambipolar diffusion, which enables the magnetic field lines to decouple from matter and pile up outside the central object as the matter accretes onto the protostar (see discussion in §\[intro\]). The piled-up magnetic field drives a C-shock, which slows down the accretion flow to a subsonic speed. The slowdown of infall leads to a high density in the postshock region, which is clearly visible in the density maps (Fig. \[ReferenceModel\]). Upon passing through the shock, the equatorial material re-accelerates towards the central object, reaching a highly supersonic infall speed at small radii. The supersonic infall clearly indicates that a rotationally supported disk (RSD hereafter for short) is not present.
The lack of an RSD is even more obvious from the rotation speed on the equator plotted in Fig. \[pressures\]. As the infalling material enters the AD shock, its rotation speed drops quickly. Over most of the postshock region, the rotation speed is nearly zero, indicating an efficient braking of the material that accretes onto the central object through the equatorial region.
Why is it that the rotation of the infalling material is braked almost completely? The answer lies in the strong magnetic field trapped interior to the AD shock ([LiMcKee1996]{}; [Contopoulos\_1998]{}; [KrasnopolskyKonigl2002]{}). The field trapping is already evident in the left panel of Fig. \[ReferenceModel\], which shows a pileup of poloidal magnetic field lines at small radii. It is shown more quantitatively in the right panel of Fig. \[pressures\], where the distribution of the magnetic pressure ($B^2/[8\pi]$) with radius is plotted along the equator. Note the rapid increase in magnetic pressure (and thus field strength) across the AD shock (near $r_s\sim 8\times 10^{15}{{\rm\,cm}}$). There is a corresponding drop in the ram pressure ($\rho v_r^2$), indicating that the strong post-shock magnetic field is trapped by the ram pressure of the pre-shock collapsing flow [@LiMcKee1996]. The further rise in the ram pressure at smaller radii is due to the gravity of the central object, which re-accelerates the equatorial material (that was temporarily slowed down to a subsonic speed behind the AD shock) to a highly supersonic speed. The “effective ion speed” in the post-shock region is much lower than that of the bulk neutral material (see the right panel of Fig. \[ReferenceModel\]). The relatively large drift velocity $\Delta {\bf v}_{AD}$, driven by a large outward magnetic force, is what enables the magnetic flux to accumulate in the postshock region in the first place: as more and more matter accretes onto the central object, more and more magnetic flux is left behind. Indeed, by the time shown in Figs. \[ReferenceModel\] and \[pressures\], about half of the total magnetic flux of the initial core is confined within the shock radius $r_s\sim 8\times
10^{15}{{\rm\,cm}}$. The postshock region, which contains a small fraction ($\sim 4\%$) of the total mass, is so strongly magnetized that it becomes highly magnetically subcritical (with a local dimensionless mass-to-flux ratio $\lambda$ well below unity) even though the whole core was significantly magnetically supercritical to begin with; the small infall speed (much below free fall value) and nearly vanishing rotation speed indicate that the material in the region is essentially magnetically supported. The strong postshock field is only part of the reason for efficient braking. Another part is that the postshock field naturally bends outwards due to strong equatorial pinching (see the left panel of Fig. \[ReferenceModel\]), which increases its lever arm and thus the braking efficiency.
Even though the equatorial rotation speed at small radii remains close to zero at late times (as shown in Figs. \[ReferenceModel\] and \[pressures\]), it does reach a substantial, supersonic, value for a brief period of time during the transition from the phase of prestellar collapse to the protostellar accretion phase. Fig. \[transition\] displays the equatorial infall and rotation speeds during the transition, for 6 times separated by $\Delta t= 4\times 10^{10}{{\rm\,s}}$. At the earliest of the times shown, the material at small radii remains static, as expected during the pre-stellar phase of core evolution for the region within one thermal Jeans length of the origin. When the central Jeans length shrinks to inside our inner boundary, rapid protostellar accretion ensures, leading to a quick spin-up of the collapsing material. The rotation speed reaches a value as high as $\sim 1 {{\rm\,km\,s^{-1}}}$ (or $\sim
5$ times the sound speed) near the inner boundary before decreasing back again. The rapid, transient spin-up is a feature that is not captured by the self-similar solutions of @KrasnopolskyKonigl2002 and @MellonLi2009. The spindown is clearly associated with the development of an ambipolar diffusion induced accretion shock (see the curves for infall speed), which strengthens as it propagates outward. The material in the post-shock region is so strongly braked that it rotates backwards for a short period of time, before settling down to the nearly non-rotating state shown in Figs. \[ReferenceModel\] and \[pressures\]. The strong magnetic braking prevents any rotationally supported disk larger than $10^{14}{{\rm\,cm}}$ (the size of our inner boundary) from forming at any time during our (long) simulation, which lasted until $t=9\times 10^{12}{{\rm\,s}}$, when $90\%$ of the core mass has been accreted by the central object.
Pure AD Runs: Grain Size, Cosmic Ray Ionization and Rotation Rates {#ADvariation}
------------------------------------------------------------------
In the reference model, the magnetic braking has clearly suppressed the formation of a rotationally supported disk (although a highly flattened, dense, magnetically supported, nearly non-rotating pseudodisk was formed). In this subsection, we explore how robust this result is, by varying the physical quantities that may affect disk formation in a lightly ionized, rotating, magnetized molecular cloud core. These include the grain size distribution and cosmic ray ionization rate, both of which affect the charge densities and thus the degree of coupling between the magnetic field and the bulk neutral matter, as well as the rate of core rotation. We discuss models where these quantities are varied over a reasonable range (see Table 1).
As discussed in §\[s:charge\], the size distribution of the grains in dense cores of molecular clouds is uncertain. We have adopted in our reference model the standard power-law MRN distribution that includes a large amount of small grains. In this subsection, we will consider an opposite limit, Model LG in Table 1, where all grains are assumed to be $1{{\,\mu\rm m}}$ in size and the small grains are completely absent. Besides grain size, the charge densities are also affected by the cosmic ray ionization rate $\zeta$. @Padovani_2009 compiled from the literature the values of $\zeta$ inferred for clouds of a wide range of column densities, from diffuse clouds to massive protostellar envelopes (see their Fig. 15). Most of the inferred values are above our reference value $\zeta=10^{-17}{{\rm\,s}}^{-1}$, although there are a few exceptions. We will consider a model with $\zeta=10^{-18}{{\rm\,s}}^{-1}$ (Model LoCR in Table 1), which is a lower limit to the inferred values. For illustration, we will also consider a case with a higher rate $\zeta=3\times
10^{-17}{{\rm\,s}}^{-1}$ (Model HiCR), close to the value inferred by @Webber1998 for the local interstellar medium. In addition, we consider the effect of varying the core rotation rate $\Omega_0$. Our reference value $\Omega_0=10^{-13}{{\rm\,s}}^{-1}$ corresponds to a ratio of rotational to gravitational binding energy of $\beta=0.025$, which is typical of the cores discussed in @Goodman_1993. A spread exists for the inferred $\beta$ values, which motivates us to consider two additional rotation rates: $\Omega_0=5\times
10^{-14}$ (Model LoROT) and $2\times 10^{-13}{{\rm\,s}}^{-1}$ (Model HiROT), corresponding to $\beta=0.006$ and $0.1$, respectively.
The different variants of the reference model are compared in Fig. \[general\], which plots the infall and rotation speeds on the equator at a common time $t=5.5\times 10^{12}{{\rm\,s}}$ for all models except Model HiROT (which has yet to form a central object at this time, because the collapse is significantly delayed by the combination of a moderately strong magnetic field and fast rotation). The most striking feature of Fig. \[general\] is that the rotation speed is essentially zero at small radii (within $\sim 10^{15}{{\rm\,cm}}$) for all of the models shown. The same is true for Model HiROT (not shown in the figure) at later times (greater than $\sim 6.2\times 10^{12}{{\rm\,s}}^{-1}$), when the fast rotating core has collapsed. Apparently the magnetic braking is strong enough to remove essentially all of the angular momentum of the equatorial material inside the AD shock for the realistic ranges of grain size distribution, cosmic ray ionization rate and core rotation rate explored here.
Suppression of RSD in some cases is easy to understand. The elimination of small grains in Model LG and the higher cosmic ray ionization rate in Model HiCR increase the densities of electrons and ions, which strengthens the coupling between the magnetic field and the bulk neutral matter. Since a better magnetic coupling is expected to make the braking more efficient, it is not surprising to find that the RSD remains suppressed as in the reference model. In these two cases, the AD shock is located at a slightly smaller radius compared with the reference case (see Fig. \[general\]). We interpret this effect as a result of the better coupling, which forces the bulk neutral material to collapse more slowly, which in turn leads to a somewhat lower central mass ($8.62\times 10^{32}{{{{\rm\,g}}}}$ for Model LG and $8.69\times 10^{32}{{{{\rm\,g}}}}$ for Model HiCR) than in the reference model at the same time ($9.29\times 10^{32}{{{{\rm\,g}}}}$). Associated with the lower mass is a lower magnetic flux accumulating outside the central object, which drives a smaller AD shock.
The situation with Model LoCR is the opposite. The lower cosmic ray ionization rate decreases the degree of ionization in the core, which weakens the coupling between the magnetic field and the bulk neutral matter. The weakening of magnetic coupling is expected to reduce the braking efficiency in principle. The braking is not weakened enough, however, to enable an RSD to form, as evidenced by the vanishing rotation speed and fast infall inside the AD shock. In this case, the shock radius is somewhat larger than that of the reference model (see Fig. \[general\]), because of a faster post-shock infall, which leads to a larger central mass ($9.76\times 10^{32}{{{{\rm\,g}}}}$ for Model LoCR) and thus a larger left-behind magnetic flux for shock driving.
Neither Model LoROT nor HiROT produced a rotationally supported disk. The former is to be expected, since the angular momentum is lower for a more slowly rotating core, and should be easier to remove by magnetic braking. The slower rotation also presents a lesser obstacle to the collapse, allowing the formation of a larger central mass ($1.24
\times 10^{33}{{{{\rm\,g}}}}$) and a larger AD shock (see Fig. \[general\]) than in the reference model at the same time. In the opposite case of HiROT, the stronger rotational support delayed the formation of a central object until a much later time. Once the central object has formed, magnetic braking is again strong enough to remove essentially all of the angular momentum from the material near the central object, preventing an RSD from forming during the protostellar accretion phase.
We conclude that disk formation is suppressed by a moderately strong magnetic field in the presence of ambipolar diffusion for the range of parameters that we consider realistic.
Ohmic Dissipation {#Ohmic}
-----------------
How does the Ohmic term in the induction equation (1) affect the core collapse and disk formation? To address this question, we repeat the reference model, but with the Ohmic dissipation included (Model REF$_{AO}$ in Table 1) in addition to the ambipolar diffusion (although not yet the Hall term, see below). It turns out that the Ohmic dissipation modifies the dynamics of the core collapse and protostellar accretion relatively little, as can be seen from Fig. \[AO\], which compares the equatorial infall and rotation speeds of Model REF$_{AO}$ and the reference model at four representative times. The speeds for the two cases are barely distinguishable over most of the space. The conclusion is that in the presence of ambipolar diffusion, Ohmic dissipation does not enable the formation of RSD.
The reason that Ohmic dissipation is ineffective in modifying the flow dynamics in the presence of ambipolar diffusion is simple: the Ohmic diffusivity $\eta_O$ is smaller than the ambipolar diffusivity $\eta_A$ by more than an order of magnitude (see Fig. \[AOvalues\]). The relatively small Ohmic diffusivity is a result of the relatively low density in the computational domain, with $\rho
\lesssim 10^{-13}{{{{\rm\,g}}}}{{\rm\,cm}}^{-3}$ (or $n_H \lesssim 4\times
10^{10}{{\rm\,cm}}^{-3}$) typically. The moderate density is a result of strong magnetic braking, which yields a relatively low-density, collapsing pseudodisk (rather than a denser, well supported RSD) in the equatorial region.
Even when ambipolar diffusion is turned off, we do not find any rotationally supported disk within our computation domain (outside a radius of $10^{14}{{\rm\,cm}}$) that is resulted from Ohmic dissipation (Model REF$_O$ in Table 1). Indeed, the overall flow dynamics in the pure Ohmic case is strikingly similar to the reference case that has only ambipolar diffusion (compare Fig. \[PureO\] to Fig. \[ReferenceModel\]). The reason for the similarity is that Ohmic resistivity does not destroy the [*net*]{} magnetic flux that passes from one hemisphere to the other through the equatorial plane (because the poloidal field lines are tied to the low density regions well above and below the equatorial plane where Ohmic dissipation is negligibly small; see discussion in [Shu\_2006]{}). Rather, it enables the poloidal field lines to diffuse radially outward, similar to ambipolar diffusion. As more and more matter accretes across the field lines into the center, the left-behind magnetic flux piles up at small radii, as shown in the left panel of Fig. \[PureO\]. The flux pile-up is qualitatively similar to the pure-AD reference case (see the left panel of Fig. \[ReferenceModel\]). It leads to a strongly magnetized equatorial region where the infall speed decreases to well below the free-fall value and the rotation is almost completely braked (see the right panel of Fig. \[PureO\]). We have repeated the Ohmic dissipation-only calculation with a reduced radius for the inner boundary (from $10^{14}{{\rm\,cm}}$ to $3\times 10^{13}{{\rm\,cm}}$ or $2{{{{\rm\,AU}}}}$), and found the same result: namely, a realistic level of (classical) Ohmic resistivity is not large enough to enable the formation of rotationally supported disks larger than several AUs, at least in 2D (axisymmetry). This is in agreement with @Krasnopolsky_2010, who showed that enhanced resistivity is needed for such disks to form.
Hall Effect {#Hall}
-----------
Under the conditions encountered in our core collapse calculations, the magnitude of the Hall diffusivity in the induction equation (1) is typically larger than the Ohmic diffusivity (see Fig. \[Diffusivities\]). We therefore expect the Hall term to have a larger effect on the collapse dynamics than the Ohmic term discussed in the last subsection. We find that this is indeed the case. Fig. \[HallN\] plots the infall and rotation speeds on the equator for Model REF$_{AHO}^-$, which includes all three nonideal MHD terms in equation (1) and an initial magnetic field that points in a direction opposite to the initial rotation axis (in the negative “z” direction, and hence the superscript “-” in the model name). Plotted are the speeds during the transition from the pre-stellar core evolution to the protostellar accretion phase, as in Fig. \[transition\] for the reference (AD only) model. Comparing Figs. \[HallN\] and \[transition\] reveals that the Hall term has relatively little effect on the equatorial infall speed. In both cases, as more and more mass accumulates at the center, an AD shock develops where the collapsing material slows down temporarily, before reaccelerating towards the origin. The effect on the rotation speed is much more pronounced. The Hall effect enabled the equatorial material in the post-AD shock region to rotate faster compared to the reference case. The difference is especially clear at later times, when the rotation inside the AD shock is almost completely braked (or even reversed) by magnetic braking in the reference case. In the presence of the Hall effect, the post-shock material rotates at a speed as high as 5 times the sound speed. The rotation speed remains much smaller than the Keplerian speed, however, indicating that a rotationally supported disk is not enabled by the Hall effect. The lack of an RSD is also evident from the fact that the post-shock region collapses at a high speed at small radii.
The significant Hall effect on the post-shock rotation speed can be understood as follows. The Hall effect depends on the current density ${\bf j}$ ($\propto \nabla\times {\bf B}$; see eq. \[1\]), which is predominantly in the toroidal direction because of strong magnetic field pinching in the equatorial region. It is particularly strong in the equatorial post-shock region, where the field is strong and is bent outward significantly (see the left panel of Fig.\[ReferenceModel\]). The toroidal current drives a twist of the field lines in the azimuthal direction, which in turn generates a torque that acts to spin up the post-shock material.
An interesting feature, pointed out for example in the disk-wind study of @WardleKonigl1993, is that when the field direction is reversed, the torque induced by the Hall effect changes direction as well [@Krasnopolsky_2011]. This is illustrated in Fig. \[HallP\]. Whereas the reference model has a post-shock region nearly completely braked in the absence of the Hall effect, both Model REF$_{AHO}^+$ and Model REF$_{AHO}^-$ have substantial, supersonic rotation in the post-shock region, although in opposite directions. In particular, in Model REF$_{AHO}^+$ where the initial magnetic field is aligned (rather than anti-aligned) with the initial rotation axis, the magnetic torque induced by the Hall effect has forced the equatorial material at small radii to rotate in a direction opposite to the material at larger distances; the resultant shear may induce instabilities in 3D that should be investigated in the future. The change in the direction of the torque from Model REF$_{AHO}^-$ to Model REF$_{AHO}^+$ is due to the change in the poloidal field direction which, for the same outward bending of field lines, produces a flip in the direction of the toroidal current. Nevertheless, in neither field orientation was the Hall spin-up strong enough to produce a rotationally supported disk, as evidenced by the rapid infall speed at small radii where the rotation speed is relatively high (but still well below the local Keplerian speed). Indeed, there is relatively little change in the infall speed with or without the Hall effect, because the radial current density is relatively small, making the Hall effect less important in that direction.
We will return to the Hall effect in section \[PureHall\], where we consider the simpler case of the collapse and spin-up of an initially non-rotating core due to Hall effect.
Weak Magnetic Fields and Potential Disk Formation {#weakfield}
=================================================
We now consider cases with a weak initial magnetic field of $B_0=10.6{{\,\mu\rm G}}$ (at an initial molecular hydrogen density of $n_{H_2}=10^5{{\rm\,cm}}^{-3}$), which is $30\%$ of the reference value. It corresponds to a dimensionless mass-to-flux ratio of $\lambda = 9.73$ for the core as a whole, and $\lambda_{\rm c}=14.6$ on the central magnetic flux tube. If the dense core were to condense more or less isotropically out of a more diffuse material of $10^{3.5}{{\rm\,cm}}^{-3}$ in density under the flux freezing condition, it would require only an unrealistically weak field of $1.06{{\,\mu\rm G}}$ in the diffuse gas for the core to have $B_0=10.6{{\,\mu\rm G}}$. For this reason, we believe that this value of $B_0$ is probably as low as, if not lower than, the minimum field strength that can be reasonably expected in the type of dense cores under consideration.
Weaker fields are expected to be less efficient in magnetic braking. This is because the braking rate involves the product of the toroidal and poloidal field strengths, and thus generally scales with the field strength as $B^2$. Indeed, for Model WREF which has the same parameters as the reference model (Model REF) except for a weaker field of $B_0=10.6{{\,\mu\rm G}}$, a small ($\sim 20{{{{\rm\,AU}}}}$) rotationally supported disk is formed temporarily early in the protostellar accretion phase, around the time $t\sim 3.68\times 10^{12}{{\rm\,s}}$, when the central mass is only $\sim 0.07{{{{\rm\,M_\odot}}}}$. This disk disappears at later times, however, because of strong magnetic braking, which drives a powerful outflow before the disk disappears (see Shang et al., in preparation).
The suppression of the rotationally supported disk at later times is illustrated in Fig. \[WeakB\_NoDisk\]. It includes a snapshot of the collapsing core (left panel) and a plot of the equatorial infall and rotation speeds (right panel), at $t=5\times 10^{12}{{\rm\,s}}$ when the central mass is $0.29{{{{\rm\,M_\odot}}}}$. At this time, the prominent polar outflow at the earlier times has disappeared. It is replaced by a polar region of strong infall, mostly along the magnetic field lines. The lack of outflow indicates the rotating disk that drives the outflow no longer exists. The disk suppression is shown clearly in the equatorial rotation speed, which is close to zero inside the AD shock (around $10^{15}{{\rm\,cm}}$). The rapid, supersonic infall at small radii further supports the lack of an RSD, as in the more strongly magnetized standard model (see Fig. \[ReferenceModel\] and \[pressures\]).
Disk suppression is not unique to Model WREF. In the left panel of Fig. \[WeakB\_others\] we show three additional examples (Models WLG, WHiCR and WLoROT in Table 1), where the disk formation is suppressed at a relatively late time $t=5\times 10^{12}{{\rm\,s}}$, when the central mass is $0.40$, $0.34$ and $0.65{{{{\rm\,M_\odot}}}}$, respectively. These models have the same weak magnetic field as Model WREF, but have either a larger grain size (Model WLG), a higher cosmic ray ionization rate (Model WHiCR), or a lower initial rotation rate (Model WLoROT). The lack of disk is not too surprising for Models WLG and WHiCR, because the magnetic fields are better coupled to the neutral matter in these models than in Model WREF. The absence of a disk in Model WLoROT is also expected, because its more slowly rotating core is more easily braked.
Some rotationally supported disks do form with other choices of parameters, however, at least at early times. The disk formation is illustrated in the right panel of Fig. \[WeakB\_others\], where the equatorial infall and rotation speeds for Models WLoCR (same as the standard weak field model but with a lower cosmic ray ionization rate), WHiROT (with a higher initial core rotation rate) and VWREF (with an unrealistically low $B_0=3.54{{\,\mu\rm G}}$) are plotted at a early time, when the central mass is only $1.79\times
10^{-2}$, $1.27\times 10^{-2}$ and $1.05\times 10^{-2}{{{{\rm\,M_\odot}}}}$, respectively. There is more mass in the rotationally supported disk than at the center, which is why the rotation curve is non-Keplerian except close to the origin. Each of the disks drives a strong, sometimes chaotic, outflow, which makes it hard to continue the non-ideal MHD simulation reliably to much later times. The disk in the weakest field case (Model VWREF) evolves into a ring, which may fragment in 3D. It is unclear whether the early disk in the other two cases can survive subsequent magnetic braking or not (it did not in the standard weak field model WREF). Once a disk becomes self-gravitating, gravitational torque will likely become important in the disk dynamics. This important effect is not captured in our axisymmetric simulations. In any case, it is clear that disk formation in the moderately weak magnetic field case is more complicated than the moderately strong field case, with the outcome depending on the core rotation rate and the degree of field-matter coupling, at least at early times. Paradoxically, the weaker field cases are more difficult to simulate because of strong, chaotic outflows. More work is needed before firmer conclusions on disk formation can be drawn.
Hall Spin-up of Non-Rotating Envelope {#PureHall}
=====================================
Of the three non-ideal MHD effects, the Hall effect is the least explored in the context of core collapse and disk formation (see, however, [Krasnopolsky\_2011]{} and [Braiding2011]{}). We have seen in §\[Hall\] that it can spin up the nearly completely braked post-AD shock material in an initially rotating core to a significant speed, and the sense of the Hall-induced rotation depends on the orientation of the magnetic field. This Hall spin-up can be illustrated even more clearly in the collapse of an initially non-rotating core, where any rotation that develops subsequently must come solely from the Hall effect.
We will concentrate on Model NoROT$_{AHO}^-$ where all three non-ideal MHD effects are included and the initial magnetic field is anti-parallel to the rotation axis. We have confirmed that the case with opposite field orientation (Model NoROT$_{AHO}^+$) produces identical results, except that the sign of the Hall-induced rotation is flipped, as expected. The left panel of Fig. \[PureHall\_vphi\] gives an overall impression of the Hall-induced rotation on an intermediate scale of $\sim 10^{16}{{\rm\,cm}}$, at a representative time $t=4.4\times 10^{12}{{\rm\,s}}$, when the central mass is $0.26{{{{\rm\,M_\odot}}}}$. Note the alternating pattern of negative and positive rotation speeds, with maximum values reaching $\sim
10^5{{\rm\,cm}}{{\rm\,s}}^{-1}$, which is much higher than the sound speed ($2\times
10^4{{\rm\,cm}}{{\rm\,s}}^{-1}$). On smaller scales, the structure is dominated by a dense flattened pseudodisk that is visible from the iso-density contours in Fig. \[PureHall\_vphi\].
The dense equatorial pseudodisk is collapsing as well as spinning in the positive azimuthal direction, as shown pictorially in the right panel of Fig. \[PureHall\_vphi\] and more quantitatively in the left panel of Fig. \[PureHall2\]. The collapse shows rapid deceleration near $\sim 2\times 10^{15}{{\rm\,cm}}$ and reacceleration interior to it, characteristic of an AD shock. The infall speed is similar to that in the pure-AD case, indicating that the Hall effect modifies relatively little the overall collapse dynamics. The difference in rotation speed is more pronounced, especially in the post-shock region, where the maximum rotation speed exceeds $10^5{{\rm\,cm}}{{\rm\,s}}^{-1}$, comparable to the peak value on the large scale shown in Fig. \[PureHall\_vphi\]. The rotational component of the “effective ion speed” (defined in equation \[4\]) is larger than the neutral rotation speed, indicating that a magnetic force is exerted in the positive azimuthal direction to drive the ion-neutral drift. It is the same force that torques up the pseudodisk. The force is particularly large in the postshock region, because the magnetic field is strong there. Nevertheless, the spin up fails far short of reaching a Keplerian rate, which is the reason why the overall collapse dynamics is little affected.
In an initially non-rotating core, any spin up in one direction must be offset by a spin up in the opposite direction, so that the total angular momentum is conserved. In the region that extends from the origin up to the AD shock, the net angular momentum is positive (see the right panel of Fig. \[PureHall2\]), because it is dominated by the positively spinning pseudodisk. Outside the AD shock, the net angular momentum is negative, dominated by the hour-glass shaped counter-rotating region shown in left panel of Fig. \[PureHall\_vphi\]. One may expect the positive and negative angular momenta to sum up to zero over the entire computational volume. However, this is not the case, because the total angular momentum is dominated by the material at large distances (near the core edge) whose rotation speed is small but non-zero (see the left panel of Fig. \[PureHall2\]). Because the mass at larger distances is larger and has a longer lever arm, it dominates the total angular momentum. The total positive angular momentum inside the computation domain is $6.52\times 10^{52}{{{{\rm\,g}}}}{{\rm\,cm}}^2{{\rm\,s}}^{-1}$ whereas the total negative angular momentum is $-9.38\times 10^{51}{{{{\rm\,g}}}}{{\rm\,cm}}^2{{\rm\,s}}^{-1}$. They do not cancel out exactly. Some of the angular momentum must have left the computation box, through torsional Alfv[é]{}n waves. Dividing the total net angular momentum by the total mass left in the simulation box ($1.47\times 10^{33}{{{{\rm\,g}}}}$) yields $3.80\times
10^{19}{{\rm\,cm}}^2{{\rm\,s}}^{-1}$, which is the average specific angular momentum. For a core size of $10^{17}{{\rm\,cm}}$, the corresponding characteristic rotation speed is $3.8\times 10^2{{\rm\,cm}}{{\rm\,s}}^{-1}$, about $2\%$ of the sound speed, much smaller than the rotation speed achieved on the $10^{16}{{\rm\,cm}}$ scale or smaller. We conclude that, despite the localized, supersonic rotation induced on small scales, the influence of the Hall effect on the global dynamics is limited.
Discussion
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Why is Protostellar Disk Formation Difficult in Magnetized Cores?
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### Ideal MHD Limit {#imhd}
The fundamental reason for the difficulty in forming protostellar disks in magnetized cores is that the protostellar collapse concentrates magnetic flux at small radii, precisely where the rotationally supported disk (RSD) tends to form in the absence of magnetic braking. The basic difficulty can be seen most clearly in the ideal MHD limit, where magnetic flux is dragged into the central star (because of flux freezing) to form a split magnetic monopole. The rapid increase in field strength towards the central object enables the (split) monopolar field to brake the circumstellar disk catastrophically, as shown analytically by @Galli_2006 ([-@Galli_2006]; see their Fig. 1 for a sketch of the expected field geometry).
A case can also be made for catastrophic disk braking in the ideal MHD limit from numerical simulations ([Allen\_2003]{}; [MellonLi2008]{}; [HennebelleFromang2008]{}), although it is less clean cut. The reason is that, as the mass of the protostar grows, more and more magnetic flux is dragged to the origin, which creates a stronger and stronger (split) magnetic monopole that squeezes more and more strongly on the material on the equator from above and below. When the oppositely directed magnetic fields above and below the equator are squeezed within a single cell of each other, numerical reconnection becomes unavoidable. The expected numerical reconnection is present in the ideal MHD simulations of @MellonLi2008, especially for relatively strongly magnetized cores (with a dimensionless mass-to-flux $\lambda$ of several or less; see their Figs. 14-16), where no RSDs form but the numerical results are complicated by reconnections. The collapse of more weakly magnetized cores of $\lambda \sim 10$ is not significantly affected by reconnections, and yet no RSDs form either. The conclusion from the ideal MHD simulations is that magnetic braking is efficient enough to suppress disk formation for $\lambda \lesssim
10$.
The above conclusion is strengthened by the resistive MHD disk-formation calculations of @Krasnopolsky_2010, which included a wide range of (prescribed) resistivity. They found that, as the resistivity is decreased below a certain value, (numerical) reconnection starts to become important (as expected), which complicates the interpretation of the numerical results. However, before the reconnection sets in, the RSD is already completely braked by a moderately strong magnetic field. Extrapolating the results for those clean, low resistivity runs without numerical reconnection to the zero resistivity limit (where numerical reconnection is unavoidable) indicates that complete disk suppression also holds true for the ideal MHD case.
@Machida_2010 appears to have come to a different conclusion. They found a $10^2$-AU scale disk in their 3D nested-grid simulations in the ideal MHD limit (their Model 4), even though their core is strongly magnetized (with a global dimensionless mass-to-flux ratio of $\lambda=1$, see their Table 1). It is unclear whether the disk is rotationally supported or not. If yes, the result would be hard to understand. For such a strongly magnetized core, the mass accumulation at the protostar should produce a strong (split) magnetic monopole in the ideal MHD limit, which is expected to trigger powerful numerical reconnection, as discussed above and shown in @MellonLi2008; we have re-run their Model 4 and found the expected episodic reconnections and no rotationally supported disk. There were no reconnections mentioned in their paper, and the apparent lack of reconnection may be an indication that considerable numerical diffusivity acts to reconnect magnetic field lines of opposite polarities efficiently and prevent magnetic flux from accumulating near the protostar to form the expected split monopole in the first place. Since the trapping of magnetic flux at small radii lies at the heart of the efficient braking that renders disk formation difficult in the ideal MHD limit, inability to do so numerically may weaken the braking efficiency artificially and lead to disk formation.
### Non-Ideal MHD Effects
In lightly ionized dense cores of molecular clouds, non-ideal MHD effects are to be expected. Non-ideal effects, particularly ambipolar diffusion and Ohmic dissipation, enable the bulk neutral matter to move across magnetic field lines, breaking the flux freezing condition that is responsible for the formation of the central split magnetic monopole which, in turn, is responsible for the catastrophic disk braking in the ideal MHD limit. The elimination of the central split monopole does not necessarily mean, however, that the magnetic braking would automatically be weakened enough for an RSD to appear. The reason is that the magnetic flux that would be trapped in the central split monopole in the ideal MHD limit is now concentrated in a small, but finite, circumstellar region instead, as first demonstrated by @LiMcKee1996 in the case of ambipolar diffusion (AD). @KrasnopolskyKonigl2002 showed semi-analytically that the AD-induced flux concentration at small radii can in principle suppress disk formation completely, just as in the ideal MHD limit. @MellonLi2009 showed numerically that RSDs are indeed suppressed by a moderately strong magnetic field (with $\lambda \sim$ several) in the presence of ambipolar diffusion for a reasonable range of cosmic ray ionization rate.
The result of @MellonLi2009 is strengthened by the simulations presented in §\[PureAD\]. We improved over their calculations by self-consistently computing charge densities including dust grains and by extending the computation to the prestellar phase of core evolution leading up to the central mass formation. The extension allows us to explore the angular momentum evolution and disk formation during the transition between the prestellar and protostellar phase of star formation. We find that the tendency to form a disk is stronger around the time of initial protostar formation than at later times (see Fig. \[transition\]). This is because there is as yet little magnetic flux accumulated near the center and it takes time for the magnetic braking to remove angular momentum. Once enough magnetic flux has accumulated near the protostar to drive a well-developed accretion (C-)shock, the field strength in the post-shock region is typically strong enough to remove most of the angular momentum of the material falling into the central object, as long as the dense core is moderately strongly magnetized to begin with (with $\lambda \sim$ a few to several).
The conclusion that the RSD is suppressed by a moderately strong magnetic field in the presence of AD is robust, because the size of the AD shock and the post-shock field strength are rather insensitive to cosmic ray ionization rate and the grain size distribution (see Fig. \[general\]). They are determined mostly by the global requirements that (1) most of the magnetic flux associated with the central stellar mass be redistributed in the post-shock region, and (2) the strong postshock magnetic field be confined by the infall ram pressure (see equations \[8\] and \[10\] of [LiMcKee1996]{}), as long as the width of the C-shock is small compared to the radius of the shock. In principle, if the ionization level is decreased by a arbitrarily large factor, the magnetic field would eventually decouple completely from the bulk neutral material and an RSD would form. In practice, however, the RSD is suppressed even for the highly conservative case of both an unrealistically low cosmic ray ionization rate of $\zeta = 10^{-18}{{\rm\,s}}^{-1}$ and an MRN grain size distribution that contains a large amount of small grains (Model LoCR in Table 1). Both the grain growth expected in dense cores and a more realistic (higher) cosmic ray ionization rate tend to make the magnetic field better coupled to the bulk neutral material and the magnetic braking more efficient.
The Ohmic dissipation does not change the above picture much, because the Ohmic diffusivity $\eta_O$ is well below the ambipolar diffusivity $\eta_{AD}$ for the density range $n_H \lesssim 10^{12}{{\rm\,cm}}^{-3}$ that is crucial for disk formation. We can estimate the ratio of the two diffusivities through $$\label{etao_etaad}
{\eta_O\over \eta_{AD} } \sim {n_c \over n_e} {1\over \beta_c \beta_e}$$ where the subscripts “$c$” and “$e$” denote, respectively, the charged species whose contribution dominates the AD term and the electrons that are mainly responsible for the Ohmic term. The (dimensionless) Hall parameter $\beta\equiv \tau \omega$ (where $\tau$ is the collisional damping time of the motion of a charged species relative to the neutral and $\omega$ the cyclotron frequency) provides a measure of how well a charge is tied to the magnetic field. The relative unimportance of Ohmic dissipation comes mainly from the fact that electrons are extremely well tied to the magnetic field in the density regime of interest, with $\beta_e \gg 1$. For example, at a representative density of $n_H=3\times 10^8{{\rm\,cm}}^{-3}$, we find $\beta_e \approx 2\times 10^5$ for the field strength-density relation given by equation (5). In the large grain (LG) case that we have considered, the AD term is dominated by metal ions, $M^+$. Their number density is close to the electron number density ($n_{M^+} \approx n_e$) and they are well tied to the magnetic field at the representative density, with $\beta_{M^+}\approx
40$. From equation (\[etao\_etaad\]), we expect $\eta_O/\eta_{AD} \sim
10^{-7}$, which is in agreement with the computed values shown in the right panel of Fig. \[Diffusivities\]. In the case of MRN grain size distribution, the AD term is dominated by the small negatively charged grains, $g^-$. At the representative density, there are about $10^2$ small negatively charged grains for each electron ($n_{g^-} \approx 10^2 n_e$) and the small charged grains are marginally tied to the magnetic field $\beta_{g^-}\sim 1$. Both the smaller electron abundance and the weaker coupling of the grains to the field tend to make Ohmic dissipation more important relative to ambipolar diffusion. Nevertheless, the electrons are so well tied to the magnetic field that, even in this case, the Ohmic diffusivity is still much smaller than the ambipolar diffusivity, by a factor of $\sim 2000$ at the representative density. This estimate is again in agreement with the computed values shown in the left panel of Fig. \[Diffusivities\].
The Hall effect is expected to be more important than Ohmic dissipation in diffusing the magnetic field in the density regime under consideration. This can be seen from the ratio of Hall and ambipolar diffusivities $${\eta_H\over \eta_{AD} } \sim {1\over \beta_c},$$ which is applicable under the conditions that the same charged species “$c$” dominates both the Hall and AD terms and $\beta_c \gtrsim
1$. These conditions are satisfied for the large grain (LG) case at the representative density $n_H=3\times 10^8{{\rm\,cm}}^{-3}$, where metal ions dominate both terms and $\beta_{M^+}\approx 40$. In this case, the Hall diffusivity is only about $2\%$ of the ambipolar diffusivity (but still much larger than the Ohmic diffusivity). It becomes comparable to the ambipolar diffusivity in the case of MRN grain size distribution, where both terms are dominated by small dust grains that are marginally coupled to the magnetic field (with a Hall parameter $\beta$ of order unity) at the representative density. At higher densities, the Hall diffusivity is expected to exceed the ambipolar diffusivity as small grains become even less well tied to the magnetic field. However, the contribution from positively charged grains start to cancel out that from negatively charged grains, leaving the Hall diffusivity comparable to the ambipolar diffusivity over a wide range of density (see the left panel of Fig. \[Diffusivities\]). The Hall effect therefore does not increase the magnetic diffusivity by more than a factor of a few. As such, it is not expected to greatly change the global flow dynamics, especially the structure of the ambipolar diffusion-induced accretion shock, which lies at the heart of the magnetic braking catastrophe. It does, however, introduce a new ingredient into the problem: it can [*actively*]{} torque up a magnetized collapsing envelope, even if the envelope is non-rotating to begin with, as first pointed out by @WardleNg1999 and demonstrated numerically in @Krasnopolsky_2011 ([-@Krasnopolsky_2011]; see also [Braiding2011]{}). On the scale of several AUs or larger that we can resolve in our non-ideal MHD simulations (§\[Hall\] and \[PureHall\]), the Hall spin-up does not reach the Keplerian speed. The angular momentum gained through Hall spin-up may, however, be conducive to the formation of RSDs on smaller scales, particularly at high enough densities where electrons begin to decouple from the magnetic field and Ohmic dissipation becomes the dominant process for field diffusion ([Machida\_2010]{}; [DappBasu2010]{}). Nevertheless, the problem of catastrophic magnetic braking that prevents the formation of a sizable RSD of tens of AUs or larger is not resolved through the three non-ideal MHD effects.
Limitations and Future Directions: How to Form RSDs?
----------------------------------------------------
We are unable to produce robust, large-scale, rotationally supported disks in our non-ideal MHD simulations for dense cores magnetized to a realistic level. And yet, rotationally supported disks are observed around young stars, at least at relatively late times, after the massive envelope has been removed (which reveals the embedded disk for direct rotation measurement; see [WilliamsCieza2011]{} for a review). Clearly, one or more assumptions made in our calculations must break down in order for the observed (late-time) RSDs to form. These include (1) assumptions made in the setup of the numerical problem because of computational constraints and (2) additional physical effects that we have not taken into account. We comment on these limitations and their relevance to RSD formation in turn.
### Numerical Limitations and Possible Ways to Form RSDs {#NumericalLimitations}
We have restricted our problem setup to 2D (assuming axisymmetry), which greatly reduced the computational demand. A potential drawback is that the imposed symmetry may have enhanced the ability of the AD shock in trapping magnetic flux near the central object, which lies at the heart of the catastrophic magnetic braking that prevents disk formation. The reason is that the high magnetic pressure may force the trapped field lines to escape along the path of least resistance when the axisymmetry is broken. It is plausible that some magnetic flux loaded with relatively little matter (recall that the post-shock region is magnetically subcritical and supported against free-fall collapse by magnetic forces) would act as a “light” fluid and escape in some azimuthal directions, allowing less magnetized fingers of “heavy” material to sink closer to the center in other directions. This type of interchange instability was considered in @LiMcKee1996. It may weaken the magnetic field strength at small radii enough to enable disk formation. Investigation of this possibility is now underway.
Another limitation of the current problem setup is that we adopted a relatively large central hole around the protostar (typically $10^{14}{{\rm\,cm}}$ or $6.7{{{{\rm\,AU}}}}$ in radius). We have experimented with smaller holes by a factor of 2–3 in a few cases and found quantitatively similar results. However, it is difficult to reduce the hole size by a much larger factor, because of the constraints on the time step, which decreases as the square of the hole size (assuming the same resolution in the $\theta$-direction) in our explicit treatment of the non-ideal MHD terms. A drawback is that we are unable to determine whether small, (sub-)AU-scale RSDs can form during the main protostellar accretion phase or not. The existence of such small RSDs is suggested by the powerful molecular outflows ubiquitously observed around deeply embedded, Class 0 protostars (e.g., [Bontemps\_1996]{}); they are generally thought to be driven by a fast primary wind launched magnetocentrifugally from the inner part of a Keplerian disk close to the central object ([Shu\_2000]{}; [KoniglPudritz2000]{}).
Small RSDs can in principle form at high enough densities where electrons start to decouple from the magnetic field. Before thermal ionization of alkali metals becomes important, the Ohmic dissipation can reduce the local current density, making it hard for the magnetic field to bend in the poloidal plane (which limits the flux accumulation at small radii) and to twist in the azimuthal direction (which weakens magnetic braking; [Shu\_2006]{}; [Machida\_2007]{}; [Krasnopolsky\_2010]{}; [DappBasu2010]{}). A worry is that, in the presence of only ambipolar diffusion and classical Ohmic dissipation, our (2D axisymmetric) calculations showed that little angular momentum is left at the inner edge of the computation domain ($6.7{{{{\rm\,AU}}}}$), making the formation of RSDs interior to it difficult. However, the Hall effect can spin up the flow that collapses through the inner boundary to a supersonic (although still locally sub-Keplerian) rotation speed, perhaps making the formation of small RSDs possible. Alternatively, the 3D magnetic interchange instability discussed earlier may weaken the magnetic braking interior to the AD shock enough to facilitate the formation of RSDs in general, and the small RSDs required for fast wind launching in particular. When and how the small RSDs grow to large RSDs observed around relatively evolved YSOs remains uncertain, and may require additional physical effects that have not been investigated in detail in this context to date.
### Additional Physical Effects for RSD Formation
An important ingredient for low-mass star formation is protostellar outflow. It may play a crucial role in disk formation. As first proposed by @MellonLi2008 ([-@MellonLi2008]; see their §6.2.2), the outflow can strip away the slowly rotating protostellar envelope, which brakes the equatorial infall material that is magnetically linked to it and that tries to spin up and form a rotationally supported disk. Part of the envelope may be removed by the core collapse process itself (see, e.g., [Machida\_2010]{}) but, if the efficiency of star formation in individual low-mass cores is of order $1/3$ ([Alves\_2007]{}; [Andre\_2010]{}), the majority of the envelope mass must be removed by some other process, most likely a (fast) protostellar wind [@MatznerMcKee2000]. Indeed, the bipolar molecular outflow, thought to be primarily the envelope material swept up by a fast wind ([Shu\_1991]{}; [Shang\_2006]{}), is observed to have a narrow jet-like appearance along the axis during the early Class 0 phase and the opening angle at the base increases as the YSO ages [@ArceSargent2006]. As the fast wind sweeps out an increasingly wider polar region in the envelope, the braking efficiency of the remaining equatorial infall region should decrease, perhaps to a low enough value that a large-scale rotationally supported disk can form.
A specific outflow-enabled large-scale disk formation scenario is as follow. We envision the early formation of a small (perhaps AU-scale) rotationally supported disk (unresolved by the current generation of instruments) during the Class 0 phase, through the processes discussed in §\[NumericalLimitations\]. Although the small disk can grow gradually through internal angular momentum redistribution (perhaps gravitational torques rather than magnetic stresses since magnetic decoupling is required for the disk to form in the first place, see discussion in [DappBasu2010]{}), we envision rapid growth in disk size (to, say, $100{{{{\rm\,AU}}}}$ or more) only during the late phase of envelope removal, when the braking of the equatorial infall material by the envelope is rendered inefficient by outflow stripping. This envelope-depletion induced rapid disk growth may occur towards the end of the main protostellar mass accretion phase, perhaps during the transition from the Class 0 to Class I phases of (low-mass) star formation. Detailed calculations and high resolution observations, perhaps using ALMA, are needed to test this scenario of late-time formation of a large-scale disk.
Another possibility for large-scale RSD formation is through enhanced magnetic diffusivity. If the diffusivity is greatly enhanced over the classical microscopic values considered in this paper by some processes, the magnetic braking may be weakened enough to allow for RSD formation. @Shu_2006 was the first to propose that enhanced Ohmic resistivity may enable RSD formation, and this was demonstrated explicitly in @Krasnopolsky_2010. The enhancement in resistivity may come from turbulence [@Kowal_2009], which is observed in dense cores from nonthermal line width, or current-driven instabilities [@NormanHeyvaerts1985], although these effects are hard to quantify at the present time (see, however, [Santos-Lima\_2010]{} who have started to quantify the so-called “turbulent reconnection diffusivity”). Similarly, @Krasnopolsky_2011 showed that if the Hall diffusivity is large enough, it can enable RSDs to form even in initially non-rotating dense cores.
Numerical diffusion may mimic to some extent enhanced magnetic diffusion of physical origins and lead to large-scale RSD formation. @Machida_2010 was able to produce large-scale disks in strongly magnetized cloud cores in both the ideal MHD limit and with a classical (microscopic) value of Ohmic dissipation. However, as we have argued in §\[imhd\], the lack of episodic reconnections expected in the ideal MHD limit indicates a considerable numerical diffusion in their calculations. Another indication is that, in the presence of only the classical Ohmic dissipation, we do not find any large-scale RSD (Model REF$_O$ in Table 1), for a good reason: the Ohmic diffusivity enables the magnetic flux that would have gone into the central object in the ideal MHD limit to accumulate in a small circumstellar region where the magnetic braking is particularly efficient (see Fig. \[PureO\]), as in the AD case. Such a magnetic flux accumulation was not obvious in @Machida_2010’s simulations, which may again indicate an enhanced magnetic diffusion, either of numerical origin or through 3D effects that are not captured by our 2D calculations.
### Weak Core Magnetization and RSD Formation
Here we comment on the possibility that dense star forming cores may be magnetized to different levels, and disks form preferentially in those that are weakly magnetized. Our calculations indicate that, in the presence of ambipolar diffusion, the core mass-to-flux ratio need to be greater than at least $\sim 10$ in order for a rotationally supported disk to form and survive to late times. Observationally, a mean value of $\lambda_{los} \approx 4.8 \pm 0.4$ is inferred by @TrolandCrutcher2008 from the line-of-sight field strength for a sample of dark cloud cores. Applying geometric corrections would reduce the value statistically by a factor of 2–3 [@Shu_1999], making it unlikely for the majority of dense cores to be magnetized as weakly as $\lambda \gtrsim 10$. Since the majority of, if not all, young stars (formed out of all dense cores) are thought to have an RSD at some point, we consider it unlikely that weak core magnetization is the main reason for RSD formation.
Observational Implications: Disk vs Pseudodisk
----------------------------------------------
We should emphasize that, in our simulations, even though large rotationally supported disks are difficult to form, highly flattened dense “disk-like” structures are prevalent (see Figs. \[ReferenceModel\], \[PureO\], \[WeakB\_NoDisk\], and \[PureHall\_vphi\]). This is not surprising because, just like rotation, the magnetic field can provide anisotropic support to the cloud core, allowing matter to settle along field lines into flattened structures [@GalliShu1993]. The fact that there are two types of forces in nature that can retard (anisotropically) the gravitational collapse naturally leads to two types of flattened structures: rotationally supported disks and magnetically induced (pseudo-)disks. If the dense cores are as strongly magnetized as indicated by the currently available observations (with a dimensionless mass-to-flux ratio of several or smaller, see §\[intro\]), then there is typically more magnetic energy than rotational energy, and a magnetically induced pseudo-disk is just as, if not more, likely to form around an accreting protostar as a rotationally supported disk, especially in view of the fact that magnetic braking hinders the formation of rotationally supported disks but not pseudodisks. It is therefore premature to conclude that dense flattened structures observed around deeply embedded protostars (such as from dust continuum observations, e.g., [Jorgensen\_2009]{}) are rotationally supported disks rather than magnetically induced pseudodisks; the latter can be just as thin as (perhaps even thinner than) rotationally supported disks, because of magnetic compression. To confuse the situation further, the pseudodisks can have a substantial rotation as well (just not enough to provide the full support against gravity) and may or may not collapse at a high speed (see again Figs. \[ReferenceModel\], \[PureO\], \[WeakB\_NoDisk\] and \[PureHall\_vphi\]). Detailed kinematic information, as well as a knowledge of the central mass, are needed to establish whether a dense flattened circumstellar structure is a rotationally supported disk or not. High resolution observations of the circumstellar magnetic field structure, such as those in @Girart_2006, will also go a long way towards testing the idea of magnetically induced pseudodisk [@Goncalves_2008].
Summary
=======
We have carried out a set of 2D axisymmetric calculations exploring non-ideal MHD effects in magnetic braking and protostellar disk formation in rotating magnetized dense cores. Our main conclusions are summarized as follows:
1\. For a realistic magnetic field of moderate strength corresponding to a core mass-to-flux ratio $\lambda \sim$ 3–4, the magnetic braking is strong enough to remove essentially all of the angular momentum of the material that accretes onto the central object in the presence of ambipolar diffusion under a wide range of conditions in 2D. Any large-scale (greater than several AUs) rotationally supported disk (RSD) is suppressed by the formation of an ambipolar diffusion-induced accretion shock, which traps a strong magnetic field near the central object, leading to efficient magnetic braking of the post-shock material.
2\. On scales greater than $\sim 10{{{{\rm\,AU}}}}$, realistic levels of Ohmic diffusivity do not enable the formation of large-scale RSDs, either by itself or in combination with ambipolar diffusion. Furthermore, Ohmic dissipation does not necessarily reduce the magnetic braking efficiency. It can make the braking more efficient by enabling magnetic flux accumulation at small radii, where the field strength is increased, similar to the case of ambipolar diffusion.
3\. The Hall effect can spin up the post-AD shock material to a significant, supersonic rotation speed, although the rotation remains too sub-Keplerian to form an RSD for the parameter space explored in this work.
4\. For an unusually weak magnetic field corresponding to a core mass-to-flux ratio $\lambda \gtrsim 10$, a small RSD often forms early in the protostellar accretion phase, when the central mass is still small. In the majority of cases, the RSD disappears at later times, braked strongly by the powerful outflow that it drives. In some cases, particularly when the cosmic ray ionization rate is unusually low and the core rotation rate is unusually high, the fate of the early disk is unknown because the simulation stops early due to numerical difficulty.
5\. We discussed several possible ways to enable the formation of large-scale RSDs: magnetic instabilities in 3D, early formation of small RSDs at high densities, outflow stripping of protostellar envelope, enhanced magnetic diffusion and weak core magnetization. The more likely of these possibilities are, in our view, the weakening AD shock in 3D through interchange instability, which is expected to decrease the field strength (and thus the braking efficiency) near the central object, outflow stripping of protostellar envelope, which may allow rapid formation of a large-scale RSD during the transition from the deeply embedded (Class 0) phase to more revealed (Class I and II) phase of low-mass star formation, and enhanced magnetic diffusivity, which may be driven by turbulence-induced reconnections.
This work was supported in part by NASA through NNG06GJ33G and NNX10AH30G, by the Theoretical Institute for Advanced Research in Astrophysics (TIARA) through the CHARMS initiative, and by the National Science Council of Taiwan through grant NSC97-2112-M-001-018-MY3.
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abstract: 'Pair production processes of neutral Higgs particles will allow us to study the trilinear Higgs couplings at future high–energy colliders. Several mechanisms give rise to multi–Higgs final states in hadron interactions. In the present paper we investigate Higgs pair production in gluon–gluon collisions. After recapitulating pair production in the Standard Model, the analysis of the cross sections is carried out in detail for the neutral Higgs particles in the minimal supersymmetric extension.'
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citexr\[\#1\]\#2[@fileswauxout citeacite[forciteb:=\#2]{}[\#1]{}]{} ‘@=12
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DESY 95–215\
December 1995\
hep-ph/9603205
[Pair Production of Neutral Higgs Particles\
in Gluon–Gluon Collisions]{}
[T. Plehn$^1$, M. Spira$^{2}$[^1] and P. M. Zerwas$^1$]{}\
$^1$ Deutsches Elektronen–Synchrotron DESY, D–22603 Hamburg, FRG\
$^2$ II. Institut für Theoretische Physik[^2], D–22761 Hamburg, FRG\
Introduction
============
The reconstruction of the Higgs potential is an experimental [ *prima facie*]{} task to establish the Higgs mechanism as the basic mechanism for generating the masses of the fundamental particles. This task requires the measurement of the self–couplings of the Higgs particles.
In the Standard Model (SM) [@0] the trilinear and the quartic couplings of the physical Higgs particle $H$ are uniquely fixed if the Higgs mass is known.
In the minimal supersymmetric extension of the Standard Model (MSSM), a large variety of couplings exists among the members $h,H,A,H^\pm$ of the Higgs quintet [@4]. \[$h$ and $H$ are the light and heavy CP–even Higgs bosons[^3], $A$ is the CP–odd (pseudoscalar) Higgs boson, and $H^\pm$ is the charged Higgs pair.\] While general two–doublet models contain three mass parameters and seven real self–couplings in CP conserving theories, the Higgs self–couplings are fixed in terms of gauge couplings in the MSSM, and the mass parameters can be expressed by the two vacuum expectation values of the neutral Higgs fields and one of the physical Higgs masses. Since the sum of the squares of the vacuum expectation values is given by the $W$ mass, tg$\beta$, the ratio of the two vacuum expectation values, and $M_A$, the mass of the CP–odd Higgs boson $A$, are generally chosen as the free parameters of the MSSM. The trilinear and quartic Higgs self–couplings are determined by those two parameters.
The measurement of the Higgs self–couplings will be a very difficult task. In the Standard Model the production cross section for $HH$ Higgs pairs are small, similarly the continuum production of Higgs pairs in the MSSM. Only if heavy MSSM Higgs bosons can decay into pairs of light Higgs bosons, the associated Higgs self–couplings can be determined fairly easily by measuring the decay branching ratios. Several aspects of multi–Higgs production have been discussed in the literature. Most analyses treat $HH$ pair production in the Standard Model only at the theoretical level [@1a; @1]. Within the MSSM, the search for the heavy Higgs boson $H$ has been simulated in the $b\bar{b}$ decay mode at the LHC [@2]. $h \to AA$ events have been searched at LEP [@3]; more general aspects of multi–Higgs production in $e^+e^-$ collisions have only recently been analyzed theoretically in Ref.[@4].
Several mechanisms give rise to the production of pairs of neutral Higgs bosons in hadron collisions. Multi–Higgs final states can be produced through Higgs–strahlung off $W$ bosons and through $WW$ fusion in proton–proton collisions. Bremsstrahlung of Higgs particles off heavy quarks can also be exploited. In the present paper we discuss the production of Higgs pairs in gluon–gluon collisions at the LHC. We have determined the cross sections for the continuum in the Standard Model $$pp \to gg \to HH$$ as well as for the continuum and resonance decays in the minimal supersymmetric theory $$pp \to gg \to \Phi_i\Phi_j \qquad \qquad \Phi_i=h,H,A$$ restricting ourselves to neutral Higgs bosons in the present report[^4]. It can be anticipated from the large number of low–$x$ gluons in high–energy proton beams that the $gg$ channel is of particular interest in the continuum for fairly low Higgs masses. In the MSSM the on–shell production of heavy Higgs bosons with subsequent Higgs cascade decays will eventually provide a copious source of light multi–Higgs final states. The cross sections are affected, besides the normal Higgs–boson couplings to gauge bosons and fermions [@5], by the trilinear Higgs couplings [@5a]: $$\begin{aligned}
&SM\hphantom{SM}:& \lambda_{HHH} = \frac{3 M_H^2}{M_Z^2}
\label{eq:3} \\
&MSSM:& \lambda_{hhh} = 3\cos(2\alpha) \sin(\beta+\alpha)
+ \frac{3 \epsilon}{M_Z^2} \frac{\cos^3\alpha}{\sin \beta} {\nonumber}\\
&& \lambda_{Hhh} =2 \sin(2\alpha) \sin(\beta+\alpha)
- \cos(2\alpha) \cos(\beta+\alpha)
+ \frac{3 \epsilon}{M_Z^2} \frac{\sin\alpha\cos^2\alpha}{\sin \beta} {\nonumber}\\
&& \lambda_{HHh} =- 2 \sin(2\alpha) \cos(\beta+\alpha)
- \cos(2\alpha) \sin(\beta+\alpha)
+ \frac{3 \epsilon}{M_Z^2} \frac{\sin^2\alpha\cos\alpha}{\sin \beta} {\nonumber}\\
&& \lambda_{HHH} = 3\cos(2\alpha) \cos(\beta+\alpha)
+ \frac{3 \epsilon}{M_Z^2} \frac{\sin^3\alpha}{\sin \beta} {\nonumber}\\
&& \lambda_{hAA} = \cos(2\beta) \sin(\beta+\alpha)
+ \frac{\epsilon}{M_Z^2} \frac{\cos\alpha\cos^2\beta}{\sin \beta} {\nonumber}\\
&& \lambda_{HAA} = - \cos(2\beta) \cos(\beta+\alpha)
+ \frac{\epsilon}{M_Z^2} \frac{\sin\alpha\cos^2\beta}{\sin \beta}
\label{eq:4}\end{aligned}$$ The couplings in the SM as well as the MSSM are normalized to $\lambda_0 = [\sqrt{2} G_F]^{1/2} M_Z^2$. The MSSM couplings depend on $\beta$ and the mixing angle $\alpha$ $$\tan 2 \alpha=
\frac{ M_{A}^2 + M_Z^2 }{ M_{A}^2 - M_Z^2 + \epsilon / \cos 2\beta}
\; \tan 2 \beta$$ in the CP–even Higgs sector; radiative corrections have been included in the leading $m_t^4$ one–loop approximation, parametrized by $$\epsilon = \frac{3 G_F}{\sqrt{2} \pi^2}
\frac{m_t^4}{\sin^2 \beta} \;
\log\left[ 1 + \frac{M_S^2}{m_t^2} \right]$$ with the common squark mass fixed to $M_S$= 1 TeV. In our numerical analysis we have included the leading two–loop corrections to the MSSM Higgs masses and couplings, taken from Ref.[@M1]. The behaviour of the couplings with $M_A$ is shown for two representative values of ${\mbox{tg$\beta$}}=1.5$ and 30 in Fig.1.
The paper is organized as follows. In the next section $HH$ pair production through gluon collisions will be discussed in the Standard Model. In the third section we will present the results for all pair combinations of neutral Higgs bosons in the MSSM. Phenomenological aspects including Higgs–strahlung and $WW$ fusion will be presented in a sequel to this paper [@8].
Higgs Pair Production in the Standard Model
===========================================
Two mechanisms contribute to the production of Higgs pairs through $gg$ collisions in the Standard Model, exemplified by the generic diagrams in Fig.2a/b. (i) Virtual Higgs bosons which subsequently decay into $HH$ final states, are coupled to gluons by the usual heavy–quark triangle [@6; @7]. (ii) The coupling is also mediated by heavy–quark box diagrams.
In the triangle diagram Fig.2a the gluons are coupled to the total spin $S_z=0$ along the collision axis. The transition matrix element associated with this mechanism can therefore be expressed by the product of one (gauge invariant) form factor $F_\triangle$, depending on the scaling variable $\tau_Q = 4 m_Q^2 /\hat{s}$, and the generalized coupling $C_\triangle$ defined as $$C_\triangle = \lambda_{HHH}~\frac{M_Z^2}{\hat s-M_H^2}$$ The coefficient $\lambda_{HHH}$ denotes the trilinear self–coupling $HHH$ in the Standard Model, cf. eq.(\[eq:3\]). $\hat{s}$ is the square of the invariant energy flow through the virtual Higgs line. The well–known form factor $F_\triangle$ [@6; @7] is given in Appendix A1.
The box diagrams in Fig.2b allow for $S_z=0$ and 2 gluon–gluon couplings so that the transition matrix element can be expressed in terms of two (gauge invariant) form factors $F_\Box$ and $G_\Box$. Fairly compact expressions of the form factors are given in Appendix A1 where the Higgs masses have to be specified to $M_c=M_d=M_H$. The couplings between the Higgs bosons and the quarks are normalized to unity by definition, $$C_\Box = 1$$ \[The overall normalization is included in the prefactors of the cross section and the form factors.\] The form factors obtained in this way for the simplified case of equal masses agree with the results in Ref.[@1].
In the two limits of light and heavy Higgs bosons with respect to the loop quark mass, very simple expressions can be derived for the three form factors $F_\triangle$ and $F_\Box$, $G_\Box$.
The form factors can be evaluated either by taking the limit $m_Q^2
\gg \hat{s} \sim M_H^2$ in the Feynman amplitude or, equivalently, by exploiting the elegant low–energy theorem $F_\Box =
m^2_Q \partial (F_\triangle/m_Q)/\partial m_Q$ for external light scalar Higgs bosons. This reduces the complexity of the calculation considerably: $$F_\triangle=\frac{2}{3}+{\cal O}\left(\hat s/m_Q^2\right)
\label{eq:13}$$ and $$\begin{aligned}
F_\Box&=&-\frac{2}{3}+{\cal O}\left( {\hat s}/m_Q^2\right)
\\ G_\Box&=&{\cal O}\left( {\hat s}/m_Q^2\right)
{\nonumber}\end{aligned}$$ Both $S_z=0$ form factors $F_\triangle$ and $F_\Box$ survive in this limit, whereas the $S_z=2$ form factor $G_\Box$ vanishes asymptotically.
In the limit of light quark masses compared with the invariant energy[^5], large logarithms of the light quark mass $\log
\hat{s}/m_Q^2$ will occur in the form factors. Detailed inspection of the Feynman amplitudes leads to the following final expressions: $$F_\triangle= - \frac{m_Q^2}{\hat s} \left[
\log\left(\frac{m_Q^2}{\hat{s}} \right) + i\pi \right]^2 + {\cal
O}\left(\frac{m_Q^2}{\hat s} \right)$$ and $$\begin{aligned}
F_\Box&=&{\cal O} \left( m_Q^2/{\hat s} \right) \label{eq:14}
\\ G_\Box&=&{\cal O}\left( m_Q^2/{\hat s} \right) {\nonumber}\end{aligned}$$ All form factors vanish to ${\cal O}(m_Q^2)$ in this limit since they are suppressed by the Yukawa coupling $\propto m_Q$; this is well–known for the quark triangle. \[This limit is not of much practical use in the SM; yet it will be relevant later in the MSSM where $b$–quark loops are dominant for large ${\mbox{tg$\beta$}}$.\]
The differential parton cross section can finally be written in the form $$\frac{d\hat\sigma (gg\to HH)}{d\hat t} =
\frac{G_F^2\alpha_s^2}
{256 (2\pi)^3} \Big[ | C_\triangle \; F_\triangle \; + C_\Box F_\Box |^2 + |
C_\Box G_\Box |^2 \Big]
\label{eq:19}$$ where $\hat{t}$ is the momentum transfer squared from one of the gluons in the initial state to one of the Higgs bosons in the final state. The total cross section for $HH$ Higgs pair production through $gg$ in $pp$ collisions[^6] can be derived by integrating (\[eq:19\]) over the scattering angle and the $gg$ luminosity $$\sigma(pp \to gg \to HH) = \int_{4 M_H^2 /s}^{1} d\tau \frac{d {\cal
L}^{gg} }{d\tau} \hat{\sigma}(\hat s = \tau s)$$ This cross section has been evaluated numerically. The analysis has been carried out for the LHC c.m. energy $\sqrt{s}$ = 14 TeV; the top quark mass has been set to $m_t$ = 175 GeV [@10]. The result is shown in Fig.3. For SM Higgs masses close to $M_H \sim$ 100 GeV the cross section is of order 10 $fb$. However, it drops rapidly with increasing Higgs mass; this is a consequence of the fast fall–off of the $gg$ luminosity with rising $\tau$ \[which is of the order of $x_g^2$ in the $pp$ collisions\]. Nevertheless, at $M_H \sim$ 100 GeV order 2,000 events will be produced for an integrated luminosity of $\int{\cal L} = 100 fb^{-1}$, giving rise to 4$b$ and $bb\tau\tau$ final states with large transverse momenta.
Neutral Higgs Pairs of the MSSM
===============================
A large variety of neutral Higgs pairs of the MSSM can be produced in gluon–gluon collisions: $$pp \to gg \to hh,hH,HH,hA,HA,AA$$ The analysis of the cross sections is in general much more involved than in the SM for two reasons. First, the masses of the two Higgs bosons in the final state are in general different, and second, besides pairs of CP–even Higgs bosons, also pairs of CP–odd and mixed CP–even/CP–odd pairs can be produced.
[**a) Pairs of CP–even Higgs bosons $\mathbf{hh,hH,HH}$**]{}
The mechanisms for the production of CP–even MSSM Higgs bosons $gg\to
H_i H_j$ $(H_i = h,H)$ are similar to the Standard Model. The generic triangle and box diagrams are shown in Figs.4a/b. All possible combinations of light and heavy Higgs bosons $h$ and $H$ can be coupled in 3–particle vertices.
The matrix element of the triangle contributions can be expressed in terms of the form factor $F_\triangle$, given in Appendix A1, and the couplings $$C_\triangle = C_\triangle^h + C_\triangle^H$$ with $$C^{h/H}_\triangle = \lambda_{H_iH_j(h/H)} ~\frac{M_Z^2}{\hat
s-M_{h/H}^2+iM_{h/H}\Gamma_{h/H}}~g_Q^{h/H} \hspace{2cm} (H_i =
h,H)$$ The couplings $g_Q^{h/H}$ denote the Higgs–quark couplings in units of the SM Yukawa coupling $[\sqrt{2}G_F]^{1/2}m_Q$. They are collected in Table \[tb:1\]. The Higgs self couplings $\lambda_{H_i H_j
(h/H)}$ can be read off eq.(\[eq:4\]).
$g^\Phi_t$ $g^\Phi_b$
------- ------------ ------------------------- -------------------------
SM $H$ 1 1
MSSM $h$ $\cos\alpha/\sin\beta$ $-\sin\alpha/\cos\beta$
$H$ $\sin\alpha/\sin\beta$ $\cos\alpha/\cos\beta$
$A$ $ 1/{\mbox{tg$\beta$}}$ ${\mbox{tg$\beta$}}$
: \[tb:1\] [*Higgs couplings in the MSSM to fermions relative to SM couplings.*]{}
For the box contributions we obtain analogous expressions, with two independent form factors $F_\Box$ and $G_\Box$ contributing. Their expressions can be found in Appendix A1. The corresponding couplings $C_\Box$ are are built up by the Higgs–quark couplings, $$C_\Box = g_Q^{H_i} g_Q^{H_j}$$
The two limiting cases in which the Higgs masses are either much smaller or much larger than the loop quark masses, are both physically relevant in this case since the top quark as well as the bottom quark play a role, with $M^2_h \ll m_t^2$ and $M^2_H \gg m_b^2$. The limits of the form factors are the same as for the SM Higgs bosons eqs.(\[eq:13\]–\[eq:14\]).
From the differential parton cross section, which includes $t$ and $b$ quark loops, $$\frac{d\hat\sigma (gg\to H_iH_j)}{d\hat t} =
\frac{G_F^2\alpha_s^2}{256 (2\pi)^3}
\Big[ | \sum_{t,b} (C_\triangle \; F_\triangle \;
+ C_\Box \; F_\Box) |^2
+ | \sum_{t,b} C_\Box \; G_\Box |^2 \Big]$$ the $pp$ cross section can be derived by integrating over the scattering angle and attaching the gluon luminosity. The result of the numerical analysis is shown for the final states $hh,hH$ and $HH$ in Figs.5a/b/c for two representative values ${\mbox{tg$\beta$}}= 1.5$ and 30. \[Mixing in the stop sector is not taken into account.\]
The final state $hh$ deserves special attention. The cross section for moderate values $M_h$ is large since the heavy CP–even Higgs boson $H$ can decay into the two light Higgs bosons (dashed lines). $H$ is produced directly in $gg$ collisions, coupled through the quark triangle, so that the matrix element involves one power of the electroweak coupling less than continuum Higgs pair production. Except for a dip due to a zero in the $Hhh$ coupling $\lambda_{Hhh}$, the cross section in the resonance range is of order $10^3$ to $10^4
fb$, dropping to the typical $hh$ continuum value of 30 $fb$ for increasing $H$ Higgs masses. For ${\mbox{tg$\beta$}}=1.5$ the curve shows a sharp threshold behavior for $M_H\sim 2 m_t$ due to on–shell top pair production in the dominant top quark triangle and, moreover, due to the sharp fall–off of the branching ratio $BR(H\to hh)$. The cross section for ${\mbox{tg$\beta$}}=30$ depends strongly on the Higgs boson masses, because resonance production $gg\to H \to hh$ is kinematically forbidden for 110 GeV $\!{\raisebox{-0.13cm}{~\shortstack{$<$ \\[-0.07cm] $\sim$}}~}M_A {\raisebox{-0.13cm}{~\shortstack{$<$ \\[-0.07cm] $\sim$}}~}210$ GeV. Above this range this channel opens up again with a small branching ratio $\sim
10^{-3}$ giving rise to the small bump in the cross section at $M_A\sim 210$ GeV.
The cross sections for $hH$ and $HH$ final states are much smaller, cf. Figs.5b/c. This is the result of the absence of any resonance effect and the phase space suppression due to the large Higgs masses in the final states. The cross section for $pp\to gg\to hH$ depends strongly on ${\mbox{tg$\beta$}}$ and it varies between very small values and $\sim
100~fb$ if the entire Higgs mass range is sweeped. For ${\mbox{tg$\beta$}}=1.5$ the Higgs mass dependence is smooth, whereas for ${\mbox{tg$\beta$}}=30$ a sharp peak develops at $M_A\sim 100$ GeV due to the strong variation of the MSSM couplings in this region. At large Higgs masses the cross section decreases strongly with increasing ${\mbox{tg$\beta$}}$. A similar picture emerges for the pair production of heavy scalar Higgs bosons $pp\to gg\to HH$. Its size increases at large Higgs masses for increasing ${\mbox{tg$\beta$}}$, where the bottom quark loops become dominant. The strong variation of the cross section for $M_A\sim 100$ GeV and ${\mbox{tg$\beta$}}=30$ is again due to the rapidly changing MSSM couplings in this mass range.
[**b) Mixed CP–even/CP–odd pairs**]{}
The analysis of the CP–mixed $hA$ and $HA$ final states is in many aspects different from the previous cases. First of all, the final state can be produced through the decay of a virtual $Z$ boson as shown in the generic diagrams of Fig.6. By evaluating the Feynman diagrams one finds the following generalized charges \[$H_i = h,H$\]
(i)
: $$C^A_\triangle = \lambda_{AAH_i}~\frac{M_Z^2}{\hat
s-M_A^2+iM_A\Gamma_A}~g_Q^A$$
(ii)
: $$C^Z_\triangle = \lambda_{ZAH_i}~\frac{M_Z^2}{\hat
s-M_Z^2+iM_Z\Gamma_Z}~a_Q$$ where $a_Q = 1 (-1)$ for top (bottom) quarks denotes the axial charge of the heavy loop quark $Q$ \[note that only the $Z$ axial coupling of the heavy loop quark can contribute\], and the trilinear couplings $\lambda_{ZAH_i}$ are given by $$\begin{aligned}
\lambda_{ZAh} & = & -\cos(\beta-\alpha) \\
\lambda_{ZAH} & = & \sin(\beta-\alpha) {\nonumber}\end{aligned}$$
(iii)
: $$C_\Box^{AH_i} = g_Q^A g_Q^{H_i}$$
The expressions of the form factors $F_\triangle^{A/Z}$ and $F_\Box$, $G_\Box$ can be found in Appendix A2.
In analogy to the CP–even case, simple expressions can be derived for the form factors in the limits of large and small Higgs masses compared with the quark masses.
The triangular form factor can be derived in this limit from the axial anomaly. The box form factor is given by the derivative of the anomaly $F_\Box = m_Q^2\partial(F_\triangle^A/m_Q)/\partial
m_Q$ [@7; @9]. Simple expressions can be obtained: $$\begin{aligned}
F^A_\triangle&=&1+{\cal O}\left( {\hat s}/m_Q^2\right) \\
F^Z_\triangle&=&{\cal O}\left( {\hat s}/m_Q^2\right) {\nonumber}\end{aligned}$$ and $$\begin{aligned}
F_\Box&=&-1+{\cal O}\left( {\hat s}/m_Q^2\right) \\
G_\Box&=&{\cal O}\left( {\hat s}/m_Q^2\right) {\nonumber}\end{aligned}$$
$$\begin{aligned}
F^A_\triangle&=& -\frac{m_Q^2}{\hat s}
\left[\log\left(\frac{m_Q^2}{\hat{s}} \right) + i\pi \right]^2 +
{\cal O}\left(\frac{m_Q^2}{\hat s} \right) {\nonumber}\\ F^Z_\triangle
&=& \frac{\hat s - M_Z^2}{\hat s}~\frac{M_{H_i}^2 - M_A^2}{M_Z^2}
\left\{ 1 + \frac{m_Q^2}{\hat s}
\left[\log\left(\frac{m_Q^2}{\hat{s}} \right) + i\pi \right]^2 +
{\cal O}\left(\frac{m_Q^2}{\hat s} \right) \right\}{\nonumber}\\
F_\Box&=&{\cal O} \left( m_Q^2/{\hat s} \right) \\
G_\Box&=&{\cal O}\left( m_Q^2/{\hat s} \right) {\nonumber}\end{aligned}$$
The differential parton cross section is determined by the form factors, including $t$ and $b$ quark loops: $$\frac{d\hat\sigma (gg\to AH_i)}{d\hat t} = \frac{G_F^2\alpha_s^2}
{256 (2\pi)^3}
\Big[ | \sum_{t,b} (C_\triangle \; F_\triangle \;
+ C_\Box \; F_\Box) |^2
+ | \sum_{t,b} C_\Box \; G_\Box |^2 \Big]$$ The results of the numerical analysis of the two $pp$ cross sections are shown in Figs.7a/b. For light masses $M_h$ the invariant mass \[$hA$\] of the final state is small so that the cross section is enhanced by the large phase space. For large Higgs masses the cross section decreases strongly with increasing ${\mbox{tg$\beta$}}$ and drops to a level of about $10^{-1}$ to $10^{-3}
fb$ at $M_A \sim 1$ TeV, where it cannot be observed anymore. The strong variation for ${\mbox{tg$\beta$}}=30$ at $M_A\sim 100$ GeV is generated by the rapid change of the MSSM couplings with $M_A$.
Due to the larger mass of the heavy scalar Higgs boson $H$, the cross section for $HA$ production turns out to be smaller than for $hA$ production.
[**c) $\mathbf{AA}$ pairs of CP–odd Higgs bosons**]{}
This case is again closely related to the CP–even pairs. In the box diagrams, Fig.8b, some terms flip just the sign when the $\gamma_5
\times \gamma_5$ couplings are reduced to ${1\hspace{-0.85mm}\mbox{l}}$ in the loop trace. As a result, the matrix elements can be expressed in terms of the couplings $$\begin{aligned}
C_\triangle & = & C^h_\triangle + C^H_\triangle\end{aligned}$$ with $$\begin{aligned}
C^{h/H}_\triangle & = & \lambda_{AA(h/H)}~ \frac{M_Z^2}{\hat
s-M_{h/H}^2+iM_{h/H}\Gamma_{h/H}}~g_Q^{h/H} {\nonumber}\\
C_\Box & = & (g_Q^A)^2\end{aligned}$$ The form factors $F_\triangle$ for the triangle contributions and $F_\Box$, $G_\Box$ for the box contribution can be found in Appendix A3 by choosing $M_c=M_d=M_A$.
In the limits of large and small Higgs masses compared with the quark masses simple expressions can be derived.
The triangle form factor coincides with the scalar expression eq.(\[eq:13\]), and the box form factor can be obtained from the derivative of the gluon self–energy \[${\cal M}(ggA^2) = \sqrt{2} G_F
m_Q \partial {\cal M}(gg)/\partial m_Q$\] [@9] leading to the results $$\begin{aligned}
F_\triangle&=&\frac{2}{3}+{\cal O}\left( {\hat s}/m_Q^2\right) \end{aligned}$$ and $$\begin{aligned}
F_\Box&=&\frac{2}{3}+{\cal O}\left( {\hat s}/m_Q^2\right) \\
G_\Box&=&{\cal O}\left( {\hat s}/m_Q^2\right) {\nonumber}\end{aligned}$$
$$F_\triangle= - \frac{m_Q^2}{\hat s} \left[ \log\left(\frac{m_Q^2}{\hat{s}}
\right) + i\pi \right]^2
+ {\cal O}\left(\frac{m_Q^2}{\hat s} \right)$$
and $$\begin{aligned}
F_\Box&=&{\cal O} \left( m_Q^2/{\hat s} \right) \\
G_\Box&=&{\cal O}\left( m_Q^2/{\hat s} \right) {\nonumber}\end{aligned}$$
The $pp$ cross section is shown in Fig.9. The large value of the cross section for small $A$ masses is due to resonance $H\to AA$ decays. When this channel is closed, the cross section becomes more and more suppressed with rising $A$ mass in the final state. For ${\mbox{tg$\beta$}}=1.5$ the cross section develops a kink at the $(t\bar t)$ threshold $M_A=2m_t$ due to $S$–wave $(t\bar t)$–resonance production. Above this mass value the phase space suppression leads to a rapidly decreasing cross section, to a level of $\sim 10^{-3} fb$ at $M_A\sim 1$ TeV. For ${\mbox{tg$\beta$}}=30$ the bottom quark loops are dominant so that the cross section depends smoothly on $M_A$. For large pseudoscalar masses the signal increases with increasing ${\mbox{tg$\beta$}}$.
Appendices
==========
[**A1. Form factors for two scalar Higgs bosons**]{} $\mathbf{ g_a g_b \to H_c H_d}$
parameter definitions: $$\hat s = (p_a+p_b)^2, \hspace{1.0cm}
\hat t = (p_c-p_a)^2, \hspace{1.0cm}
\hat u = (p_c-p_b)^2$$ $$S = {\hat s}/m_Q^2, \hspace{1.0cm}
T = {\hat t}/m_Q^2, \hspace{1.0cm}
U = {\hat u}/m_Q^2$$ $$\rho_c = M_c^2/m_Q^2, \hspace{1.0cm}
\rho_d = M_d^2/m_Q^2, \hspace{1.0cm}
\tau_Q = 4/S$$ $$T_1 = T - \rho_c, \hspace{1.0cm}
U_1 = U - \rho_c, \hspace{1.0cm}
T_2 = T - \rho_d, \hspace{1.0cm}
U_2 = U - \rho_d$$
Scalar integrals: $$\begin{aligned}
C_{ij}\!\!\!\! & = & \!\!\!\!\!\int \frac{d^4q}{i\pi^2}~\frac{1}
{(q^2-m_Q^2)\left[{\phantom{\frac{1}{1}}\!\!\!\!}(q+p_i)^2-m_Q^2\right]
\left[{\phantom{\frac{1}{1}}\!\!\!\!}(q+p_i+p_j)^2-m_Q^2\right]} \\ \\
D_{ijk}\!\!\!\! & = & \!\!\!\!\!\int \frac{d^4q}{i\pi^2} \frac{1}
{(q^2-m_Q^2)\left[{\phantom{\frac{1}{1}}\!\!\!\!}(q+p_i)^2-m_Q^2\right]
\left[{\phantom{\frac{1}{1}}\!\!\!\!}(q+p_i+p_j)^2-m_Q^2\right]\left[{\phantom{\frac{1}{1}}\!\!\!\!}(q+p_i+p_j+p_k)^2-m_Q^2\right]}\end{aligned}$$ The analytic expressions can be found in Ref.[@7b].
$$\begin{aligned}
F_\triangle & = & \frac{2}{S} \left\{ 2+(4 - S) m_Q^2 C_{ab} \right\}
= \tau_Q \left[{\phantom{\frac{1}{1}}\!\!\!\!}1+(1-\tau_Q) f(\tau_Q)\right]\end{aligned}$$
$$\begin{aligned}
f(\tau_Q)=\left\{
\begin{array}{ll} \displaystyle
\arcsin^2\frac{1}{\sqrt{\tau_Q}} & \tau_Q \geq 1 \\
\displaystyle -\frac{1}{4}\left[ \log\frac{1+\sqrt{1-\tau_Q}}
{1-\sqrt{1-\tau_Q}}-i\pi \right]^2 \hspace{0.5cm} & \tau_Q<1
\end{array} \right.\end{aligned}$$
$$\begin{aligned}
F_\Box & = & \frac{1}{S^2}\left\{ {\phantom{\frac{1}{1}}\!\!\!\!}4 S + 8 S m_Q^2 C_{ab}
-2S(S+\rho_c+\rho_d-8)m_Q^4 (D_{abc}+D_{bac}+D_{acb}) \right. \\
& + & \left. (\rho_c+\rho_d-8)m_Q^2\!\left[ {\phantom{\frac{1}{1}}\!\!\!\!}T_1 C_{ac} + U_1 C_{bc}
+ U_2 C_{ad} + T_2 C_{bd} - (TU-\rho_c\rho_d) m_Q^2 D_{acb}\right] \right\} \\
G_\Box & = & \frac{1}{S(TU-\rho_c\rho_d)}\left\{ {\phantom{\frac{1}{1}}\!\!\!\!}(T^2+\rho_c\rho_d-8T)
m_Q^2 \left[ {\phantom{\frac{1}{1}}\!\!\!\!}S C_{ab} + T_1 C_{ac} +T_2 C_{bd} - ST m_Q^2 D_{bac} \right]
\right. \\
& & \hspace{2.55cm}+(U^2+\rho_c\rho_d-8U)m_Q^2\left[ {\phantom{\frac{1}{1}}\!\!\!\!}S C_{ab} + U_1 C_{bc}
+U_2 C_{ad} - SU m_Q^2 D_{abc} \right] \\
& & \hspace{2.55cm} - (T^2+U^2-2\rho_c\rho_d) (T+U-8) m_Q^2 C_{cd} \\
& & \hspace{2.55cm} \left. - 2(T+U-8)(TU-\rho_c\rho_d) m_Q^4
(D_{abc}+D_{bac}+D_{acb}) {\phantom{\frac{1}{1}}\!\!\!\!}\right\}\end{aligned}$$
$$\begin{aligned}
S_z=0&:&\qquad A_1^{\mu\nu} = g^{\mu\nu}-\frac{p_a^\nu p_b^\mu }{(p_a p_b)} \\
S_z=2&:&\qquad A_2^{\mu \nu} = g^{\mu \nu}
+\frac{p_c^2 p_a^\nu p_b^\mu}{p_T^2 (p_a p_b)}
-\frac{2 (p_b p_c) p_a^\nu p_c^\mu}{p_T^2 (p_a p_b)}
-\frac{2 (p_a p_c) p_b^\mu p_c^\nu}{p_T^2 (p_a p_b)}
+\frac{2 p_c^\mu p_c^\nu}{p_T^2} \\ \\
&&\text{with} \qquad \qquad p_T^2 =
2\frac{(p_ap_c)(p_bp_c)}{(p_ap_b)}-p_c^2 \\ \\
&&A_1 \cdot A_2 = 0 , \qquad A_1 \cdot A_1 = A_2 \cdot A_2 = 2\end{aligned}$$
$$\begin{aligned}
{\cal M}\left(g_ag_b \to H_cH_d \right)&=&
{\cal M}_\triangle^h
+{\cal M}_\triangle^H
+{\cal M}_\Box \\
{\cal M}_\triangle^{h/H}&=&
\frac{G_F \alpha_s {\hat{s}}}{2 \sqrt{2} \pi} \;
C_\triangle^{h/H} \; F_\triangle {A_1}_{\mu \nu} \;
\epsilon_a^\mu \epsilon_b^\nu \; \delta_{ab} \\
{\cal M}_\Box&=&
\frac{G_F \alpha_s {\hat{s}}}{2 \sqrt{2} \pi} \;
C_\Box \; ( F_\Box {A_1}_{\mu \nu} + G_\Box {A_2}_{\mu \nu} ) \;
\epsilon_a^\mu \epsilon_b^\nu \; \delta_{ab} \end{aligned}$$
[**A2. Form factors for a mixed scalar–pseudoscalar pair**]{} $\mathbf{g_a g_b \to A_c H_d}$
$$\begin{aligned}
F_\triangle^A & = & -2 {m_Q^2}C_{ab} = \tau_Q f(\tau_Q) \\
F_\triangle^Z & = & \left( 1-\frac{{\hat{s}}}{M_Z^2} \right) \frac{\rho_c-\rho_d}{S}
\left[ {\phantom{\frac{1}{1}}\!\!\!\!}1 + 2 m_Q^2 C_{ab} \right]
= \left( 1-\frac{{\hat{s}}}{M_Z^2} \right) \frac{\rho_c - \rho_d}{S}
\left[ {\phantom{\frac{1}{1}}\!\!\!\!}1 - \tau_Q f(\tau_Q) \right]\end{aligned}$$
$$\begin{aligned}
F_\Box & = & \frac{1}{S^2}\left\{ {\phantom{\frac{1}{1}}\!\!\!\!}-2S(S+\rho_c-\rho_d) m_Q^4 (D_{abc}+D_{bac}+D_{acb}) \right. \\
& & \left. + (\rho_c-\rho_d)m_Q^2 \left[ {\phantom{\frac{1}{1}}\!\!\!\!}T_1 C_{ac} + U_1 C_{bc}
+ U_2 C_{ad} + T_2 C_{bd} - (TU-\rho_c\rho_d) m_Q^2 D_{acb}\right] \right\} \\
G_\Box & = & \frac{1}{S(TU-\rho_c\rho_d)}\left\{ {\phantom{\frac{1}{1}}\!\!\!\!}(U^2-\rho_c\rho_d)
m_Q^2 \left[ {\phantom{\frac{1}{1}}\!\!\!\!}S C_{ab} + U_1 C_{bc} + U_2 C_{ad} - SU m_Q^2 D_{abc} \right]
\right. \\
& & \hspace{2.60cm} -(T^2-\rho_c\rho_d) m_Q^2
\left[ {\phantom{\frac{1}{1}}\!\!\!\!}S C_{ab} + T_1 C_{ac} + T_2 C_{bd} - ST m_Q^2 D_{bac} \right] \\
& & \hspace{2.60cm} +\left[{\phantom{\frac{1}{1}}\!\!\!\!}(T+U)^2-4\rho_c\rho_d\right](T-U)m_Q^2C_{cd} \\
& & \hspace{2.60cm} \left. + 2(T-U)(TU-\rho_c\rho_d) m_Q^4
(D_{abc}+D_{bac}+D_{acb}) {\phantom{\frac{1}{1}}\!\!\!\!}\right\}\end{aligned}$$
$$\begin{aligned}
S_z=0&:&\qquad A_1^{\mu\nu} = \frac{1}{(p_a p_b)} \epsilon^{\mu \nu p_a p_b} \\
S_z=2&:&\qquad A_2^{\mu \nu} = \frac{ p_c^\mu \epsilon^{\nu p_a p_b p_c}
+p_c^\nu \epsilon^{\mu p_a p_b p_c}
+(p_b p_c) \epsilon^{\mu \nu p_a p_c}
+(p_a p_c) \epsilon^{\mu \nu p_b p_c} }
{(p_a p_b) p_T^2} \\ \\
&&\text{with} \qquad \qquad p_T^2 =
2\frac{(p_ap_c)(p_bp_c)}{(p_ap_b)}-p_c^2 \\ \\
&&A_1 \cdot A_2 = 0 \qquad A_1 \cdot A_1 = A_2 \cdot A_2 = 2\end{aligned}$$
$$\begin{aligned}
{\cal M}\left(g_ag_b \to A_cH_d \right)&=&
{\cal M}_\triangle^A
+{\cal M}_\triangle^Z
+{\cal M}_\Box \\
{\cal M}_\triangle^{A/Z}&=&
\frac{G_F \alpha_s {\hat{s}}}{2 \sqrt{2} \pi} \;
C_\triangle^{A/Z} \; F_\triangle^{A/Z} {A_1}_{\mu \nu} \;
\epsilon_a^\mu \epsilon_b^\nu \; \delta_{ab} \\
{\cal M}_\Box&=&
\frac{G_F \alpha_s {\hat{s}}}{2 \sqrt{2} \pi} \;
C_\Box \; ( F_\Box {A_1}_{\mu \nu} + G_\Box {A_2}_{\mu \nu} ) \;
\epsilon_a^\mu \epsilon_b^\nu \; \delta_{ab} \end{aligned}$$
[**A3. Form factors for two pseudoscalar Higgs bosons**]{}[^7] $\mathbf{ g_a g_b \to A_c A_d}$
$$\begin{aligned}
F_\triangle & = & \frac{2}{S} \left\{ 2+(4 - S) m_Q^2 C_{ab} \right\}
= \tau_Q \left[{\phantom{\frac{1}{1}}\!\!\!\!}1+(1-\tau_Q) f(\tau_Q)\right]\end{aligned}$$
$$\begin{aligned}
\AA\end{aligned}$$
$$\begin{aligned}
S_z=0&:&\qquad A_1^{\mu\nu} = g^{\mu\nu}-\frac{p_a^\nu p_b^\mu }{(p_a p_b)} \\
S_z=2&:&\qquad A_2^{\mu \nu} = g^{\mu \nu}
+\frac{p_c^2 p_a^\nu p_b^\mu}{p_T^2 (p_a p_b)}
-\frac{2 (p_b p_c) p_a^\nu p_c^\mu}{p_T^2 (p_a p_b)}
-\frac{2 (p_a p_c) p_b^\mu p_c^\nu}{p_T^2 (p_a p_b)}
+\frac{2 p_c^\mu p_c^\nu}{p_T^2} \\ \\
&&\text{with} \qquad \qquad p_T^2 =
2\frac{(p_ap_c)(p_bp_c)}{(p_ap_b)}-p_c^2 \\ \\
&&A_1 \cdot A_2 = 0 \qquad A_1 \cdot A_1 = A_2 \cdot A_2 = 2\end{aligned}$$
$$\begin{aligned}
{\cal M}\left(g_ag_b \to A_cA_d \right)&=&
{\cal M}_\triangle^h
+{\cal M}_\triangle^H
+{\cal M}_\Box \\
{\cal M}_\triangle^{h/H}&=&
\frac{G_F \alpha_s {\hat{s}}}{2 \sqrt{2} \pi} \;
C_\triangle^{h/H} \; F_\triangle {A_1}_{\mu \nu} \;
\epsilon_a^\mu \epsilon_b^\nu \; \delta_{ab} \\
{\cal M}_\Box&=&
\frac{G_F \alpha_s {\hat{s}}}{2 \sqrt{2} \pi} \;
C_\Box \; ( F_\Box {A_1}_{\mu \nu} + G_\Box {A_2}_{\mu \nu} ) \;
\epsilon_a^\mu \epsilon_b^\nu \; \delta_{ab}\end{aligned}$$
[99]{} P.W. Higgs, Phys. Rev. Lett. [**12**]{} (1964) 132 and Phys. Rev. [**145**]{} (1966) 1156; F. Englert and R. Brout, Phys. Rev. Lett. [**13**]{} (1964) 321; G.S. Guralnik, C.R. Hagen and T.W. Kibble, Phys. Rev. Lett. [**13**]{} (1964) 585. A. Djouadi, H.E. Haber and P.M. Zerwas, Report DESY 95–214. W.–Y. Keung, Mod. Phys. Lett. [**A2**]{} (1987) 762; D.A. Dicus, K.J. Kallianpur and S.S.D. Willenbrock, Phys. Lett. [**B200**]{} (1988) 187; O.J.P. Eboli, G.C. Marques, S.F. Novaes and A.A.Natale, Phys. Lett. [**B197**]{} (1987) 269. E.W.N. Glover and J.J. van der Bij, Nucl. Phys. [**B309**]{} (1988) 282. J. Dai, J.F. Gunion and R. Vega, Phys. Rev. Lett. [**71**]{} (1993) 2699; D. Froidevaux and E. Richter–Was, Z. Phys. [**C67**]{} (1995) 213; J. Dai, J.F. Gunion and R. Vega, Preprint UCD–95–25. For a review see J.–F. Grivaz, Proceedings, International Europhysics Conference on High Energy Physics, Brussels 1995. A. Djouadi, J. Kalinowski and P.M. Zerwas, Report DESY 95–211. V. Barger, M. Berger, A. Stange and R. Phillips, Phys. Rev. [**D45**]{} (1992) 4128; Z. Kunszt and F. Zwirner, Nucl. Phys. [**B385**]{} (1992) 3. M. Carena, J. Espinosa, M. Quiros and C.E.M. Wagner, Phys. Lett. [**B355**]{} (1995) 209. T. Plehn, M. Spira and P.M. Zerwas, in preparation. H. Georgi, S. Glashow, M. Machacek and D.V. Nanopoulos, Phys. Rev. Lett. [**40**]{} (1978) 692. M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, Nucl. Phys. [**B453**]{} (1995) 17. J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. [**B106**]{} (1976) 292; A.I. Vainshtain, M.B. Voloshin, V.I. Sakharov and M.A. Shifman, Sov. J. Nucl. Phys. [**30**]{} (1979) 711. B.A. Kniehl and M. Spira, Z. Phys. [**C69**]{} (1995) 77. A. Menzione, Proceedings, International Europhysics Conference on High Energy Physics, Brussels 1995. G. Passarino and M. Veltman, Nucl. Phys. [**B160**]{} (1979) 151; B.A. Kniehl, Phys. Rev. [**D42**]{} (1990) 3100. M. Glück, E. Reya and A. Vogt, Z. Phys. [**C53**]{} (1992) 127. S. Bethke, Proceedings QCD 94, Montpellier 1994; Report RWTH Aachen, PITHA–94–30.
[**FIGURE CAPTIONS**]{}\
[**Fig.1:**]{} Trilinear Higgs couplings in the MSSM as a function of the pseudoscalar Higgs mass $M_A$ for two representative values ${\mbox{tg$\beta$}}=1.5$ and 30 \[no mixing in the stop sector, c.f. Ref.[@M1]\].\
[**Fig.2:**]{} Generic diagrams describing SM Higgs pair production in gluon–gluon collisions: a) triangle and b) box contributions.\
[**Fig.3:**]{} Total cross section of SM Higgs pair production at the LHC \[$\sqrt{s} = 14$ TeV\]. The top mass has been chosen to be $m_t
= 175$ GeV and the bottom mass $m_b = 5$ GeV. The GRV parametrizations [@11] of the parton densities have been adopted. The factorization and renormalization scales are identified with the invariant mass of the Higgs boson pair. The world average value $\alpha_s(M_Z) = 0.118$ [@12] has been chosen for the QCD coupling.\
[**Fig.4:**]{} Generic diagrams contributing to pair production of CP–even MSSM Higgs bosons in gluon–gluon collisions $gg\to hh,
hH,HH$: a) triangle and b) box contributions.\
[**Fig.5:**]{} Total cross sections for pair production of CP–even MSSM Higgs bosons in gluon–gluon collisions at the LHC: a) $hh$, b) $hH$ and c) $HH$ pair production for ${\mbox{tg$\beta$}}=1.5$ and 30. The secondary axes present the scalar Higgs masses $M_{h/H}$ corresponding to the pseudoscalar masses $M_A$ for both values of ${\mbox{tg$\beta$}}$. The same parameters and parton densities as in Fig.3 have been adopted. The dashed curves in Fig.5a represent the resonant contributions $gg\to H\to hh$.\
[**Fig.6:**]{} Diagram of the $Z$ boson $s$–channel contribution to mixed CP–even/CP–odd MSSM Higgs pair production in gluon–gluon collisions and box diagram.\
[**Fig.7:**]{} Total cross sections for mixed CP–even/CP–odd pair production of MSSM Higgs bosons in gluon–gluon collisions at the LHC for ${\mbox{tg$\beta$}}=1.5$ and 30: a) $hA$ and b) $HA$ production. The secondary axes show the scalar Higgs boson masses corresponding to the pseudoscalar mass $M_A$ for both values of ${\mbox{tg$\beta$}}$. The same parameters as in Fig.3 have been chosen.\
[**Fig.8:**]{} Generic triangle and box diagrams for pair production of CP–odd MSSM Higgs bosons in gluon–gluon collisions.\
[**Fig.9:**]{} Total cross section of CP–odd MSSM Higgs pair production in gluon–gluon collisions at the LHC for ${\mbox{tg$\beta$}}=1.5$ and 30. The parameters are the same as in Fig.3. The dashed curves represent the resonant contributions $gg\to H\to AA$.
[^1]: Address after Jan.1,1996: TH Division, CERN, CH–1211 Geneva 23, Switzerland
[^2]: Supported by Bundesministerium für Bildung und Forschung (BMBF), Bonn, under Contract 05 6 HH 93P (5), and by EU Program [*Human Capital and Mobility*]{} through Network [*Physics at High Energy Colliders*]{} under Contract CHRX–CT93–0357 (DG12 COMA).
[^3]: Following standard notations we have taken care that no confusion will arise from using the same symbol for the SM and the heavy CP–even MSSM Higgs particle.
[^4]: The derivation of the SM cross sections has nicely been described in Ref.[@1]. We will briefly discuss this case again to exemplify the techniques for the simplest scenario of Higgs pair production, before generalizing the analysis for unequal masses and pseudoscalar Higgs particles in the MSSM.
[^5]: In the leading logarithmic approximation the scale is set effectively by the Higgs mass since the invariant energy $\sqrt{\hat s}$ is of the same order as $M_H$ for the cross sections we shall study.
[^6]: In analogy to single Higgs production via gluon fusion, QCD corrections are expected to enhance the event rate significantly. However, they are known only for the triangle [@7] and not for the box; therefore they are not taken into account in the numerical analysis.
[^7]: We present the formulae for different masses of the two pseudoscalar particles so that they can also be applied to non–minimal SUSY extensions in which several pseudoscalar particles in general belong to the Higgs spectrum.
|
---
abstract: 'Much work has been done in understanding human creativity and defining measures to evaluate creativity. This is necessary mainly for the reason of having an objective and automatic way of quantifying creative artifacts. In this work, we propose a regression-based learning framework which takes into account quantitatively the essential criteria for creativity like novelty, influence, value and unexpectedness. As it is often the case with most creative domains, there is no clear ground truth available for creativity. Our proposed learning framework is applicable to all creative domains; yet we evaluate it on a dataset of movies created from IMDb and Rotten Tomatoes due to availability of audience and critic scores, which can be used as proxy ground truth labels for creativity. We report promising results and observations from our experiments in the following ways : 1) Correlation of creative criteria with critic scores, 2) Improvement in movie rating prediction with inclusion of various creative criteria, and 3) Identification of creative movies.'
author:
- |
Disha Shrivastava$^\dagger$ Saneem Ahmed CG$^\dagger$ Anirban Laha$^\dagger$ Karthik Sankaranarayanan$^\dagger$\
$^\dagger$IBM Research India,\
{dishriva,saneem.cg,anirlaha,kartsank}@in.ibm.com
bibliography:
- 'sigproc.bib'
title: A Machine Learning Approach for Evaluating Creative Artifacts
---
|
---
abstract: 'The purely mathematical root of the dequantization constructions is the quest for a sheafification needed for presheaves on a noncommutative space. The moment space is constructed as a commutative space, approximating the noncommutative space appearing as a dynamical space, via a stringwise construction. The main result phrased is purely mathematical, i.e. the noncommutative stalks of some sheaf on the noncommutative space can be identified to stalks of some sheaf associated to it on the commutative geometry (topology) of the moment space. This may be seen as a (partial) inverse to the deformation–quantization idea, but in fact with a much more precise behaviour of stalks of sheaves. The method, based on minimal axiomatics necessary to rephrase continuity principles in terms of partial order (noncommutative topology) exclusively, leads to the appearance of objects like strings and (M–)branes. Also, spectral families and observables may be defined and studied as separated filtrations on the noncommutative topology! We highlight the relation with pseudo–places and generalized valuation theory. Finally, we hint at a new notion of “space” as a dynamical system of noncommutative topologies with the same commutative shadow (4 dimensional space–time if you wish) and variable (but isomorphic) moment spaces, which for special choices may be thought of as a higher dimensional brane–space.'
author:
- |
Freddy Van Oystaeyen\
Department of Mathematics & Computer Science\
University of Antwerp\
Middelheimcampus\
Middelheimlaan 1, 2020 Antwerpen, Belgium\
e-mail : [email protected]
title: |
Dequantization of Noncommutative Spaces\
and Dynamical Noncommutative Geometry
---
Introduction {#introduction .unnumbered}
============
Noncommutative spaces obtained by quantization-deformation may to some extent be traced back to the base space they have been deformed from. A noncommutative topology in the sense of \[VO1\] has a so-called commutative shadow that turns out to be a lattice. However, the lack of geometric points in a noncommutative space contrasts the set-theory based commutative geometry with its function theory and corresponding analysis. Looking for a converse construction for quantization, a better commutative approximation of a noncommutative space is needed; this construction should allow tracing of physical aspects like observables, spectral families etc...
Different notions of noncommutative space have been introduced e.g. noncommutative manifolds \[Co\], noncommutative “quantized” algebras \[ATV\], \[VdB\], general quantization-deformations \[Ko\]. In these theories the actual geometric object is left as a virtual object and results deal usually with noncommutaive algebras or categories thought of as rings of functions or categories of modules.
In the author’s approach, started in \[VO1\], a noncommutative topology is at hand and then sheaf theory should to some extent replace function theory. Applied to Physics this model seems to allow some explanation of quantum phenomena but for explicite calculations and geometric reasoning a commutative model is unavoidable because at some point coordinatization and the use of real or complex numbers becomes necessary. In the model we propose these commutative techniques may be carried out in the dequantized space. The construction of this commutative approximation of noncommutative space derives from a dynamical version of noncommutative geometry that may also be seen as a noncommutaive topological blow-up and blow-down construction. Restricting assumptions to bare necessities one finds that all structural properties only depend on a few intuitive axioms at the level of ordered (partial) structures. Respecting this philosophy we started from a totally ordered set (not even necessarily a group), $T$ and suitably connected noncommutative spaces $\Lambda_t, t\in T$. A stringwise construction then yields a new spectral space, called the [**moment space**]{}, at each $t\in T$, with its topology in the classical sense i.e. a commutative space.
Sheaves on lattices were already termed quantum sheaves, but now we view (pre-)sheaves on noncommutative topologies fitting together in a kind of dynamical geometry and connect them to sheaves on the moment spaces. The idea that commutative moment spaces are good approximations of noncommutative spaces is reflected in the main result : the stalks at points of the moment space at $t\in T$ equal stalks at a point at some $\Lambda_{t'}$, $t'\in T$ for some $t'$ in a prescribed closed $T$-interval containing $t$. The philosophy here is that there may not exist enough geometric points in the noncommutative geometry $(\Lambda_t)$ at $t\in T$ but it holds in the moment space at $t$ because it encodes dynamical information in some $T$-interval containing $t$. The construction is abstract and rather general, but when looking for applications in Physics one is only interested in one case ... reality ? Then one would think of $T$ as a multidimensional irreversible “time” with “dimension” big enough to make up for the missing geometruc points in the noncommutative space $\Lambda_t$ at $t\in T$, i.e. the $T$-dimension explains the difference in dimension between the noncommutative space, or the moment space, and that of its space of points (the commutative shadow). Whereas the commutative shadow represents the abstract space where the mathematics takes place, e.g. space-time, the moment space represents the space where the mathematics of observed and measured items takes place, and because observing or measuring takes time the latter space has to encode dynamical data. Stated as a slogan : “Observation Creates Space from Time”. Moreover, when observing a smallest possible entity, we would like to think of this as an observed point i.e. a point in the moment space, at $t$ say. Tracing this via the dynamical noncommutative geometry to the commutative shadow (thinking of this as the fixed space we calculate in) the observed point appears as a string (it may be open or closed). On the other hand the point in the moment space is via a string of temporal points (not geometric points) in the noncommutative space also traceable to a possibly “higher dimensional string” in the moment space, in fact a tube-like string connecting opens in (the topologies of the noncommutative spaces blown down to) the spectral topology of the moment space (also identified to one commutative space for example). Of course this suggests that a version of $M$-theory already appears in the mathematical formalism explaining the transfer between commutative shadow and moment space.
Moreover the incertainty principle also marks the difference between the two commutative spaces, in some sense expressing a quantum-commutativity of the noncommutative spaces, for example phrased in sheaf theoretical terms.
We have aimed at a rather self contained presentation of basic facts and definitions of the new mathematical objects in particular ... generalized Stone spaces and noncommutative topology.
The methods of this paper can be adapted to other types of noncommutative spaces e.g.quantales,... . Further generalization, e.g. to noncommutative Grothendieck topologies, has not been included, this would be an exercise using some material from \[VO1\]. We include some comments about specific examples and further mathematical elaborations using deformations of affine spaces or other nice classical (algebraic) varieties is left as work in progress.
Preliminaries on Noncommutative Topology
========================================
We consider (partially ordered) $\Lambda$ with $0$ and $1$. When $\Lambda$ is equipped with operations $\wedge$ and $\vee$, we say that $\Lambda$ is a noncommutative topology if the following axioms hold :
for $x,y\in\Lambda, x\wedge y\le x$ and $x\wedge y\le y$ and $x\wedge
1=1\wedge x=x$, $x\wedge 0=0\wedge x=0$, moreover $x\wedge\ldots \wedge x=0$ if and only if $x=0$.
For $x,y,z\in\Lambda$, $x\wedge y\wedge z=(x\wedge y)\wedge z=x\wedge (y\wedge
z)$, and if $x\le y$ then $z\wedge x\le z\wedge y, x\wedge z\le y\wedge z$.
Properties similar to i. and ii. with respect to $\vee$, in particular $x\vee
x\vee\ldots\vee x=1$ if and only if $x=1$.
Let ${\rm
id}_{\wedge}(\Lambda)=\{\lambda\in\Lambda,\lambda\wedge\lambda=\lambda\}$; for $x\in{\rm id}_{\wedge}(\Lambda)$ and $x\le z$ we have : $x\vee(x\wedge z)\le
(x\vee x)\wedge z$, $x\vee (z\wedge x)\le (x\vee z)\wedge x$.
For $x\in\Lambda$ and $\lambda_1,\ldots,\lambda_n\in\Lambda$ such that $1=\lambda_1\vee\ldots\vee\lambda_n$, we have $x=(x\wedge\lambda_1)\vee\ldots\vee(x\wedge\lambda_n)=(\lambda_1\wedge
x)\vee\ldots\vee(\lambda_n\wedge x)$.
There are left (right) versions of this definition as introduced in \[VO.1\]., but we do not go into this here. In fact we restrict attention to the situation where $\vee$ is a commutative operation and $\Lambda$ is $\vee$-complete i.e. for an arbitrary family ${\cal F}$ of elements in $\Lambda$, $\vee{\cal F}$ exists in $\Lambda$, where $\vee{\cal F}$ is characterized by the property : $\lambda\le \vee{\cal F}$ for all $\lambda\in{\cal F}$ and if $\lambda\le\mu$ for all $\lambda\in{\cal F}$ then $\vee{\cal F}\le\mu$.
In case only v) is dropped we call $\Lambda$ a [**skew topology**]{}; this is sometimes interesting e.g. the lattice $L(H)$ of a Hilbert space $H$ is a skew topology and condition v) does not hold ! Here $L(H)$ is the lattice of closed linear subspaces of $H$ with respect to intersection and closure of the sum. The restriction to abelian $\vee$ (most interesting examples are like this) entails that ${\rm id}_{\wedge}(\Lambda)$ is closed under the operation $\vee$. Moreover on ${\rm id}_{\wedge}(\Lambda)$ we introduce a new operation defined by $\sigma$$\tau=\vee\{\gamma\in{\rm
id}_{\wedge}(\Lambda),\gamma\le\sigma\wedge\tau\}$ for $\sigma,\tau\in{\rm
id}_{\wedge}(\Lambda)$. Let us write $SL(\Lambda)$ for the set ${\rm
id}_{\wedge}(\Lambda)$ with the operations and $\vee$; then $SL(\Lambda)$ is easily checked to be again a skew topology (with $\vee$ commutative and being $\vee$-complete).
Lemma (cf. \[VO.1\] or \[VO.2\] 2.2.3. and 2.2.5)
-------------------------------------------------
If $\Lambda$ is a skew topology $\vee$-complete with respect to a commutative operation $\vee$, then $SL(\Lambda)$ is a lattice satisfying the modular inequality.
A subset $X \subset \Lambda$ is [**directed**]{} if for every $x,y\in X$ there is a $z\in X$ such that $z\le x, z\le y$; we say that $X$ is a [**filter**]{} in $\Lambda$ if it is directed and for $x\in X$, $x\le y$ yields $y\in X$. Two directed sets $A$ and $B$ in $\Lambda$ are equivalent if they define the same filter $\Aol=\Bol$ where for any directed set $A$ we put $\Aol=\{\lambda\in\Lambda$, there is an $a\in A$ such that $a\le\lambda\}$. Let $D(\Lambda)$ be the set of directed subsets in $\Lambda$ and we write $A\sim B$ of the directed subsets $A$ and $B$ are equivalent, we write $[A]$ for the equivalence class of $A$ and let $C(\Lambda)$ be the set of classes of directed subsets of $\Lambda$. We introduce a partial order in $C(\Lambda)$ by putting $A\le B$ if $\Bol\subset \Aol$ and $[A]\le [B]$ induced by the partial order on ${\cal D}(\Lambda)$ defined by the foregoing. For $A$ and $B$ in ${\cal D}(\Lambda)$ define $A\dot{\wedge} B=\{a\wedge b, a\in A, b\in
B\}, A\dot{\vee}B=\{a\vee b, a\in A, b\in B\}$ and : $[A]\wedge
[B]=[A\dot{\wedge}B],[A]\vee[B]=[A\dot{\vee}B]$.
Lemma
-----
If $\Lambda$ is a skew topology, resp. a noncommutative topology, then so is $C(\Lambda)$ with respect to the partial order and operations $\wedge$ and $\vee$ as defined above. The canonical map $\Lambda\r
C(\Lambda),\lambda\mapsto [\{\lambda\}]$ is a map of skew topologies.
We simplify notations by writing $[\{\lambda\}]=[\lambda]$ and call such an element [**classical**]{} in $C(\Lambda)$.
A directed set $A$ in $\Lambda$ is [**idempotently directed**]{} if for every $a\in A$ there exists an $a'\in A\cap {\rm id}_{\wedge}(\Lambda)$ such that $a'\le a$; in this case $[A]\in i_{\wedge}(C(\Lambda))$ but these elements of $C(\Lambda)$ may be thought of as obtained from a directed set having a cofinal subset of “commutative” opens (the idempotents belonging to the commutative shadow $S((\Lambda))$. We write ${\rm Id}_{\wedge}(C(\Lambda))$ for the subset of ${\rm id}_{\wedge}(C(\Lambda))$ consisting of the classes of idempotently directed subsets of $\Lambda$; the elements $[A]$ of ${\rm
Id}_{\wedge}({\rm Id}_{\wedge}(C(\Lambda))$ are called [**strongly idempotents**]{}. We identify $\Lambda$ and the image of $\Lambda\r C(\Lambda)$, then observe that ${\rm Id}_{\wedge}(C(\Lambda))\cap \Lambda={\rm
id}_{\wedge}(\Lambda)$. We shall write $\prod(\Lambda)$ for the skew (noncommutative) topology obtained by talking $\wedge$-finite bracketed expression $P(\wedge,\vee,x_i)$ in terms of strong idempotents $x_i\in{\rm
Id}_{\wedge}(C(\Lambda))$; similarly we write $T(\Lambda)$ for the skew (noncommutative) topology obtained by taking $\wedge$-finite bracketed expressions $p(\wedge,\vee,\lambda_i)$ in idempotents $\lambda_i\in {\rm
id}_{\wedge}(\Lambda)$.
It is not hard to verify : $\prod(\Lambda)=\prod(T(\Lambda))$. Moreover $C(T(\Lambda))$ satifies the same axions (i. to v.) as $\Lambda$ and $T(\Lambda)$ but with respect to ${\rm Id}_{\wedge}(C(\Lambda))$. The “strong” commutative shadow of $\prod$ is obtained by restricting on ${\rm id}_{\wedge}(C(\Lambda))$ to ${\rm
Id}_{\wedge}(C(\Lambda))$ and viewing $SL_{s}(\prod)$ as the lattice structure induced on ${\rm Id}_{\wedge}(C(\Lambda))$.
Lemma
-----
If $\Lambda$ is a $\vee$-complete noncommutative topology such that $\vee$ is commutative then : $SL_s(\prod)=C(SL(\Lambda))$.
Definition. Generalized Stone Topology
--------------------------------------
Consider a skew topology $\Lambda$ and $C(\Lambda)$. For $\lambda\in\Lambda$, let $O_{\lambda}\subset C(\Lambda)$ be given by $O_{\lambda}=\{[A],\lambda\in\Aol\}$. It is very easy to verify : $O_{\lambda\wedge\mu}\subset O_{\lambda}\cap O_{\mu},O_{\lambda\vee\mu}\supset
O_{\lambda}\cup O_{\mu}$, hence the $O_{\lambda}$ define a basis for a topology on $C(\Lambda)$ termed : generalized Stone topology. This definition obtains a more classical meaning when related to suitable point-spectra constructed in $C(\Lambda)$.
We say that $[A]$ in $C(\Lambda)$ is a [**minimal point of $\Lambda$**]{} if $\Aol$ is a maximal filter, i.e. $\Aol\neq \Lambda$ but if $\Aol\subsetneqq B$ where $B$ is a filter then $B=\Lambda$; it follows that a minimal point is necessarily in ${\rm id}_{\wedge}(C(\Lambda))$ and it is indeed a minimal nonzero element of the poset $C(\Lambda)$. An [**irreducible point**]{} $[A]$ of $\Lambda$ is characterized by either one of the followingequivalent properties :
$[A]\le \vee\{[A_{\alpha}],\alpha\in{\cal A}\}$ yields $[A]\le[A_{\alpha}]$ for some $\alpha\in{\cal A}$.
If $\vee\{\lambda_{\alpha},\alpha\in{\cal A}\}\in\Aol$ then $\lambda_{\alpha}\in \Aol$ for some $\alpha\in{\cal A}$.
More general types of points may be considered, e.g.the elements of a so-called quantum-basis, cf \[VO.2\], but we need not go into this here. Under some suitable condition often (but not always) present in examples the irreducible points in ${\rm Id}_{\wedge}(C(\Lambda))$ are exactly those that are $\vee$-irreducible in $C(\Lambda)$ (e.g. if $\Lambda$ satisfies the weak FDI property, cf. \[VO.1\], Proposition 5.9.).
Define the (irreducible) [**point-spectrum**]{} by putting : ${\rm
Sp}(\Lambda)=\{[p],[p]$ an (irreducible) point of $\Lambda\}$. Put $p(\lambda)=\{[p]\in {\rm Sp}(\Lambda),[p]\le [\lambda]\}$ for $\lambda\in\Lambda$, then $p(\lambda\wedge\mu)\subset p(\lambda)\cap p(\mu),
p(\lambda\vee \mu)=p(\lambda)\cup p(\mu)$. Thus the $p(\lambda)$ define a basis for a topology on ${\rm Sp}(\lambda)$ called the point-topology. Write $SP(\Lambda)$ for $Sp(\Lambda)\cap {\rm Id}_{\wedge}(C(\Lambda))$ and refer to this as the [**Point-spectrum**]{} (capital $P$). For $\lambda\in \Lambda$ we consider $P(\lambda)=\{[P]\in {\rm SP}(\Lambda),[P]\le [\lambda]\}$ and this indices the Point-topology on ${\rm SP}(\Lambda)$; this time we even have $P(\lambda\wedge\mu)=P(\lambda)\cap P(\mu)$ and this time we even have $P(\lambda\wedge\mu)=P(\lambda)\cap P(\mu)$ and this defines a topology on ${\rm SP}({\rm SL}(C(\Lambda)))$. Similar constructions may be applied to the minimal point-spectrum $Q{\rm Sp}(\Lambda)$ on ${\rm Sp}(\Lambda)$ the generalized Stone topology is nothing but the point-topology. In the foregoing $\Lambda$ may be replaced by $T(\Lambda)$ or $\prod(\Lambda)$ with topologies induced by the generalized Stone topology on the point spectra always again being called : generalized Stone topology. Finally the generalized Stone topology can also be defined on the commutative shadow ${\rm
SL}(\Lambda)$, which is a modular lattice, and then the topology induced on $Q{\rm SP}({\rm SL}(\Lambda))$ is exactly the Stone topology of the Stone space of ${\rm SL}(\Lambda)$.
In the special case $\Lambda=L(H)$ (only a skew topology) the generalized Stone space defined on $Q{\rm SP}(L(H))$ is exactly the classical Stone space as used in Gelfand duality theory for $L(H)$. Warning : $L(H)$ does not satisfy the axiom v. and whereas $Q{\rm SP}(L(H))$ is rather big, ${\rm
Sp}(L(H))={\rm SP}(L(H))$ is empty ! Moreover over $L(H)$ there are no sheaves but there will be many sheaves over $C(L(H))$ making sheaffication of a separated presheaf over $L(H)$ possible over $Q{\rm SP}(L(H))$.
The foregoing fixes a context for results in the sequel, however the methods in Sections 2-5 are rather generic and can be applied to other notions of noncommutative spaces.
Dynamical Noncommutative Topology
=================================
The realization of a relation between the static and the dynamic goes far back, at least to D’Alembert but the notion that the first may be used to study the latter may be not completely well founded. Almost existentialistic problems defy the correctness of a mathematical description of so-called physical reality. So we choose for a converse approach constructing space as a dynamical noncommutative topological space and defining geometrical objects as existing over some parameter-intervals. Noncommutative continuity is introduced via the variation of an external parameter in a totally ordered set $T$ (if one wants to consider this as a kind of time, fine... but then this time is an index, not a dimension). This is philosophically satisfying because momentary observations which are only abstractly possible (real measurement takes time) put us in the discrete-versus-continuous situation of noncommutative geometry !
Let $T$ be a totally ordered set and for every $t\in T$ we give a noncommutative space $\Lambda_t$. This can have several meanings, in the sequel we take this to mean that $\Lambda_t$ is the generalized Stone space constructed on $C(X_t)$ for some skew topology $X_t$ as in Section 1. This is just to fix ideas, in fact one could just as well restrict to topologies induced by the genaralized Stone topology on point spectra of any type, see also Section 1 or \[VO1\], or take pattern topologies as introduced in \[VO1,2\], or go to other theories and take quantales etc... For $t\le t'$ in $T$ we have $\varphi_{tt'}:\Lambda_t\r \Lambda_{t'}$ which are poset maps respecting $\wedge$ and $\vee$; when $t=t'$ we take $\varphi_{tt}=1_{\Lambda_t}$ to be the identity of $\Lambda_t$ and when $t\le t'\le t''$ then we demand that $\varphi_{t't''}\circ \varphi_{tt'}=\varphi_{tt''}$, where notation for composition of maps is conventional i.e. writing the one acting last frst. If $A_t\subset \Lambda_t$ is a directed set then $\varphi_{tt'}(A_t)\subset
\Lambda_{t'}$, for $t\le t'$, is again a directed set.
It is easily verified that if we start from a system $\{X_t,\Psi_{tt'},T\}$ defined as above, we obtain a similar system $\{C(X_t),\Psi^e_{tt'},T\}$ where $\Psi^e_{tt'}:C(X_t)\r C(X_{t'})$ is given by putting : $\Psi^e_{tt'}([A])=[\Psi_{tt'}(A)]$, for $[A]\in C(X_t)$ and $t\in
t'$ in $T$. In case it is interesting to view $\Lambda_t$ as coming from some $X_t$ via $C(X_t)$ we may restrict attention to systems given by $\varphi_{tt'}$, $t\le t'$ in $T$, stemming from $\psi_{tt'}$ on $X_t$ by extension as indicated above. Note that not every system $\{C(X_t),\varphi_{tt'},T\}$ has to derive from a system $\{X_t,\psi_{tt'},T\}$ in general.
Lemma
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Any system of poset maps $\varphi_{tt'},t\le t'$ in $T$, defines a system of poset maps $\varphi^e_{tt'},t\le t'$ in $T$. If the maps $\varphi_{tt'}$ respect the operations $\wedge$ and $\vee$ in the $\Lambda_t$ then so does $\varphi^e_{tt'}$ for $C(\Lambda_t), t\in T$. In this situation $\varphi_{tt'}$ maps $\wedge$-idempotent elements of $\Lambda_t$ to $\wedge$-idempotent elements of $\Lambda_{t'}$ (also $\vee$-idempotent to $\vee$-idempotent) moreover if $[A_t]$ is strongly idempotent in $C(\Lambda_t)$ then $[\varphi_{tt'}(A_t)]$ is a strongly idempotent element of $C(\Lambda_{t'})$, for every $t\le t'$ in $T$,
#### Proof
First statements follow obviously from : for directed sets $A$ and $B$, $$\displaylines{
\varphi^e_{tt'}([A]\wedge[B])=\varphi^e_{tt'}([A\dot{\wedge}B])
=[\varphi_{tt'}(A\dot{\wedge} B)]\hfill \cr
\hfill =[\varphi_{tt'}(A)\dot{\wedge}\varphi
_{tt'}(B)]=[\varphi_{tt'}(A)]\wedge [\varphi_{tt'}(B)]}$$ for $t\le t'$ in $T$. Similar with respect to $\vee$, using $\dot{\vee}$. In case $\lambda\in\Lambda_t$ is idempotent in $\Lambda_t$ then $\varphi_{tt'}(\lambda)\wedge
\varphi_{tt'}(\lambda)=\varphi_{tt'}(\lambda\wedge
\lambda)=\varphi_{tt'}(\lambda$ for $t\le t'$ in $T$. Finally if $A$ is idempotently directed look at $\varphi_{tt'}(a)$ for $a\in A_t$; by assumption there exists some $\mu\in{\rm id}_{\wedge}(\Lambda_t)$ such that $\mu\le a$, thus $\varphi_{tt'}(\mu)\le \varphi_{tt'}(a)$ and $\varphi_{tt'}(\mu)\in
{\rm id}_{\wedge}(\varphi_{tt'}(A_t))$, for $t\le t'$ in $T$. Consequently $\varphi_{tt'}(A_t)$ is idempotently directed too.
The skew topology $\prod_t$, introduced after Lemma 1.2. is called the pattern topology of $X_t$, i.e. it is obtained by taking all $\wedge$-finite bracketed expressions with respect to $\vee$ and $\wedge$ in the letters of ${\rm Id}_{\wedge}(C(\Lambda_t))$.
Corollary
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The system $\{\Lambda_t,\varphi_{tt'}T\}$ induces a system $\{\prod_t,\varphi_{tt'}|\prod_t,T\}$, satisfying the same properties, on the pattern topologies.
In general the $\varphi_{tt'},t\le t'$, do not map poits of $\Lambda_t$ to points of $\Lambda_{t'}, t\le t'$, neither does $\varphi_{tt'}$ respect the operation of the commutative shadow $SL(\Lambda_t)$, i.e. the $\varphi_{tt'}$ do not necessarily induce a system on the commutatve shadows !
Axioms for Dynamical Noncommutative Topology (DNT)
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A system $\{\Lambda_t,\varphi_{tt'},T\}$ is called a DNT if the following five conditions are satisfied :
- Writing $0$, resp. 1, for the minimal, resp. maximal element of $\Lambda_t$ (we shall assume these exist throughout) then $\varphi_{tt'}(0)=0$, $\varphi_{tt'}(1)=1$ for all $t\le t'$ in $T$.
- For all $t\in T, \varphi_{tt}=1_{\Lambda_t}$ and for $t\le t'\le t''$ in $T$, $\varphi_{t't''}\circ \varphi_{tt'}=\varphi_{tt''}$. Moreover, all $\varphi_{tt'}$ preserve $\wedge$ and $\vee$. Hence DNT.1. and 2. just restate that $\{\Lambda_t,\varphi_{tt'},T\}$ is as before.
- If for some $t\in T$, $0<x<y$ in $\Lambda_t$, then there is a $t<t_1$ in $T$ such that for $z_1\in \Lambda_{t_1}$ satisfying $\varphi_{tt_1}(x)<z_1<\varphi_{tt_1}(y)$ there is a $z\in \Lambda_t$, $x<z<y$, such that $\varphi_{tt_1}(z)=z_1$.
- For every $t\in T$ and $0<x<z<y$ in $\Lambda_t$ there exist $t_1,t_2\in T$ such that $t_1<t<t_2$ and for every $t'\in ]t_1,t_2[$ we have either $t\le t'$ and $\varphi_{tt'}(x)<\varphi_{tt'}(z)<\varphi_{tt'}(y)$, or $t'\le t$ and then if $x'< y'$ in $\Lambda_{t'}$ exist such that $\varphi_{t't}(x')=x,\varphi_{t't}(y')=y$ then there also exist $z'$ in $\Lambda_{t'}$ such that $x'<z'<y'$ and $\varphi_{t't}(z')=z$. Taking the special case $y=1$ and $y'=1$ then we see that a nontrivial strict relation in $\Lambda_t$ is alive in an open $T$-interval containing $t$.
- [**$T$-local unambiguity**]{}. In the situation of DNT.3, resp. DNT.4, the $t_1\in T$, resp. $t_1$ and $t_2$, may be chosen such that $z\in \Lambda_t$ is the unique element such that $\varphi_{tt_1}(z)=z_1$, resp. $x',y',z'$ in $\Lambda_t$, are the unique elements such that $\varphi_{t't}(z')=z$, $\varphi_{t't}(y')=y$, $\varphi_{t't}(x')=x$, or when $t\le t'$ the $x,y,z,$ are unique elements mapping to $\varphi_{tt'}(z),\varphi_{tt'}(y),y_{tt'}(x)$ resp.
Since we are able to take finite intersection of open $T$-intervals in the totally ordered set $T$, we may extend the foregoing to finite chains of $0<x_1<x_2<\ldots <x_n,n\ge 3$.
Observe that the axioms allow that non-interacting elements, i.e. $x$ such that $0 < x < 1$ is the only orderrelation it is involved in, may appear and disappear momentarily. here disappearing means going to 0 under all $\varphi_{tt'}$, $t\le t'$, if $x\in\Lambda_t$.
Very often properties studied are only preserved in some $T$-interval, in particular this happens often when trying to derive a property of a related system from another one that may be globally defined for $T$. This leads to an interesting phenomenon, already encoding some aspect of the moment-spaces to be defined later.
Definition of Observed Truth
----------------------------
A statement in terms of finitely many ingredients of a DNT and depending on parametrization by $t\in T$ is said to be an [**observed truth**]{} at $t_0\in T$ if there is an open $T$-interval $]t_1,t_2[$ containing $t_0$, such that the statement holds for parameter values in this interval.
It seems that mathematical statements about a DNT turn into “observed truth” when one tries to check them in the commutative shadow, meaning on the negative side that many global (over $T$) properties of a DNT cannot be established globally over $T$ in the commutative world !
The noncommutative topologies $\Lambda_t$ considered in the sequel will be such that $\vee$ is commutative and $\vee$ of arbitrary families exist; in fact one may restrict to so-called “virtual topologies” as introduced in \[VO2\]; here we do not need to assume axiom ($\nu$) with respect to global covers, we may want to restrict to this case when needed. We refer to the special case mentioned as a DVT.
Proposition
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Let $\{\Lambda_t,\varphi_{tt'},T\}$ be a DVT and let $SL(\Lambda_t)$ be the commutative shadow of $\Lambda_t$ with maps $\varphi_{tt'}:SL(\Lambda_t)\r
SL(\Lambda_{t'})$, $t\le t'$ in $T$, just being the restrictions of the $\varphi_{tt'}$ (using same notation). Then the statement that $\{SL(\Lambda_t),\varphi_{tt'},T\}$ is a DVT too is an observed truth at every $t_0\in T$.
#### Proof
All $\varphi_{tt'}$ map $\wedge$-idempotents to $\wedge$-idempotents, cf. Lemma 2.1., so DNT.1. is obvious. For DNT.2 we have to check that $\varphi_{tt'}$ preserves on ${\rm id}_{\wedge}(\Lambda_t)$, since $=\vee$ now we have nothing to check for . Look at $t_0\in T$, $\varphi_{t_0t}:\Lambda_{t_0}\r \Lambda_t$ and $\sigma,\tau$ in ${\rm id}_{\wedge}(\Lambda_{t_0})$. If $\sigma<\tau$ then $\varphi_{t_0t}(\sigma)\le \varphi_{t_0t}(\tau)$ and $\varphi_{t_0t}(\sigma)$$\varphi_{t_0t}(\tau)=\varphi_{t_0t}(\sigma)=\varphi_{t_0t}(\sigma$$\tau)$, interchange the role of $\sigma$ and $\tau$ in case $\tau <\sigma$. So we may assume $\sigma$ and $\tau$ to be incomparable. Restricting $t$ to a suitable $T$-interval (DNT 5) we may assume that $\varphi_{t_0t}(\sigma)\neq\varphi_{t_0t}(\tau)$. Assume $\varphi_{t_0t}(\sigma$$\tau)<\varphi_{t_0t}(\sigma)$ $\varphi_{t_0t}(\tau)$.
If $\varphi_{t_0t}(\sigma$$\varphi_{t_0t}(\tau)=
\varphi_{t_0t}(\sigma_)$ (a similar argument will hold if $\sigma$ and $\tau$ are interchanged) then $\varphi_{t_0t}(\sigma)\le\varphi_{t_0t}(\tau)$, hence $\varphi_{t_0t}(\sigma)<\varphi_{t_0t}(\tau)$. Using DNT.5 again, taking $t$ close enough to $t_0$, we obtain $\sigma$$\tau <\sigma_1<\tau$ such that $\varphi_{t_0t}(\sigma$$\tau)<\varphi_{t_0t}(\sigma_1)
=\varphi_{t_0t}(\sigma)<\varphi_{t_0t}(\tau)$. Passing to $[t_0,t]$ small enough in order to have unambiguity for $\varphi_{t_0t}(\sigma)$, we arrive at $\sigma_1=\sigma$, contradicting the incomparability of $\sigma$ and $\tau$. Therefore we arrive at strict relations : $\varphi_{t_0t}(\sigma$$\tau)<\varphi_{t_0t}(\sigma)$ $\varphi_{t_0t}(\tau)<\varphi_{t_0t}(\sigma),\varphi_{t_0t}(\tau)$. We may moreover assume (DNT.3) that $t$ is close enough to $t_0$ so that there is a $z\in\Lambda_{t_0}$ such that $\sigma$$\tau<\sigma,\tau$ and $\varphi_{t_0t}(z)=\varphi_{t_0t}(\sigma)$$\varphi_{t_0t}(\tau)$. If $z$ is not $\wedge$-idempotent, then $\sigma$$\tau<z\wedge
z<\sigma$ would lead to $\varphi_{t_0t}(z\wedge
z)=\varphi_{t_0t}(\sigma)$$\sigma_{t_0t}(\tau)$ because $\varphi_{t_0t}$ respects $\wedge$ and the latter is idempotent in $\Lambda_t$; then $\varphi_{t_0t}(z)=\varphi_{t_0t}(z\wedge z)$ but the unambiguity guaranteed by the choice of $t$ close enough to $t_0$ (DNT.5) then yields $z=z\wedge z$ or $z\in {\rm id}_{\wedge}(\Lambda_{t_0})$. Thus $z=\sigma$$\tau$ by definition, a contradiction. Consequently, for $t$ in some small enough $T$-interval containing $t_0$ we have obtained :$\varphi_{t_0t}(\sigma$$\tau)=\varphi_{t_0t}(\sigma)$$\varphi_{t_0t}(\tau)$, thus DNT.2 is an observed truth. To check DNT.3, take $\sigma < \tau$ in ${\rm id}_{\wedge}(\Lambda_{t_1})$, $t<t_1$ such that $z_1\in {\rm id}_{\wedge}(\Lambda_{t_1})$ exists such that we have $\varphi_{tt_1}(\sigma)<z_1<\varphi_{tt_1}(\tau)$. Now by DNT.3. for $\{\Lambda_t,\varphi_{tt'},T\}$ there is a $z\in\Lambda_t, z < \tau$, such that $\varphi_{tt_1}(z)=z_1$, and DNT.5 for $(\Lambda_t,\varphi_{tt'},T\}$, used as in foregoing part of the proof, yields $\varphi_{tt_1}(z)=z_1$ with $z$ also $\wedge$-idempotent in $\Lambda_{t}$, for $t_1$ close enough to $t$. The proof of DNT.4 follows in the same way and DNT.5 is equally obvious because unambiguity in a suitable $T$-interval allows to pull back idempotency. Therefore all DNT-axioms hold for $\{SL(\Lambda_t),\varphi_{tt'},\tau)$ in a suitable $T$-interval, hence we have the observed truth statement that $\{SL(\Lambda_t),\varphi_{tt'},T\}$ is DNT.Now fix a notion of point i.e. either minimal point or irreducible point as in Section 1. We say that $\lambda_t\in \Lambda_t$ is a [**temporal point**]{} if $t\in ]t_0,t_1[$ such that for some $t'\in ]t_0,t_1[$ there is a point $p_{t'}\in\Lambda_{t'}$ such that : either $t\le t'$ and $\varphi_{tt'}(\lambda_t)=p_{t'}$, or $t'\le t$ and $\varphi_{t't}(p_{t'})=\lambda_t$; in the first case we say $\lambda_t$ is a [**future point**]{}, in the second case a [**past point**]{}. The system $\{\Lambda_t,\varphi_{tt'},T\}$ is said to be [**temporally pointed**]{} if for every $t\in T$ and $\lambda_t,\in\Lambda_t$ there exists a family of temporal points $\{p_{\alpha,t};\alpha\in{\cal T}\}$ in $\Lambda_t$ such that $\lambda_t$ is covered by it, i.e. $\lambda_t=\vee\{p_{\alpha,t},\alpha\in{\cal A}\}$. Write $T{\cal
P}(\Lambda_t)$ for the set of temporal points of $\Lambda_t$, if we write ${\rm Spec}(\Lambda_t)=\{p_{t'}$ point in $\Lambda_{t'},p_{t'}$ defines a temporal point of $\Lambda_t\}$ then $T{\cal P}(\Lambda_t)$ may also be written as $T{\rm Spec}(\Lambda_t)$ (note $T{\rm Spec}(\Lambda_t)$ is in $\Lambda_t$ but ${\rm Spec}(\Lambda_t)$ not).
We need to build in more “continuity” aspects in the DVT-axioms without using functions or extra assumptions on $T$ e.g. that it should be a group. A temporary poited system $\{\Lambda_t,\varphi_{tt'},T\}$ is a [**space continuum**]{} if the following conditions hold :
- There is a minimal closed interval $T_t$ containing $t$ in $T$ such that $T{\rm Spec}(\Lambda_t)$ has support in $I_t$. The set of points in $\Lambda_{t'}$ with $t'\in I_t$, representing temporary points in $\Lambda_t$ is then called the [**minimal spectrum**]{} for $T{\rm Spec}(\Lambda_t)$, denoted by ${\rm Spec}(\Lambda_t,I_t)$.
- For any open $T$-interval $I$ such that $I_t\subset I$ there exists an open $T$-interval $I^*_t$ with $t\in I^*_t$, such that for $t'\in I^*_t$ we have $I_{t'}\subset I$.
- For intervals $[t_1,t_2]$ and $[t_3,t_4]$ we write $[t_1,t_2]<[t_3,t_4]$ if $t_1\le t_3$ and $t_2\le t_4$ (similarly for open intervals). If $t\le t'$ in $T$ then $I_t < I_{t'}$. This provides an “orientation” of the variation of the minimal spectra !
- [**Local Preservation of Directed Sets**]{} For given $t\le t'$ in $I_t$ and any directed set $A_t$ in $\Lambda_t$, the subset $\{\gamma_t\in
A_t$, there exists $\xi_t<\gamma_t$ in $A_t$ such that $\varphi_{tt'}(\xi_t)<\varphi_{tt'}(\gamma_t)\}$ is cofinal in $A_t$ (defines the same limit $[A_t]$). For $t''\le t$ in $I_t$ there is a directed set $A_{t''}$ in $\Lambda_{t''}$ mapped by $\varphi_{t''t}$ to a cofinal subset of $A_t$.
A subset $J$ of $T$ is [**relative open around $t\in T$**]{} if it is intersection of $I_t$ and an open $T$-interval. For $x=(\ldots,x_t,\ldots)\in\prod_{t\in T}\Lambda_t$ we put ${\rm sup}(x)=\{t\in
T,x_t\neq 0\}$. We say that such an $x$ is [**topologically accessible**]{} if all $x_{t'}t\in {\rm sup}(x)$, are classical i.e. $x_t=[\chi_t]$ (for some $\chi_t\in X_t$ and $\Lambda_t=C(X_t)$. In case we do not consider $\Lambda_t$ as coming from some $X_t$ the condition becomes void. An $x$ as before is said to be [**$t$-accessible**]{} if ${\rm sup}(x)=J$ is relative open around $t$ and for all $t' \le t''$ in $J$ we have $\varphi_{t't''}(x_{t'})\le x_{t''}$. When $\Lambda_t$ has enough points i.e. if $I_t=\{t\}$, then the points in an open for the point topology would be charactericed by $\{p,p\le [\chi_t]\}=U(\chi_t)$ for some $\chi_t\in X_t$. When $\Lambda_t$ does not have enough points then we have to modify the definition of point spectrum and point topology correspondingly. If $x=(\ldots,x_t,\ldots)$ is $t$ accessible and $p_{t'}\in{\rm
Spec}(\Lambda_t,I_t)$ then we say $p_{t'}\in x$ if $t'\in J={\rm sup}(x)$ and there exists an open $T$-interval $J_1\subset J$ with $t' \in J_1$ such that for $t''\in J_1$ we have : if $t'\le t''$ then $p_{t''}=\varphi_{t't''}(p_{t'})\le x_{t''}$, or if $t''\le t'$ there is a $p_{t''}\in\Lambda_{t''}$ such that $\varphi_{t''t'}(p_{t''})=p_{t'},p_{t''}\le x_{t''}$, i.e. $\{p_{t''},t''\in
J_1\}$ is the restriction of a temporal point representing $p_{t'}$ defined over a bigger $T$-interval $]t_0,t_1[$ containing both $t'$ and $t$ (note : $J_1$ need not contain $t$).
Theorem
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The empty set together with the sets $U_t(x)=\{p_{t'},p_{t'}\in x$ for some $t'\in I_t\}\subset {\rm Spec}(\Lambda_t,I_t)$, $x$ being $t$-accessible in $\prod_{t\in T}\Lambda_t$, form a topology on ${\rm Spec}(\Lambda_t,I_t)$, called [**spectral topology**]{} at $t\in T$.
#### Proof
Consider $x\neq y$ both $t$-accessible with respective $T$-intervals $J$, resp. $J'$ contained in $I_t$. If $p_{t'}\in U_t(x)\cap U_t(y)$ then $t'\in
J\cap J'$ and for every $t_1\in J$, $p_{t_1}\le x_{t_1}$, for every $t_2\in J'$, $p_{t_2}\le y_{t_2}$. Of course the interval $J\cap J'$ is relative open around $t$. If $t'\le t''$ with $t''\in J\cap J'$ then $o_{t''}=\varphi_{t't''}(p_{t'})$ is idempotent in $\Lambda_{t''}$ because $p_{t'}$ is in $\Lambda_{t'}$ as it is a point. Hence we obtain : $$p_{t''}=p_{t''}\wedge p_{t''}\le x_{t''}\wedge y_{t''}$$ Obviously for all $t''\le t'''$ in $J\wedge J'$ we do have : $\varphi_{t''t'''}(x_{t''}\wedge y_{t''})\le x_{t'''}\wedge y_{t'''}$. On the other hand, for $t''\le t'$ we obtain : $\varphi_{t''t'}(p_{t''})=p_{t'}$ and therefore $p_{t'}\le \varphi_{t''t'}(x_{t''})\le x_{t'}$, as well as $p_{t'}\le \varphi_{t''t'}(y_{t''})\le y_{t'}$. Hence, again by idempotency of $p_{t'}$ in $\Lambda_{t'}$ we arrive at $p_{t'}\le x_{t'}\wedge y_{t'}$. By restricting $J\cap J'$ to the interval obtained by allowing only those $t''\le t'$ which belong to an (open) unambiguity interval for $p_{t'}$ we arrive at a relative open around $t$, say $J''\subset J\cap J'$, containing $t'$.
Now for $p_{t''}$ with $t''\in J''$ it follows that $p_{t''}$ is idempotent because both $p_{t''}$ and $p_{t''}\wedge p_{t''}$ map to $p_{t'}$ via $\varphi_{t''t'}$ for $t''\le t'$ (other $t''$ in $J''$ are no problem). Thus for $t''$ in $J''$ we do arrive at $p_{t''}\le x_{t''}\wedge y_{t''}$. Difine $w$ by putting $w_{t''}=x_{t''}\wedge y_{t''}$ for $t''\in J''$. Clearly, $w$ is $t$-accessible and $p_{t'}\in U_t(w)$. Conversely if $p_{t'}\in U_t(w)$ then $p_{t'}\in U_t(x)\cap U_t(y)$ is clear because $J''$ used in the definition of $w$ is open in $J\cap J'$. Now we look at a union of $U_{i,t}=U_t(x_i)$ for $i\in J$ and each $x_i$ being $A$-accessible with corresponding relative open interval $J_i$ in $I_t$. Define $w$ over the “interval” $J=\cup_i\{J_i,i\in {\cal J}\}$ by putting $w_t=\vee\{x_{i,t},i\in {\cal J}\}$ for $t\in J$. It is clear that $J$ is relative open around $t$ and for all $t_1\le t_2$ in $J$ we have $\varphi_{t_1t_2}(w_{t_1})\le w_{t_2}$ because $\varphi_{t,t_2}$ respects arbitrary $\vee$. Now $p_{t'}\in w$ means that $p_{t''}\le \vee
\{x_{i,t''},i\in {\cal J}\}$ for $t''$ in some relative open containiing $t'$, say $J_1\subset J$. We use relative open sets in $T$ because $I_t$ was closed and there are two situations to consider concerning $t'\in I_t$. First if $t'$ is the lowest element of $I_t$ then for all $t''\in J_1$ we have that $p_{t''}=\varphi_{t't''}(p_{t'})\le \varphi_{t't''}(\vee\{x_{i,t'},i\in {\cal
J}\})$ and for all $t'\le t_1\le t''$ we also obtain : $p_{t_1}\le
\varphi_{t't_1}(\vee\{x_{i,t'},i\in {\cal J}\})$ and $p_{t''}\le
\varphi_{t_1t''}(\vee\{x_{t,t_1},i\in {\cal J}\})$. Otherwise, if $t'$ is not the lowest element of $I_t$ then we may restrict $J_1$ to be an open interval $]t_0,t'_0[$ containing $t'$ with $t_0\in J$. The same reasoning as in the first case yields for all $t''\in ]t_0,t'_0[$ so that : $p_{t''}\le
\varphi_{t_0t''}(\vee\{x_{i,t_0},\tau\in{\cal J}\})$ and for any $t'\le t_1\le
t''$ $p_{t''}\le \varphi_{t_1t''}(\vee\{x_{i,t_1},i\in{\cal
J}\})=\vee\{\varphi_{t_1t''}(x_{i,t_1}),i\in{\cal J}\}$. Since $t'\in J_1$ we obtain $p_{t'}\le \vee\{\varphi_{t_1t'}(x_{i,t_1}),i\in J\}$ for all $t_1\in
[t_0,t']$.
Since $p_{t'}$ is a point in $\Lambda_{t'}$ there is an $i_0\in J$ such that $p_{t'}\le\varphi_{t_0t'}(x_{i_0,t_0})$ and therefore we have that $p_{t'}\le
\varphi_{t_1t'}(x_{i_0,t_1})$ with $t_1\in [t_0,t']$, the gain being that $i_0$ does not depend on $t_1$ here ! Now for $t''\ge t'$ in $J_1\cap J_{i_0}$ (note that this is not empty because $x_{i_0}$ is nonzero at $t_0$ because $p_{t'}\le \varphi_{t_0t''}(x_{i_0,t_0})$ would then make $p_{t'}$ zero and we do not look at the zero (the empty set) as a point of $\Lambda_t$) we obtain : $$p_{t''}=\varphi_{t't''}(p_{t'})\le \varphi_{t't''}(x_{i_0,t'})\le
x_{i_0,t''}\leqno {(*)}$$ In the other situation $t''\le t'$ in $J_1\cap J_{i_0}$ we have $\varphi_{t''t'}(p_{t''})=p_{t'},\varphi_{t''t'}(x_{i_0, t''})\le x_{i_0,t'}$. By restricting $J_1\cap J_{i_0}$ further so that the $t''\le t'$ are only varying in an (open) unambiguity interval for $p_{t'}$, say $J_2\subset
J_1\cap J_{i_0}$, we arrive at one of two cases : either $p_{t''}=x_{i_0,t''}$ or else $p_{t''}\neq x_{i_0,t''}$ and also $p_{t'}< x_{i_0,t'}$. In the first case $p_{t'}\in x_{i_0}$ follows because $p_{t_1}=\varphi_{t''t_1}(x_{i_0,
t''})\le x_{i_0,t_1}$ for $t_1$ in $]t'',1[\cap J_2$, the latter interval containing $t'$ is relative open again. In the second case we may look at $p_{t'}< x_{i_0,t'}<1$, hence there exists a $z_{t''}$ such that $p_{t''}<z_{t''}<1$ and $\varphi_{t''t'}(z_{t''})=x_{i_0,t'}$. Again we have to distinguish two cases, first $\varphi_{t''t'}(x_{i_0t''})=x_{i_0,t'}$ or $\varphi_{t''t'}(x_{i_0,t''})<x_{i_0,t'}$.
In the first case $z_{t''}$ and $x_{i_0,t''}$ map to the same element via $\varphi_{t''t'}$, hence up to restricting the interval further such that $t''$ stays within an unambiguity interval for $x_{i_0,t'}$, we may conclude $z_{t''}=x_{i_0,t''}$ in this case and then $p_{t''}< x_{i_0,t''}$. In the second case we may look at : $p_{t'}\le \varphi_{t''t'}(x_{i_0,t''})<
x_{i_0,t'}< 1$ (where the first inequality stems from (\*) above). Again restricting the interval further (but open) we find a $z'_{t''}$ in $\Lambda_{t''}$ such that $x_{i,t''} < z'_{t''} < 1$ such that $\varphi_{t''t'}(z'_{t''})=x_{i_0,t'}$. Since we are dealing with the case $p_{t''}\neq x_{i_0,t''}$ and we are in an unambiguity interval for $p_{t'}$ it follows that $p_{t'} <\varphi _{t''t'}(x_{i_0,t''})$. Look at : $p_{t'}<\varphi_{t''t'}(x_{i_0,t''})< x_{i_0,t'}$ with $\varphi_{t''t'}(p_{t''})=p_{t'}$ and $\varphi_{t''t'}(z'_{t''})=x_{i_0,t'}$; by restructing the interval (open) further if necessary we obtain the existence of $z''_{t''}$ such that, $p_{t''} < z''_{''}< z'_{t''}$ such that $\varphi_{t''t'}(z''_{t''})=\varphi_{t''t'}(x_{i_0,t''})$. Finally restricting again the $t''\le t'$ to vary in an unambiguity interval for $\varphi_{t''t'}(x_{i_0,t''})$ it follows that $z''_{t''}=xz_{i_0,t''}$ and hence $z''_{t''}\ge p_{t''}$ yields $x_{i_0,t''}\ge p_{t''}$ for $t''$ in a suitable relative open around $t$ containing $t'$. This also in the case we arrive at $p_{t'}\in x_{i_0}$ or $p_{t'}\in U_t(x_{i_0})$. It follows that ${\cal U}_t(w)=U\{U_{i,t},i\in {\cal J}\}$ establishing that arbitrary unions of opens are open. By taking ${\cal U}_t(1)$ we obtain the whole spectrum at $t$ as an open too.
A Possible Relation to $M$-Theory ?
-----------------------------------
In noncommutative topology and derived point topologies the gen-topology appears naturally (and it is a classical i.e. commutative topology (cf. \[VO1\]). Moreover continuity in the gen-topology also appears naturally in noncommutative geometry of associative algebras but we did not ask the $\varphi_{tt'}$ in the DNT-axioms to be continuous in the gen topology. However one may prove that in general “continuity of the $\varphi_{tt'}$ in the gen-topologies of $\Lambda_t$ resp. $\Lambda_{t'}$ is an observed truth ! Consequently for $t'$ close enough to $t$ the $\varphi_{tt'}$ is continuous with respect to the gen-topologies (cfr. \[VO2\]).
In the mathematical theory all $\Lambda_t$ may be different and there is no reason to aim at $Sl(\Lambda_t)$ nor ${\rm Spec}(\Lambda_t,I_t)$ to be invariant under $t$-variation. From the point of view of Physics one may reason that only one case is important i.e. the case we see as “ reality”. This being the utmost special case it is then not far-fetched to assume that the dynamic noncommutative space has as commutative shadow the abstract mathematical frame we reason in about reality. for example identified with 3 or 4 dimensional space or space-time, and also an observational mathematical frame we measure in that is ${\rm Spec}(\Lambda_t,I_t)$ identified to an $11$-dimensional space (for $M$-theory) or any other one fitting physical interpretations in some theory one chooses to believe in. An observed point in ${\rm Spec}(\Lambda_t,I_t)$ is then given by a string of elements say $p_{t'}\in \Lambda_{t'}$, $t'\in J\subset I_t$ with $p_t\in\Lambda_t$ a temporal point. If $p_{t'}$ is a point then for all $t'\le t''$, $\varphi_{t't''}(p_{t'})$ is idempotent so appears in the commutative shadow $SL(\Lambda_{t''})$. Hence an observed point in ${\rm Spec}(\Lambda_t,I_t)$ appears as a string in the base space $SL(\Lambda_{t''})(t'\le t'')$, identified with $n$-dimensioal space but the string may “start after” $t$ when the point was “observed”. On the other hand, the assumption that the system $(\Lambda_t,\varphi_{tt'},T\}$ is temporally pointed leads to a decomposition of of every $p_{t'}$ into temporal points of $\Lambda_{t'}$ realizing it as an open of ${\rm Spec}[\Lambda_{t'},I_{t'}]$. Thus in the spectral space (identified with a certain $m$-dimensional space say), the observed point appears as a “string” connecting opens i.e. a possibly more dimensional string that can be thought of as a tube. The difference between the dimensions $m$ and $n$ has to be accounted for by the “rank” of $T$ (e.g. if $T$ were a group like $\mathbb{R}^d_+$, $d$ would be the rank) which allowed to create the extra points in ${\rm Spec}(\Lambda_t,I_t)$ when compared to $(SL(\Lambda_t)$. Note that even when the $\varphi_{tt'}$ do not necessarily define maps between ${\rm Spec}(\Lambda_t,I_t)$ and ${\rm
Spec}(\Lambda_{t'},I_{t'})$ or between $SL(\Lambda_t)$ and $SL(\Lambda_{t'})$, the given strings at the $\Lambda_t$-level do define sequences of elements or opens in the ${\rm Spec}(\Lambda_{t},I_t)$ resp. $SL(\Lambda_t)$ that may be viewed as strings, resp. tubes. Two more intriguing observations :
Identifying ${\rm Spec}(\Lambda_t,I_t)$ to one fixed commutative world and $SL(\Lambda_t)$ to another allows strings and tubes to be open or closed.
Only temporal points corresponding to future points can be non-idempotent, therefore all noncommutativity is due to future points and uncertainty may be seen as an effect of the possibility that the interval needed to realize the temporal point $p_t$ by a point $p_{t''}\in\Lambda_{t''}$ for $t\le t''$ is [**larger**]{} than the unambiguity interval for $p_{t''}$ ! Passing from a commutative frame $(SL(\Lambda))$ to noncommutative (dynamical) geometry and phrasing theories and calculations in ${\rm Spec}(\Lambda_t,I_t)$ at the price of having to work in higher dimension seems to fit quantum theories. Of course this is at the level of mathematical formalism, for suitable interpretations within physics the physical entity connected to the notion of point in ${\rm Spec}(\Lambda_t,I_t)$ should be the smallest possible, i.e. a kind of building block of everything, so that observing it as a point in the moment spaces is acceptable; that these points are mathematically described as strings or higher dimensional strings via the noncommutative geometry is a “Deus ex Machina” pointing at an unsuspected possibility of embedding $M$-theory in our approach. No further speculation about this here, perhaps specialists in string theory may be interested in investigating further this formal incidence.
Moment Presheaves and Sheaves
=============================
Continuing the point of view put forward in the short introduction to Section 2, points or more precisely functions defined in a set theoretic spirit, should be replaced by a generalization of “germs of functions” obtained from limit constructions in classical topology terms to noncommutative structures. Thus the notion of point is replaced by an atavar of the notion “stalk” of a pregiven sheaf, more correctly when different (pre)sheaves over a noncommutative space are being considered, say with values in some nice category of objects $\ul{\cal C}$, then a “point” is a suitable limit functor on $\ul{\cal C}$-objects generalizing the classical construction of localization (functor) at a point. Assuming that a suitable topological space and satisfactory sheaf of “functions” on it have been identified, satisfactory in the sense that it allows to study the desired geometric phenomena one is aiming at, then the notion of point via stalks should be suitable too. For example, prime ideals would be identified via stalks of the structure sheaf of a commutative Noetherian ring without having to check a primeness condition of the corresponding localization functor. More on the definition of noncommutative geometry via (localization) functors can be found in \[VO1\] where it is introduced as a functor geometry over a noncommutative topologyee also \[MVO\] and \[VOV\].
In this section we fix a category $\ul{\cal C}$ allowing limits and colimits; we might restrict to Abelian or even Grothendieck categories but that is not essential. In fact, the reader who wants to fix ideas on a concrete situation may choose to work in the category of abelian groups.
For every $t\in T, \Gamma_t$ is a presheaf over $\Lambda_t$ and for $t\le t'$ in $T$ there is a $\phi_{tt'}:T_t\r T_{t'}$, defined by morphisms in $\ul{\cal
C}$ as follows :
For $\lambda_t\in\Lambda_t$ there is a $\phi_{tt'}(\lambda_t):\Gamma_t(\lambda_t)\r
\Gamma_{t'}(\varphi_{tt'}(\lambda_t))$
for $\mu_t\le \lambda_t$ in $\Lambda_t$ we have commutative diagrammes in $\ul{\cal C}$ : $$\begin{diagram}
\Gamma_t(\lambda_t)\rto^{\phi_{tt'}(\lambda_t)}\quad
\dto^{\rho^{t}_{\lambda_t,\mu_t}}&
\qquad\Gamma_{t'}(\varphi_{tt'}(\lambda_t))
\dto^{\rho^{t'}_{\lambda_{t'},\mu_{t'}}}\\
\Gamma_t(\mu_t)\rto_{\phi_{tt'}(\mu_t)} \qquad &
\qquad\Gamma_{t'}(\varphi_{tt'}(\mu_t))\\
\end{diagram}$$ where we have written $\lambda_{t'},\mu_{t'}$ for $\varphi_{tt'}(\lambda_t)$, resp. $\varphi_{tt'}(\mu_t)$ and $\rho^t_{\lambda_{t'}\mu_t}$ for the restriction morphism of $\Gamma_t$.
$\phi_{tt}(\lambda_t)=I_{\Gamma_t(\lambda_t)}$ for all $t\in T$, and for $t\le
t'$ and let $t'\le t''$ we have $\phi_{t't''}(\phi_{tt'}(T_t(\lambda_t)))=\phi_{tt''}(T_t(\lambda_t))$ for all $\lambda_t\in \Lambda_t$.
The system $\{\Gamma_t,\phi_{tt'},T\}$ is called a (global) dynamical presheaf over the DNT $\{\Lambda_t,\varphi_{tt'},T\}$.
Since sheaves on a noncommutative topology do not form a topos it is a problem to define a suitable sheafification i.e. : can a presheaf $\Gamma$ on $\Lambda$ be sheafified to a sheaf $\ul{\ul{a}}\Gamma$ on the same $\Lambda$ via a suitable notion of “stalk”, then allowing interpretations in terms of “points” ? In fact, the axioms of DNT allow to give a solution to the sheafification problem at the price that the sheaf $\ul{\ul{a}}\Gamma_t$ has to be constructed over ${\rm Spec}(\Lambda_t,I_t)$ !
From hereon we let $\ul{\Lambda}=\{\Lambda_t, \varphi_{tt'},T\}$ be a temporally pointed system which is a space continuum. We refer to $Y_t={\rm
Spec}(\Lambda_t,I_t)$ with its spectral topology as the [**moment space**]{} at $t\in T$.
For $p_{t'}\in Y_t$ we may calculate (in $\ul{\cal C}$): $\Gamma_{t',p_{t'}}=\mathop{\rm lim}\limits_{\longrightarrow}\Gamma_{t'}(x_t)$ where $\mathop{\rm lim}\limits_{\longrightarrow}$ is over $x_{t'}\in
\Lambda_{t'}$ such that $p_{t'}\le x_{t'}$, and where $x=(\ldots,x_t,\ldots)$ is $t$-accessible, in fact we have $p_{t'}\in x$. In the foregoing we did not demand $\ul{\Lambda}$ to derive from a system $\Xul$ and passing from $X_t$ to $\Lambda_t$ as a generalized Stone space or pattern topology via $C(X_t)$,. We preferred not to dwell upon the formal comparison of dynamical theories for the $X_t$ and the $\Lambda_t$. In order to keep trace of a possible original $X_t$ if it was considered in the construction of $\Lambda_t$ one may if desired use the following.
Definition
----------
We say that $u_t\in\Lambda_t$ is [**classical**]{} if $u_t=[\chi_t]$ for $\chi_t\in X_t$. If $u_t$ is classical then there is an open interval containing $t$ in $T$, say $L$, such that for every $t'\in L$ we have that $u_{t'}$ is classical, where for $t\le t'$ we have $u_{t'}=\varphi_{tt'}(u_t)$ and for $t'\le t, u_{t'}$ is a suitably chosen representative for $u_t$, $\varphi_{t't}(u_{t'})=u_t$. Restricting further to an unambiguity interval of $u_t$, the representations $u_{t'}$ for $t'$ in that interval are unique. Since points of $X_t$ are by definition elements in $C(X_t)$, the filter in $\Lambda_t$ defined by that point has a cofinal directed subset of classical elements. When in $\{\Lambda_t,\varphi_{tt'},T\}$ we restrict attention to classical $x$, i.e. every $x_t$ classical $\Lambda_t$, then we say that we look at a [**traditional system**]{}.
Lemma
-----
For a traditional space continuum with dynamical presheaf $\{\Gamma_t,\phi_{tt'},T\}$, the stalk for $p_{t'}\in Y_t$ of $\Gamma_{t'}$ is exactly $\Gamma_{t',p_{t'}}$ as defined above.
#### Proof
In calculating $\mathop{\rm lim}\limits_{\longrightarrow}
\{\Gamma_{t'}(u_{t'}),p_{t'}\le u_{t'}\}$ we may restrict to classical $u_{t'}$ in $\Lambda_{t'}$. It now suffices to establish the existence of a $t$-accessible $y$ such that $p_{t'}\in y$ and $y_{t'}\le u_{t'}$. From $p_{t'}\in U_t(x)$ we obtain $(\ldots,x_{t'},\ldots)$ with a relative open $T$-interval $J,t'\in J$, such that $p_{t''}\le x_{t''}$ for every $t''\in J$. Since $u_{t'}$ and $x_{t'}$ are classical, so is $u_{t'}\wedge x_{t'}$ and moreover $p_{t'}\le u_{t'}\wedge x_{t'}$ because $p_{t'}$ is a point in $\Lambda_{t'}$ (hence idempotent !). Let $J_1$ be an open $T$-interval containing $t'$ such that $u_{t'}\wedge x_{t'}$ has a representative $u_{t''}$ in $\Lambda_{t''}$ such that $\varphi_{t''t'}=u_{t'}\wedge x_{t'}$. Since $x_{t'}\neq u_{t'}\wedge x_{t'}$ may be assumed (otherwise put $y=x$) we arrive at $p_{t''}\le u_{t''}<
x_{t''}$. Using the intersection of $J_1$ and the interval iaround $t'$ allowing to select classical $u_{t''}$, call this interval $J_2$, we put $y_{t''}=u_{t''}$ for $t''\le t'$ in $J_2$ and $y_{t_1}=x_{t_1}$ for $t'<t_1$ in $J$. Then $y$ is $t$-accesible with respect to the relative open $T$-interval around $t$ just defined : we have $y_{t'}\le u_{t'}$ and $p_{t'}\in y$. Consequently : $\mathop{\rm lim}\limits_{\mathop{\longrightarrow}\limits_{p_{t'} \le u_{t'}}}
\Gamma_{t'}(u_{t'}) = \mathop{\rm lim}\limits_{\mathop{\longrightarrow}
\limits_{p_{t'}\in x}}\Gamma_{t'}(x_{t'})$.
In the sequel we assume objects in $\ul{\cal C}$ are at least sets but let us restrict to abelian groups. Again let $\{\Gamma_t, \phi_{tt'},T\}$ be a dynamical presheaf over a traditional space continuum. On $Y_t$ we define a presheaf, with respect to the spectral topology, by taking for ${\cal P}(U_t(x))$ the abelian group in $\coprod_{t'\in I_t}\Gamma_{t'}(x_{t'})$formed by strings over ${\rm sup}(x)=\{t'\in I_t,x_{t'}\neq 0\}$. i.e. $\{\gamma_{t'}, t'\in {\rm sup}(x),\phi_{t''t'}(\gamma_{t''})=\gamma_{t'}$ for $t''\le t'$ in ${\rm sup}(x)\}$. Let us write $x<y$ if $x_{t'}\le y_{t'}$ for all $t'$ in $I_t$, in particular $x < y$ means ${\rm sup}(x)\subset {\rm sup}(y)$. In sheaf theory one usually omits the empty set, so here the $o\in\Lambda_t$ at every $t$, and it makes sense to do that here as well. However one may define at every $t'\in T$, $\Gamma_{t'}(0)=\mathop{\rm lim}\limits_{\longrightarrow}
\{\Gamma_{t'}(x_{t'}),x_{t'}$ classical in $\Lambda_{t'}\}$ and all statements made in the sequel will remain consistent.
If $x<y$ then we have restriction morphisms $\rho^{t'}_{y_{t'},x_{t'}},
: \Gamma_{t'}(y_{t'})\r \Gamma_{t'}(x_{t'})$. Commutativity of the diagrams in the beginning of the section yield corresponding morphisms on the strings over the respective supports : $\rho_{y,x}:{\cal P}(U_t(y))\r {\cal P}(U_t(x))$. For a point $p_{t'}$ we let $\eta(p_{t'})$ be the set of $U_t(x)$ such that we have $p_{t'}\in U_t(x)$ i.e. $p_{t'}\in x$; in particular $t'\in J_x$ where $J_x$ is the relative open around $t$ in the definition of $x$ and consequently : $t'\in \cap \{{\rm sup}(x),\eta(p_{t'})\ {\rm contains}\
U_t(x)\}$. For the dynamical sheaf theory we may want to impose coherence conditions on the system assuming some relations between $\Gamma_{t''}$ and $\Gamma_{t'}$ if $t'$ and $t''$ are close enough in $T$. We shall restrict here to only one extra assumption, in some sense dual to the unambiguity interval assumption for the underlying DNT.
Definition
----------
The dynamical presheaf $\{\Gamma_t,\phi_{tt'},T\}$ on a traditional space continuum is locally temporaly flabby at $t\in T$ if for $t$-accessible $x$ such that $p_{t'}\in x$ and $s_{t'}\in\Gamma_{t'}(x_{t'})$ there exists a $t$-accessible $y<x$ with $p_{t'}\in y$ and a string $\sol\in {\cal P}(U_t(y))$ such that $\sol_{t'}=\rho^{t'}_{x_{t'},y_{t'}}(s_{t'})$.
Theorem
-------
To a dynamical presheaf on a traditional space continuum there corresponds for every $t\in T$ a presheaf ${\cal P}_t$ on the moment space ${\rm
Spec}(\Lambda_t,I_t)$ with its spectral topology given by the $U_t(x)$ for $t$ accessible $x$. In case all $\Gamma_{t'},t'\in I_t$, are separated presheaves then ${\cal P}_t$ is separated too. The sheafification $\ul{\ul{a}}{\cal P}_t$ of ${\cal P}_t$ on the moment space ${\rm Spec}(\Lambda_t,I_t)=Y_t$ is called the [**moment sheaf**]{} of [**spectral sheaf**]{} at $t\in T$. In case the dynamical presheaf is locally temporally flabby (LTF) then for any point $p_{t'}\in Y_t$ the stalk ${\cal P}_{t,p_{t'}}$ may be identified with $\Gamma_{t',p_{t'}}$.
#### Proof
At every $t\in T$, ${\cal P}_t$ is the spectral presheaf constructed on ${\rm
Spec}(\Lambda_t,I_t)$ with its spectral topology. Now suppose all $\Gamma_t$ are separated presheaves and look at a finite cover $U_t(x)=U_t(x_1)\cup\ldots\cup U_t(x_n)$ and a $\gamma\in\Gamma_t(U_t(x))$ such that for $i=1,\ldots,n$, $\rho_{x.x_i}(\gamma)=0$. We have seen before that the union $U_t(x_1)\cup\ldots\cup U_t(x_n)$ corresponds to the $t$ accessible element $x_1\wedge\ldots\wedge x_n$ obtained as the string over ${\rm
sup}(x_1)\wedge\ldots\wedge{\rm sup}(x_n)$ given by the $x_{1,t'}\cup\ldots\cup
x_{n.t'}$ in $\Lambda_{t'}$. For all $t'\in {\rm sup}(x)$ we obtain, in view of the compatibility diagrams for restrictions and $\phi_{t't''},t'\le
t^n:\rho^{t'}_{x_{t'},x_{i,t'}}(\gamma_{t'})=0$, for $i=1,\ldots,n$. The assumed separatedness of $\Gamma_{t'}$, for all $t'$ then leads to $\gamma_{t'}=0$ for all $t'\in{\rm sup}(x)$ and therefore $\gamma=0$ as a string over ${\rm
sup}(x)$. Consequently ${\cal P}_t$ is separated, for all $t\in T$. In order to calculate the stalk at $p_{t'}\in{\rm Spec}(\Lambda_t,I_t)$ for ${\cal P}_t$ we have to calculate : $\mathop{\rm lim}\limits_{\mathop{\longrightarrow}
\limits_{p_{t'}\in x}}\Gamma_t(U_t(x))=E_{t'}$.
Starting with $p_{t'}\in x$ for some $t$-accessible $x$ we have a representative $\gamma_x\in \Gamma_t(U_t(x))$ being a string over ${\rm
sup}(x)$ and the latter containing a relative open $J(x)$ around $t$ containing $t'$. So an element $e_{t'}$ in $E_{t'}$ may be viewed as given by a direct family $\{\gamma_x,p_{t'}\in x, \rho_{x,y}(\gamma_x)=\gamma_y\
{\rm for}\ y<x\}$. At $t'$, which is in ${\rm sup}(x)$ for all $x$ appearing in the forgoing family (as $U_t(x)$ varies over $\eta(p_{t'}))$, we obtain $\{(\gamma_x)_{t'}, p_{t'}\le
x_{t'},\rho^{t'}_{x_{t'},y_{t'}}((\gamma_x)_{t'})=(\gamma_y)_{t'}$ which defines an element of $\Gamma_{t',p_{t'}}$, say $\eol_{t'}$. We have a well-defined map $\pi_{t'}:E_{t'}\r \Gamma_{t',p_{t'}},e_{t'}\mapsto
\eol_{t'}$. Without further assumptions we therefore arrive at a sheaf $\ul{\ul{a}}{\cal P}_t$ with stalk $E_{t'}$ at $p_{t'}$ and a presheaf map ${\cal P}_t\r\ul{\ul{a}}{\cal P}_t$ which is “injective” in case all $\Gamma_{t''}$ are separated. Now we have to make use of the locally temporally flabbyness (LTF). Look at a germ $s_{t'}\in (\Gamma_{t'})_{p_{t'}}$. In view of Lemma 3.2. there exists a $t$-accessible $x$ such that $s_{t'}\in\Gamma_{t'}((x_{t'})$ with $p_{t'}\in x$, in particular $p_{t'}\le
x_{t'}$.
The LTF-condition allows to select a $t$-accessible $y<x$ with $p_{t'}\in y$ together with a string, $\vec{s(y)}\in {\cal P}(U_t(y))$ such that $\vec{s_{t'}}(y)=\rho^{t'}_{x_{t'},y_{t'}}(s_{t'})$. The element $e_{t'}$ in $E_{t'}$ defined by the directed family obtained by taking restrictions of $\vec{s_{t'}}(y)$ has $\eol_{t'}$ exactly $s_{t'}$ (note that $t'$ supports all the restrictions of $\vec{s_{t'}}(y)$ because $y$ varies in $\eta(p_{t'})$). Thus $\pi_{t'}:E_{t'}\r \Gamma_{t',p_{t'}}$ is epimorphic. If $e_{t'}$ and $e'_{t'}$ have the same image under $\pi_{t'}$ then there is a $t$-accessible $y$ such that $e_{t'}-e'_{t'}$ is represented by the zero-string over ${\rm sup}(y)$; in fact this follows by taking $s_{t'}=0$ in the foregoing. Leading to a $t$-accessible $y$ as above that may be restricted to a $t$-accessible $y'$ defined by taking for ${\rm sup}(y')$ the relative open $J$ containing $t'$ in the support of $y$ where $\vec{s_{t'}}(y)=0$. Therefore, $\pi_{t'}$ is also injective.
Can one avoid a condition like $LTF$ in the foregoing theorem ? It seems that the idea of “germ” appearing in the notion of stalks spatially needs an extension in the temporal direction, so probably some condition close to $LTF$ is really necessary here.
Some Remarks on Spectral families and Observables
=================================================
Let $\Gamma$ be any totally ordered abelian group. On a noncommutative topology $\Lambda$ we define a [**$\Gamma$-filtration**]{} by a family $\{\lambda_{\alpha},\alpha\in\Gamma\}$ such that for $\alpha\le \beta$ in $\Gamma$, $\lambda_{\alpha}\le \lambda_{\beta}$ in $\Lambda$ and $\vee\{\lambda_{\alpha},\alpha\in\Gamma\}=1$. A $\Gamma$-filtration is said to be separated if from $\gamma={\rm inf}\{\gamma_{\alpha},\alpha\in{\cal
A}\}$ in $\Gamma$ it follows that $\lambda_{\gamma}=\wedge\{\lambda_{\alpha},\alpha\in{\cal A}\}$ and $0=\wedge\{\lambda_{\gamma},\gamma\in\Gamma\}$. [**A $\Gamma$-spectral family in $\Lambda$**]{} is just a separated $\Gamma$-filtration, it may be seen as a map $F:\Gamma\r \Lambda,\gamma\mapsto \lambda_{\gamma}$, where $F$ is a poset map satisfying the separatedness condition. Note that by definition the order in $\wedge \{\lambda_{\gamma},\gamma\in \Gamma\}$ does not matter but $\lambda_{\gamma_{\alpha}}$ need not be idempotent in $\Lambda$. Taking $\Gamma=\mathbb{R}_+$ and $\Lambda=L(H)$ the lattice of a Hilbert space $H$, we recover the usual notion of a spectral family. We say that a $\Gamma$-spectral family on $\Lambda$ is idempotent if $\lambda_{\gamma}\in{\rm id}_{\wedge}(\Lambda)$ for every $\gamma\in\Gamma$.
#### Observation
If $\Gamma$ is indiscrete, i.e. for all $\gamma\in\Gamma$, $\gamma={\rm inf}\{\tau,\gamma < \tau\}$ (example $\Gamma=\mathbb{R}^n_+$), then every $\Gamma$-spectral family is idempotent.
Proposition
-----------
Let us consider a $\Gamma$-spectral family on $\Lambda$, then for $\gamma,\tau\in\Gamma:\lambda_{\gamma}\wedge
\lambda_{\tau}=\lambda_{\tau}\wedge\lambda_{\gamma}=\lambda_{\delta}$, where $\delta={\rm min}\{\tau,\gamma\}$.
If the $\Gamma$-spectral family is idempotent then for $\gamma,\tau\in
\Gamma$, $\lambda_{\gamma}\wedge\lambda_{\tau}=\lambda_{\tau}\wedge\lambda_{\gamma}$ and the $\Gamma$-spectral family on $\Gamma$ is in fact a $\Gamma$-spectral family of the commutative shadow $SL(\Lambda)$.
#### Proof
Easy enough.
A filtration $F$ on a noncommutative $\Lambda$ is said to be [**right bounded**]{} if $\lambda_{\gamma}=1$ for some $\gamma\in\Gamma$, $F$ is [**left bounded**]{} if $\lambda_{\delta}=0$ for some $\delta\in\Gamma$.
For a right bounded $\Gamma$-filtration $F:\Gamma\r \Lambda$ we may define for every $\mu\in\Lambda$ the induced filtration $F|_{\mu}:\Gamma\r \Lambda(\mu)$ where we use $\mu=1_{\wedge(\mu)}$, $\Lambda(\mu)=\{\lambda\in\Lambda,\lambda\le\mu\}$. Note that $F|_{\mu}$ need [**not**]{} be separated whenever $F$ is, indeed if $\delta={\rm
inf}\{\delta_{\alpha},\alpha\in A\}$ in $\Gamma$ then $\lambda_{\delta}=\wedge\{\lambda_{\delta_{\alpha}},\alpha\in {\cal A}\}$ in $\Gamma$ then $\lambda_{\delta}=\wedge\{\lambda_{\delta_{\alpha}},\alpha\in{\cal A}\}$ but $\mu \wedge\lambda_{\delta}$, and $\wedge\{\mu\wedge\lambda_{\alpha},\alpha\in{\cal A}\}$ need not be lequal in general.
Proposition
-----------
If $F$ defines a right bounded $\Gamma$-spectral family on $\Lambda$ then $F|_{\mu}$ is a spectral family of $\Lambda(\mu)$ in each of the following cases :
- $\mu\in{\rm id}_{\wedge}(\Lambda)$ and $\mu$ commutes with all $\lambda_{\alpha},\alpha\in\Gamma$.
- $\mu\wedge\lambda_{\alpha}$ is idempotent for each $\alpha\in\Gamma$.
#### Proof
Easy and straightforward.
An element $\mu$ with property a. as above is called an [**$F$-centralizer**]{} of $\Lambda$.
Corollary
---------
In case $\Lambda$ is a lattice then for every $\mu\in\Lambda$ a right bounded $\Gamma$-spectral family of $\Lambda$ induces a right bounded spectral family on $\Lambda(\mu)$.
Let $F$ be a $\Gamma$-spectral family on a noncommutative $\Lambda$. To $\lambda\in\Lambda$ associate $\sigma(\lambda)\in \Gamma\cup \{\infty\}$ where $\sigma(\lambda)={\rm inf}\{\gamma,\lambda\le\lambda_{\gamma}\}$ and we agree to write ${\rm inf}\phi=\infty$. The map $\sigma:\Lambda\r
\Gamma\cup\{\infty\}$ is a generalization of the principal symbol map in the theory of filtered rings and their associated graded rings. We refer to $\sigma$ as the [**observable function**]{} of $F$. $$\begin{array}{r}
{\rm Clearly~} : \sigma(\lambda\wedge \mu)\le {\rm
min}\{\sigma(\lambda),\sigma(\mu)\}\\
\sigma(\lambda\vee\mu)\le {\rm max}\{\sigma(\lambda),\sigma(\mu)\}\\
\end{array}$$ The [**domain of $\sigma$**]{} is $\cup\{[0,\lambda_{\gamma}],\gamma \in \Gamma\}$ and observe that $\vee\{\lambda_{\gamma},\gamma\in\Gamma\}=1$ does not imply that $D(\sigma)=\Lambda$ (can even be checked for $\Gamma=\mathbb{R}_+,\Lambda=L(H)$).
If $F:T\r L(H)$ is a $\Gamma$-spectral family and $V\subset H$ a linear subspace, then we may define $\gamma_{\vee}\in\Gamma$, $\gamma_{V}={\rm
inf}\{\gamma\in\Gamma,\subset L(H)_{\gamma}\}$, again putting ${\rm
inf}\phi=\infty$. The map $\rho:L(H)\r \Gamma\cup\{\infty\}, U\mapsto
\rho(U)=\gamma_U$ is well-defined. One easily verifies for $\cup$ and $\vee$ in $H$ : $$\begin{array}{l}
\rho(U+V)\le {\rm max}\{\rho(U),\rho(V)\}\\
\rho(U\cap V)\le {\rm min}\{\rho(U),\rho(V)\}\\
\end{array}$$ The function $\rho$ defines a $\ul{\rho}$ defined on $H$ by putting $\rho(x)=\rho(\mathbb{C}x)$. We denote $\ul{\rho}$ again by $\rho$ and call it the [**pseudo-place of the $\Gamma$-spectral family**]{}. Then any $\Gamma$-spectral family defines a function on the projective Hilbert space $\mathbb{P}(H)$ described on the lines in $H$ by $\ol{\rho}:\mathbb{P}(H)\r
\Gamma,\ul{\mathbb{C}v}\mapsto \rho(\mathbb{C}v)$, where we wrote $\ul{\mathbb{C}v}$ for $\mathbb{C}v$ as an object in $\mathbb{P}(H)$.
The pseudo-place aspect of $\rho$ translates to $\ol{\rho}$ in the following sense : $\mathbb{C}w\subset \mathbb{C}v+\mathbb{C}u$ we have $\ul{\rho}(\ul{\mathbb{C}w})\le {\rm max}\{\ol{\rho}(\ul{\mathbb{C}v}),\ol{\rho}(\ul{\mathbb{C}u})\}$.
A linear subspae $U\subset H$ such that $\mathbb{P}(U)\subset
\ol{\rho}^{-1}(]-\infty,\gamma])$ allows for $u\neq 0$ in $U:\rho(\mathbb{C}u)\le \gamma$ i.e. $u\in L(H)_{\gamma}$. Hence, the largest $U$ in $H$ such that $\mathbb{P}(U)$ is in $\ol{\rho}^{-1}(]-\infty,\gamma])$ is exactly $L(H)_{\gamma}$; this means that the filtration $F$ may be reconstructed from the knowledge of $\ol{\rho}$. One easily recovers the classical result that maximal abelian Von Neumann regular subalgebras of ${\cal L}(H)$ correspond bijectively to maximal distributive lattices in $L(H)$. Since any $\Gamma$-spectral family is a directed set in $\Lambda$ it defines an element of $C(\Lambda)$ which we call a [**$\Gamma$-point**]{}. The set of $\Gamma$-points of $\Lambda$ is denoted $[\Gamma]\subset C(\Lambda)$. We may for example think of $[\mathbb{R}]\subset C(L(H))$ as being identified via the Riemann-Stieltjens integral to the set of self-adjoint operators on $H$.
Let $\sigma:\Lambda\r \Gamma\cup\{\infty\}$ be the observable functions of a $\Gamma$-spectral family on $\Lambda$ defined by $F:\Gamma\r\Lambda$. Put ${\cal F}:C(\Lambda)\r \Gamma\cup\{\infty\}$,$[A]\mapsto{\rm
inf}\{\gamma\in\Gamma,\lambda_{\gamma}\in\Aol\}$, $\Aol$ the filter of $A$. Then $\widehat{\sigma}$ is the observable corresponding to the $\Gamma$-filtration on $C(\Lambda)$ defined by $[A]_{\gamma}$, where for $\gamma\in\Gamma$,$[A]_{\gamma}$ is the class of the smallest filter containing $\lambda_{\gamma}$ i.e. the filter $\{\mu\in\Lambda,\lambda_{\gamma}\le\mu\}$. This is clearly a $\Gamma$-spectral family because in fact $[A]_{\gamma}<[\lambda_{\gamma}]$. We define $[\Gamma]\cap {\rm Sp}(\Lambda)
=\Gamma{\rm -Sp}(\Lambda),[\Gamma]\cap Q Sp(\Lambda)[\Gamma]\cap QSp(\Lambda)
=\Gamma-QSp(\Lambda)$ and similarly with $p$ replaced by $P$ when ${\rm Id}_{\wedge}(C[\Lambda])$ is considered instead of ${\rm id}_{\wedge}(C(\Lambda_-))$ (see section 1).
In view of Proposition 4.1.i. a $\Gamma$-spectral family is contained in a sublattice (that is with commutative $\wedge$) of the noncommutative $\Lambda$, in fact $\{\lambda_{\gamma},\gamma\in\Gamma\}$ is such a sublattice. If $Ab(\Lambda)$ is the set of maximal commutative sublattices of $\Lambda$ then every $\Gamma$-spectral family in $\Lambda$ is a $\Gamma$-spectral family in some $B\in Ab(\Lambda)$ ($B$ refers to Boulean sector in case $\Gamma=\mathbb{R}_+,\Lambda=L(H)$). The above remarks may be seen as a generalization of the result concerning maximal commutative Von Neumann regular subalgebras in ${\cal L}(H)$ quoted above.
$\Gamma$-spectral families may be defined on the moment spaces ${\rm
Spec}(\Lambda_t,T_t)$ in exactly the way described above as filtrations $\{U_t(x_{\gamma}),\gamma\in\Gamma\}$, where each $x_{\gamma}$ is $t$-accessible, defining a separated $\Gamma$-filtration. For $t''\in I_t$ we may look at $V_t(x_{\gamma})=\{p_{t''},p_{t''}\in U_t(x_{\gamma})\}$, agaiin $p_{t''}=\varphi_{t't''}(p_{t'})$ or $\varphi_{t't''}(p_{t'})=p_{t''}$ depending whether $t'\le t''$ or $t''\le t'$. The family $\{V_t(x_{\gamma}),\gamma\in\Gamma\}$ need not (!) be a $\Gamma$-spectral family at $t''\in T$. A stronger notion of [**dynamical spectral family**]{} may be obtained by demanding the existence of stringwise spectral families in a relative open $T$-interval $J$ around $t$. Then indeed at $t''\in J\subset
I(t)$ such a stringwise $\Gamma$-spectral family induces a $\gamma$-spectral famly in $\Lambda_{t''}$ but not immediately on ${\rm
Spec}(\Lambda_{t''},I_{t^n})$ unless a more stringent relation is put on $I_{t^n}$ and its comparison with respect to $I_t$. We just point out the interesting problems arising with respect to observables when passing to moment spaces but this is work in progress.
References {#references .unnumbered}
==========
- F. Van Oystaeyen, [*Algebraic Geometry for Associative Algebras*]{}, M. Dekker, Math. Monographs, Vol. 232, New York, 2000.
- A. Connes, [*$C^*$-Algeèbres et Géométries Differentielle*]{}, C. R. Acad. Sc. Paris, 290, 1980, 599-604.
- \[\[CoD\] A. Connes, M. Dubois-Violette, [*Noncommutative Finite Dimensional Manifolds I, Spherical Maniforls and related Examples*]{}, Notes IHES, MO1, 32, 2001.
- M. Artin, J. Tate, M. Van den Bergh, [*Modules over Regular Algebras of Dimension 3*]{}, Invent. Nath. 106, 1991, 335-388.
- M. Van den Bergh, [*Blowing up of Noncommutative Smooth Surfaces*]{}, Mem. Amer. Math. Soc., 154, 2001, no. 734.
- M. Kontsevich, [*Deformation, Quantization of Poissin Manifolds*]{}, q-alg, 9709040, 1997.
- F. Van Oystaeyen, [*Virtual Topology amd Functor Geometry*]{}, lecture Notes UA, Submitted.
- D. Murdoch, F. Van Oystaeyen, [*Noncommutative Localization and Sheaves,*]{}. J. of Algebra, 35, 1975, 500-525.
- J.-P. Serre, [*Faisceaux Algébriques Cohérents*]{}, Ann. Math. 61, 1955, 197-278.
- F. Van Oystaeyen, A. Verschoren, [*Noncommutative Algebraoc Geometry*]{}, LNM 887, Springer Verlag, berlin, 1981.
|
---
abstract: 'Generalized Galois numbers count the number of flags in vector spaces over finite fields. Asymptotically, as the dimension of the vector space becomes large, we give their exponential growth and determine their initial values. The initial values are expressed analytically in terms of theta functions and Euler’s generating function for the partition numbers. Our asymptotic enumeration method is based on a Demazure module limit construction for integrable highest weight representations of affine Kac-Moody algebras. For the classical Galois numbers, that count the number of subspaces in vector spaces over finite fields, the theta functions are Jacobi theta functions. We apply our findings to the asymptotic number of linear $q$-ary codes, and conclude with some final remarks about possible future research concerning asymptotic enumerations via limit constructions for affine Kac-Moody algebras and modularity of characters of integrable highest weight representations.'
address: |
Stavros Kousidis, Institute for Theoretical Physics\
ETH Zurich\
Wolfgang–Pauli–Strasse 27\
CH-8093 Zurich\
Switzerland
author:
- Stavros Kousidis
bibliography:
- 'galoiskacmoody.bib'
title: 'Asymptotics of generalized Galois numbers via affine Kac-Moody algebras'
---
Introduction
============
The generalized Galois numbers $G_N^{(r)}(q)$ count the number of flags $0 = V_0 \subseteq V_1 \subseteq \cdots \subseteq V_r = \mathbf{F}_q^N$ of length $r$ in an $N$-dimensional vector space over a field with $q$ elements [@v10]. In particular, when $r=2$ these are the classical Galois numbers studied by Goldman and Rota [@MR0252232] which give the total number of subspaces in $\mathbf{F}_q^N$.
We show that the generalized Galois numbers grow asymptotically, as $r$ is fixed and $N \rightarrow \infty$, exponentially with factor $O(N^2)$ in logarithmic “time” scale: $$G_N^{(r)}(q) \sim I_r(q) \cdot e^{O(N^2) \log (q)}
.$$ Here, “time” equals the cardinality of the finite field. Our main result is the explicit description of the initial values $I_r(q)$ via theta functions and Euler’s generating function for the partition numbers.
This investigation serves three purposes. First, the generalized Galois numbers are of independent interest as they enumerate points in fundamental geometric objects defined over finite fields. For example, by definition the classical Galois numbers $$G_N(q)=G_N^{(2)}(q) = \sum_{k=0}^N |\mathrm{Gr}(k,N)(\mathbf{F}_{q})|$$ count the number of $\mathbf{F}_q$-rational points in Grassmann varieties. The numbers of solutions of the set of equations for $\mathrm{Gr}(k,N)$ in extension fields $\mathbf{F}_{p^n}$ of $\mathbf{F}_{p}$ are in turn subject to the study of local zeta-functions $Z(\mathrm{Gr}(k,N),t) = \exp ( \sum_{n \geq 1} |\mathrm{Gr}(k,N)(\mathbf{F}_{p^n})| \frac{t^n}{n} )$ in number theory. Let us mention that a generating function for the local zeta-function $Z(\mathrm{Gr}(k,N),t)$ can be given by $$Z(\mathrm{Gr}(k,N),t) = \frac{1}{(1-t)^{b_0}(1-pt)^{b_1}\ldots(1-p^{k(N-k)}t)^{b_{k(N-k)}}} ,$$ where the $b_i = \dim H_{2i}(\mathrm{Gr}(k,N)(\mathbf{C}),\mathbf{Z})$ are the even topological Betti numbers of the complex Grassmannian. Consequently, the study of Galois numbers reflects upon many subjects.
Second, the Galois numbers enumerate asymptotically the number of equivalence classes of linear $q$-ary codes in algebraic coding theory as recently shown by Hou and Wild [@MR2177491; @MR2307131; @MR2492098; @MR1755766; @MR2191288]. For example, the asymptotic number $N_{n,q}^{\mathfrak{S}}$ of linear $q$-ary codes under permutation equivalence is $$N_{n,q}^{\mathfrak{S}} \sim \frac{G_n(q)}{n!}
.$$ We apply our findings to those asymptotic equivalences, and derive considerable simplifications of the asymptotic enumeration of linear $q$-ary codes ().
Third, our investigation serves the demonstration of the asymptotic enumeration method itself (). We identify the generalized Galois numbers $G_N^{(r)}(q)$ as the basic specialization of the Demazure modules $V_{-N\omega_1}(\Lambda_0)$ of the affine Kac-Moody algebra $\widehat{\mathfrak{sl}}_r$ (see ). Those characters pass via a graded limit construction [@MR2323538; @MR894387; @MR932325; @MR980506] to the characters of the fundamental representations of our affine Kac-Moody algebra: $$\lim_{n \rightarrow \infty} \chi (V_{-(rn+j)\omega_1}(\Lambda_0)) = \chi (V(\Lambda_j))
.$$ By a symmetry argument, Kac’s [@MR513845] character formula for the basic representation $$\chi (V(\Lambda_0)) = \sum_{k=0}^\infty p^{(r-1)}(k) e^{\Lambda_0 -k\delta} \cdot \sum_{\gamma \in Q} e^{- (h || \gamma ||^2\delta + \gamma )}$$ then allows us to prove our main result:
Consider the generalized Galois number $G_N^{(r)}(q)$. For any prime power $e^\delta= p^m$ (in fact for any complex number $e^\delta$ where $\delta \in -2\pi i \mathbf{H}$) and $0 \leq j <r$ we have the limit $$\begin{aligned}
\label{eq:gen galois values}
\lim_{n \rightarrow \infty} G^{(r)}_{rn + j}(e^\delta) \cdot e^{- u_j(r,n) \delta} & = \frac{ \Theta_{F_j} (-\frac{\delta}{2 \pi i})}{\phi(e^{-\delta})^{r-1}} .
\end{aligned}$$ Here, $\phi(x)^{-1} = \prod_{m=1}^\infty (1-x^{m})^{-1}$ denotes Euler’s generating function for the partition numbers, and $\Theta_{F_j}(z) = \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{2 \pi i z F_j({\bf k})} $ are theta functions associated to the quadratic forms $F_0 , F_1 , \ldots , F_{r-1}$ on the lattice $\mathbf{Z}^{r-1}$ given by $$\begin{aligned}
F_0 (k_1,\ldots , k_{r-1}) & = \sum_{l=1}^{r-1} k_l^2 - \sum_{l=1}^{r-2} k_l k_{l+1} , \\
F_j (k_1,\ldots , k_{r-1}) & = \left( k_j + \frac 12 \right)^2 + \sum_{l=1, l \neq j}^{r-1} k_l^2 - \sum_{l=1}^{r-2} k_l k_{l+1} .
\end{aligned}$$ The exponents $u_0,u_1,\ldots ,u_{r-1}$ are $$\begin{aligned}
u_0(r,n) & = \frac{r(r-1)n^2}{2} , \\
u_j(r,n) & = \frac{(rn+j)(rn+j-1)}{2} - \frac{rn(rn+2j-r)}{2r} + \frac 14
.
\end{aligned}$$
For the classical Galois numbers our theta functions turn out to be Jacobi theta functions (see ).
Let us conclude the introduction with the following remark on our asymptotic enumeration method. In the case of generalized Galois numbers we do not make use of the modularity of characters of integrable highest weight modules, since the prime powers $p^{-m} < 1$ lie in the region of convergence of our modular forms. However, we will discuss, in , an important eventual application of our asymptotic enumeration method where modularity has to be exploited.
Notation and Background {#sec:notation}
=======================
The generalized Galois number $G_N^{(r)}(q) \in \mathbf{N}[q]$ can be defined as the specialization of the generalized $N$-th Rogers-Szegő polynomial at $({\bf 1},q)$ [@v10]: $$G_N^{(r)}(q) = H_N^{(r)}({\bf 1},q)
.$$ The $N$-th generalized Rogers-Szegő polynomial $H_N^{(r)}({\bf z},q) \in \mathbf{N}[z_1,\ldots ,z_{r},q]$ [@r1893a; @r1893b; @s1926] (see [@MR1634067] for an account) is defined as the generating function of the $q$-multinomial coefficients: $$H^{(r)}_N({\bf z} , q)
= \sum_{\substack{{\bf k} = (k_1,\ldots,k_r) \in \mathbf{N}^{r}\\k_1 + \ldots + k_r =N}}{\genfrac{[}{]}{0pt}{}{N}{{\bf k}}}_q {\bf z^k}
.$$ Recall from [@v10] that the $q$-multinomial coefficient ${\genfrac{[}{]}{0pt}{}{N}{k_1, \ldots , k_r}}_q$ counts the number of flags $0 = V_0 \subseteq \cdots \subseteq V_r = \mathbf{F}_q^N$ subject to the conditions $\dim(V_i) = k_1 + \cdots + k_i$.
For general facts about affine Kac-Moody algebras and their representation theory we refer the reader to [@MR2188930; @MR1104219], and for Demazure modules to [@MR2323538]. Let us briefly fix the notation we will use throughout. We consider the affine Kac-Moody algebra $\widehat{\mathfrak{sl}}_{r}$. We denote the simple roots by $\alpha_0 , \alpha_1, \ldots , \alpha_{r-1}$, the highest root by $\theta = \alpha_1 + \ldots + \alpha_{r-1}$ and the imaginary root by $\delta = \alpha_0 + \theta$. The affine root lattice is then defined as $\widehat{Q} = \mathbf{Z} \alpha_0 \oplus \mathbf{Z} \alpha_1 \oplus \ldots \oplus \mathbf{Z} \alpha_{r-1}$ and the real span of the simple roots is given by $\widehat{\mathfrak{h}}_{\mathbf{R}}^\ast = \mathbf{R} \otimes_{\mathbf{Z}} \widehat{Q}$. We have a non-degenerate symmetric bilinear form on $\widehat{\mathfrak{h}}_{\mathbf{R}}^\ast$ by $\langle \alpha_i , \alpha_j \rangle = c_{ij}$ where $C = (c_{ij})$ is the Cartan matrix of $\widehat{\mathfrak{sl}}_{r}$, and define $|| \cdot ||^2 = (2h)^{-1} \langle \cdot , \cdot \rangle$ where $h=r$ is the Coxeter number of $\widehat{\mathfrak{sl}}_{r}$. For a dominant integral weight $\Lambda = m_1\Lambda_0 + m_2\Lambda_1 + \ldots + m_{r-1} \Lambda_{r-1}$ we let $V(\Lambda)$ be the integrable highest weight representation of weight $\Lambda$ of $\widehat{\mathfrak{sl}}_{r}$ and $\chi(V(\Lambda))$ its character. The $\Lambda_0, \Lambda_1, \ldots , \Lambda_{r-1}$ are called fundamental weights, the $V(\Lambda_l)$ the fundamental representations and $V(\Lambda_0)$ the basic representation. As for the Demazure modules, we will only consider the translations $t_{-k\omega_1} = (s_1 s_2 \ldots s_{r-1} \sigma^{r-1})^k$ in the extended affine Weyl group of $\widehat{\mathfrak{sl}}_{r}$, where $\omega_1 = \Lambda_1 - \Lambda_0$. Here, $\sigma$ denotes the automorphism of the Dynkin diagram of $\widehat{\mathfrak{sl}}_{r}$ which sends $0$ to $1$, and $s_1, \ldots , s_{r-1}$ are the simple reflections associated to the simple roots $\alpha_1, \ldots , \alpha_{r-1}$. We denote the Demazure module associated to those translations by $V_{-k\omega_1}(\Lambda)$ and its character by $\chi(V_{-k\omega_1}(\Lambda))$. We write the monomials in the characters of our modules as $e^{\lambda}$, the coefficient $k$ in the monomial $e^{-k\alpha_0}$ is referred to as the degree.
$\mathbf{H}$ will denote the upper half plane in $\mathbf{C}$. We write $\sim$ for asymptotic equivalence, that is for $f,g : \mathbf{N} \rightarrow \mathbf{R}_{>0}$ we write $f(n) \sim g(n)$ if $\lim_{n \rightarrow \infty} f(n)/g(n) =1$.
Asymptotics of generalized Galois numbers {#sec:asymptotics}
=========================================
Let us start with a direct consequence of Kac’s character formula [@MR513845 (3.37)].
Consider the basic representation $V(\Lambda_0)$ of $\widehat{\mathfrak{sl}}_{r}$. Let $Q$ be the lattice $Q = \widehat{Q}/\mathbf{Z}\alpha_0 = \mathbf{Z}\alpha_1 \oplus \ldots \oplus \mathbf{Z} \alpha_{r-1} \cong \mathbf{Z}^{r-1}$. Then, $$\begin{aligned}
\label{character basic representation}
\chi (V(\Lambda_0)) & = \frac{e^{\Lambda_0}}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{\frac 14 \theta} \prod_{l=0}^{r-1} e^{-F_l({\bf k}) \alpha_l}
.
\end{aligned}$$ Here, $\phi(x)^{-1} = \prod_{i=1}^m (1-x^m)^{-1}$ is Euler’s generating function for the partition numbers, and the $F_0,F_1,\ldots ,F_{r-1}$ are quadratic forms on the lattice $Q \cong \mathbf{Z}^{r-1}$ defined as $$\begin{aligned}
F_0 (k_1,\ldots , k_{r-1}) & = \sum_{l=1}^{r-1} k_l^2 - \sum_{l=1}^{r-2} k_l k_{l+1} ,\\
F_j (k_1,\ldots , k_{r-1}) & = \left( k_j + \frac 12 \right)^2 + \sum_{l=1, l \neq j}^{r-1} k_l^2 - \sum_{l=1}^{r-2} k_l k_{l+1} .
\end{aligned}$$
Due to Kac [@MR513845 (3.37)] we have the following character formula vor the basic representation $V(\Lambda_0)$: $$\begin{aligned}
\label{eq:Kac character}
\chi (V(\Lambda_0)) = \sum_{k=0}^\infty p^{(r-1)}(k) e^{\Lambda_0 -k\delta} \cdot \sum_{\gamma \in Q} e^{- (h || \gamma ||^2\delta + \gamma )}
\end{aligned}$$ The function $p^{(r-1)}(k)$ is defined via its generating function $\sum_{k=0}^\infty p^{(r-1)}(k) x^k = \phi(x)^{-r+1}$. Let $f({\bf k})$ be the quadratic form $f({\bf k}) = f(k_1 , \ldots , k_{r-1}) = \sum_{i=1}^{r-1} k_i^2 - \sum_{i=1}^{r-2} k_i k_{i+1}$ on the lattice $Q \cong \mathbf{Z}^{r-1}$. If we express $\gamma \in Q$ as the linear combination $\gamma = k_1 \alpha_1 + \ldots + k_{r-1} \alpha_{r-1}$, we have $h|| \gamma ||^2 = f({\bf k})$. Then, $$\begin{aligned}
\chi (V(\Lambda_0)) & = \sum_{k=0}^\infty p^{(r-1)}(k) e^{\Lambda_0 -k\delta} \cdot \sum_{\gamma \in Q} e^{- (h || \gamma ||^2\delta + \gamma )} \\
& = \frac{e^{\Lambda_0}}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{\gamma \in Q} e^{- (h || \gamma ||^2\delta + \gamma) } \\
& = \frac{e^{\Lambda_0}}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{- f({\bf k}) \delta} e^{-k_1\alpha_1 } \ldots e^{-k_{r-1} \alpha_{r-1}} \\
& = \frac{e^{\Lambda_0}}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{- f({\bf k}) \alpha_0} e^{-(f({\bf k}) +k_1)\alpha_1} \ldots e^{-(f({\bf k}) +k_{r-1})\alpha_{r-1}}
.
\end{aligned}$$ Note that $F_0({\bf k}) = f({\bf k})$ and $F_j ({\bf k}) - \frac 14 = f ({\bf k}) +k_j$. This finishes the proof.
We are ready to prove our main result.
\[thm:gen galois values\] Consider the generalized Galois number $G_N^{(r)}(q)$. For any prime power $e^\delta= p^m$ (in fact for any complex number $e^\delta$ where $\delta \in -2\pi i \mathbf{H}$) and $0 \leq j <r$ we have the limit $$\begin{aligned}
\label{eq:gen galois values}
\lim_{n \rightarrow \infty} G^{(r)}_{rn + j}(e^\delta) \cdot e^{- u_j(r,n) \delta} & = \frac{ \Theta_{F_j} (-\frac{\delta}{2 \pi i})}{\phi(e^{-\delta})^{r-1}} .
\end{aligned}$$ Here, $\phi(x)^{-1} = \prod_{m=1}^\infty (1-x^{m})^{-1}$ denotes Euler’s generating function for the partition numbers, and $\Theta_{F_j}(z) = \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{2 \pi i z F_j({\bf k})} $ are theta functions associated to the quadratic forms $F_0 , F_1 , \ldots , F_{r-1}$ on the lattice $\mathbf{Z}^{r-1}$ given by $$\begin{aligned}
F_0 (k_1,\ldots , k_{r-1}) & = \sum_{l=1}^{r-1} k_l^2 - \sum_{l=1}^{r-2} k_l k_{l+1} , \\
F_j (k_1,\ldots , k_{r-1}) & = \left( k_j + \frac 12 \right)^2 + \sum_{l=1, l \neq j}^{r-1} k_l^2 - \sum_{l=1}^{r-2} k_l k_{l+1} .
\end{aligned}$$ The exponents $u_0,u_1,\ldots ,u_{r-1}$ are $$\begin{aligned}
u_0(r,n) & = \frac{r(r-1)n^2}{2} , \\
u_j(r,n) & = \frac{(rn+j)(rn+j-1)}{2} - \frac{rn(rn+2j-r)}{2r} + \frac 14
.
\end{aligned}$$
Let $N = rn+j$ for $0\leq j<r$ and consider the Demazure module $V_{-N\omega_1}(\Lambda_0)$ associated to the translation $t_{-N\omega_1} = (s_1 s_2 \ldots s_{r-1} \sigma^{r-1})^N$. By Sanderson [@MR1771615] we can describe its character $\chi(V_{-N\omega_1}(\Lambda_0))$ via a certain specialization of a symmetric Macdonald polynomial (see [@MR1354144 Chapter VI] for their definition and properties). That is, let $[N] = (N,0,\ldots,0) \in \mathbf{N}^{r}$ denote the one-row Young diagram, and $\eta_N$ the smallest composition of degree $N$, i.e. since $N = rn +j$ we have $\eta_N = ((n)^{r-j},(n+1)^j) \in \mathbf{N}^{r}$. Following [@MR1771615 §2] we have $[N] = t_{-N\omega_1} \cdot \eta_N$ with the convention $\sigma \cdot \eta_N = \eta_N$. Furthermore, by computing the expression $u([N])-u(\eta_{[N]})$ in [@MR1771615 Theorem 6], the maximal occurring degree in $\chi(V_{-N\omega_1}(\Lambda_0))$ is given by $$\begin{aligned}
d_r(N) = d_r(rn+j) = \frac{(rn+j)(rn+j-1)}{2} - \frac{rn(rn+2j-r)}{2r} .
\end{aligned}$$ Note that $d_r(rn) = u_0 (r,n)$ and $d_r(rn+j) = u_j (r,n) - \frac 14$ for $j = 1,\ldots ,r-1$. Let ${\bf z} = (e^{\Lambda_1 - \Lambda_0}, e^{\Lambda_2 -\Lambda_1}, \ldots , e^{\Lambda_{r-1} - \Lambda_{r-2}}, e^{\Lambda_0 -\Lambda_{r-1}})$. Then, by [@MR1771615 Theorem 6 and 7][^1] we have $$\begin{aligned}
\label{demazure macdonald}
\chi(V_{-N \omega_1}(\Lambda_0)) = e^{\Lambda_0-d_r(N)\delta} \cdot P_{[N]}({\bf z};e^\delta,0)
,
\end{aligned}$$ where $P_{[N]}({\bf z};q,0)$ denotes the specialized symmetric Macdonald polynomial associated to the partition $[N]$. Furthermore, by Hikami [@MR1457389 Equation (3.4)] this Macdonald polynomial equals the $N$-th generalized Rogers-Szegő polynomial: $$\begin{aligned}
\label{macdonald rogerszegoe}
P_{[N]}({\bf z};q,0) = H_N^{(r)}({\bf z} , q)
.
\end{aligned}$$ Combining and we obtain $$\begin{aligned}
\label{demazure rogersszegoe}
\chi (V_{-N \omega_1}(\Lambda_0)) = e^{\Lambda_0-d_r(N)\delta} \cdot H_N^{(r)}({\bf z},e^\delta)
.
\end{aligned}$$ Consequently, the basic specialization at $e^{-\alpha_1} = \ldots = e^{-\alpha_{r-1}} =1$ of the Demazure character on the left-hand side of gives the generalized Galois number $G_N^{(r)}$ up to translation: $$\begin{aligned}
\label{demazure basic specialization galois}
\left. \chi (V_{-N\omega_1}(\Lambda_0)) \right\vert_{(e^{-\alpha_1} = \ldots = e^{-\alpha_{r-1}} =1)} = e^{\Lambda_0-d_r(N)\delta}\cdot G_N^{(r)} (e^\delta)
.
\end{aligned}$$
Now, let us proceed to the limit considerations. By Fourier and Littelmann [@MR2323538 Theorem D] (which is based on work by Mathieu and Kumar [@MR894387; @MR932325; @MR980506]) the characters of our Demazure modules pass, as $N \rightarrow \infty$, as functions in $(e^{-\alpha_0},e^{-\alpha_1}, \ldots , e^{-\alpha_{r-1}})$ to the characters of the fundamental representations $V(\Lambda_0), V(\Lambda_1), \ldots , V(\Lambda_{r-1})$ of $\widehat{\mathfrak{sl}}_{r}$ as follows $$\begin{aligned}
\label{limit demazure weyl-kac}
\lim_{n \rightarrow \infty} \chi (V_{-(rn+j)\omega_1}(\Lambda_0)) & = \chi (V(\Lambda_j))
.
\end{aligned}$$
We are ready to prove our claimed identity in the case $j=0$. Recall that $\delta = \alpha_0 + \theta$ where $\theta = \alpha_1 + \ldots + \alpha_{r-1}$, and $d_r(rn) = u_0 (r,n)$. Then, the equations , and the character formula for the basic representation $V(\Lambda_0)$ imply $$\begin{aligned}
\lim_{n \rightarrow \infty} G_{rn}^{(r)} (e^\delta) \cdot e^{- u_0(r,n)\delta}
& = e^{-\Lambda_0} \cdot \left. \chi (V(\Lambda_0)) \right\vert_{(e^{-\alpha_1} = \ldots = e^{-\alpha_{r-1}} =1)} \\
& = \frac{1}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{-F_0({\bf k}) \delta} \\
& = \frac{1}{\phi(e^{-\delta})^{r-1}} \cdot \Theta_{F_0} \left(-\frac{\delta}{2 \pi i} \right).
\end{aligned}$$
There is a subtlety to our deduction in the cases $j=1,\ldots ,r-1$. The characters of the representations $V(\Lambda_0), V(\Lambda_1), \ldots , V(\Lambda_{r-1})$ are symmetrical in the sense that they subsequently differ by an application of the automorphism $\sigma$ that sends $0$ to $1$ in the Dynkin diagram of $\widehat{\mathfrak{sl}}_{r}$. An application of $\sigma$ cyclically shifts the fundamental weights $\Lambda_i \mapsto \Lambda_{i+1}$ and the simple roots $\alpha_i \mapsto \alpha_{i+1}$ (by cyclic we mean $\Lambda_{r}=\Lambda_0$ and $\alpha_{r} = \alpha_0$). Consequently, it leaves $\delta$ invariant $\sigma(\delta) = \delta$ and $\sigma^j (\theta) = \alpha_0 + \theta - \alpha_{j}$ for $j=1,\ldots ,r-1$. To be precise, Kac’s character formula for the fundamental representations $V(\Lambda_1),\ldots ,V(\Lambda_{r-1})$ reads as follows $$\begin{aligned}
\chi (V(\Lambda_j)) & = \frac{e^{\Lambda_j}}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{\frac 14 (\alpha_0 + \theta - \alpha_{j})} \prod_{l=0}^{r-1} e^{-F_{l+r-j}({\bf k}) \alpha_l}
.
\end{aligned}$$ Recall that $u_j(r,n) = d_r(rn+j) + \frac 14$. Therefore, we obtain $$\begin{aligned}
\lim_{n \rightarrow \infty} G^{(r)}_{rn + j}(e^\delta) \cdot e^{- u_j(r,n) \delta} & = e^{-\frac 14 \delta} \cdot \lim_{n \rightarrow \infty} G^{(r)}_{rn + j}(e^\delta) \cdot e^{- d_r(rn+j) \delta} \\
& = e^{-\frac 14 \delta} \cdot e^{-\Lambda_j} \cdot \left. \chi (V(\Lambda_j)) \right\vert_{(e^{-\alpha_1} = \ldots = e^{-\alpha_{r-1}} =1)} \\
& = \frac{e^{-\frac 14 \delta}}{\phi(e^{-\delta})^{r-1}} \cdot \sum_{{\bf k} \in \mathbf{Z}^{r-1}} e^{\frac 14 \delta} e^{-F_j({\bf k})\delta} \\
& = \frac{1}{\phi(e^{-\delta})^{r-1}} \cdot \Theta_{F_j} \left(-\frac{\delta}{2 \pi i} \right)
.
\end{aligned}$$ This establishes the theorem.
\[intrinsic quadratic forms\] Motivated by Kac’s character formula we write the quadratic forms on $Q = \mathbf{Z}\alpha_1 \oplus \ldots \oplus \mathbf{Z}\alpha_{r-1} \cong \mathbf{Z}^r$ intrinsically in terms of data associated to our affine Kac-Moody algebra $\widehat{\mathfrak{sl}}_{r}$ as follows. For $j =1,\ldots ,r-1$ one has $$\begin{aligned}
F_0 (\gamma) & = h || \gamma ||^2 \\
F_j (\gamma) & = h || \gamma ||^2 + \langle \Lambda_j , \gamma \rangle +\frac 14.
\end{aligned}$$ The exponents $u_0 , u_1 , \ldots , u_{r-1}$ can be described via the translation formula [@MR1104219 (6.5.3)].
One can phrase asymptotically as $$\begin{aligned}
G_{rn+j}^{(r)}(e^\delta) \sim \frac{\Theta_{F_j}(-\frac{\delta}{2 \pi i})}{\phi(e^{-\delta})^{r-1}} \cdot (e^{\delta})^{u_j(n,r)}
\end{aligned}$$ Note that for fixed $r$ the exponents $u_j(n,r)$ lie in $O(n^2)$.
For $j=1,\ldots,r-1$ the limits in coincide. In fact, the quadratic forms $F_1,\ldots,F_{r-1}$ differ only by a cyclic shift of the coordinates. Summation over the complete lattice $\mathbf{Z}^{r-1}$ produces equality.
Let us summarize the implications of for the classical Galois numbers $G_N(q) = G_N^{(2)}(q)$ that count the number of subspaces in $\mathbf{F}_q^N$.
\[galois values\] Consider the classical Galois numbers $G_N(q)$. For any prime power $q = p^m$ (in fact for any complex number $|q| > 1$) we have $$\begin{aligned}
\label{galois values odd}
g_{2\infty +1}(q) & = \lim_{n \rightarrow \infty} G_{2n+1}(q) \cdot q^{-\frac{(2n+1)^2 }{4}} = \frac{\vartheta_2(0,q^{-1})}{\phi(q^{-1})} , \\
\label{galois values even}
g_{2\infty}(q) & = \lim_{n \rightarrow \infty} G_{2n}(q) \cdot q^{- \frac{(2n)^2}{4}} = \frac{\vartheta_3(0,q^{-1})}{\phi(q^{-1})}
.
\end{aligned}$$ Here, $\phi(x)^{-1} = \prod_{m=1}^\infty (1-x^{m})^{-1}$ denotes Euler’s generating function for the partition numbers, and $\vartheta_2,\vartheta_3$ are the Jacobi theta functions $$\begin{aligned}
\vartheta_2(z,q) & = \sum_{k=-\infty}^{\infty} q^{(k+ \frac 12)^2} e^{(2k+1)iz} , \\
\vartheta_3(z,q) & = \sum_{k=-\infty}^{\infty} q^{k^2} e^{2kiz}
.
\end{aligned}$$ The limits differ by $$\begin{aligned}
\label{difference galois values}
g_{2\infty}(q) - g_{2\infty +1}(q) = \frac{\vartheta_4(0,q^{-\frac 14})}{\phi(q^{-1})} ,
\end{aligned}$$ where $\vartheta_4$ is the Jacobi theta function $$\begin{aligned}
\vartheta_4(z,q) = \sum_{k=-\infty}^\infty (-1)^k q^{k^2}e^{2kiz} .
\end{aligned}$$
It remains to prove . This follows from the identity $\vartheta_4(z,q) = \vartheta_3(2z,q^4) - \vartheta_2(2z,q^4)$ [@MR0178117 pp. 464].
For completeness, we take a closer look at the numbers $g_{2\infty +1}(q)$, $g_{2\infty}(q)$ and their differences.
\[closer asymptotic galois values\] If $q$ is a prime power (in fact $q \geq 2$) $$\begin{aligned}
\label{comparison}
g_{2\infty}(2) \geq g_{2\infty}(q) > g_{2\infty+1}(q) > 0.
\end{aligned}$$ For large values of $q$ we have $$\begin{aligned}
\label{asymptotic large primes odd galois values}
g_{2\infty+1}(q) \cdot q^{\frac{(2n+1)^2}{4}} & \sim 2q^{n(n-1)} ,\\
\label{asymptotic large primes even galois values}
g_{2\infty}(q) \cdot q^{n^2} & \sim q^{n^2} ,
\end{aligned}$$ and consequently $$\begin{aligned}
\label{large primes odd galois values}
\lim_{q \rightarrow \infty} g_{2\infty+1}(q) & = 0 , \\
\label{large primes even galois values}
\lim_{q \rightarrow \infty} g_{2\infty}(q) & = 1 .
\end{aligned}$$
Our statements can be deduced from the Jacobi triple product identity. Namely, we have $\vartheta_4(0,q^{-1}) = \prod_{m=1}^\infty (1-q^{-2m})(1-q^{-(2m-1)})^2$ which is $>0$ if $q>1$. Since $\phi(q^{-1})>0$ for $q>1$, the strict inequality $g_{2\infty}(q) > g_{2\infty+1}(q)$ is established. For the obviously sharp bound $g_{2\infty}(2) \geq g_{2\infty}(q)$ we look at $$\begin{aligned}
\label{triple product galois even}
\frac{\vartheta_3(0,q^{-1})}{\phi(q^{-1})} & = \frac{\prod_{m=1}^\infty (1-q^{-2m}) (1+q^{-(2m-1)})^2}{\prod_{m=1}^\infty (1-q^{-m})} \\
\notag
& = \prod_{m=1}^\infty (1+q^{-m})(1+q^{-(2m-1)})^2
\end{aligned}$$ which is a product of compositions of monotonic functions on $q > 1$. This identity also shows and . To prove $g_{2\infty+1}(q) > 0$, and one considers $$\begin{aligned}
\label{triple product galois odd}
\frac{\vartheta_2(0,q^{-1})}{\phi(q^{-1})} & = \frac{ 2q^{-\frac 14} \prod_{m=1}^\infty (1-q^{-2m}) (1+q^{-2m})^2}{\prod_{m=1}^\infty (1-q^{-m})} \\
\notag
& = 2 q^{-\frac 14} \prod_{m=1}^\infty (1+q^{-m})(1+q^{-2m})^2 .
\qedhere
\end{aligned}$$
For some prime powers $q$, the numbers $g_{2\infty+1}(q)$, $g_{2\infty}(q)$ and their differences have been listed in . The table has been produced in Mathematica with the following functions (up to a $10$ digit precision: `N[ ,10]`) for $g_{2\infty+1}(q)$, $g_{2\infty}(q)$ and $g_{2\infty}(q) - g_{2\infty+1}(q)$, respectively. $$\begin{tabular}{l}
\verb+f[q_]:=N[EllipticTheta[2,0,1/q]1/QPochhammer[1/q,1/q],10]+ \\
\verb+g[q_]:=N[EllipticTheta[3,0,1/q]1/QPochhammer[1/q,1/q],10]+ \\
\verb+h[q_]:=N[EllipticTheta[4,0,q^(-1/4)]1/QPochhammer[1/q,1/q],10]+
\end{tabular}$$
Certainly, our allows an implementation for evaluating the asymptotic initial values of generalized Galois numbers.
------------- ------------------------------- ------------------ -------------------------------------
$q$ $g_{2\infty+1}(q)$ $g_{2\infty}(q)$ $g_{2\infty}(q) - g_{2\infty+1}(q)$
$2$ $7.371949491$ $7.371968801$ $0.0000193107$
$3$ $3.018269046$ $3.019783846$ $0.0015147993$
$5$ $1.829548122$ $1.845509008$ $0.0159608865$
$7$ $1.499386995$ $1.537469387$ $0.0380823915$
$11$ $1.229171217$ $1.312069129$ $0.0828979124$
$13$ $1.155207999$ $1.258137150$ $0.1029291515$
$17$ $1.054013475$ $1.191906557$ $0.1378930825$
$19$ $1.016940655$ $1.170103722$ $0.1531630663$
$23$ $0.9584786871$ $1.138621162$ $0.1801424752$
$29$ $0.8947912163$ $1.108510891$ $0.2137196747$
$29^{2011}$ $1.203473556 \cdot 10^{-735}$ $1.000000000$ $1.0000000000$
------------- ------------------------------- ------------------ -------------------------------------
: Asymptotic Galois numbers[]{data-label="table galois numbers"}
can be derived by the character formula of Feingold and Lepowsky [@MR509801 Theorem 4.5] for the basic representation of $\widehat{\mathfrak{sl}}_2$. That is, $$\begin{aligned}
\label{feingold lepowsky 0}
\chi (V(\Lambda_0)) = \sum_{k=0}^\infty p(k) e^{\Lambda_0 - k\delta} \sum_{l=-\infty}^\infty e^{-l^2 \alpha_0} e^{-l(l+1)\alpha_1}
,
\end{aligned}$$ where $p(k)$ is the partition function that counts the number of ways to write $k$ as a sum of positive integers. In fact, Kac’s character formula [@MR513845 (3.37)] reduces to this expression (see [@MR513845 (3.39)]), and our proof of reduces to this setting.
Applications to linear $q$-ary codes {#sec:applications}
====================================
To describe the asymptotic number of non-equivalent binary $n$-codes in terms of the classical Galois numbers $G_n(2)$, Wild [@MR1755766; @MR2191288] examines numbers $d_1(q),d_2(q)$ (see Lemma 1 in both articles) which, in the notation of , are defined as $$\begin{aligned}
d_1(q) & = g_{2\infty +1}(q) , \\
d_2(q) & = g_{2\infty}(q) .
\end{aligned}$$ He proves that they are positive constants (depending on $q$) less than $32$, gives a numerical evaluation method by use of the recursion formula of Goldman and Rota [@MR0252232 (5)], evaluates $d_1(q)$, $d_2(q)$ numerically for $q=2$, and shows $d_1(q) < d_2(q)$ for general $q$. Now, the detailed analytic behavior of those numbers can be extracted from and (see also for examples).
For a general prime power $q$, Hou [@MR2177491; @MR2492098] derives asymptotic equivalences for the numbers of linear $q$-ary codes under three notions of equivalence. That is, the permutation equivalence ($\mathfrak{S}$), the monomial equivalence ($\mathfrak{M}$), and semi-linear monomial equivalence ($\Gamma$). He proves $$\begin{aligned}
N_{n,q}^{\mathfrak{S}} & \sim \frac{G_n(q)}{n!} , \\
N_{n,q}^{\mathfrak{M}} & \sim \frac{G_n(q)}{n!(q-1)^{n-1}} , \\
N_{n,q}^{\Gamma} & \sim \frac{G_n(q)}{n!(q-1)^{n-1}a} ,
\end{aligned}$$ where $a = \left| \mathrm{Aut}(\mathbf{F}_q) \right| = \log_p(q)$ with $p = \mathrm{char}(\mathbf{F}_q)$. The asymptotic equivalence $N_{n,2}^{\mathfrak{S}} \sim \frac{G_n(2)}{n!}$ concerns binary codes and is previously derived by Wild [@MR1755766; @MR2191288]. Based on their results, the transitivity of $\sim$ and our produce the following list.
\[q-ary codes asymptotic\] The asymptotic numbers of linear $q$-ary codes, as $q$ is fixed and $n \rightarrow \infty$, under the three notions of equivalence $(\mathfrak{S})$, $(\mathfrak{M})$ and $(\Gamma)$ are given by $$\begin{aligned}
N_{2n+1,q}^{\mathfrak{S}} & \sim \frac{\vartheta_2(0,q^{-1})}{\phi(q^{-1})} \cdot \frac{q^{\frac{(2n+1)^2}{4}}}{(2n+1)!} , \\
N_{2n,q}^{\mathfrak{S}} & \sim \frac{\vartheta_3(0,q^{-1})}{\phi(q^{-1})} \cdot \frac{q^{n^2}}{(2n)!} , \\
N_{2n+1,q}^{\mathfrak{M}} & \sim \frac{\vartheta_2(0,q^{-1})}{\phi(q^{-1})} \cdot \frac{q^{\frac{(2n+1)^2}{4}}}{(2n+1)!(q-1)^{2n}} , \\
N_{2n,q}^{\mathfrak{M}} & \sim \frac{\vartheta_3(0,q^{-1})}{\phi(q^{-1})} \cdot \frac{q^{n^2}}{(2n)!(q-1)^{2n-1}} , \\
N_{2n+1,q}^{\Gamma} & \sim \frac{\vartheta_2(0,q^{-1})}{\phi(q^{-1})} \cdot \frac{q^{\frac{(2n+1)^2}{4}}}{(2n+1)!(q-1)^{2n}a} , \\
N_{2n,q}^{\Gamma} & \sim \frac{\vartheta_3(0,q^{-1})}{\phi(q^{-1})} \cdot \frac{q^{n^2}}{(2n)!(q-1)^{2n-1}a} .
\end{aligned}$$ Furthermore, for large prime powers $q$ one has $$\begin{aligned}
\frac{\vartheta_2(0,q^{-1})}{\phi(q^{-1})} \cdot q^{\frac{(2n+1)^2}{4}} & \sim 2q^{n(n-1)} , \\
\frac{\vartheta_3(0,q^{-1})}{\phi(q^{-1})} \cdot q^{n^2} & \sim q^{n^2} .
\end{aligned}$$
For the last two statements see .
Conclusion {#sec:conclusion}
==========
The asymptotic enumeration method presented in this article can be summarized as follows. Once a certain specialization of Demazure characters has been identified with an interesting combinatorial function, the limit construction for affine Kac-Moody algebras can be used to carry it along towards the character of the integrable highest weight module, and derive asymptotic identities. There are at least two bottlenecks that one has to pass. First, a suitable character formula (for the limiting integrable highest weight representation) has to be available, that performs well with the chosen specialization. Fortunately, there is a great number of results and literature available, e.g. [@MR509801; @MR513845; @MR585190; @MR750341] (see also [@MR2534107; @MR2719689; @MR1810948]). Second, the domain, of the combinatorial function, that enumerates the objects in question must lie in the region of convergence of the limiting expressions. For example, Demazure modules specialize to tensor products of representations of the underlying finite-dimensionsal Lie algebra. Unfortunately, the analytic string functions limiting the tensor product multiplicities cannot be simply evaluated, for reasons of (non-)convergence, at the value $1$. A much finer analysis of their asymptotic behavior when $q \rightarrow 1$ is needed, that has to exploit the fact that we deal with modular forms [@MR750341]. Such an asymptotic analysis must take the maximal weights in the integrable highest weight module into account where those string functions emerge. Possibly borrowing and mimicking terminology from stochastic analysis like the central limit region, moderate and strong deviations region, and region of rare events. An investigation of tensor product multiplicities along those lines is planned in a future publication.
An interesting alternative project could be to re-interpret our asymptotic enumeration method geometrically through the geometric realization of Demazure and integrable highest weight modules via cohomology of Schubert and flag varieties [@MR1923198].
Acknowledgements
================
This work was supported by the Swiss National Science Foundation (grant PP00P2-128455), the National Centre of Competence in Research ‘Quantum Science and Technology’, and the German Science Foundation (SFB/TR12, and grants , ).
I would like to thank Matthias Christandl for his kind hospitality at the ETH Zurich, Peter Littelmann who drew my attention towards the Demazure module limit construction a couple of years ago, Thomas Bliem who pointed me towards Rogers-Szegő polynomials, and Ghislain Fourier for many helpful conversations.
[^1]: There seems to be a missprint in [@MR1771615 §4]. Namely, the image $\pi(q)$ should equal $q = e^\delta$, not $q = e^{-\delta}$.
|
---
author:
- 'Prasanta RUDRA\'
date:
title: |
[BULLETIN DE L’ACADEMIE POLONAISE DES SCIENCES]{}\
[Serie des sciences math., ast. et phys. – Vol. XXV, No. 5, p. 521, 1977]{}\
\
Irreducible Tensor Operators and the Wigner-Eckart Theorem for Finite Magnetic Groups\
by
---
The transformation properties of irreducible tensor operators and the applicability of the Wigner-Eckart theorem to finite magnetic groups have been studied.
Introduction {#intro}
============
Selection rules and ratio of intensities for transitions between different states of a physical system are obtained from the appropriate matrix elements of operators between the initial and the final states of the system [@rose; @judd]. The calculation of the matrix elements becomes simplified if one invokes the Wigner-Eckart theorem [@wigner; @eckart; @fano]. This theorem introduces the concept of a set of operators that transforms according to some irreducible representation of the appropriate symmetry group of the system. For compact and for finite groups, if the Kronecker inner direct product of two irreducible representations contains any irreducible representation only once, the matrix element of such an operator between states belonging to irreducible representations will be proportional to the corresponding Clebsch-Gordan (CG) coefficient, the proportionality constant being called the reduced matrix element. The proof of this theorem depends on the fact that the matrix element when transformed by a symmetry element of the system has the same value as the untransformed matrix element. If the symmetry group of the system is a magnetic group, then antilinear elements [@wigner] are present and for these elements the transformed matrix elements are complex conjugate of the untransformed value. For this reason the Wigner-Eckart theorem is not in general valid in the case of magnetic groups. Recently Backhouse [@backhouse] and Doni and Paravicini [@doni] have investigated the theory of selection rules in magnetic crystals whose symmetry group contains antilinear elements. We have here investigated the conditions when the Wigner-Eckart theorem is valid for symmetry groups containing antilinear elements. To this end we have studied the transformation laws of irreducible tensor operators (both linear and antilinear) for magnetic groups. In order that the results can be applied to spinor cases as well, projective corepresentations [@janssen; @bradley] have been considered. Previously Aviran and Zak [@zak] have investigated this problem. Their results are somewhat complicated because a quadratic relationship between the CG coefficients was used in their analysis, whereas a linear relationship has been used here.
Irreducible tensor operators {#tensor}
============================
Here we give transformation laws of irreducible tensor operators for a magnetic group [@janssen; @bradley] $$M(G) = G \cup a_0 G, ~~~a_0^2\in G, \label{ito1}$$ where $a_0$ is an antilinear element and $G$ is a group of linear elements. The corepresentation $D^{\lambda}\left(\alpha\right),~~\alpha \in M\left(
G\right)$, belonging to the cofactor system $\lambda\left(\alpha,\beta
\right),~~\alpha,\beta\in M\left(G\right)$ satisfies [@janssen; @bradley; @rudra1] $$\begin{aligned}
D^{\lambda}\left(\alpha\right)D^{\lambda}\left(\beta\right)^{\left[\alpha
\right]} & = & \lambda\left(\alpha,\beta\right)^{\left[\alpha\beta\right]}
D^{\lambda}\left(\alpha\beta\right), \nonumber \\
\lambda\left(\alpha,\beta\right)^{\left[\gamma\right]}\lambda\left(
\alpha\beta,\gamma\right) & = & \lambda\left(\alpha,\beta\gamma\right)
\lambda\left(\beta,\gamma\right), \label{ito2} \\
|\lambda\left(\alpha,\beta\right)| & = & 1. \nonumber \end{aligned}$$ We have used the square bracket symbol $\left[\alpha\right]$ everywhere, so that $$A^{\left[\alpha\right]} = \left\{\begin{array}{l} A,~{\rm if}~\alpha~
{\rm is~linear}, \\ A^{\ast},~{\rm if}~\alpha~{\rm is~antilinear,}
\end{array} \right.$$ where $A$ is a matrix, an operator, or a complex number.
We define the Wigner operator [@wigner] $O_{\alpha},~\alpha\in
M\left(G\right)$, by the relation $$O_{\alpha}O_{\beta}^{\left[\alpha\right]} = \lambda\left(\alpha,\beta
\right)^{\left[\alpha\beta\right]}O_{\alpha\beta}. \label{ito3}$$ This relation is satisfied when $O_{\alpha}$s operate on the bases belonging to the appropriate cofactor system. For proper rotations characterized by the Eulerian angles $\left(\alpha, \beta, \gamma\right)$ $$O_{\left(\alpha,\beta,\gamma\right)}=\exp\left(-i\alpha J_z\right)
\exp\left(-i\beta J_y\right)\exp\left(-i\gamma J_z\right), \label{ito4}$$ where $J_i$s are the usual angular momentum operators. The relation (\[ito3\]) will be automatically satisfied if we take the appropriate bases belonging to the vector corepresentation or the spinor corepresentation. For the time reversal operator $\theta$, which is antilinear, its action on the spin states $|j,m\rangle$ will be given by $$O_{\theta}|j,m\rangle = \left(-1\right)^{j-m} |j,-m\rangle. \label{ito5}$$ The $m$-th component of any tensor operator belonging to the $\mu$-th irreducible corepresentation of the cofactor system $\lambda\left(\alpha,
\beta\right)$, which may be either linear or antilinear, will transform as $$T_m^{\lambda\mu}\left(\alpha\right) = \lambda\left(\alpha^{-1},\alpha
\right)^{\ast\left[T\right]}O_{\alpha}T_m^{\lambda\mu}O_{\alpha^{-1}
}^{\left[\alpha\right]} = \sum_n D_{nm}^{\lambda\mu}\left(\alpha
\right)^{\ast\left[T\right]}T_n^{\lambda\mu}. \label{ito6}$$ This relation will also cover the case when $T$ is antilinear. For the sake of completeness we write here the transformation relation of the bases belonging to the $\mu$-th irreducible corepresentation of the same factor system $\lambda\left(\alpha,\beta\right)$ $$\left|\right.\Phi_m^{\lambda\mu}\left(\alpha\right)\left.\right\rangle =
O_{\alpha} \left|\right.\Phi_m^{\lambda\mu}\left.\right\rangle = \sum_n
D_{nm}^{\lambda\mu}\left(\alpha\right)\left|\right.\Phi_n^{\lambda\mu}
\left. \right\rangle. \hspace{5.5cm} \left(6{\rm a}\right)$$ Thus $$T_m^{\lambda\mu}\left(\alpha\right)\left|\right.\Phi\left(\alpha\right)
\left.\right\rangle = O_{\alpha}T_m^{\lambda\mu}\left|\right.\Phi
\left.\right\rangle. \hspace{7.8cm} \left(6{\rm b}\right)$$
The result of successive action of two operators $O_{\beta}$ and $O_{\alpha}$ will be given by $$O_{\alpha}O_{\beta}T_m^{\lambda\mu}O_{\beta^{-1}}^{\left[\beta\right]}
O_{\alpha^{-1}}^{\left[\alpha\right]} = \lambda\left(\alpha,\beta
\right)^{\left[\alpha\beta\right]}\frac{\lambda\left(\alpha^{-1},\alpha
\right)\lambda\left(\beta^{-1},\beta\right)^{\left[\alpha\right]}}{\lambda
\left(\beta^{-1}\alpha^{-1},\alpha\beta\right)}\cdot O_{\alpha\beta}
T_m^{\lambda\mu}O_{\beta^{-1}\alpha^{-1}}^{\left[\alpha\beta\right]}.
\label{ito7}$$ The proof is a straightforward application of Eq. \[ito2\] forthe choice $\lambda\left(\alpha,e\right)=\lambda\left(e,\alpha\right)=1$.
The irreducible tensors $T_m^{\lambda\mu}$ can be obtained by the operation of the projection operator $P_m^{\lambda\mu}$ on an arbitrary tensor $T$ $$T_m^{\lambda\mu} = P_m^{\lambda\mu}T = \sum_{\alpha\in M}
D_{mm_0}^{\lambda\mu}\left(\alpha\right)^{\left[T\right]}\lambda\left(
\alpha^{-1},\alpha\right)^{\ast\left[T\right]} O_{\alpha}T
O_{\alpha^{-1}}^{\left[\alpha\right]}, \label{ito8}$$ where $m_0$ is any fixed index.
Incidentally, the projection operator $P_m^{\lambda\mu}$, which operating on an arbitrary state $\left|\right.\Phi\left.\right\rangle$ will give the $m$-th basis of the $\mu$-th irreducible corepresentation belonging to the cofactor system $\lambda\left(\alpha,\beta\right)$ has the same form [@rudra2] as for the vector corepresentation. $$P_m^{\lambda\mu} = \sum_{\alpha\in M}D_{mm_0}^{\lambda\mu}\left(\alpha
\right)^{\ast}O_{\alpha}. \label{ito9}$$
Wigner-Eckart theorem {#wet}
=====================
For groups with linear elements the Wigne-Eckart theorem [@wigner] states that under the restriction given in Sec. \[intro\] $$\left\langle\Phi_{m_3}^{\lambda_1\mu_1}\left|T_{m_2}^{\lambda_2\mu_2}
\right|\Phi_{m_3}^{\lambda_3\mu_3}\right\rangle = \frac{1}{d_{\lambda_3
\mu_3}}\left\langle\lambda_1 \mu_1\left|\left|\lambda_2\mu_2\right|\right|
\lambda_3\mu_3\right\rangle\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2
m_2\left|\right.\lambda_3\mu_3 m_3\right\rangle, \label{wet10}$$ where $\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2 m_2\left | \right.
\lambda_3\mu_3 m_3\right\rangle$ is the CG coefficient defined by the relation $$\left|\right.\Phi_{m_3}^{\lambda_3\mu_3}\left.\right\rangle = \sum_{m_1m_2}
\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2 m_2\left|\right.\lambda_3
\mu_3 m_3\right\rangle\left|\right.\Phi_{m_1}^{\lambda_1\mu_1}\left.
\right\rangle \left.\right|\Phi_{m_2}^{\lambda_2\mu_2}\left.\right\rangle.
\label{wet11}$$ The reduced matrix element is defined by $$\left\langle\lambda_1\mu_1\left|\right|\lambda_2\mu_2\left|\right|
\lambda_3\mu_3\right\rangle = \sum_{n_1n_2n_3}
\left\langle\lambda_1\mu_1n_1;\lambda_2\mu_2 n_2\left.\right|\lambda_3
\mu_3 n_3\right\rangle^{\ast}\left\langle\Phi_{n_1}^{\lambda_1\mu_1}
\left|T_{n_2}^{\lambda_2\mu_2}\right|\Phi_{n_3}^{\lambda_3\mu_3}\right\rangle
\label{wet12}$$ and $d_{\lambda_3\mu_3}$ = the dimension of the irreducible corepresentation $D^{\lambda_3\mu_3}$.
The CG coefficient is zero unless $\lambda_3\left(\alpha,\beta\right) =
\lambda_1\left(\alpha,\beta\right)\lambda_2\left(\alpha,\beta\right)$.
For magnetic groups no such simple relation is, in general, true. We now investigate the conditions for the validity of such a theorem for magnetic groups. We note that $$\begin{aligned}
\left\langle\Phi_{m_1}^{\lambda_1\mu_1}\left(\alpha\right)\left|
T_{m_2}^{\lambda_2\mu_2}\left(\alpha\right)\right|\Phi_{m_3}^{\lambda_3\mu_3}
\left(\alpha\right)\right\rangle & = & \sum_{n_1n_2n_3}\left\langle
\Phi_{n_1}^{\lambda_1\mu_1}\left|T_{n_2}^{\lambda_2\mu_2}\right|
\Phi_{n_3}^{\lambda_3\mu_3}\right\rangle\times \nonumber \\
& & \hspace{0.3cm} D_{n_1m_1}^{\lambda_1\mu_1}\left(\alpha\right)^{\ast}
D_{n_2m_2}^{\lambda_2\mu_2}\left(\alpha\right)^{\ast\left[T\right]}
D_{n_3m_3}^{\lambda_3\mu_3}\left(\alpha\right)^{\left[T\right]}.
\label{wet13}\end{aligned}$$
Case 1. $T$ is a linear operator.\
In this case the matrix element on the left-hand side of Eq. (\[wet13\]) is zero unless $$\lambda_3\left(\alpha,\beta\right) = \lambda_1\left(\alpha,\beta\right)
\lambda_2\left(\alpha,\beta\right),~~~~\forall \alpha,\beta\in M\left(
G\right). \label{wet14}$$ Using the transformation relation (\[ito6\]) and (6a) for $T_m^{\lambda\mu}\left(\alpha\right)$ and $\Phi_m^{\lambda\mu}\left(
\alpha\right)$ and the linear equations (Eq. (27) of Ref. [@rudra1]) satisfied by the CG coefficients of a magnetic group $$\begin{aligned}
\sum_{m_1m_2}\left[\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2\left.
\right|\lambda_3\mu_3 m_3\right\rangle\sum_{u\in G}
D_{i_1m_1}^{\lambda_1 \mu_1}\left(u\right)D_{i_2m_2}^{\lambda_2\mu_2}
\left(u\right)D_{i_3m_3^{\prime}}^{\lambda_3\mu_3}\left(u\right)^{\ast}
\right. ~~~~~& & \nonumber \\
~~~~+\left.\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2 m_2
\left.\right| \lambda_3\mu_3 m_3^{\prime}\right\rangle^{\ast}\sum_{a\in M-G}
D_{i_1m_1}^{\lambda_1 \mu_1}\left(a\right)D_{i_2m_2}^{\lambda_2\mu_2}
\left(a\right)D_{i_3m_3}^{\lambda_3\mu_3}\left(a\right)^{\ast}
\right] & & \nonumber \\
~~~~=~\frac{\left|M\right|}{d_{\lambda_3\mu_3}}\delta_{m_3,m_3^{\prime}}
\left\langle\lambda_1\mu_1i_1;\lambda_2\mu_2i_2\left.\right|\lambda_3
\mu_3i_3\right\rangle, & & \label{wet15}\end{aligned}$$ we get $$\begin{aligned}
\sum_{m_1m_2}
\left[\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2m_2\left.
\right|\lambda_3\mu_3 m_3^{\prime}\right\rangle^{\ast}\sum_{u\in G}
\left\langle\Phi_{m_1}^{\lambda_1 \mu_1}\left(u\right)\left|
T_{m_2}^{\lambda_2\mu_2} \left(u\right)\right|\Phi_{m_3}^{\lambda_3\mu_3}
\left(u\right)\right\rangle
\right. ~~~~~& & \nonumber \\
~~~~+ \left.\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2m_2\left.
\right|\lambda_3\mu_3 m_3\right\rangle\sum_{a\in M-G}
\left\langle\Phi_{m_1}^{\lambda_1 \mu_1}\left(a\right)\left|
T_{m_2}^{\lambda_2\mu_2} \left(a\right)\right|
\Phi_{m_3^{\prime}}^{\lambda_3\mu_3} \left(a\right)\right\rangle
\right] & & \nonumber \\
~~~~=~\frac{\left|M\right|}{d_{\lambda_3\mu_3}}\delta_{m_3,m_3^{\prime}}
\sum_{n_1n_2n_3}\left\langle\lambda_1\mu_1n_1;\lambda_2\mu_2n_2\left.
\right|\lambda_3 \mu_3n_3\right\rangle^{\ast}\left\langle
\Phi_{n_1}^{\lambda_1\mu_1}\left|T_{n_2}^{\lambda_2\mu_2}\right|
\Phi_{n_3}^{\lambda_3\mu_3}\right\rangle, & & \label{wet16}\end{aligned}$$ where $\left|M\right| = {\rm the~order~of~the~magnetic~group}~M\left(G
\right)$.
In expressing $$\left\langle\Phi_{m_1}^{\lambda_1\mu_1}\left(\alpha\right)\left|
T_{m_2}^{\lambda_2\mu_2}\left(\alpha\right)\right|\Phi_{m_3}^{\lambda_3
\mu_3}\left(\alpha\right)\right\rangle$$ in Eq. \[wet16\] we shall use the identity [@rudra1] $$\left\langle O_{\alpha_1}z_1\Phi_1\left|\right.O_{\alpha_2}z_2\Phi_2
\right\rangle = z_1^{\ast\left[\alpha_1\right]}z_2^{\left[\alpha_2\right]}
\left\langle\Phi_1\left|O_{\alpha_1^{-1}\alpha_2}\right|\Phi
\right\rangle^{\left\{\alpha_1,\alpha_2\right\}}$$ with $$\left\langle\Phi_1\left|O_{\alpha_1^{-1}\alpha_2}\right|\Phi_2
\right\rangle^{\left\{\alpha_1,\alpha_2\right\}} = \left\{\begin{array}{l}
\left\langle\Phi_2\left|O_{\alpha_2^{-1}\alpha_1}\right|\Phi_1\right\rangle
~~~{\rm if~both}~\alpha_1,\alpha_2\in M-G \\
\left\langle\Phi_1\left|O_{\alpha_1^{-1}\alpha_2}
\right|\Phi_2\right\rangle^{\left[\alpha_1\right]}
~~~{\rm otherwise}\end{array} \right. \label{wet17}$$ where $\Phi_i$s are state vectors and $z_i$s are complex numbers. We observe that the expression on the left-hand side of Eq. (\[wet16\]) is real and we can write $$\begin{aligned}
\frac{1}{2}\sum_{m_1m_2}
\left[\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2m_2\left.
\right|\lambda_3\mu_3 m_3^{\prime}\right\rangle^{\ast}
\left\langle\Phi_{m_1}^{\lambda_1 \mu_1}\left|
T_{m_2}^{\lambda_2\mu_2} \right|\Phi_{m_3}^{\lambda_3\mu_3}
\right\rangle\right. ~~~~~& & \nonumber \\
~~~~+ \left.\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2m_2\left.
\right|\lambda_3\mu_3 m_3\right\rangle
\left\langle\Phi_{m_1}^{\lambda_1 \mu_1}\left|
T_{m_2}^{\lambda_2\mu_2} \right|
\Phi_{m_3^{\prime}}^{\lambda_3\mu_3} \right\rangle
\right] & & \nonumber \\
~~~~=~\frac{1}{d_{\lambda_3\mu_3}}\delta_{m_3,m_3^{\prime}}
\left\langle\lambda_1\mu_1\left|\left|\lambda_2\mu_2\right|
\right|\lambda_3 \mu_3\right\rangle_L, & & \hspace{2.0cm} (17a) \nonumber \end{aligned}$$ where for a linear operator $T$ the reduced matrix element is given by $$\left\langle\lambda_1\mu_1\left|\left|\lambda_2\mu_2\right|\right|
\lambda_3\mu_3\right\rangle_L = \sum_{n_1n_2n_3}\left\langle\lambda_1
\mu_1n_1;\lambda_2\mu_2n_2\left|\right.\lambda_3\mu_3n_3
\right\rangle^{\ast}\left\langle\Phi_{n_1}^{\lambda_1\mu_1}\left|
T_{n_2}^{\lambda_2\mu_2}\right|\Phi_{n_3}^{\lambda_3\mu_3}\right\rangle.
\label{wet18}$$
These will be the equations satisfied by the matrix elements. The CG coefficients satisfy the orthogonality relations [@rudra1] $$\begin{aligned}
\sum_{m_1m_2m_1^{\prime}m_2^{\prime}}\left\langle\lambda_1\mu_1m_1;
\lambda_2\mu_2m_2\left|\right.\lambda_3\mu_3m_3\right\rangle^{\ast}
\left\langle\lambda_1\mu_1m_1^{\prime};\lambda_2\mu_2m_2^{\prime}
\left|\right.\lambda_3\mu_3^{\prime}m_3^{\prime}\right\rangle^{\ast}
\times & & \nonumber \\
\left\langle\lambda_1\mu_1m_1^{\prime}\left|\right.\lambda_1\mu_1m_1
\right\rangle
\left\langle\lambda_2\mu_2m_2^{\prime}\left|\right.\lambda_2\mu_2m_2
\right\rangle
= \delta_{\mu_3\mu_3^{\prime}}\left\langle\lambda_3\mu_3m_3^{\prime}
\left|\right.\lambda_3\mu_3m_3\right\rangle & & \label{wet19}\end{aligned}$$ It should be noted that the bases belonging to either a type (a) or a type (c) corepresentation [@wigner] are all orthogonal [@rudra1]. Thus if none of the 3 corepresntations $\mu_1,\mu_2:\mu_3$ are of Wigner type (b), then $$\left\langle\Phi_{m_1}^{\lambda_1\mu_1}\left|T_{m_2}^{\lambda_2\mu_2}
\right|\Phi_{m_3}^{\lambda_3\mu_3}\right\rangle = \frac{1}{d_{\lambda_3
\mu_3}}\left\langle\lambda_1 \mu_1\left|\left|\lambda_2\mu_2\right|\right|
\lambda_3\mu_3\right\rangle_L\left\langle\lambda_1\mu_1 m_1;\lambda_2\mu_2
m_2\left|\right.\lambda_3\mu_3 m_3\right\rangle, \label{wet20}$$ is a solution of Eq. (\[wet18\]). If any of the 3 corepresentations appearing in the matrix element is of type (b), then the corresponding matrix element is a linear combination of terms proportional to the CG coefficients. Even when they are valid, Eq. (\[wet20\]) will be unique only if the CG coefficients obtained from Eq. (\[wet15\]) are unique [@rudra1]—[@sakata]. The CG coefficients are unique if and only if $$\det \left|L\left(i_1i_2i_3,m_1m_2m_3\right)+A\left(i_1i_2i_3,m_1m_2m_3
\right)-\delta_{i_1m_1}\delta_{i_2m_2}\delta_{i_3m_3}\right| =0$$ and $$\det \left|L\left(i_1i_2i_3,m_1m_2m_3\right)-A\left(i_1i_2i_3,m_1m_2m_3
\right)-\delta_{i_1m_1}\delta_{i_2m_2}\delta_{i_3m_3}\right| =0$$ where $$\begin{aligned}
L\left(i_1i_2i_3,m_1m_2m_3\right) & = & \frac{d_{\lambda_3\mu_3}}{\left|
M\right|}\sum_{u\in G} D_{i_1m_1}^{\lambda_1\mu_1}\left(u\right)
D_{i_2m_2}^{\lambda_2\mu_2}\left(u\right) D_{i_1m_1}^{\lambda_3\mu_3}
\left(u\right)^{\ast} \nonumber \\
{\rm and} & & \nonumber \\
A\left(i_1i_2i_3,m_1m_2m_3\right) & = & \frac{d_{\lambda_3\mu_3}}{\left|
M\right|}\sum_{a\in M-G} D_{i_1m_1}^{\lambda_1\mu_1}\left(a\right)
D_{i_2m_2}^{\lambda_2\mu_2}\left(a\right) D_{i_1m_1}^{\lambda_3\mu_3}
\left(a\right)^{\ast}. \label{wet21} \end{aligned}$$
In the case of groups having no antilinear operators if we replace the second summand on the left-hand side of Eq. (17a) by the first summand we get the set of linear equations satisfied by the matrix elements. Since in these cases the bases are all orthogonal and the CG coefficients are essentially unique Eq. (\[wet20\])is an exact relation [@harter; @rudra2].
In the previous analysis we have assumed that in the expansion of the Qronecer inner direct product of two irreducible corepresentations $D^{\lambda_1\mu_1}$ and $D^{\lambda_2\mu_2}$ the irreducible corepresentation $D^{\lambda_3\mu_3}$ occurs only once. When there are more than one repetition, the different repetitions of $D^{\lambda_3
\mu_3}$ in the decomposition of the inner product representation are characterized by $\left\langle\lambda_1\mu_1i_1;\lambda_2\mu_2i_2
\left|\right.\tau_3\lambda_3\mu_3i_3\right\rangle$. As has been shown in [@rudra1] the CG coefficients for different $\tau_3$s satisfy equations similar to Eq. (\[wet15\]). So similar considerations will be valid for $\left\langle\Phi_{i_1}^{\lambda_1
\mu_1}\left|T_{i_2}^{\lambda_2\mu_2}\right|\Phi_{i_3}^{\tau_3
\lambda_3\mu_3}\right\rangle$.
But in general the most we can tell about a quantum mechanical state $\left|\right.\Psi_{i_3}^{\lambda_3\mu_3}$ is that it transforms as the $i-3$-th component of the irreducible corepresentation $D^{\lambda_3\mu_3}$. In this case $$\left|\right.\Psi_{i_3}^{\lambda_3\mu_3}\left.\right\rangle =
\sum_{\tau_3}a_{\tau_3} \left|\right.\Phi_{i_3}^{\tau_3\lambda_3\mu_3}
\left.\right\rangle$$ and $$\left\langle\Phi_{i_3}^{\lambda_1\mu_1}\left|T_{i_2}^{\lambda_2\mu_2}
\right|\Psi_{i_3}^{\lambda_3\mu_3}\right\rangle = \sum_{\tau_3}
a_{\tau_3}\left\langle\Phi_{i_1}^{\lambda_1\mu_1}\left|T_{i_2}^{\lambda_2
\mu_2}\right|\Phi_{i_3}^{\tau_3\lambda_3\mu_3}\right\rangle.$$ The quantum mechanical matrix element for the transition probability is thus a linear combination of terms each of which is a product of a reduced matrix element $\left\langle\lambda_1\mu_1\left|\left|\lambda_2\mu_2
\right|\right|\tau_3\lambda_3\mu_3\right\rangle$ and the corresponding CG coefficient $\left\langle\lambda_1\mu_1i_1;\lambda_2\mu_2\left|\right.
\tau_3\lambda_3\mu_3i_3\right\rangle$.
Case 2. $T$ is an antilinear operator.
In this case the matrix element on the left-hand side of Eq. (\[wet13\]) is zero unless $$\lambda_2\left(\alpha,\beta\right) = \lambda_1\left(\alpha,\beta\right)
\lambda_3\left(\alpha,\beta\right). \label{wet22}$$ An analysis similar to that for Case 1 will show that the matrix elements will satisfy the relations $$\begin{aligned}
\frac{1}{2}\sum_{m_1m_3}
\left[\left\langle\lambda_1\mu_1 m_1;\lambda_3\mu_3m_3\left.
\right|\lambda_2\mu_2 m_2^{\prime}\right\rangle^{\ast}
\left\langle\Phi_{m_1}^{\lambda_1 \mu_1}\left|
T_{m_2}^{\lambda_2\mu_2} \right|\Phi_{m_3}^{\lambda_3\mu_3}
\right\rangle\right. ~~~~~& & \nonumber \\
~~~~+ \left.\left\langle\lambda_1\mu_1 m_1;\lambda_3\mu_3m_3\left.
\right|\lambda_2\mu_2 m_2\right\rangle
\left\langle\Phi_{m_1}^{\lambda_1 \mu_1}\left|
T_{m_2^{\prime}}^{\lambda_2\mu_2} \right|
\Phi_{m_3}^{\lambda_3\mu_3} \right\rangle
\right] & & \nonumber \\
~~~~=~\frac{1}{d_{\lambda_2\mu_2}}\delta_{m_2,m_2^{\prime}}
\left\langle\lambda_1\mu_1\left|\left|\lambda_2\mu_2\right|
\right|\lambda_3 \mu_3\right\rangle_{AL}, & & \nonumber \end{aligned}$$ where the reduced matrix element for an antilinear operator $T$ is $$\left\langle\lambda_1\mu_1\left|\left|\lambda_2\mu_2\right|\right|
\lambda_3\mu_3\right\rangle_{AL} = \sum_{n_1n_2n_3}\left\langle\lambda_1
\mu_1n_1;\lambda_3\mu_3n_3\left|\right.\lambda_2\mu_2n_2
\right\rangle^{\ast}\left\langle\Phi_{n_1}^{\lambda_1\mu_1}\left|
T_{n_2}^{\lambda_2\mu_2}\right|\Phi_{n_3}^{\lambda_3\mu_3}\right\rangle.
\label{wet23}$$ When none of these corepresentations are of type (b) according to Wigner’s classification [@wigner], then $$\left\langle\Phi_{m_1}^{\lambda_1\mu_1}\left|T_{m_2}^{\lambda_2\mu_2}
\right|\Phi_{m_3}^{\lambda_3\mu_3}\right\rangle = \frac{1}{d_{\lambda_2
\mu_2}}\left\langle\lambda_1 \mu_1\left|\left|\lambda_2\mu_2\right|\right|
\lambda_3\mu_3\right\rangle_{AL}\left\langle\lambda_1\mu_1 m_1;\lambda_3\mu_3
m_3\left|\right.\lambda_2\mu_2m_2\right\rangle, \label{wet24}$$ is a solution. The condition for essential uniqueness of this factorization is given by a relation similar to Eq. (\[wet21\]) if the indices $2$ and $3$ there are interchanged. For linear groups again, the relation (\[wet24\]) is exact.
In case of reprtition of $D^{\lambda_2\mu_2}$ in the decomposition of the inner direct product representation $D^{\lambda_1\mu_1}\otimes
D^{\lambda_3\mu_3}$ the same considerations as in the case of linear $T$ will hold.\
\
This work has been financed by the Department of Atomic Energy, India,\
Project No. BRNS/Physics/14/74.\
\
DEPARTMENT OF PHYSICS, UNIVERSITY OF KALYANI,\
KALYANI, WEST BENGAL, 741235 (INDIA)
[99]{} M. E. Rose, [*Angular Momentum*]{}, Wiley, New York, 1971 B. R. Judd, [*Operator Technique in Atomic Spectroscopy*]{}, McGraw-Hill, New York, 1963 E. P. Wigner, [*Group Theory*]{}, Academic Press, New York, 1959 C. Eckart Rev. Mod. Phys., [**2**]{}, 305 (1930) U. Fano, G. Racah, [*Irreducible Tensorial Set*]{}, Academic Press, New York, 1959 N. B. Backhouse, J. Math. Phys., [**15**]{}, 119 (1974) E. Doni, G. Pastori Paravicini, J. Phys., [**C6**]{}, 2859 (1973); [**C7**]{}, 1786 (1974) T. Janssen, J. Math. Phys., [**13**]{}, 342 (1972) C. J. Bradley, B. L. Davies, Rev. Mod. Phys., [**40**]{}, 359 (1968) A. Aviran, J. Zak, J. Math. Phys., [**9**]{}, 2138 (1968) P. Rudra, ibid., [**15**]{}, 2031 (1974) A. Aviran, D. B. Litvin, ibid., [**14**]{}, 1491 (1973) J. N. Kotzev, Kristallographiys, [**19**]{}, 549 (1974);\
English translation, Soviet Phys. Cryst., [**19**]{}, 286 (1974) I. Sakata, J. Math. Phys., [**15**]{}, 1702, 1710 (1974) W. G. Harter, ibid., [**10**]{}, 739 (1969) P. Rudra, ibid., [**7**]{}, 935 (1966)
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---
abstract: 'An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both form odd subgraphs. In this paper we give a necessary and sufficient condition for the decomposability of a multigraph into two odd subgraphs. We also present a polynomial time algorithm for finding such a decomposition or showing its non-existence. We also deal with the case of the decomposability into an even subgraph and an odd subgraph.'
author:
- 'Mikio Kano[^1]'
- 'Gyula Y Katona[^2]'
- 'Kitti Varga[^3]'
title: Decomposition of a graph into two disjoint odd subgraphs
---
Introduction
============
In this paper we mainly consider [*multigraphs*]{}, which may have multiple edges but have no loops. A graph without multiple edges or loops is called a [*simple graph*]{}. Let $G$ be a multigraph with vertex set $V(G)$ and edge set $E(G)$. The number of vertices in $G$ is called its [*order*]{} and denoted by $|G|$, and the number of edges in $G$ is called its [*size*]{} and denoted by $e(G)$. Let $EvenV(G)$ denote the set of vertices of even degree and $OddV(G)$ denote the set of vertices of odd degree. For a vertex set $U$ of $G$, the subgraph of $G$ induced by $U$ is denoted by $\langle U \rangle _G$. For two disjoint vertex sets $U_1$ and $U_2$ of $G$, the number of edges between $U_1$ and $U_2$ is denoted by $e_G(U_1, U_2)$. For a vertex $v$ of $G$, the degree of $v$ in $G$ is denoted by $\deg_G(v)$. Moreover, when some edges of $G$ are colored with red and blue, for a vertex $v$, the number of red edges incident with $v$ is denoted by $\deg_{\textrm{red}}(v)$, and the number of red edges in $G$ is denoted by $e_{\textrm{red}}(G)$. Analogously, $\deg_{\textrm{blue}}(v)$ and $e_{\textrm{blue}}(G)$ are defined.
An odd (resp. even) subgraph of $G$ is a subgraph in which every vertex has odd (resp. even) degree. An odd factor of $G$ is a spanning odd subgraph of $G$. It is obvious by the handshaking lemma that every connected multigraph containing an odd factor has even order. This condition is also sufficient as follows.
\[oddfactor\] A multigraph $G$ has an odd factor if and only if every component of $G$ has even order.
Moreover, such an odd factor, if it exists, can be found in polynomial time (Problem $42$ of §7 in [@lovasz]). Consider a connected multigraph of even order on the vertices $v_1, \ldots, v_{2m}$ and for any $i \in \{1,2, \ldots,m\}$, fix a path $P_i$ connecting $v_i$ and $v_{i+m}$. Then the edges appearing odd times in the paths $P_1, \ldots, P_m$ forms an odd factor. The above proposition also follows from the fact that for a tree $T$ of even order, the set $$\{e\in E(T): \mbox{$T-e$ consists of two odd components}\}$$ forms an odd factor of $T$.
We say that $G$ can be decomposed into $n$ odd subgraphs if its edge set can be partitioned into $n$ sets $E_1, \ldots, E_n$ so that for every $i \in \{ 1, \ldots, n \}$, $E_i$ forms an odd subgraph. Some authors say that in this case $G$ is odd $n$-edge-colorable.
Our main result gives a criterion for a multigraph to be decomposed into two odd subgraphs, and proposes a polynomial time algorithm for finding such a decomposition or showing its non-existence.
We begin with some known results related to ours.
\[pyber-1\] Every simple graph can be decomposed into four odd subgraphs.
This upper bound is sharp, for example, the wheel of four spokes ($W_4$, see Figure 1) cannot be decomposed into three odd subgraphs. In [@matrai] Mátrai constructed an infinite family of graphs with the same property.
\[pyber-2\] Every forest can be decomposed into two odd subgraphs.
\[pyber-3\] Every connected simple graph of even order can be decomposed into three odd subgraphs.
Since every connected simple graph $G$ of even order has an odd factor, if we take an odd factor $F$ with maximum size, then $G-E(F)$ becomes a forest, and it can be decomposed into two odd subgraphs by Theorem \[pyber-2\]. Thus Theorem \[pyber-3\] follows.
\[luzar\] Every connected multigraph can be decomposed into six odd subgraphs. And equality holds if and only if the multigraph is a Shannon triangle of type $(2,2,2)$ (see Figure 1).
\[petrusevski\] Every connected multigraph can be decomposed into four odd subgraphs except for the Shannon triangles of type $(2,2,2)$ and $(2,2,1)$ (see Figure 1).
=\[draw,circle,fill=black,minimum size=3,inner sep=0\]
\(w) at (0,0) ; (v1) at (0:1) ; (v2) at (90:1) ; (v3) at (180:1) ; (v4) at (270:1) ;
\(w) – (v1); (w) – (v2); (w) – (v3); (w) – (v4); (v1) – (v2) – (v3) – (v4) – (v1);
at (-1,1) [$W_4$]{};
(a1) at (90:1.25) ; (a2) at (210:1.25) ; (a3) at (330:1.25) ;
(a1) to \[bend right=30\] (a2); (a2) to \[bend right=30\] (a3); (a3) to \[bend right=30\] (a1); (a1) to \[bend left=30\] (a2); (a2) to \[bend left=30\] (a3); (a3) to \[bend left=30\] (a1); (a1) to node\[pos=0.5, fill=white, rotate=-30\] [odd]{} (a2); (a2) to node\[pos=0.5, fill=white, rotate=-90\] [odd]{} (a3); (a3) to node\[pos=0.5, fill=white, rotate=30\] [odd]{} (a1);
(a1) at (90:1.25) ; (a2) at (210:1.25) ; (a3) at (330:1.25) ;
(a1) to \[bend right=30\] (a2); (a2) to \[bend right=30\] (a3); (a3) to \[bend right=30\] (a1); (a1) to \[bend left=30\] (a2); (a2) to \[bend left=30\] (a3); (a3) to \[bend left=30\] (a1); (a1) to node\[pos=0.5, fill=white, rotate=-30\] [even]{} (a2); (a2) to node\[pos=0.5, fill=white, rotate=-90\] [odd]{} (a3); (a3) to node\[pos=0.5, fill=white, rotate=30\] [odd]{} (a1);
(a1) at (90:1.25) ; (a2) at (210:1.25) ; (a3) at (330:1.25) ;
(a1) to \[bend right=30\] (a2); (a2) to \[bend right=30\] (a3); (a3) to \[bend right=30\] (a1); (a1) to \[bend left=30\] (a2); (a2) to \[bend left=30\] (a3); (a3) to \[bend left=30\] (a1); (a1) to node\[pos=0.5, fill=white, rotate=-30\] [even]{} (a2); (a2) to node\[pos=0.5, fill=white, rotate=-90\] [even]{} (a3); (a3) to node\[pos=0.5, fill=white, rotate=30\] [odd]{} (a1);
(a1) at (90:1.25) ; (a2) at (210:1.25) ; (a3) at (330:1.25) ;
(a1) to \[bend right=30\] (a2); (a2) to \[bend right=30\] (a3); (a3) to \[bend right=30\] (a1); (a1) to \[bend left=30\] (a2); (a2) to \[bend left=30\] (a3); (a3) to \[bend left=30\] (a1); (a1) to node\[pos=0.5, fill=white, rotate=-30\] [even]{} (a2); (a2) to node\[pos=0.5, fill=white, rotate=-90\] [even]{} (a3); (a3) to node\[pos=0.5, fill=white, rotate=30\] [even]{} (a1);
We say that $G$ can be covered by $n$ odd subgraphs if its edge set can be covered by $n$ sets $E_1, \ldots, E_n$ (not necessarily disjointly) so that for every $i \in \{ 1, \ldots, n \}$, $E_i$ forms an odd subgraph.
Every connected multigraph of odd order can be covered by three odd subgraphs.
In this paper we study the decomposability of a multigraph into an even subgraph and an odd subgraph, and into two odd subgraphs. We also remark that the case of decomposing into two even subgraphs is trivial.
\[even+odd\] A multigraph $G$ can be decomposed into an even subgraph and an odd subgraph if and only if every component of $\langle OddV(G) \rangle_G$ has even order.
Such a decomposition exists if and only if there is an odd factor in $\langle OddV(G) \rangle_G$, since all edges incident with any vertex of even degree must belong to the even subgraph. So by Proposition \[oddfactor\], the proposition follows.
Since an odd factor can be found in polynomial time, we can conclude the following.
\[even+odd\_poly\] There is a polynomial time algorithm for decomposing a multigraph into an odd subgraph and an even subgraph or showing the non-existence of such a decomposition.
The case of the decomposability into two even subgraphs is trivial: a multigraph can be decomposed into two even subgraphs if and only if every vertex of the multigraph has even degree.
The following two theorems are our main results.
\[n&s\_condition\] Let $G$ be a multigraph and let $\mathcal{X}$ denote the set of components of $\langle OddV(G) \rangle_G$, and let $\mathcal{Y}$ and $\mathcal{Z}$ denote the sets of components of $\langle EvenV(G) \rangle_G$ with odd order and even order, respectively. Now $G$ can be decomposed into two odd subgraphs if and only if for every $\mathcal{S} \subseteq \mathcal{Y} \cup \mathcal{Z}$ with $|\mathcal{S} \cap \mathcal{Y}|$ odd, there exists $X \in \mathcal{X}$ that has neighbors in odd number of components of $\mathcal{S}$.
\[2odd\_poly\] There is a polynomial time algorithm for decomposing a multigraph into two odd subgraphs or showing the non-existence of such a decomposition.
Proofs of Theorems \[n&s\_condition\] and \[2odd\_poly\]
=========================================================
We begin with a definition and a proposition on it.
Let $G$ be a multigraph and $T \subseteq V(G)$. A subgraph $J$ of $G$ is called a $T$-join if $OddV(J)=T$.
\[Tjoin\] Let $G$ be a multigraph and $T \subseteq V(G)$. There exists a $T$-join in $G$ if and only if every component of $G$ contains an even number of vertices of $T$.
The following theorem gives another necessary and sufficient condition for a multigraph to be decomposed into two odd subgraphs.
\[partition-thm\] Let $G$ be a multigraph and $\mathcal{Y}$ and $\mathcal{Z}$ denote the sets of components of $\langle EvenV(G) \rangle_G$ with odd order and even order, respectively. Then $G$ can be decomposed into two odd subgraphs if and only if there exists a partition $\mathcal{R} \cup \mathcal{B}$ of the components of $\langle OddV(G) \rangle_G$ such that
1. $e_G(R,Y)$ and $e_G(B,Y)$ are both odd for every $Y \in \mathcal{Y}$, and
2. $e_G(R,Z)$ and $e_G(B,Z)$ are both even for every $Z \in \mathcal{Z}$,
where $R$ and $B$ are the sets of vertices that belong to the components in $\mathcal{R}$ and $\mathcal{B}$, respectively.
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Suppose that $G$ can be decomposed into two odd subgraphs, and color the edges of one with red and with blue the other. Obviously, if a vertex of $G$ has odd degree, then all edges incident with it must have the same color. Consider an arbitrary component $X$ of $\langle OddV(G) \rangle_G$. Then all edges that have at least one endpoint in $X$ have the same color. Let $\mathcal{R}$ and $\mathcal{B}$ denote the set of those components of $\langle OddV(G) \rangle_G$ in which the edges are red and blue, respectively. Let $Y \in \mathcal{Y}$. Then $$\sum_{v \in Y} \deg_{\textrm{red}}(v) = 2 e_{\textrm{red}}(\langle Y\rangle_G) + e_G(R,Y) \text{.}$$ Since $|Y|$ is odd and $\deg_{\textrm{red}}(v)$ is odd for every $v \in Y$, the left side of the equation is odd, and so $e_G(R,Y)$ must be odd. Similarly, $e_G(B,Y)$ is also odd, and $e_G(R,Z)$ and $e_G(B,Z)$ are both even. Therefore, the conditions (i) and (ii) hold.
Next assume that there exists a partition $\mathcal{R} \cup \mathcal{B}$ satisfying (i) and (ii). Then color all the edges incident with any vertex of $R$ red and all the edges incident with any vertex of $B$ blue. Note that no edge of $\langle EvenV(G)\rangle_G$ is colored now, and there exist no edges between $R$ and $B$. Let $T \subseteq EvenV(G)$ be the set of vertices having even red-degree in this stage.
Now we show that every component of $\langle EvenV(G) \rangle_G$ contains an even number of vertices of $T$. Let $Y \in \mathcal{Y}$. Then by condition (i), $$\sum_{v \in Y}\deg_{\textrm{\textrm{red}}}(v) = \underbrace{\sum_{v \in Y \cap T} \underbrace{\deg_{\textrm{red}}(v)}_{\text{even}}}_{\text{even}} + \sum_{v \in Y\setminus T} \underbrace{\deg_{\textrm{red}}(v)}_{\text{odd}} = \underbrace{e_G(R,Y)}_{\text{odd}} \text{.}$$ Hence $|Y\setminus T|$ is odd, and since $|Y|$ is odd, $|Y \cap T|$ must be even. By the same argument given above, for any $Z \in \mathcal{Z}$, it follows that $|Z|$ is even and $e_G(R,Z)$ is even by the condition (ii), and thus $|Z\setminus T|$ is even and $|Z \cap T|$ is even. So by Proposition \[Tjoin\], there exists a $T$-join in $\langle EvenV(G) \rangle_G$. Color all the edges of this $T$-join red, and all the remaining edges blue. Now the resulting red subgraph and blue subgraph are odd subgraphs and form a partition of $E(G)$.
Now we prove Theorem \[n&s\_condition\].
*Proof of Theorem \[n&s\_condition\].* Let $\mathcal{X}$ denote the set of components of $\langle OddV(G) \rangle_G$, and let $\mathcal{Y}$ and $\mathcal{Z}$ denote the sets of components of $\langle EvenV(G) \rangle_G$ with odd order and even order, respectively.
Consider the bipartite graph $G^*$, whose vertices correspond to the elements of $\mathcal{X}$ and $\mathcal{Y} \cup \mathcal{Z}$, and an element of $\mathcal{X}$ and that of $\mathcal{Y} \cup \mathcal{Z}$ is joined by an edge if and only if there are odd number of edges of $G$ between the corresponding components. Then it is easy to see that every vertex of $Y\in \mathcal{Y}$ and $Z\in\mathcal{Z}$ has even degree in $G^*$.
Our goal is to give a system of linear equations that is solvable if and only if $G$ is decomposable into two odd subgraphs and its solutions describe partitions satisfying the properties of Theorem \[partition-thm\]. For every $X_i \in \mathcal{X}$, we assign a binary variable $x_i$ which decides whether $X_i \in \mathcal{R}$ or not. If $x_i =1$, then $X_i \in \mathcal{R}$, and if $x_i = 0$, then $X_i \in \mathcal{B}$. Since we want $e_G (R,Y)$ to be odd for every $Y \in \mathcal{Y}$ and $e_G (R,Z)$ to be even for every $Z \in \mathcal{Z}$, consider the following system of linear equations over the binary field $GF(2)=\{0,1\}$. $$\begin{aligned}
\sum_{X_i \in N_{G^*}(Y)} x_i & = 1 \qquad \mbox{for all}~~ Y \in \mathcal{Y} \\
\sum_{X_i \in N_{G^*}(Z)} x_i & = 0 \qquad \mbox{for all}~~ Z \in \mathcal{Z}
\end{aligned}$$
By Theorem \[partition-thm\], the multigraph $G$ is decomposable into two odd subgraphs if and only if this system has a solution. The system is solvable if and only if one of the following three equivalent statements holds.
- There is no collection of equations such that the sum of the left-hand sides is 0 and the sum of the right-hand sides is 1 (over the binary field).
- For any subset of the equations if the sum of the right-hand sides is 1, then there exists a variable $x_i$ which appears odd times in these equations.
- For any $\mathcal{S} \subseteq \mathcal{Y} \cup \mathcal{Z}$ for which $|\mathcal{S} \cap \mathcal{Y}|$ is odd, there exists $X \in \mathcal{X}$ such that $|N_{G^*}(X) \cap \mathcal{S}|$ is odd.
Note that statement (iii) is a graph presentation of statement (ii).
Since a system of linear equations over the binary field can be solved in polynomial time, Theorem \[2odd\_poly\] follows.
However, it is worth translating the algorithm to the language of graphs. The steps of the Gauss-elimination can be followed in the auxiliary bipartite graph $\widehat{G}^*$ which is a slight modification of the graph $G^*$ used in the proof of Theorem \[n&s\_condition\]. In the following we will use ${}^*$ as an operation that contracts components into single vertices. So the color classes of $G^*$ are the vertex sets $\mathcal{X}^*$ and $\mathcal{Y}^*\cup \mathcal{Z}^*$, and our goal is to partition $\mathcal{X}^*$ into $\mathcal{R}^*$ and $\mathcal{B}^*$. To obtain $\widehat{G}^*$ a new vertex $b$ is added to $G^*$ and it is connected to all vertices in $\mathcal{Y}^*$. This vertex $b$ corresponds to the constant 1 on the right side in the linear equations.
To start the Gauss-elimination we need to select a variable that has a non-zero coefficient (i.e. 1) in at least two equations and pick one of these equations. Therefore in $\widehat{G}^*$ we choose an edge $x_iw$ with $|N_{\widehat{G}^*}(x_i)|\ge 2$, $x_i\in \mathcal{X}^*$ and $w \in \mathcal{Y}^* \cup \mathcal{Z}^*$. Now in the Gauss-elimination, we add the equation corresponding to $w$ to all the equations corresponding to any element of $N_{\widehat{G}^*}(x_i)- \{w \}$ to make the coefficient of $x_i$ zero in these equations. Then the resulting system of linear equations corresponds to the bipartite graph $\widehat{G}^*_1$ that is obtained from $\widehat{G}^*$ by replacing the induced subgraph $\langle N_{\widehat{G}^*}(w) \cup \left( N_{\widehat{G}^*}(x_i) - \{
w \} \right) \rangle_{\widehat{G}^*}$ with its complement. So $x' \in N_{\widehat{G}^*}(w)$ and $w'\in N_{\widehat{G}^*}(x_i) - \{ w \}$ are adjacent in $\widehat{G}^*_1$ if and only if $x'$ and $w'$ are not adjacent in $\widehat{G}^*$. The other edges are not changed. Notice that the degree of $x_i$ in $\widehat{G}^*_1$ will be one.
Next we repeat this procedure by choosing an other edge $x_jw'$ in $\widehat{G}^*_1$ that satisfies the same conditions. Since the degree of $x_i$ is already one, $x_j$ will automatically differ from the previously chosen vertices, but we also choose $w'$ to be different from all previously chosen vertices. If there are no more such edges then the procedure stops.
Consider the graph of the final stage. At this point we can obtain the desired partition of the edge set into two odd subgraphs or show the non-existence of such a partition as follows.
- If a vertex $w \in\mathcal{Y}^*$ is connected only to the vertex $b$, then the graph $G$ cannot be decomposed into two odd subgraphs, since this means that adding up some equations results $0$ on the left-hand side and $1$ on the right-hand side.
So we may assume that no vertex $w \in\mathcal{Y}^*$ is connected only to the vertex $b$. In this case we obtain a solution as follows.
- If a vertex $x_i \in\mathcal{X}^*$ has degree at least two, then let $x_i\in \mathcal{B}^*$ and remove all the edges incident with $x_i$. This means that the variable $x_i$ is a free variable, so it can be set to 0. Thus we may assume that every $x\in \mathcal{X}^*$ is adjacent to at most one vertex of $\mathcal{Y}^*\cup \mathcal{Z}^* $. Removing these edges makes $x_i$ an isolated vertex, but note that other vertices in $\mathcal{X}^*$ cannot be isolated.
If there is a vertex in $\mathcal{Y}^* \cup \mathcal{Z}^*$ that is adjacent to $b$ and has more than one neighbors in $\mathcal{X}^*$ (that are all leaves), then let one of these neighbors be in $\mathcal{R}^*$ and all the others in $\mathcal{B}^*$. This means that we set one variable to 1 and all the others to 0, so their sum is equal to 1.
- Otherwise, if $x_i$ is in the same component as $b$, then let $x_i\in \mathcal{R}^*$, meaning that $x_i$ was set to 1 in the solution.
- If $x_i$ is not in the component of $b$, then let $x_i \in \mathcal{B}^*$, meaning that $x_i$ was set to 0 in the solution.
The above graph operation gives us a partition of $\mathcal{X}^*$ into $\mathcal{R}^*\cup \mathcal{B}^*$ and the corresponding partition of $\mathcal{X}$ satisfies the conditions in Theorem \[partition-thm\], and hence $G$ is decomposed into two odd subgraphs.
Acknowledgment
==============
The research of the first author was supported by JSPS KAKENHI Grant Number 16K05248. The research of the second author was supported by National Research, Development and Innovation Office NKFIH, K-116769 and K-124171. The research of the third author was supported by National Research, Development and Innovation Office NKFIH, K-124171.
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
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abstract: 'We clarify a Wess-Zumino-Wtten-like structure including Ramond fields and propose one systematic way to construct gauge invariant actions: Wess-Zumino-Witten-like complete action $S_{\rm WZW}$. We show that Kunitomo-Okawa’s action proposed in arXiv:1508.00366 can obtain a topological parameter dependence of Ramond fields and belongs to our WZW-like framework. In this framework, once a WZW-like functional $\mathcal{A}_{\eta } = \mathcal{A}_{\eta } [\Psi ]$ of a dynamical string field $\Psi $ is constructed, we obtain one realization of $S_{\rm WZW}$ parametrized by $\Psi $. On the basis of this way, we construct an action $\widetilde{S}$ whose on-shell condition is equivalent to the Ramond equations of motion proposed in arXiv:1506.05774. Using these results, we provide the equivalence of two these theories: arXiv:1508.00366 and arXiv:1506.05774.'
author:
- '[^1]'
date: |
${}^{\ast }$Yukawa Institute of Theoretical Physics, Kyoto University,\
Kyoto 606-8502, Japan
title: |
**[Comments on complete actions for\
open superstring field theory]{}**
---
Introduction
============
Recently, a field theoretical formulation of superstrings has been moved toward its new phase: An action and equations of motion including the Neveu-Schwarz and Ramond sectors were constructed[@Kunitomo:2015usa; @Erler:2015lya]. With recent developments[@Erler:2015rra; @Erler:2015uba; @Erler:2015uoa; @Goto:2015hpa; @Konopka:2015tta; @Sen:2015hia; @Sen:2015uoa; @Sen:2015uaa], we have gradually obtained new and certain understandings of superstring field theories. In the work of [@Kunitomo:2015usa], a gauge invariant action including the NS and R sectors was constructed without introducing auxiliary Ramond fields or self-dual constraints. They started with the Wess-Zumino-Witten-like action[^2] of the NS Berkovits theory[@Berkovits:1995ab] and coupled it to the R string field $\Psi ^{\rm R}$ in the restricted small Hilbert space: $XY \Psi ^{\rm R} = \Psi ^{\rm R}$. The dynamical string field is an amalgam of the NS large field $\Phi ^{\rm NS}$ and the R restricted small[^3] field $\Psi ^{\rm R}$. While the complete action [@Kunitomo:2015usa] is given by one extension of WZW-like formulation[@Berkovits:1995ab; @Berkovits:1998bt; @Okawa:2004ii; @Berkovits:2004xh; @Matsunaga:2013mba; @Matsunaga:2014wpa], the other one, the Ramond equations of motion [@Erler:2015lya], is a natural extension of $A_{\infty }$ formulation for the NS sector[@Erler:2013xta; @Erler:2014eba]. The $A_{\infty }$ formulation provides a systematic regularization procedure of superstring field theory[@Witten:1986qs; @Wendt:1987zh] in the early days. This procedure was extended to the case including Ramond fields and the Ramond equations of motion was constructed by introducing the concept of Ramond number projections in [@Erler:2015lya].
In this paper, we focus on these two important works [@Kunitomo:2015usa] and [@Erler:2015lya], and discuss some interesting properties based on Wess-Zumino-Witten-like point of view. Particularly, we investigate the following three topics and obtain some exact results.
1. We show that one can add the $t$-dependence of Ramond string fields into the complete action proposed in [@Kunitomo:2015usa] and make the $t$-dependence of the action “topological”, which leads us a natural idea of Wess-Zumino-Witten-like structure including Ramond fields.
2. We clarify a Wess-Zumino-Witten-like structure including Ramond fields and propose a Wess-Zumino-Witten-like complete action. Then, it is proved that one can carry out all computation of our action using the properties of pure-gauge-like and associated fields only. The action proposed in [@Kunitomo:2015usa] gives one realization of our WZW-like complete action.
3. On the basis of this WZW-like framework, we construct an action whose equations of motion gives the Ramond equations of motion proposed in [@Erler:2015lya]. As well as the action proposed in [@Kunitomo:2015usa], this action also gives another realization of our WZW-like complete action: different parameterization of the same WZW-like structure and action.
These facts provide the equivalence of two (WZW-like) theories [@Kunitomo:2015usa] and [@Erler:2015lya] on the basis of the same discussion demonstrated in [@Erler:2015rra]. Then, we can also read the relation giving a field redefinition of NS and R string fields with a partial gauge fixing or a trivial uplift by the same way used in [@Erler:2015rra; @Erler:2015uba] or [@Erler:2015uoa] for the NS sector of open superstrings without stubs.
This paper is organized as follows. First, we introduce a $t$-dependence of Ramond string fields and transform the complete action proposed in [@Kunitomo:2015usa] into the form which has topological $t$-dependence in section 1.1. Then, we clarify a Wess-Zumino-Witten-like structure including Ramond fields. In section 2, we propose a Wess-Zumino-Witten-like complete action. We show that our WZW-like complete action has so-called topological parameter dependence in section 2.1 and is gauge invariant in 2.2. In particular, these properties all can be proved by computations based only on the properties of pure-gauge-like fields and associated fields, which is a key point of our construction. In other words, to obtain the variation of the action, equations of motion, and gauge invariance, one does NOT need explicit form or detailed properties of $F$ giving $F \eta F^{-1} = D^{\rm NS}_{\eta }$ and $F \Xi $ satisfying $D^{\rm NS}_{\eta } F \Xi + F\Xi D^{\rm NS}_{\eta } =1$, which would heavily depend on the set up of theory. In section 3, we construct an action reproducing the same equations of motion as that proposed in [@Erler:2015lya]. For this purpose, it is shown that a Wess-Zumino-Witten-like structure naturally arises from $A_{\infty }$ relations and $\eta $-exactness of the small Hilbert space in section 3.2. As well as the action proposed in [@Kunitomo:2015usa], this action also gives another realization of our WZW-like complete action. Utilizing these facts, we discuss the equivalence of two theories [@Kunitomo:2015usa] and [@Erler:2015lya] in section 3.3. We end with some conclusions. Some proofs are in appendix A.
Complete action and topological $t$-dependence
----------------------------------------------
In this section, we use the same notation as [@Kunitomo:2015usa]. First, we show that one can add a parameter dependence of R string fields into Kunitomo-Okawa’s action, and that the resultant action has topological parameter dependence. Next, from these computations, we identify a pure-gauge-like field $A^{\rm R}_{\eta }$ and an associated field $A^{\rm R}_{d}$ of the Ramond sector. We end this section by introducing a Wess-Zumino-Witten-like form of Kunitomo-Okawa’s action.
First, we introduce the large and small Hilbert spaces. The large Hilbert space $\mathcal{H}$ is the state space whose superconformal ghost sector is spanned by $\xi (z)$, $\eta (z)$, and $\phi (z)$. We write $\eta $ for the zero mode $\eta _{0}$ and $\mathcal{H}_{S}$ for the kernel of $\eta \equiv \eta _{0}$. We call this subspace $\mathcal{H}_{\rm S} \subsetneq \mathcal{H}$ the small Hilbert space, whose superconformal ghost sector is spanned by $\beta (z)$ and $\gamma (z)$. Let $P_{\eta }$ is a projector onto the $\eta $-exact states: we can write $\mathcal{H}_{\rm S} = P_{\eta } \mathcal{H}$ because $\eta $-complex is exact in $\mathcal{H}$. Following the commutation relation $\eta \xi = 1 - \xi \eta $ for $\xi = \xi _{0}$ or $\Xi $ of [@Kunitomo:2015usa], we define a projector $P_{\xi } \equiv 1 - P_{\eta }$ onto the not $\eta $-exact states. Note also that for any state $\Phi \in \mathcal{H}$, these projectors act as $$\begin{aligned}
\nonumber
P_{\eta } + P_{\xi } = 1 , \hspace{5mm} P_{\eta }^{2} = P_{\eta }, \hspace{5mm} P_{\xi }^{2} = P_{\xi }, \hspace{5mm} P_{\eta } P_{\xi } = P_{\xi }P_{\eta } = 0 , \end{aligned}$$ by definition, and that $P_{\eta }$ acts as the identity operator $1$ on $\Phi \in \mathcal{H}_{\rm S}$ because of $\mathcal{H}_{\rm S} \subset P_{\eta } \mathcal{H}_{\rm S}$.
Next, we consider the restriction of the state space. Let $X$ be a picture-changing operator which is a Grassmann even and picture number $1$ operator defined by $X = \delta ( \beta _{0} ) G_{0} - b_{0} \delta ^{\prime } (\beta _{0} ) $, and let $Y$ be an inverse picture-changing operator which is a Grassmann even and picture number $-1$ operator defined by $Y = c_{0} \delta ^{\prime }(\gamma _{0})$. These operator satisfy $$\begin{aligned}
\nonumber
X Y X = X , \hspace{5mm} Y X Y = Y , \hspace{5mm} Q X = X Q , \hspace{5mm} \eta X = X \eta . \end{aligned}$$ The restricted space is the state space spanned by the states $\Psi \in \mathcal{H}$ satisfying $XY \Psi = \Psi $, on which the operator $XY$ becomes a projector $(XY)^{2} = XY$. The restricted small space $\mathcal{H}_{R}$ is the space spanned states $\Psi $ satisfying $$\begin{aligned}
\nonumber
X Y \, \Psi = \Psi , \hspace{7mm} \eta \, \Psi = 0. \end{aligned}$$ We use this restricted small Hilbert space $\mathcal{H}_{R}$ as the state space of the Ramond string field. See also [@Kazama:1985hd; @Terao:1986ex; @Yamron:1986nb; @Kugo:1988mf]. One can quickly check that when $\Psi $ is in $\mathcal{H}_{R}$, $Q \Psi $ is also in $\mathcal{H}_{R}$. See (2.25) of [@Kunitomo:2015usa].
Let $\Phi ^{\rm NS}$ be a Neveu-Schwarz open string field of the Berkovits theory, which is a Grassmann even and ghost-and-picture number $(0|0)$ state in the large Hilbert space $\mathcal{H}$, and let $\Psi ^{\rm R}$ be a Ramond open string field of [@Kunitomo:2015usa], which is a Grassmann odd and ghost-and-picture number $(1|-\frac{1}{2})$ state in the restricted small Hilbert space $\mathcal{H}_{\rm R}$. The kinetic term is given by $$\begin{aligned}
\nonumber
S_{0} = - \frac{1}{2} \langle \Phi ^{\rm NS} , \, Q \eta \Phi ^{\rm NS} \rangle - \frac{1}{2} \langle \xi Y \Psi ^{\rm R} , \, Q \Psi ^{\rm R} \rangle , \end{aligned}$$ where $Q$ is the BRST operator of open superstrings, $\langle A , B \rangle $ is the BPZ inner product of $A,B\in \mathcal{H}$ in the large Hilbert space $\mathcal{H}$. As explained in [@Kunitomo:2015usa], we can use both $\xi = \xi _{0}$ and $\xi = \Xi $ for the above $\xi $ in the BPZ inner product. Utilizing NS WZW-like functionals $A^{\rm NS}_{\eta } = ( \eta e^{\Phi ^{\rm NS}} ) e^{-\Phi ^{\rm NS} }$, $A^{\rm NS}_{t} = ( \partial _{t} e^{\Phi ^{\rm NS}}) e^{-\Phi ^{\rm NS}}$ and the invertible linear map $F$, the full action is given by $$\begin{aligned}
\nonumber
S = - \frac{1}{2} \langle \xi Y \Psi ^{\rm R} , \, Q \Psi ^{\rm R} \rangle - \int_{0}^{1} dt \, \langle A^{\rm NS}_{t} (t) , Q A^{\rm NS}_{\eta } (t) + m_{2} ( F(t) \Psi ^{\rm R} , F (t) \Psi ^{\rm R} ) \rangle , \end{aligned}$$ where $m_{2}$ is the Witten’s star product $m_{2} (A , B ) \equiv A \ast B$ [@Witten:1985cc] and as well as $A^{\rm NS}_{\eta } (t)$ or $A^{\rm NS}_{t} (t)$, $F(t)$ also satisfies $F(t=0) = 0$ and $F (t=1)=F$. Note that $F$ has no ghost-and-picture number and satisfies $F\eta F^{-1}=D^{\rm NS}_{\eta }$ and $D^{\rm NS}_{\eta } F \Xi + F \Xi D^{\rm NS}_{\eta } =1$. In this paper, we do not need the explicit form of $F$. See [@Kunitomo:2015usa] or appendix A for the explicit form of $F$.
In this Kunitomo-Okawa’s action, only the NS field $\Phi ^{\rm NS}(t)$ has a parameter dependence and the R field $\Phi ^{\rm R}$ has not: $\partial _{t} \Phi ^{\rm NS} ( t) \not= 0$ and $\partial _{t} \Psi ^{\rm R} = 0$, where a $t$-parametrized NS field $\Phi ^{\rm NS}(t)$ is a path on the state space satisfying $\Phi ^{\rm NS}(t=0) = 0$ and $\Phi ^{\rm NS}(t=1) = \Phi ^{\rm NS}$. We show that a complete action of open superstring field theory proposed in [@Kunitomo:2015usa] can be written as $$\begin{aligned}
\label{S KO}
S = - \int_{0}^{1} dt \, \Big( \langle \xi Y \partial _{t} \Psi ^{\rm R} (t) , Q F(t) \Psi ^{\rm R} (t) \rangle + \langle A^{\rm NS}_{t} (t) , Q A^{\rm NS}_{\eta } (t) + m_{2} \big( F(t) \Psi ^{\rm R} (t) , F(t) \Psi ^{\rm R} (t) \big) \rangle \Big) \end{aligned}$$ using a $t$-parametrized R field $\Psi ^{\rm R} (t)$ satisfying $\Psi ^{\rm R} (t=0) = 0$ and $\Psi ^{\rm R} (t=1) = \Psi ^{\rm R}$. Then, we also show that $t$-dependence of (\[S KO\]) is topological $$\begin{aligned}
\label{delta S KO}
\delta S = - \langle \xi Y \delta \Psi ^{\rm R} , \, Q F \Psi ^{\rm R} \rangle - \langle A^{\rm NS}_{\delta } ,Q A^{\rm NS}_{\eta } + m_{2} ( F \Psi ^{\rm R} , F \Psi ^{\rm R} ) \rangle . \end{aligned}$$
Let $P_{\eta}$ be a projector onto the space of $\eta $-exact states and let $P_{\xi }$ be a projector defined by $P_{\xi } \equiv 1-P_{\eta }$. For example, one can use $P_{\eta } = \eta \xi $ and $P_{\xi } = \xi \eta $. Note that these projectors $P_{\eta }$ and $P_{\xi }$ satisfy $P_{\eta } + P_{\xi } = 1$ on the large Hilbert space $\mathcal{H}$. One can check that $$\begin{aligned}
\label{R Kinetic KO}
\langle \xi Y \partial _{t} \Psi ^{\rm R} (t) , Q F (t) \Psi ^{\rm R} (t) \rangle & = \langle \xi Y \partial _{t} \Psi ^{\rm R} (t) , Q ( P_{\eta } + P_{\xi } ) F (t) \Psi ^{\rm R} (t) \rangle
{\nonumber\\}& = \langle \xi Y \partial _{t} \Psi ^{\rm R} (t) , Q \Psi ^{\rm R} (t) \rangle + \langle \xi Y \partial _{t} \Psi ^{\rm R} (t) , \eta X F (t) \Psi ^{\rm R} (t) \rangle
{\nonumber\\}& = \frac{\partial }{\partial t} \Big( \frac{1}{2} \langle \xi Y \Psi ^{\rm R} (t) , Q \Psi ^{\rm R} (t) \rangle \Big) - \langle \partial _{t} \Psi ^{\rm R} (t) , F (t) \Psi ^{\rm R} (t) \rangle \end{aligned}$$ using $P_{\eta } ( F \Psi ^{\rm R} ) = \Psi ^{\rm R}$ and $P_{\eta } Q P_{\xi } = \eta X$. See appendix A for the BPZ properties.
Let ${ [ \hspace{-0.6mm} [ }A , B { ] \hspace{-0.6mm} ] }$ be a graded commutator of the star product $m_{2}$: $$\begin{aligned}
\nonumber
{ [ \hspace{-0.6mm} [ }A , B { ] \hspace{-0.6mm} ] }\equiv m _{2} ( A , B ) - (-)^{AB} m_{2} ( B ,A ) . \end{aligned}$$ The upper index of $(-)^{A}$ denotes the grading of the state $A$, namely, its ghost number. Then, we can write ${ [ \hspace{-0.6mm} [ }F \Psi ^{\rm R} , F \Psi ^{\rm R} { ] \hspace{-0.6mm} ] }= 2 m_{2} ( F \Psi ^{\rm R} , F \Psi ^{\rm R} )$ and ${ [ \hspace{-0.6mm} [ }D^{\rm NS}_{\eta } , F \Xi { ] \hspace{-0.6mm} ] }= 1$. Some computations are in appendix A. Similarly, as (\[R Kinetic KO\]), using the $D^{\rm NS}_{\eta }$-exactness $F \Psi ^{\rm R} = D^{\rm NS}_{\eta } F \Xi \Psi ^{\rm R}$ and $$\begin{aligned}
\nonumber
\partial _{t} \big( F (t) \Psi ^{\rm R} (t) \big) = F (t) \partial _{t} \Psi ^{\rm R} (t) + F (t) \Xi D^{\rm NS}_{\eta } (t) { [ \hspace{-0.6mm} [ }A^{\rm NS}_{t} (t) , F (t) \Psi ^{\rm R} (t) { ] \hspace{-0.6mm} ] }, \end{aligned}$$ one can also check that $$\begin{aligned}
\langle A^{\rm NS}_{t} , m_{2} \big( F(t) \Psi ^{\rm R} (t) , F(t) \Psi ^{\rm R} (t) \big) \rangle & = - \frac{1}{2} \langle F (t) \Psi ^{\rm R} (t) , { [ \hspace{-0.6mm} [ }A^{\rm NS}_{t} (t) , F (t) \Psi ^{\rm R} (t) { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = - \frac{1}{2} \langle \Psi ^{\rm R} (t) , F (t) \Xi D^{\rm NS}_{\eta } (t) { [ \hspace{-0.6mm} [ }A^{\rm NS}_{t} (t) , F (t) \Psi ^{\rm R} (t) { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}\nonumber
& = -\frac{1}{2} \langle \Psi ^{\rm R} (t) , \partial _{t} \big( F (t) \Psi ^{\rm R} (t) \big) - F (t) \partial _{t} \Psi ^{\rm R} (t) \rangle . \end{aligned}$$ Furthermore, since $\eta \Psi ^{\rm R} = 0$ and thus $\langle \partial _{t} \Psi ^{\rm R} , F \Psi ^{\rm R} \rangle + \langle F \partial _{t} \Psi ^{\rm R} , \Psi ^{\rm R} \rangle = \langle \partial _{t} \Psi ^{\rm R} , \Psi ^{\rm R} \rangle = 0$, we find $$\begin{aligned}
\label{R Interaction KO}
\langle A^{\rm NS}_{t} (t) , m_{2} \big( F(t) \Psi ^{\rm R} (t) , F(t) \Psi ^{\rm R} (t) \big) \rangle & = -\frac{1}{2} \langle \Psi ^{\rm R} (t) , \partial _{t} \big( F (t) \Psi ^{\rm R} (t) \big) \rangle + \frac{1}{2} \langle \partial _{t} \Psi ^{\rm R} (t) , F (t) \Psi ^{\rm R} (t) \rangle
{\nonumber\\}& = \frac{\partial }{\partial t} \Big( - \frac{1}{2} \langle \Psi ^{\rm R} (t) , F (t) \Psi ^{\rm R} (t) \rangle \Big) + \langle \partial _{t} \Psi ^{\rm R} (t) , F (t) \Psi ^{\rm R} (t) \rangle . \end{aligned}$$ Therefore using $\langle \Psi ^{\rm R} , F \Psi ^{\rm R} \rangle = - \langle \xi Y \Psi ^{\rm R} , \eta X F \Psi ^{\rm R} \rangle $, we obtain $$\begin{aligned}
\nonumber
(\ref{R Kinetic KO}) + (\ref{R Interaction KO})
= \frac{1}{2} \Big( \langle \xi Y \Psi ^{\rm R} , \, Q \Psi ^{\rm R} + \eta X F \Psi ^{\rm R} \rangle \Big)
= \frac{1}{2} \langle \xi Y \Psi ^{\rm R} , \, Q F \Psi ^{\rm R} \rangle . \end{aligned}$$
As a result, our $t$-integrated action (\[S KO\]) becomes $$\begin{aligned}
\label{Original KO}
S & = - \frac{1}{2} \langle \xi Y \Psi ^{\rm R} , Q F \Psi ^{\rm R} \rangle - \int_{0}^{1} dt \, \langle A^{\rm NS}_{t} (t) , \, Q A^{\rm NS}_{\eta }(t) \rangle ,
{\nonumber\\}& = - \frac{1}{2} \langle \xi Y \Psi ^{\rm R} , Q \Psi ^{\rm R} \rangle - \int_{0}^{1} dt \, \langle A^{\rm NS}_{t} (t) , \, Q A^{\rm NS}_{\eta } (t) + m_{2} \big( F(t) \Psi ^{\rm R} , F(t) \Psi ^{\rm R} \big) \rangle . \end{aligned}$$ The second line is the original form used in [@Kunitomo:2015usa] but we do not need the second line expression to show that the variation of the action (\[S KO\]) is given by (\[delta S KO\]). Translation from the second line to the first line is in appendix A. Since the variation of the first term of the first line in (\[Original KO\]) is $$\begin{aligned}
\nonumber
\delta \Big( \frac{1}{2} \langle \xi Y \Psi ^{\rm R} , Q F \Psi ^{\rm R} \rangle \Big) = \langle \xi Y \delta \Psi ^{\rm R} , \, Q F \Psi ^{\rm R} \rangle + \langle A^{\rm NS}_{\delta } , m_{2} ( F \Psi ^{\rm R} , F \Psi ^{\rm R}) \rangle , \end{aligned}$$ we obtain (\[delta S KO\]) and the action $S$ has topological $t$-dependence.
Let $\Omega ^{\rm NS}$ and $\Lambda ^{\rm NS}$ be ghost-and-picture number $( -1 | -1 )$ and $( -1 | 0 )$ NS states of the large Hilbert space $\mathcal{H}$ respectively, and let $\lambda ^{\rm R}$ be a ghost-and-picture number $( 0 | -\frac{1}{2} )$ R state of the restricted small Hilbert space $\mathcal{H}_{R}$: $\eta \lambda ^{\rm R} = 0$ and $XY \lambda ^{\rm R} = \lambda ^{\rm R}$. These states naturally appear in gauge transformations of the action. The action $S$ has gauge invariances: $\delta _{\Omega ^{\rm NS}}S = 0$ with $\Omega ^{\rm NS}$-gauge transformations $$\begin{aligned}
A^{\rm NS}_{\delta _{\Omega ^{\rm NS}} } = \eta \Omega ^{\rm NS} - { [ \hspace{-0.6mm} [ }A^{\rm NS}_{\eta } \Omega ^{\rm NS} { ] \hspace{-0.6mm} ] }, \hspace{5mm} \delta _{\Omega ^{\rm NS}} \Psi ^{\rm R} = 0 ,\end{aligned}$$ $\delta _{\Lambda ^{\rm NS}} S = 0$ with $\Lambda ^{\rm NS}$-gauge transformations $$\begin{aligned}
\nonumber
A^{\rm NS}_{\delta _{\Lambda ^{\rm NS}} } = Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }F \Psi ^{\rm R} , F \Xi { [ \hspace{-0.6mm} [ }F \Psi ^{\rm R} , \Lambda ^{\rm NS} { ] \hspace{-0.6mm} ] }{ \big] \hspace{-1.1mm} \big] },
\hspace{5mm} \delta _{\Lambda ^{\rm NS}} \Psi ^{\rm R} = X \eta F \Xi D^{\rm NS}_{\eta } { [ \hspace{-0.6mm} [ }F \Psi ^{\rm R} , \Lambda ^{\rm NS} { ] \hspace{-0.6mm} ] },\end{aligned}$$ and $\delta _{\lambda ^{\rm R}} S = 0$ with $\lambda ^{\rm R}$-gauge transformations $$\begin{aligned}
\nonumber
A^{\rm NS}_{\delta _{\lambda ^{\rm R}} } = - { \big[ \hspace{-1.1mm} \big[ }F \Psi ^{\rm R} , F \Xi \lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] },
\hspace{5mm} \delta _{\lambda ^{\rm R}} \Psi ^{\rm R} = Q \lambda ^{\rm R} - X \eta F \Xi \lambda ^{\rm R} .\end{aligned}$$
Note also that using a new gauge parameter $\Lambda ^{\rm R}$ defined by $$\begin{aligned}
\Lambda ^{\rm R} \equiv F \Xi \Big( - \lambda ^{\rm R} + { \big[ \hspace{-1.1mm} \big[ }F \Psi ^{\rm R} , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ where $\Lambda ^{\rm R}$ belongs to the large Hilbert space and has ghost-and-picture number $(-1|\frac{1}{2})$, we obtain a simpler expression of $\Lambda $-gauge transformations as follows $$\begin{aligned}
A^{\rm NS}_{\delta _{\Lambda }} & = Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }A^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] },
\\
\delta _{\Lambda } \Psi ^{\rm R} & = - P_{\eta } Q \Big( \eta \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }A^{\rm NS}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }F \Psi ^{\rm R} , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) . \end{aligned}$$
In the rest of this section, we identify a pure-gauge-like field $A^{\rm R}_{\eta }$ and associated fields $A^{\rm R}_{d}$ in the Ramond sector and rewrite the action (\[S KO\]) into our Wess-Zumino-Witten-like form.
We write $A^{\rm R}_{\eta }$ for $F \Psi ^{\rm R}$, which is one realization of [*a Ramond pure-gauge-like field*]{}: $$\begin{aligned}
A^{\rm R}_{\eta } & \equiv F \Psi ^{\rm R} . \end{aligned}$$ By definition, the R pure-gauge-like field $A^{\rm R}_{\eta }$ satisfies $D^{\rm NS}_{\eta } A^{\rm R}_{\eta } = 0$, namely, $$\begin{aligned}
\label{R eta}
\eta A^{\rm R}_{\eta } - { [ \hspace{-0.6mm} [ }A^{\rm NS}_{\eta } , A^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }= 0 . \end{aligned}$$ As we will explain, the $\eta $-exact component $P_{\eta } A^{\rm R}_{\eta }$ appears in the action and its properties is important. Since the linear map $F$ satisfies $\xi F = \xi $ for $\xi = \Xi$, we quickly find that $$\begin{aligned}
\nonumber
P_{\eta } A^{\rm R}_{\eta } & = \Psi ^{\rm R} ,
\\ \nonumber
d ( P_{\eta } A^{\rm R}_{\eta } ) & = d \Psi ^{\rm R} , \end{aligned}$$ where $P_{\eta } = \eta \xi $ is a projector onto the small Hilbert space $\mathcal{H}_{\rm S}$ and $d = Q, \partial _{t} , \delta $ is a derivation operator commuting with $\eta $. Then, using $P_{\eta } A^{\rm R}_{\eta }$, we can express [*the $XY$-projection invariance*]{} of Ramond string fields $X Y \Psi ^{\rm R} = \Psi ^{\rm R}$ by $$\begin{aligned}
\label{R eta XY}
X Y ( P_{\eta } A^{\rm R}_{\eta } ) & = P_{\eta } A^{\rm R}_{\eta } . \end{aligned}$$ Note that $P_{\eta } A^{\rm R}_{\eta } \in \mathcal{H}_{R}$. Similarly, we introduce [*a Ramond associated field*]{} $A^{\rm R}_{d}$ by $$\begin{aligned}
(-)^{d} A^{\rm R}_{d} \equiv F \Xi \Big( d \Psi ^{\rm R} - (-)^{d} { [ \hspace{-0.6mm} [ }A^{\rm NS}_{d} , F \Psi ^{\rm R} { ] \hspace{-0.6mm} ] }- (-)^{d} \eta { [ \hspace{-0.6mm} [ }d , \Xi { ] \hspace{-0.6mm} ] }F \Psi ^{\rm R} \Big) . \end{aligned}$$ Using properties of $F$, one can check that the R associated field $A^{\rm R}_{d}$ satisfies $$\begin{aligned}
\label{R d}
(-)^{d} d A^{\rm R}_{\eta } = \eta A^{\rm R}_{d} - { [ \hspace{-0.6mm} [ }A^{\rm NS}_{\eta } , A^{\rm R}_{d} { ] \hspace{-0.6mm} ] }- { [ \hspace{-0.6mm} [ }A^{\rm R}_{\eta } , A^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }, \end{aligned}$$ or equivalently, $(-)^{d} d A^{\rm R}_{\eta } = D^{\rm NS}_{\eta } A^{\rm R}_{d}-{ [ \hspace{-0.6mm} [ }A^{\rm R}_{\eta } , A^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }$. See appendix A. Then, we obtain $$\begin{aligned}
\nonumber
\langle \xi Y \partial _{t} \Psi ^{\rm R} , Q F \Psi ^{\rm R} \rangle + \langle A^{\rm NS}_{t} , m_{2} ( F \Psi ^{\rm R} , F \Psi ^{\rm R} ) \rangle = \langle \xi Y \partial _{t} ( P_{\eta } A^{\rm R}_{\eta } ) , Q A^{\rm R}_{\eta } \rangle + \langle A^{\rm NS}_{t} , m_{2} ( A^{\rm R}_{\eta } , A^{\rm R}_{\eta } ) \rangle . \end{aligned}$$
Utilizing these expressions, the action becomes $$\begin{aligned}
S = - \int_{0}^{1} dt \Big( \langle \xi Y \partial _{t} ( P_{\eta } A^{\rm R}_{\eta } ) , \, Q A^{\rm R}_{\eta } \rangle + \langle A^{\rm NS}_{t} , \, Q A^{\rm NS}_{\eta } + m_{2} ( A^{\rm R}_{\eta } , A^{\rm R}_{\eta } ) \rangle \Big) . \end{aligned}$$ Note also that gauge transformation parametrized by $\Omega = \Omega ^{\rm NS} + \Omega ^{\rm R}$ is given by $$\begin{aligned}
\nonumber
A^{\rm NS}_{\delta _{\Omega } } = \eta \Omega ^{\rm NS} - { \big[ \hspace{-1.1mm} \big[ }A^{\rm NS}_{\eta } , \Omega ^{\rm NS} { \big] \hspace{-1.1mm} \big] },
\hspace{5mm}
\delta _{\Omega } ( P_{\eta } A^{\rm R}_{\eta } ) = 0 . \end{aligned}$$ and gauge transformations parametrized by $\Lambda = \Lambda ^{\rm NS} + \Lambda ^{\rm R}$ is given by $$\begin{aligned}
\nonumber
A^{\rm NS}_{\delta _{\Lambda }} & = Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }A^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] },
\\ \nonumber
\delta _{\Lambda } ( P_{\eta } A^{\rm R}_{\eta } ) & = - P_{\eta } Q \Big( \eta \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }A^{\rm NS}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }A^{\rm R} , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ where we use a R state $\Lambda ^{\rm R}$, a redefined R gauge parameter, $$\begin{aligned}
\nonumber
\Lambda ^{\rm R} \equiv F \Xi \Big( - \lambda ^{\rm R} + { \big[ \hspace{-1.1mm} \big[ }A^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ which belongs to the large Hilbert space $\mathcal{H}$ and has ghost-and-picture number $( -1 | \frac{1}{2} )$.
In the work of [@Kunitomo:2015usa], all computations of the variation of the action, equations of motion, and gauge invariance heavily depend on the explicit form or properties of the linear map $F$. However, as we will show in the next section, these all computations are derived from WZW-like properties of the Ramond sector: (\[R eta\]), (\[R eta XY\]), and (\[R d\]).
Wess-Zumino-Witten-like complete action
=======================================
We first summarize Wess-Zumino-Witten-like relations of the NS sector and the R sector separately, and show that these relations indeed provide the topological parameter dependence of the action. Second, coupling NS and R, we define a Wess-Zumino-Witten-like complete action and prove that the gauge invariance of the action is also derived from the WZW-like relations. Lastly, we introduce a notation unifying separately given results of NS and R sectors.
WZW-like structure and $XY$-projection
--------------------------------------
An NS pure-gauge-like field ${\cal A}^{\rm NS}_{\eta }$ is a ghost-and-picture number $(1|-1)$ state satisfying $$\begin{aligned}
\label{NS pure-gauge-like equation}
\eta {\cal A}^{\rm NS}_{\eta } - \frac{1}{2} { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm NS}_{\eta } { ] \hspace{-0.6mm} ] }= 0 . \end{aligned}$$ Let $d$ be a derivation operator satisfying ${ [ \hspace{-0.6mm} [ }d , Q { ] \hspace{-0.6mm} ] }= 0$, and let $(d_{\rm g} | d_{\rm p})$ be ghost-and-picture number of $d$. An NS associated field ${\cal A}^{\rm NS}_{d}$ is a ghost-and-picture number $(d_{\rm g} | d_{\rm p} )$ state satisfying $$\begin{aligned}
\label{NS associated}
(-)^{d} d {\cal A}^{\rm NS}_{\eta } & = \eta {\cal A}^{\rm NS}_{d} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }{\nonumber\\}& \equiv D^{\rm NS}_{\eta } {\cal A}^{\rm NS}_{d} . \end{aligned}$$ By definition of $(\ref{NS pure-gauge-like equation})$ and $(\ref{NS associated})$, one can check that the relation $$\begin{aligned}
\label{NS bianchi}
D^{\rm NS}_{\eta } \Big( d_{1} {\cal A}^{\rm NS}_{d_{2}} - (-)^{d_{1} d_{2}} d_{2} {\cal A}^{\rm NS}_{d_{1} } - (-)^{d_{1} d_{2}} { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{d_{1}} , {\cal A}^{\rm NS}_{d_{2}} { \big] \hspace{-1.1mm} \big] }\Big) = 0 \end{aligned}$$ holds when two derivation $d_{1}$ and $d_{2}$ satisfy ${ [ \hspace{-0.6mm} [ }d_{1} , d_{2} { ] \hspace{-0.6mm} ] }\equiv d_{1} d _{2} - (-)^{d_{1} d_{2}} d_{2} d_{1} = 0$.
Utilizing these fields, an NS action is given by $$\begin{aligned}
S^{\rm NS} & = - \int_{0}^{1} dt \, \langle {\cal A}^{\rm NS}_{t} , Q {\cal A}^{\rm NS}_{\eta } \rangle . \end{aligned}$$ It is known that the variation of the NS action is given $$\begin{aligned}
\delta S^{\rm NS} = - \langle {\cal A}^{\rm NS}_{\delta } , \, Q {\cal A}^{\rm NS}_{\eta } \rangle , \end{aligned}$$ which we call the topological parameter dependence of WZW-like action. See [@Berkovits:2004xh; @Erler:2015rra; @Erler:2015uoa; @GM2].
An R pure-gauge-like field ${\cal A}^{\rm R}_{\eta }$ is a ghost-and-picture number $(1|-\frac{1}{2} )$ state satisfying $$\begin{aligned}
\label{R pure-gauge-like equation}
\eta {\cal A}^{\rm R}_{\eta } - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }= 0 , \end{aligned}$$ or equivalently, $D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\eta } = 0$. Let $d$ be a derivation operator satisfying ${ [ \hspace{-0.6mm} [ }d , \eta { ] \hspace{-0.6mm} ] }= 0$, and let $(d_{\rm g} | d_{\rm p})$ be ghost-and-picture number of $d$. An R associated field ${\cal A}^{\rm R}_{d}$ is a ghost-and-picture number $( d_{\rm g} | d_{\rm p} + \frac{1}{2} )$ state satisfying $$\begin{aligned}
\label{R associated equation}
(-)^{d} d {\cal A}^{\rm R}_{\eta } & = \eta {\cal A}^{\rm R}_{d} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm R}_{d} { ] \hspace{-0.6mm} ] }- { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{d} { ] \hspace{-0.6mm} ] },\end{aligned}$$ namely, $(-)^{d} d {\cal A}^{\rm R}_{\eta } = D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{d} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }$.
As we will show, utilizing these fields and assuming $XY$-projection invariance of $P_{\eta } {\cal A}^{\rm R}_{\eta }$ $$\begin{aligned}
\label{XY=1}
X Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) = P_{\eta } {\cal A}^{\rm R}_{\eta } , \end{aligned}$$ one can construct a gauge invariant action Wess-Zumino-Witten-likely, whose parameter dependence is topological. We propose that an R action is given by $$\begin{aligned}
S^{\rm R} & = - \int_{0}^{1} dt \Big( \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle + \langle {\cal A}^{\rm NS}_{t} , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle \Big) . \end{aligned}$$ This $S^{\rm R}$ is Wess-Zumino-Witten-like. In other words, $S^{\rm R}$ has topological $t$-dependence $$\begin{aligned}
\label{delta Swzw}
\delta S^{\rm R} = - \langle \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle - \langle {\cal A}^{\rm NS}_{\delta } , m_{2}( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle . \end{aligned}$$
First, we consider the variation of the first term of $S^{\rm R}$. This term consists of two ingredients: $$\begin{aligned}
\nonumber
\int_{0}^{1} dt \, \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle = \int_{0}^{1} dt \, \Big( \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle - \langle \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , {\cal A}^{\rm R}_{\eta } \rangle \Big) . \end{aligned}$$ We can quickly find that the first part has topological $t$-dependence $$\begin{aligned}
\delta \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle & = \langle \frac{\partial }{\partial t} \big{\{ } \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \big{\} } , Q ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle + \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}\nonumber
& = \frac{\partial }{\partial t} \langle \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle \end{aligned}$$ since using (\[XY=1\]), $\xi Q - X = - Q \xi $, and $\langle \partial _{t}( P_{\eta } {\cal A}^{\rm R}_{\eta }) , \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta }) \rangle = 0$, the following relation holds $$\begin{aligned}
\langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle & = \langle ( \xi Q - X) Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , X Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}& = \langle Q \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}\nonumber
& = \langle \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \frac{\partial }{\partial t} \big{\{ } Q ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \big{\} } \rangle. \end{aligned}$$ Note however that the variation of the second ingredient provides an extra term $$\begin{aligned}
\delta \langle \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , {\cal A}^{\rm R}_{\eta } \rangle & = \langle \frac{\partial }{\partial t} \big{\{ } \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \big{\} } , {\cal A}^{\rm R}_{\eta } \rangle + \langle \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \delta {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}& = \frac{\partial }{\partial t} \langle \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , {\cal A}^{\rm R}_{\eta } \rangle - \langle \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle + \langle \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \delta {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}\nonumber
& = \frac{\partial }{\partial t} \langle \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , {\cal A}^{\rm R}_{\eta } \rangle - \langle \delta {\cal A}^{\rm R}_{\eta } , \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle . \end{aligned}$$ Here, we used $P_{\eta } + P_{\xi } =1$ and $\langle \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \delta {\cal A}^{\rm R}_{\eta } \rangle = \langle \partial _{t} {\cal A}^{\rm R}_{\eta } , \delta ( P_{\xi } {\cal A}^{\rm R}_{\eta } ) \rangle = - \langle \delta ( P_{\xi } {\cal A}^{\rm R}_{\eta } ) , \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle$. As a result, the variation of the first term of $S^{\rm R}$ is given by $$\begin{aligned}
\label{1st term}
\delta \int_{0}^{1} dt \, \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle = \langle \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle + \int_{0}^{1} dt \, \langle \delta {\cal A}^{\rm R}_{\eta } , \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle . \end{aligned}$$
Second, we compute the variation of the second term of $S^{\rm R}$. Using (\[NS bianchi\]), (\[R associated equation\]) for $d = \partial _{t} , \delta $, and Jacobi identities of the commutator, we obtain $$\begin{aligned}
\label{2nd term}
\delta \langle {\cal A}^{\rm NS}_{t} , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle & = \langle \delta {\cal A}^{\rm NS}_{t} , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle + \langle {\cal A}^{\rm NS}_{t} , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , \delta {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \langle \partial {\cal A}^{\rm NS}_{\delta } , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle + \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{t} , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle + \langle {\cal A}^{\rm NS}_{t} , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , \delta {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \frac{\partial }{\partial t} \langle {\cal A}^{\rm NS}_{\delta } , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle
- \langle {\cal A}^{\rm NS}_{\delta } , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , \partial _{t} {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& \hspace{20mm} - \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\delta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }, m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle + \langle \delta {\cal A}^{\rm R}_{\eta } , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \frac{\partial }{\partial t} \langle {\cal A}^{\rm NS}_{\delta } , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle
- \frac{1}{2} \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\delta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}\nonumber
& \hspace{30mm} + \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle
+ \langle \delta {\cal A}^{\rm R}_{\eta } , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \frac{\partial }{\partial t} \langle {\cal A}^{\rm NS}_{\delta } , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle
- \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}\nonumber
& \hspace{25mm} + \Big( \langle D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\delta } , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle + \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{t} { ] \hspace{-0.6mm} ] }\rangle \Big) \end{aligned}$$ In particular, from the forth line to the last line, we applied $$\begin{aligned}
\langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle & =
\langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{t} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{t} { ] \hspace{-0.6mm} ] }\rangle
- \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle .
{\nonumber\\}[2mm]
\langle \delta {\cal A}^{\rm R}_{\eta } , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle & = \langle D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\delta } - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{t} \rangle
- \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle .
{\nonumber\\}[2mm] \nonumber
- \frac{1}{2} \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\delta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle & = \frac{1}{2} \langle {\cal A}^{\rm NS}_{\delta } , { \big[ \hspace{-1.1mm} \big[ }{ [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }, {\cal A}^{\rm NS}_{t} { \big] \hspace{-1.1mm} \big] }\rangle
= - \langle {\cal A}^{\rm NS}_{\delta } , { \big[ \hspace{-1.1mm} \big[ }{ [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{t} { ] \hspace{-0.6mm} ] }, {\cal A}^{\rm NS}_{\eta } { \big] \hspace{-1.1mm} \big] }\rangle
{\nonumber\\}\nonumber
& = \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle . \end{aligned}$$ As a result, the variation of the second term of $S^{\rm R}$ is given by $$\begin{aligned}
\delta \int_{0}^{1} dt \, \langle {\cal A}^{\rm NS}_{t} , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\eta } ) \rangle = \langle {\cal A}^{\rm NS}_{\delta } , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle
- \int_{0}^{1} dt \, \langle \delta {\cal A}^{\rm R}_{\eta } , \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle \end{aligned}$$ with the following relation $$\begin{aligned}
\langle \delta {\cal A}^{\rm R}_{\eta } , \partial _{t} {\cal A}^{\rm R}_{\eta } \rangle & = \langle D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\delta } - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{t} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}\nonumber
& = \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle - \Big( \langle D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\delta } , { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{t} { ] \hspace{-0.6mm} ] }\rangle + \langle { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{\delta } { ] \hspace{-0.6mm} ] }, D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{t} { ] \hspace{-0.6mm} ] }\rangle \Big) . \end{aligned}$$ Hence, (\[1st term\]) plus (\[2nd term\]) provides that $S^{\rm R}$ has topological $t$-dependence (\[delta Swzw\]).
WZW-like complete action
------------------------
We propose a Wess-Zumino-Witten complete action and show its gauge invariance on the basis of WZW-like relations (\[NS pure-gauge-like equation\] - \[NS bianchi\]) and (\[R pure-gauge-like equation\] - \[XY=1\]).
We propose that a Wess-Zumino-Witten-like complete action is given by $$\begin{aligned}
S_{\rm WZW } & \equiv S^{\rm NS} + S^{\rm R}
{\nonumber\\}& = \int_{0}^{1} dt \Big( \langle \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle + \langle {\cal A}^{\rm NS}_{t} , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle \Big) . \end{aligned}$$ Since $S^{\rm NS}$ and $S^{\rm R}$ have topological $t$-dependence, the variation of the action $S_{\rm WZW}$ is given by $$\begin{aligned}
\delta S_{\rm WZW} = \langle \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle + \langle {\cal A}^{\rm NS}_{\delta } , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle . \end{aligned}$$ We therefore obtain the equations of motion $$\begin{aligned}
\label{NS EOM}
{\rm NS} \, : & \hspace{5mm} Q {\cal A}^{\rm NS} + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) = 0 ,
\\ \label{R EOM}
{\rm R} \, : & \hspace{5mm} P_{\eta } \big( Q {\cal A}^{\rm R}_{\eta } \big) = 0 . \end{aligned}$$
Let $\Lambda ^{\rm NS}$, $\Lambda ^{\rm R}$, and $\Omega ^{\rm NS}$ be NS, R, and NS gauge parameter fields which have ghost-and-picture number $(-1|0)$, $(-1|\frac{1}{2})$, and $(-1|1)$, respectively. These $\Lambda ^{\rm NS}$, $\Lambda ^{\rm R}$, and $\Omega ^{\rm NS}$ all belong to the large Hilbert space. The action is invariant under two types of gauge transformations: the gauge transformations parametrized by $\Lambda = \Lambda ^{\rm NS} + \Lambda ^{\rm R}$ $$\begin{aligned}
\label{delta Lambda NS}
{\cal A}^{\rm NS}_{\delta _{\Lambda }} & = Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] },
\\
\label{delta Lambda R}
\delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) & = - P_{\eta } Q \Big( \eta \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ and the gauge transformations parametrized by $\Omega = \Omega ^{\rm NS}$ $$\begin{aligned}
\label{delta Omega}
{\cal A}^{\rm NS}_{\delta _{\Omega } } = \eta \Omega ^{\rm NS} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , \Omega ^{\rm NS} { \big] \hspace{-1.1mm} \big] },
\hspace{5mm}
\delta _{\Omega } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) = 0 . \end{aligned}$$
The $\Lambda $-gauge transformations of the action is given by $$\begin{aligned}
\label{delta Lambda}
\delta _{\Lambda } S_{\rm WZW} = \langle \xi Y \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle + \langle {\cal A}^{\rm NS}_{\delta _{\Lambda }} , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle . \end{aligned}$$ We show that $\delta _{\Lambda } S_{\rm WZW } = 0$ with $\Lambda$-gauge transformations of fields $$\begin{aligned}
{\cal A}^{\rm NS}_{\delta _{\Lambda }} & = Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] },
{\nonumber\\}\nonumber
\delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) & = - P_{\eta } Q \Big( \eta \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ where $\Lambda ^{\rm NS}$ is an NS gauge parameter carrying ghost-and-picture number $(-1 | 0)$ and $\Lambda ^{\rm R}$ is a R gauge parameter carrying ghost-and-picture number $(-1 | \frac{1}{2})$. Note that these $\Lambda ^{\rm NS}$ and $\Lambda ^{\rm R}$ belong to the large Hilbert space $\mathcal{H}$ but $\delta _{\Lambda } (P_{\eta } {\cal A}^{\rm R}_{\eta } )$ has to be in the restricted one $\mathcal{H}_{R}$.
First, we consider the first term of (\[delta Lambda\]) with (\[delta Lambda R\]). This term consists of two ingredients $$\begin{aligned}
\label{R part of delta Lambda}
\langle \xi Y \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle & = \langle \xi Y \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , \, Q (P_{\eta } + P_{\xi } ) {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}& = \langle \xi Y \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q X Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle - \langle \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}& = \langle \xi Q \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle - \langle \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle . \end{aligned}$$ Here, we used $\delta ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) = P_{\eta } ( \delta P_{\eta } {\cal A}^{\rm R}_{\eta } ) $, $X Y ( \delta P_{\eta } {\cal A}^{\rm R}_{\eta } ) = \delta ( X Y P_{\eta } {\cal A}^{\rm R}_{\eta })$, and $X Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) = P_{\eta } {\cal A}^{\rm R}_{\eta }$. Since the first ingredient of (\[R part of delta Lambda\]) with $\Lambda $-gauge transformations (\[delta Lambda R\]) becomes $$\begin{aligned}
\langle \xi Q \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle &
= - \langle \xi Q P_{\eta } Q \big( D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\big) , Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}& = \langle P_{\xi } Q \xi Q \big( D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\big) , Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}\nonumber
& = \langle Q \big( D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\big) , P_{\eta } {\cal A}^{\rm R}_{\eta } \rangle \end{aligned}$$ and the second ingredient of (\[R part of delta Lambda\]) with $\Lambda $-gauge transformations (\[delta Lambda R\]) becomes $$\begin{aligned}
- \langle \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle & = \langle P_{\eta } Q \big( D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\big) , P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}\nonumber
& = \langle Q \big( D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\big) , P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle , \end{aligned}$$ we obtain $$\begin{aligned}
\label{R delta Lambda 1}
\langle \xi Y \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle & = \langle \xi Q \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Y ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) \rangle - \langle \delta _{\Lambda } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}& = \langle Q \big( D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\big) , P_{\eta } {\cal A}^{\rm R}_{\eta } + P_{\xi } {\cal A}^{\rm R}_{\eta } \rangle
{\nonumber\\}&= - \langle D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }, Q {\cal A}^{\rm R}_{\eta } \rangle . \end{aligned}$$
Next, we compute the second term of (\[delta Lambda\]) with (\[delta Lambda NS\]). Using $Q^{2} = 0$, ${ \big[ \hspace{-1.1mm} \big[ }{ [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }, {\cal A}^{\rm R}_{\eta } { \big] \hspace{-1.1mm} \big] }= 0$, and $D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\eta } = 0$, we quickly find that $$\begin{aligned}
\langle {\cal A}^{\rm NS}_{\delta _{\Lambda }} , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle & = \langle Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }, Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}& = \langle { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }, Q {\cal A}^{\rm NS}_{\eta } \rangle + \langle Q \Lambda ^{\rm NS} , m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle
{\nonumber\\}& = - \langle { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }, D^{\rm NS}_{\eta } {\cal A}^{\rm NS}_{Q} \rangle - \langle \Lambda ^{\rm NS} , { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , Q {\cal A}^{\rm R}_{\eta } { \big] \hspace{-1.1mm} \big] }\rangle
{\nonumber\\}\nonumber
& = \langle D^{\rm NS}_{\eta } \Lambda ^{\rm R} , { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{Q} { \big] \hspace{-1.1mm} \big] }\rangle - \langle { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }, Q {\cal A}^{\rm R}_{\eta } \rangle . \end{aligned}$$ The property (\[R associated equation\]) of the R pure-gauge-like field $- Q {\cal A}^{\rm R}_{\eta } = D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{Q} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{Q} { \big] \hspace{-1.1mm} \big] }$ gives $$\begin{aligned}
\nonumber
D^{NS}_{\eta } \Big( Q {\cal A}^{\rm R}_{\eta } - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{Q} { \big] \hspace{-1.1mm} \big] }\Big) = 0 . \end{aligned}$$ Hence, we obtain $$\begin{aligned}
\label{R delta Lambda 2}
\langle {\cal A}^{\rm NS}_{\delta _{\Lambda }} , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle = \langle D^{\rm NS}_{\eta } \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }, Q {\cal A}^{\rm R}_{\eta } \rangle , \end{aligned}$$ which just cancels (\[R delta Lambda 1\]), and we conclude $ \delta _{\Lambda} S_{\rm WZW} = (\ref{R delta Lambda 1}) + (\ref{R delta Lambda 2}) = 0$ with (\[delta Lambda NS\]) and (\[delta Lambda R\]).
The $\Omega $-gauge transformation of the action $S_{\rm WZW}$ is given by $$\begin{aligned}
\delta _{\Omega } S_{\rm WZW} = \langle \xi Y \delta _{\Omega } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) , Q {\cal A}^{\rm R}_{\eta } \rangle + \langle {\cal A}^{\rm NS}_{\delta _{\Omega }} , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle . \end{aligned}$$ One can show that $\delta _{\Omega } S_{\rm WZW} = 0$ with $\Omega $-gauge transformations of fields $$\begin{aligned}
\label{delta Omega NS}
{\cal A}^{\rm NS}_{\delta _{\Omega } } & = \eta \Omega ^{\rm NS} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , \Omega ^{\rm NS} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm R}_{\eta } , \Omega ^{\rm R} { \big] \hspace{-1.1mm} \big] },
\\
\label{delta Omega R}
\delta _{\Omega } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) & = - P_{\eta } Q \Big( \eta \Omega ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , \Omega ^{\rm R} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ where $\Omega ^{\rm NS}$ is an NS gauge parameter carrying ghost-and-picture number $(-1 | 1)$, $\Omega ^{\rm R}$ is a R gauge parameter carrying ghost-and-picture number $(-1 | \frac{1}{2})$, and both $\Omega ^{\rm NS}$ and $\Omega ^{\rm R}$ belong to the large Hilbert space. Note, however, that since R gauge parameters $\Omega ^{\rm R}$ and $\Lambda ^{\rm R}$ have the same ghost-and-picture number $(-1 | \frac{1}{2} )$, we can not distinguish these two parameters. As a result, $\Omega ^{\rm R}$-gauge transformation is absorbed into $\Lambda ^{\rm R}$-gauge transformation (\[delta Lambda R\]) and $\Omega $-gauge transformations (\[delta Omega NS\]) and (\[delta Omega R\]) reduces to (\[delta Omega\]): $$\begin{aligned}
\nonumber
{\cal A}^{\rm NS}_{\delta _{\Omega } } = \eta \Omega ^{\rm NS} - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , \Omega ^{\rm NS} { \big] \hspace{-1.1mm} \big] },
\hspace{5mm}
\delta _{\Omega } ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) = 0 . \end{aligned}$$ Then, using $Q {\cal A}^{\rm NS}_{\eta } = - D^{\rm NS}_{\eta } {\cal A}^{\rm NS}_{Q}$ and $D^{\rm NS}_{\eta } {\cal A}^{\rm R}_{\eta } = 0$, we quickly find that $$\begin{aligned}
\nonumber
\delta _{\Omega } S_{\rm WZW} = \langle D^{\rm NS}_{\eta } \Omega ^{\rm NS} , Q {\cal A}^{\rm NS}_{\eta } + m_{2} ( {\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } ) \rangle = 0. \end{aligned}$$ Therefore, the action $S_{\rm WZW}$ is invariant under $\Omega $-gauge transformations (\[delta Omega\]).
Unified notation
----------------
We introduce a notation which is useful to unify the results of NS and R sectors. Then, the concept of Ramond number projections proposed in [@Erler:2015lya] naturaly appears. We say Ramond number of the $k$-product $M_{k}$ is $n$ when number of R imputs of $M_{k}$ minus number of R output of $M_{k}$ equals to $n$. The symbol $M_{k}|_{n}$ denotes the $k$-product projected onto Ramond number $n$. For example, R number $0$ and $2$ projection of the star product $m_{2}$ are $$\begin{aligned}
&\langle {\rm NS} + {\rm R} , \, m_{2}|_{0}({\rm NS} + {\rm R}, {\rm NS} + {\rm R}) \rangle = \langle {\rm NS} , \, m_{2}( {\rm NS} , {\rm NS} )\rangle + \langle {\rm R} , \, m_{2} ( {\rm NS} , {\rm R} )+m_{2}( {\rm R} , {\rm NS} )\rangle ,
{\nonumber\\}[2mm] \nonumber
& \hspace{30mm} \langle {\rm NS} + {\rm R} , \, m_{2}|_{2} ({\rm NS} + {\rm R} , {\rm NS} +{\rm R} )\rangle = \langle {\rm NS} , \, m_{2} ( {\rm R} , {\rm R} )\rangle . \end{aligned}$$ It is helpful to specify whether the (output) state ${\cal A}$ is NS or R. We write ${\cal A}|^{\rm NS}$ for the NS (output) state and ${\cal A}|^{\rm R}$ for the R (output) state. For example, for the sum of NS and R states $$\begin{aligned}
\nonumber
( {\rm NS} + {\rm R} )|^{\rm NS} = {\rm NS } , \hspace{10mm } ({\rm NS} + {\rm R})|^{\rm R} = {\rm R} .
$$ Then, we can write as follows: $$\begin{aligned}
& m_{2}({\rm NS} + {\rm R}, {\rm NS} + {\rm R} ) \big{|}^{\rm NS}_{0} = m_{2}( {\rm NS} , {\rm NS} ),
\hspace{5mm} m_{2} ({\rm NS} + {\rm R}, {\rm NS} + {\rm R} ) \big{|}^{\rm R}_{0} = { \big[ \hspace{-1.1mm} \big[ }{\rm NS} , {\rm R} { \big] \hspace{-1.1mm} \big] },
{\nonumber\\}[2mm] \nonumber
& \hspace{10mm} m_{2} ({\rm NS} + {\rm R} , {\rm NS} +{\rm R} ) \big{|}^{\rm NS}_{2} = m_{2} ( {\rm R} , {\rm R} ) , \hspace{5mm} m_{2} ({\rm NS} + {\rm R} , {\rm NS} +{\rm R} ) \big{|}^{\rm R}_{2} = 0 . \end{aligned}$$
We can introduce a pure-gauge-like field including both sector $$\begin{aligned}
{\cal A}_{\eta } \equiv {\cal A}^{\rm NS}_{\eta } + {\cal A}^{\rm R}_{\eta }\end{aligned}$$ such that ${\cal A}_{\eta }$ satisfies $$\begin{aligned}
D_{\eta } {\cal A}_{\eta } \equiv \eta {\cal A}_{\eta } - m_{2}|_{0} ( {\cal A}_{\eta } , {\cal A}_{\eta } ) = 0 . \end{aligned}$$ Note that NS and R out-puts of $D_{\eta } {\cal A}_{\eta } = 0$ give $$\begin{aligned}
\nonumber
{\rm NS} \, &: \hspace{5mm} \big( D_{\eta } {\cal A}_{\eta } \big) \big{|} ^{\rm NS} \equiv \eta {\cal A}^{\rm NS}_{\eta } - m_{2} \big( {\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm NS}_{\eta } \big) = 0 ,
\\ \nonumber
{\rm R} \, &: \hspace{5mm} \big( D_{\eta } {\cal A}_{\eta } \big) \big{|} ^{\rm R} \equiv \eta {\cal A}^{\rm R}_{\eta } - { \big[ \hspace{-1.1mm} \big[ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm R}_{\eta } { \big] \hspace{-1.1mm} \big] }= 0 , \end{aligned}$$ which are just the defining equations of NS and R pure-gauge-like fields (\[NS pure-gauge-like equation\]) and (\[R pure-gauge-like equation\]) respectively. Similarly, we can also define an associated field of $d$ including both sector $$\begin{aligned}
{\cal A}_{d} \equiv {\cal A}^{\rm NS}_{d} + {\cal A}^{\rm R}_{d} \end{aligned}$$ such that ${\cal A}_{d}$ satisfies $$\begin{aligned}
(-)^{d} d \, {\cal A}_{\eta } = D_{\eta } {\cal A}_{d} , \end{aligned}$$ whose NS out-put $((-)^{d} d {\cal A}_{\eta } = D_{\eta } {\cal A}_{d})|^{\rm NS}$ and R out-put $((-)^{d} d {\cal A}_{\eta } = D_{\eta } {\cal A}_{d})|^{\rm R}$ just provide the defining equations of NS and R pure-gauge-like fields (\[NS pure-gauge-like equation\]) and (\[R pure-gauge-like equation\]) respectively $$\begin{aligned}
\nonumber
{\rm NS} \, &: \hspace{5mm} (-)^{d} d {\cal A}^{\rm NS}_{\eta } = \eta {\cal A}^{\rm NS}_{d} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm NS}_{d} { ] \hspace{-0.6mm} ] },
\\ \nonumber
{\rm R} \, &: \hspace{5mm} (-)^{d} d {\cal A}^{\rm R}_{\eta } = \eta {\cal A}^{\rm R}_{d} - { [ \hspace{-0.6mm} [ }{\cal A}^{\rm NS}_{\eta } , {\cal A}^{\rm R}_{d} { ] \hspace{-0.6mm} ] }- { [ \hspace{-0.6mm} [ }{\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }. \end{aligned}$$
In this notation, our Wess-Zumino-Witten-like complete action is given by $$\begin{aligned}
\label{WZW-like complete action}
S_{\rm WZW} = \int_{0}^{1} dt \, \langle \mathcal{A}^{\ast }_{t} \, , \, Q \mathcal{A}_{\eta } + m_{2}|_{2} ( \mathcal{A}_{\eta } , \mathcal{A}_{\eta } ) \rangle , \end{aligned}$$ where the conjugate associated field ${\cal A}^{\ast }_{t}$ is defined by $$\begin{aligned}
\mathcal{A}^{\ast }_{t} \equiv {\cal A}^{\rm NS}_{t} + \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) . \end{aligned}$$ Note that the projection onto Ramond number $2$ states implies $m_{2}|_{2} ({\cal A}_{\eta } , {\cal A}_{\eta }) = m_{2} ({\cal A}^{\rm R}_{\eta } , {\cal A}^{\rm R}_{\eta } )$ for ${\cal A}^{\rm NS}_{t}$ and $m_{2}|_{2} ( {\cal A}_{\eta } , {\cal A}_{\eta } ) = 0$ for $\xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta })$. Then, the variation of the action becomes $\delta S_{\rm WZW} = \langle {\cal A}^{\ast }_{\delta } \, , \, Q {\cal A}_{\eta } + m_{2}|_{2} ({\cal A}_{\eta } , {\cal A}_{\eta } ) \rangle $ with ${\cal A}^{\ast }_{\delta } \equiv {\cal A}^{\rm NS}_{\delta } + \xi Y \delta ( P_{\eta } {\cal A}^{\rm R}_{\eta })$ and the equations of motion is given by $$\begin{aligned}
Q \mathcal{A}_{\eta } + m_{2}|_{2} ( \mathcal{A}_{\eta } , \mathcal{A}_{\eta } ) = 0 , \end{aligned}$$ which reproduces NS and R equations of motion (\[NS EOM\]) and (\[R EOM\]) by NS and R out-puts projections respectively. When we consider another parametrization of the action and its relation to the parametrization given in section 1.1, this notation would be useful.
Another parametrization
=======================
We use the same notation as [@Erler:2015lya] in this section. Readers who are unfamiliar with $A_{\infty }$ algebras or coalgebraic computations see, for example, [@Erler:2015lya; @Erler:2015rra; @Erler:2015uba; @Erler:2015uoa; @GM2; @Erler:2013xta; @Erler:2014eba; @Gaberdiel:1997ia] or other mathematical manuscripts[@Bar; @complex; @Penkava; @Kajiura:2003ax]. In the work of [@Erler:2015lya], the on-shell conditions of superstring field theories are proposed. For open superstring field theory, it is given by $$\begin{aligned}
\label{EKS eom}
\pi _{1} \big( {\boldsymbol Q} + {\boldsymbol m}_{2}|_{2} \big) \, \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } = 0 ,\end{aligned}$$ where $\widetilde{\Psi } = \widetilde{\Psi }^{\rm NS} + \widetilde{\Psi }^{\rm R}$ is an NS plus R string field, $Q$ is the BRST operator, and $m_{2}|_{2}$ denotes the star product $m_{2}$ with R number 2 projection. Note that NS and R out-puts of (\[EKS eom\]) are given by $$\begin{aligned}
\label{EKS NS eom}
{\rm NS} \, & : \hspace{5mm} Q \, \pi _{1} \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm NS} + m_{2} \Big( \pi _{1} \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm R} , \pi _{1} \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm R} \Big) = 0 ,
\\
\label{EKS R eom}
{\rm R} \, & : \hspace{30mm} Q \, \pi _{1} \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm R} = 0 . \end{aligned}$$ Note also that in general, the cohomomorphism $\widehat{\bf G}$ is constructed by the path-ordered exponential (with direction) of a gauge product ${\boldsymbol \mu }(t)$, a coderivation, as follows $$\begin{aligned}
\label{exp}
\widehat{\bf G} \equiv \mathcal{P} \exp{\Big[ \int_{0}^{t} d t' \, {\boldsymbol \mu } (t') \Big] } . \end{aligned}$$
In this paper, we always use $\widehat{\bf G}$ given in [@Erler:2015lya]. In [@Erler:2015lya], the gauge product ${\boldsymbol \mu }(t)$ consists of R number $0$ projected objects. Therefore, $\pi _{1} \widehat{\bf G}$ has at most one Ramond state in its in-puts.
Another parametrization of the WZW-like complete action
-------------------------------------------------------
In this section, we define pure-gauge-like and associated fields parametrized by $\widetilde{\Psi }= \widetilde{\Psi }^{\rm NS} + \widetilde{\Psi }^{\rm R}$ and construct a gauge invariant action, whose equations of motion equals to (\[EKS eom\]), the Ramond equations of motion proposed in [@Erler:2015lya]. The proofs of required properties are in section 3.2.
We can construct an NS pure-gauge-like field $\widetilde{A}^{\rm NS}_{\eta }$ by $$\begin{aligned}
\label{NS pure-gauge-like tilde}
\widetilde{A}^{\rm NS}_{\eta } \equiv \pi _{1} \widehat{\bf G} \, \frac{1}{1- \widetilde{\Psi }} \Big{|}^{\rm NS}
= \pi _{1} \widehat{\bf G} \, \frac{1}{1- \widetilde{\Psi }^{\rm NS}} , \end{aligned}$$ and a R pure-gauge-like field $\widetilde{A}^{\rm R}_{\eta }$ by $$\begin{aligned}
\label{R pure-gauge-like tilde}
\widetilde{A}^{\rm R}_{\eta } \equiv \pi _{1} \widehat{\bf G} \, \frac{1}{1- \widetilde{\Psi }} \Big{|}^{\rm R}
= \pi _{1} \widehat{\bf G} \, \Big( \frac{1}{1-\widetilde{\Psi }^{\rm NS}} \otimes \widetilde{\Psi }^{\rm R} \otimes \frac{1}{1- \widetilde{\Psi }^{\rm NS}} \Big) . \end{aligned}$$ These pure-gauge-like fields are parametrized by NS and R string field $\widetilde{\Psi } = \widetilde{\Psi }^{\rm NS} + \widetilde{\Psi }^{\rm R}$. While the NS string field $\widetilde{\Psi }$ is a Grassmann odd and ghost-and-picture number $(1|-1)$ state in the small Hilbert space $\mathcal{H}_{\rm S}$, the R string field $\widetilde{\Psi }^{\rm R}$ is a Grassmann odd and ghost-and-picture number $(1|-\frac{1}{2})$ state in the restricted small Hilbert space $\mathcal{H}_{R}$. Hence, $\widetilde{\Psi }^{\rm NS} \in \mathcal{H}_{\rm S}$ and $\widetilde{\Psi }^{\rm R} \in \mathcal{H}_{R}$ satisfy $$\begin{aligned}
\nonumber
\eta \, \widetilde{\Psi }^{\rm NS} = 0, \hspace{5mm} \eta \, \widetilde{\Psi }^{\rm R} = 0, \hspace{5mm} XY \widetilde{\Psi }^{\rm R} = \widetilde{\Psi }^{\rm R} . \end{aligned}$$ Note that $\widetilde{A}^{\rm NS}_{\eta }$ has ghost-and-picture number $(1 | -1)$ and $\widetilde{A}^{\rm R}_{\eta }$ has ghost-and-picture number $(1|-\frac{1}{2} )$ by construction. As we will see in section 3.2, one can check that $\widetilde{A}^{\rm NS}_{\eta }$ and $\widetilde{A}^{\rm R}_{\eta }$ satisfy the defining properties of pure-gauge-like fields: $$\begin{aligned}
\label{EKS NS pure-gauge-like eq.}
{\rm NS} \, & : \hspace{13mm} \eta \widetilde{A}^{\rm NS}_{\eta } - m_{2} ( \widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm NS}_{\eta } ) = 0 ,
\\[2mm]
\label{EKS R pure-gauge-like eq.}
{\rm R} \, & : \hspace{5mm} \eta \widetilde{A}^{\rm R}_{\eta } - m_{2} ( \widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{\eta } ) - m_{2} ( \widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm NS}_{\eta } ) = 0 . \end{aligned}$$ Let $d$ be a derivation operator commuting with $\eta $. Then, with these pure-gauge-like fields parametrized by small Hilbert space string fields $\widetilde{\Psi }$, an NS associated field $\widetilde{A}^{\rm NS}_{d}$ defined by $$\begin{aligned}
\label{NS associated tilde}
\widetilde{A}^{\rm NS}_{d} & \equiv \pi _{1} \widehat{\bf G} \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm NS} \end{aligned}$$ and a R associated field $\widetilde{A}^{\rm R}_{d}$ defined by $$\begin{aligned}
\label{R associated tilde}
\widetilde{A}^{\rm R}_{d} & \equiv \pi _{1} \widehat{\bf G} \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm R} \end{aligned}$$ satisfy the defining properties of associated fields: $$\begin{aligned}
\label{NS associated eq. tilde}
(-)^{d} d \, \widetilde{A}^{\rm NS}_{\eta } & = \eta \widetilde{A}^{\rm NS}_{d} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm NS}_{d} { \big] \hspace{-1.1mm} \big] },
\\[2mm] \label{R associated eq. tilde}
(-)^{d} d \, \widetilde{A}^{\rm R}_{\eta } & = \eta \widetilde{A}^{\rm R}_{d} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{d} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm NS}_{d} { \big] \hspace{-1.1mm} \big] }, \end{aligned}$$ which we prove in section 3.2. Once the defining properties (\[EKS NS pure-gauge-like eq.\]), (\[EKS R pure-gauge-like eq.\]), (\[NS associated eq. tilde\]), and (\[R associated eq. tilde\]) are proved using pure-gauge-like fields $\widetilde{A}^{\rm NS}_{\eta }$, $\widetilde{A}^{\rm R}_{\eta }$ defined by (\[NS pure-gauge-like tilde\]), (\[R pure-gauge-like tilde\]) and associated fields $\widetilde{A}^{\rm NS}_{d}$, $\widetilde{A}^{\rm R}_{d}$ defined by (\[NS associated tilde\]), (\[R associated tilde\]), we can construct a gauge invariant action on the basis of Wess-Zumino-Witten-like framework proposed in section 2.
To apply our WZW-like framework, we need the $XY$-projection invariance of $P_{\eta } \widetilde{A}^{\rm R}_{\eta }$ $$\begin{aligned}
\label{XY projection tilde}
X Y ( P_{\eta } \widetilde{A}^{\rm R}_{\eta } ) = P_{\eta } \widetilde{A}^{\rm R}_{\eta } . \end{aligned}$$ Unfortunately, for any choice of cohomomorphism $\widehat{\bf G}$, the R pure-gauge-like field $\widetilde{A}^{\rm R}_{\eta }$ defined by (\[R pure-gauge-like tilde\]) does not satisfy this property. Note, however, that if we can take $\widehat{\bf G}$ satisfying $$\begin{aligned}
\label{xi G = xi}
{\boldsymbol \xi } \, \widehat{\bf G} = {\boldsymbol \xi } , \end{aligned}$$ then the R pure-gauge-like field defined by (\[R pure-gauge-like tilde\]) automatically satisfy (\[XY projection tilde\]) because $$\begin{aligned}
X Y ( P_{\eta } \widetilde{A}^{\rm R}_{\eta } ) & = X Y P_{\eta } \, \pi _{1} \widehat{\bf G} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm R} = X Y \eta \, \pi _{1} {\boldsymbol \xi } \widehat{\bf G} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm R}
{\nonumber\\}\nonumber
& = X Y P_{\eta } \, \pi _{1} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm R}
= X Y P_{\eta } \widetilde{\Psi }^{\rm R}
= P_{\eta } \widetilde{\Psi }^{\rm R} = P_{\eta } \widetilde{A}^{\rm R}_{\eta } , \end{aligned}$$ with $X Y \widetilde{\Psi }^{\rm R} = \widetilde{\Psi }^{\rm R}$ and $P_{\eta } = \eta \xi$ where $\xi = \xi _{0}$ for NS states and $\xi = \Xi $ for R states. Recall that $\widehat{\bf G}$ is constructed by the path-ordered exponential of a gauge product ${\boldsymbol \mu }(t)$ as (\[exp\]). When we take this gauge product as $\xi $-exact one ${\boldsymbol \mu } (t) \equiv {\boldsymbol \xi } {\boldsymbol M} (t)$, the cohomomorphism $\widehat{\bf G}$ is given by $$\begin{aligned}
\nonumber
\widehat{\bf G} \equiv \mathcal{P} e^{\int dt \, {\boldsymbol \xi } {\boldsymbol M} (t) } = \mathbb{I} + {\boldsymbol \xi } \Big( M_{2} + \frac{1}{2}M_{3} + \frac{1}{2} M_{2} \xi M_{2} + \dots \Big) , \end{aligned}$$ and it satisfies (\[xi G = xi\]). This $\xi $-exact choice of the gauge product is always possible by using ambiguities of the construction of (intermediate) gauge products ${\boldsymbol \mu }(t)$ or setting the initial condition of the defining equations of the $A_{\infty }$ products in [@Erler:2015lya]. Note that although a naive choice of $\xi $-exact gauge products ${\boldsymbol \mu } = {\boldsymbol \xi } {\boldsymbol M}$ breaks the ciclic property of the $A_{\infty }$ products $\widetilde{\boldsymbol M} = \widehat{\bf G}^{-1} \, ({\boldsymbol Q} + {\boldsymbol m}_{2}|_{2} ) \, \widehat{\bf G}$, it is no problem in our Wess-Zumino-Witten-like framework.
We would like to emphasize that it does not necessitate the ciclic property of $\widehat{\bf G}$ or $\widetilde{\boldsymbol M}$ to construct the Wess-Zumino-Witten-like complete action. We need the ciclic property of $D_{\eta }$ only, which holds for any choice of $\widehat{\bf G}$. Hence, we can always impose (\[XY projection tilde\]) in a consistent way with the definitions of pure-gauge-like fields (\[NS pure-gauge-like tilde\]), (\[R pure-gauge-like tilde\]).
Utilizing pure-gauge-like and associated fields satisfying (\[EKS NS pure-gauge-like eq.\]), (\[EKS R pure-gauge-like eq.\]), (\[NS associated eq. tilde\]), (\[R associated eq. tilde\]), and (\[XY projection tilde\]), we construct the Wess-Zumino-Witten-like complete action $$\begin{aligned}
\label{S tilde}
\widetilde{S} = \int_{0}^{1} dt \Big( \langle \xi Y \partial _{t} ( P_{\eta } \widetilde{A}^{\rm R}_{\eta } ) , Q \widetilde{A}^{\rm R}_{\eta } \rangle + \langle \widetilde{A}^{\rm NS}_{t} , Q \widetilde{A}^{\rm NS}_{\eta } + m_{2} ( \widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm R}_{\eta } ) \rangle \Big) , \end{aligned}$$ which is parametrized by small Hilbert space string fields $\widetilde{\Psi } = \widetilde{\Psi }^{\rm NS} + \widetilde{\Psi }^{\rm R}$. As we found in section 2, the variation of the action is given by $$\begin{aligned}
\label{delta S tilde}
\delta \widetilde{S} = \langle \xi Y \delta ( P_{\eta } \widetilde{A}^{\rm R}_{\eta } ) , Q \widetilde{A}^{\rm R}_{\eta } \rangle + \langle \widetilde{A}^{\rm NS}_{\delta } , Q \widetilde{A}^{\rm NS}_{\eta } + m_{2} ( \widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm R}_{\eta } ) \rangle , \end{aligned}$$ and the action is invariant under two types of gauge transformations: the gauge transformations parametrized by $\Lambda = \Lambda ^{\rm NS} + \Lambda ^{\rm R}$ $$\begin{aligned}
\nonumber
\widetilde{A}^{\rm NS}_{\delta _{\Lambda }} & = Q \Lambda ^{\rm NS} + { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm R}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] },
\\ \nonumber
\delta _{\Lambda } ( P_{\eta } \widetilde{A}^{\rm R}_{\eta } ) & = - P_{\eta } Q \Big( \eta \Lambda ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \Lambda ^{\rm R} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm R}_{\eta } , \Lambda ^{\rm NS} { \big] \hspace{-1.1mm} \big] }\Big) , \end{aligned}$$ and the gauge transformations parametrized by $\Omega = \Omega ^{\rm NS}$ $$\begin{aligned}
\nonumber \widetilde{A}^{\rm NS}_{\delta _{\Omega } } = \eta \Omega ^{\rm NS} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \Omega ^{\rm NS} { \big] \hspace{-1.1mm} \big] },
\hspace{5mm}
\delta _{\Omega } ( P_{\eta } \widetilde{A}^{\rm R}_{\eta } ) = 0 . \end{aligned}$$ Here NS, R, and NS gauge parameter fields $\Lambda ^{NS}$, $\Lambda ^{\rm R}$, and $\Omega ^{\rm NS}$ have ghost-and-picture number $(-1|0)$, $(-1|\frac{1}{2})$, and $(-1|1)$, respectively, and these fields all belong to the large Hilbert space. Note, however, that the gauge transformation $\delta _{\Lambda } (P_{\eta } \widetilde{A}^{\rm R}_{\eta })$ has to be in the restricted small Hilbert space $\mathcal{H}_{R}$.
Since the action $\widetilde{S}$ has topological $t$-dependence and its variation is given by (\[delta S tilde\]), we obtain the equations of motion $$\begin{aligned}
\label{NS eom tilde}
{\rm NS} \, : & \hspace{5mm} Q \widetilde{A}^{\rm NS}_{\eta } + m_{2} ( \widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm R}_{\eta } ) = \pi _{1} \big( {\boldsymbol Q} + {\boldsymbol m}_{2}|_{2} \big) \, \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } \Big{|}^{\rm NS} = 0 ,
\\ \label{R eom tilde}
{\rm R} \, : & \hspace{10mm} P_{\eta } \big( Q \widetilde{A}^{\rm R}_{\eta } \big) = P_{\eta } \Big( \pi _{1} \big( {\boldsymbol Q} + {\boldsymbol m}_{2}|_{2} \big) \, \widehat{{\bf G}} \frac{1}{1 - \widetilde{\Psi } } \Big) \Big{|}^{\rm R} = 0 , \end{aligned}$$ which is equivalent to (\[EKS eom\]). While the NS out-put of the equations of motion (\[NS eom tilde\]) is the same as (\[EKS NS eom\]), the R out-put (\[R eom tilde\]) is equal to the small Hilbert space component of (\[EKS R eom\]). Note that $P_{\xi } ( Q \widetilde{A}^{\rm R}_{\eta } )$ can not be determined from the action because it vanishes in the inner product and does not affect the value of the action. We thus set $P_{\xi } ( Q \widetilde{A}^{\rm R}_{\eta } ) = 0$ and obtain (\[EKS R eom\]).
It is interesting to compare kinetic terms of (\[S tilde\]) and (\[S KO\]). In the present parametrization of (\[S tilde\]), the kinetic term of $\widetilde{S}$ is given by $$\begin{aligned}
\nonumber
- \frac{1}{2} \langle \xi \widetilde{\Psi }^{\rm NS} , Q \widetilde{\Psi }^{\rm NS} \rangle - \frac{1}{2} \langle \xi Y \widetilde{\Psi }^{\rm R} , Q \widetilde{\Psi }^{\rm R} \rangle . \end{aligned}$$ Note that the Ramond kinetic term is just equal to that of Kunitomo-Okawa’s action. Similarly, we quickly check that the NS kinetic term is equivalent to that of Kunitomo-Okawa’s action with the (trivial emmbeding) condition $\widetilde{\Psi }^{\rm NS} = \eta \Phi ^{\rm NS}$ or the (linear partial gauge fixing) condition $\Phi ^{\rm NS} = \xi \widetilde{\Psi }^{\rm NS}$. Therefore, the kinetic term of $\widetilde{S}$ has the same spectrum as that of [@Kunitomo:2015usa].
WZW-like relation from $A_{\infty }$ and $\eta $-exactness
----------------------------------------------------------
Let ${\boldsymbol \eta }$ be the coderivation constructed from $\eta $, which is nilpotent ${\boldsymbol \eta }^{2} = 0$, and let ${\boldsymbol a }$ be a nilpotent coderivation satisfying ${\boldsymbol a} {\boldsymbol \eta } = -(-)^{{\boldsymbol a} {\boldsymbol \eta }} {\boldsymbol \eta } {\boldsymbol a}$ and ${\boldsymbol a}^{2} = 0$. Then, we assume that $\widehat{\bf G}^{-1} : (\mathcal{H} , {\boldsymbol a}) \rightarrow (\mathcal{H}_{\rm S} , {\boldsymbol D}_{\boldsymbol a})$ is an $A_{\infty }$-morphism, where $\mathcal{H}$ is the large Hilbert space, $\mathcal{H}_{\rm S}$ is the small Hilbert space, and ${\boldsymbol D}_{\boldsymbol a} \equiv \widehat{\bf G}^{-1} {\boldsymbol a} \, \widehat{\bf G}$. Note that ${\boldsymbol D}_{\boldsymbol a}$ is nilpotent: ${\boldsymbol D}_{\boldsymbol a}^{2} = ( \widehat{\bf G}^{-1} {\boldsymbol a} \, \widehat{\bf G} ) ( \widehat{\bf G}^{-1} {\boldsymbol a} \, \widehat{\bf G} ) = \widehat{\bf G}^{-1} {\boldsymbol a}^{2} \, \widehat{\bf G} = 0$. For example, one can use ${\boldsymbol Q}$, ${\boldsymbol Q}+{\boldsymbol m}_{2}|_{2}$, and so on for ${\boldsymbol a}$, and various $\widehat{\bf G}$ appearing in [@Erler:2013xta; @Erler:2014eba; @Erler:2015lya] for $\widehat{\bf G}$. Suppose that the coderivation ${\boldsymbol D}_{\boldsymbol a}$ also commutes with ${\boldsymbol \eta }$, which means $$\begin{aligned}
\label{[D_a,eta]=0}
( {\boldsymbol D}_{\boldsymbol a} )^{2} = 0 , \hspace{5mm} { \big[ \hspace{-1.1mm} \big[ }{\boldsymbol D}_{\boldsymbol a} , {\boldsymbol \eta } { \big] \hspace{-1.1mm} \big] }= 0. \end{aligned}$$ Then, we can introduce a dual $A_{\infty }$-products ${\boldsymbol D}_{\boldsymbol \eta }$ defined by $$\begin{aligned}
{\boldsymbol D}^{\boldsymbol \eta } \equiv \widehat{\bf G} \, {\boldsymbol \eta } \, \widehat{\bf G}^{-1} . \end{aligned}$$ Note that the pair of nilpotent maps $( {\boldsymbol D}^{\boldsymbol \eta } ,{\boldsymbol a})$ have the same properties as $( {\boldsymbol D}_{\boldsymbol a} , {\boldsymbol \eta })$: $$\begin{aligned}
\label{[D^eta,a]=0}
( {\boldsymbol D}^{\boldsymbol \eta } ) ^{2} = 0 , \hspace{5mm} { \big[ \hspace{-1.1mm} \big[ }{\boldsymbol D}^{\boldsymbol \eta } , {\boldsymbol a} { \big] \hspace{-1.1mm} \big] }= 0 . \end{aligned}$$ We can quickly find when the $A_{\infty }$-products ${\boldsymbol D}_{\boldsymbol a}$ commutes with the coderivation ${\boldsymbol \eta }$ as (\[\[D\_a,eta\]=0\]), its dual $A_{\infty }$-product ${\boldsymbol D}^{\boldsymbol \eta }$ and coderivation ${\boldsymbol a}$ also satisfies (\[\[D\^eta,a\]=0\]) as follows $$\begin{aligned}
{\boldsymbol a} {\boldsymbol D}^{\boldsymbol \eta } &= \big( \widehat{\bf G} \, \widehat{\bf G}^{-1} \big) \, {\boldsymbol a} \, \big( \widehat{\bf G} \, {\boldsymbol \eta } \, \widehat{\bf G}^{-1} \big) = \widehat{\bf G} \, {\boldsymbol D}_{\boldsymbol a} \, {\boldsymbol \eta } \, \widehat{\bf G}^{-1}
{\nonumber\\}\nonumber
& = (-)^{{\boldsymbol a}{\boldsymbol \eta }} \widehat{\bf G} \, {\boldsymbol \eta } \, {\boldsymbol D}_{\boldsymbol a} \, \widehat{\bf G}^{-1}
= (-)^{{\boldsymbol a}{\boldsymbol \eta }} \, \widehat{\bf G} \, {\boldsymbol \eta } \, \widehat{\bf G}^{-1} \, {\boldsymbol a} \, \widehat{\bf G} \, \widehat{\bf G}^{-1}
= (-)^{{\boldsymbol a}{\boldsymbol \eta }} {\boldsymbol D}^{\boldsymbol \eta } \, {\boldsymbol a} .\end{aligned}$$
In this paper, as these coderication ${\boldsymbol a}$ and $A_{\infty }$-morphism $\widehat{\bf G}$, we always use ${\boldsymbol a} \equiv {\boldsymbol Q} + {\boldsymbol m}_{2}|_{2}$ and $\widehat{\bf G}$ introduced in (\[xi G = xi\]), namely, a gauge product $\widehat{\bf G}$ given by [@Erler:2015lya] with the choice satisfying ${\boldsymbol \xi} \, \widehat{\bf G} = {\boldsymbol \xi }$. Therefore, the dual $A_{\infty }$ products is always given by $$\begin{aligned}
\label{D^eta}
{\boldsymbol D}^{\boldsymbol \eta } = {\boldsymbol \eta } - {\boldsymbol m}_{2}|_{0} , \end{aligned}$$ and the symbol ${\boldsymbol D}^{\boldsymbol \eta }$ always denotes (\[D\^eta\]) in the rest. (See section 6.2 of [@Erler:2015lya].) Then, the Maurer-Cartan element of ${\boldsymbol D}^{\boldsymbol \eta } = {\boldsymbol \eta } - {\boldsymbol m}_{2}|_{0}$ is given by $$\begin{aligned}
{\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1-\widetilde{A}} & = \frac{1}{1-\widetilde{A}} \otimes \pi _{1} \Big( {\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1-\widetilde{A}} \Big) \otimes \frac{1}{1-\widetilde{A}}
{\nonumber\\}\nonumber
& = \frac{1}{1-\widetilde{A}} \otimes \Big( {\boldsymbol \eta } \widetilde{A} - {\boldsymbol m}_{2}|_{0} ( \widetilde{A} , \widetilde{A} ) \Big) \otimes \frac{1}{1-\widetilde{A} } , \end{aligned}$$ where $\widetilde{A} = \widetilde{A}^{\rm NS} + \widetilde{A}^{\rm R}$ is a state of the large Hilbert space $\mathcal{H}$ and $\pi _{1}$ is an natural $1$-state projection onto $\mathcal{H}$. Hence, the solution of the Maurer-Cartan eqiation ${\boldsymbol D}^{\boldsymbol \eta } (1-\widetilde{A})^{-1} = 0$ is given by a state $\widetilde{A}_{\eta } = \widetilde{A}^{\rm NS}_{\eta } + \widetilde{A}^{\rm R}_{\eta }$ satisfying $$\begin{aligned}
\label{EKS pure-gauge-like eq.}
{\boldsymbol \eta } \widetilde{A}_{\eta } - {\boldsymbol m}_{2}|_{0} ( \widetilde{A}_{\eta } , \widetilde{A}_{\eta } ) = 0 , \end{aligned}$$ and vice versa, the solution $\widetilde{A}_{\eta }= \widetilde{A}^{\rm NS}_{\eta } + \widetilde{A}^{\rm R}_{\eta }$ satisfies (\[EKS pure-gauge-like eq.\]), or equivalently, $$\begin{aligned}
\nonumber
{\rm NS} \, & : \hspace{13mm} \eta \widetilde{A}^{\rm NS}_{\eta } - m_{2} ( \widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm NS}_{\eta } ) = 0 ,
\\ \nonumber
{\rm R} \, & : \hspace{5mm} \eta \widetilde{A}^{\rm R}_{\eta } - m_{2} ( \widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{\eta } ) - m_{2} ( \widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm NS}_{\eta } ) = 0 , \end{aligned}$$ which is just equivalent to the condition (\[EKS NS pure-gauge-like eq.\]) and (\[EKS R pure-gauge-like eq.\]) characterizing NS and R pure-gauge-like fields $\widetilde{A}^{\rm NS}_{\eta }$ and $\widetilde{A}^{\rm R}_{\eta }$. As a result, we obtain one of the most important fact the solutions of the Maurer-Cartan equation of ${\boldsymbol D}^{\boldsymbol \eta } = {\boldsymbol \eta } - {\boldsymbol m}_{2}|_{0}$ gives desired NS and R pure-gauge-like fields.
When the ${\boldsymbol \eta }$-complex $( \mathcal{H} , {\boldsymbol \eta } )$ is exact, there exist ${\boldsymbol \xi }$ such that ${ [ \hspace{-0.6mm} [ }{\boldsymbol \eta } , {\boldsymbol \xi } { ] \hspace{-0.6mm} ] }= {\boldsymbol 1}$ and $\mathcal{H}$, the large Hilbert space, is decomposed into the direct sum of $\eta $-exacts and $\xi $-exacts $\mathcal{H} = P_{\eta } \mathcal{H} \oplus P_{\xi }\mathcal{H}$, where $P_{\eta }$ and $P_{\xi }$ are projector onto $\eta $-exact and $\xi $-exact states respectively.[^4] Note that since the small Hilbert space $\mathcal{H}_{\rm S}$ is defined by $\mathcal{H}_{\rm S} \equiv P_{\eta } \mathcal{H}$ and satisfies $\mathcal{H}_{\rm S} \subset P_{\eta } \mathcal{H}_{\rm S}$, all the states $\widetilde{\Psi }$ belonging to $\mathcal{H}_{\rm S}$ satisfy $P_{\eta } \widetilde{\Psi } = \widetilde{\Psi }$ and $P_{\xi } \widetilde{\Psi } = 0$, or simply, $$\begin{aligned}
\nonumber
{\boldsymbol \eta } \, \Psi = 0. \end{aligned}$$ Using this fact, we can construct desired pure-gauge-like fields $\widetilde{A}^{\rm NS}_{\eta }$ and $\widetilde{A}^{\rm R}_{\eta }$ as solutions of the Maurer-Cartan equation of ${\boldsymbol D}^{{\boldsymbol \eta }} = {\boldsymbol \eta } - {\boldsymbol m}_{2}|_{0}$. Note that the Maurer-Cartan equation consists of NS and R out-puts $$\begin{aligned}
{\rm NS} \, & : \hspace{5mm}
\pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1- \widetilde{A}_{\eta }} \Big{|}^{\rm NS} = \pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1- \widetilde{A}^{\rm NS}_{\eta }} = 0 ,
\hspace{5mm }
{\nonumber\\}\nonumber
{\rm R} \, & : \hspace{6.5mm}
\pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1- \widetilde{A}_{\eta }} \Big{|}^{\rm R} = \pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \Big( \frac{1}{1- \widetilde{A}^{\rm NS}_{\eta }} \otimes \widetilde{A}^{\rm R}_{\eta } \otimes \frac{1}{1- \widetilde{A}^{\rm NS}_{\eta }} \Big) = 0 , \end{aligned}$$ where the upper index of $|$ denotes the NS or R projection: for any state $\widetilde{A} = \widetilde{A}^{\rm NS} + \widetilde{A}^{\rm R} \in \mathcal{H}$, the NS projection $\widetilde{A}|^{\rm NS}$ is defined by $\widetilde{A}|^{\rm NS} \equiv \widetilde{A}^{\rm NS}$ and the R projection $\widetilde{A}|^{\rm R}$ is defined by $\widetilde{A}|^{\rm R} \equiv \widetilde{A}^{\rm R}$.
An NS pure-gauge-like field $\widetilde{A}^{\rm NS}_{\eta }$ is given by $$\begin{aligned}
\nonumber
\widetilde{A}^{\rm NS}_{\eta } \equiv \pi _{1} \widehat{\bf G} \, \frac{1}{1- \widetilde{\Psi }} \Big{|}^{\rm NS}
= \pi _{1} \widehat{\bf G} \, \frac{1}{1- \widetilde{\Psi }^{\rm NS}} \end{aligned}$$ because it becomes a trivial NS state solution of the Maurer-Cartan equation as follows $$\begin{aligned}
{\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1- \widetilde{A}^{\rm NS}_{\eta }}
& = {\boldsymbol D}^{\boldsymbol \eta } \frac{1}{1- \pi _{1} \widehat{\bf G} \, \frac{1}{1-\widetilde{\Psi }^{\rm NS} } } = {\boldsymbol D}^{\boldsymbol \eta } \, \widehat{\bf G} \, \frac{1}{1-\widetilde{\Psi }^{\rm NS}} = \widehat{\bf G} \, {\boldsymbol \eta } \frac{1}{1-\widetilde{\Psi }^{\rm NS}}
{\nonumber\\}\nonumber
& = \widehat{\bf G} \, \Big( \frac{1}{1- \widetilde{\Psi }^{\rm NS} } \otimes {\boldsymbol \eta } \widetilde{\Psi }^{\rm NS} \otimes \frac{1}{1-\widetilde{\Psi }^{\rm NS} } \Big) = 0 . \end{aligned}$$ Note that $\pi _{1} {\boldsymbol D}^{\boldsymbol \eta } (1- \widetilde{A}^{\rm NS}_{\eta })^{-1} = 0$ is equal to $$\begin{aligned}
\nonumber
\eta \, \widetilde{A}^{\rm NS}_{\eta } - m_{2} ( \widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm NS}_{\eta } ) = 0 . \end{aligned}$$
Similarly, a R pure-gauge-like field $\widetilde{A}^{\rm R}_{\eta }$ is given by $$\begin{aligned}
\nonumber
\widetilde{A}^{\rm R}_{\eta } \equiv \pi _{1} \widehat{\bf G} \, \frac{1}{1- \widetilde{\Psi }} \Big{|}^{\rm R}
= \pi _{1} \widehat{\bf G} \, \Big( \frac{1}{1-\widetilde{\Psi }^{\rm NS}} \otimes \widetilde{\Psi }^{\rm R} \otimes \frac{1}{1-\widetilde{\Psi }^{\rm NS}} \Big) \end{aligned}$$ because it becomes a trivial R state solution of the Maurer-Cartan equation as follows $$\begin{aligned}
{\boldsymbol D}^{\boldsymbol \eta } \Big( \frac{1}{1- \widetilde{A}^{\rm NS}_{\eta }} \otimes \widetilde{A}^{\rm R}_{\eta } \otimes \frac{1}{1- \widetilde{A}^{\rm NS}_{\eta }} \Big)
& = {\boldsymbol D}^{\boldsymbol \eta } \, \widehat{\bf G} \Big( \frac{1}{1- \widetilde{\Psi }^{\rm NS}} \otimes \widetilde{\Psi }^{\rm R} \otimes \frac{1}{1- \widetilde{\Psi }^{\rm NS} } \Big)
{\nonumber\\}\nonumber
& = \widehat{\bf G} \, {\boldsymbol \eta } \, \Big( \frac{1}{1- \widetilde{\Psi }^{\rm NS}} \otimes \widetilde{\Psi }^{\rm R} \otimes \frac{1}{1- \widetilde{\Psi }^{\rm NS} } \Big) = 0 . \end{aligned}$$ Hence, the R state solution $\widetilde{A}^{\rm R}_{\eta }$ satisfies $$\begin{aligned}
\nonumber
\eta \, \widetilde{A}^{\rm R}_{\eta } - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{\eta } { \big] \hspace{-1.1mm} \big] }= 0 . \end{aligned}$$ Note that ${ [ \hspace{-0.6mm} [ }\widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }= m_{2} ( \widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{\eta } ) + m_{2} ( \widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm NS}_{\eta } )$.
We introduce the $\widetilde{A}_{\eta }$-shifted products $[ B_{1} , \dots , B_{n} ]^{\eta }_{\widetilde{A}_{\eta }}$ defined by $$\begin{aligned}
\big[ B_{1} , \dots , B_{n} \big] ^{\eta }_{\widetilde{A}_{\eta }} \equiv \pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes B_{1} \otimes \frac{1}{1-\widetilde{A}_{\eta } } \otimes \dots \otimes \frac{1}{1-\widetilde{A}_{\eta } } \otimes B_{n} \otimes \frac{1}{1-\widetilde{A}_{\eta } } \Big) . \end{aligned}$$ Note that higher shifted products all vanish $[ B_{1} , \dots , B_{n>2} ]^{\eta }_{\widetilde{A}_{\eta }} =0$ because now we consider ${\boldsymbol D}^{\boldsymbol \eta } \equiv \eta - m_{2}|_{0}$. In particular, we write $D_{\eta } B$ for $[ B ]_{\eta }$: $$\begin{aligned}
D_{\eta } B \equiv \pi _{1} {\boldsymbol D}^{{\boldsymbol \eta }} \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes B \otimes \frac{1}{1-\widetilde{A}_{\eta } } \Big) , \end{aligned}$$ or equivalently, for $\widetilde{A}_{\eta } = \widetilde{A}^{\rm NS}_{\eta } + \widetilde{A}^{\rm R}_{\eta }$ and $B = B^{\rm NS} + B^{\rm R}$, $$\begin{aligned}
\nonumber
{\rm NS} \, & : \hspace{5mm} D_{\eta } B \big{|}^{\rm NS} = \eta B^{\rm NS} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , B^{\rm NS} { \big] \hspace{-1.1mm} \big] },
\\ \nonumber
{\rm R} \, & : \hspace{6.5mm} D_{\eta } B \big{|}^{\rm R} = \eta B^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , B^{\rm R} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm R}_{\eta } , B^{\rm NS} { \big] \hspace{-1.1mm} \big] }. \end{aligned}$$ When $\widetilde{A}_{\eta }$ gives a solution on the Maurer-Cartan equation of ${\boldsymbol D}_{\boldsymbol \eta}$, these $\widetilde{A}_{\eta }$-shifted products also satisfy $A_{\infty }$-relations, which implies that the linear operator $D_{\eta }$ becomes nilpotent. We find $$\begin{aligned}
( D_{\eta } )^{2 } B & = \pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes \pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes B \otimes \frac{1}{1-\widetilde{A}_{\eta } } \Big) \otimes \frac{1}{1-\widetilde{A}_{\eta } } \Big)
{\nonumber\\}\nonumber
& = \pi _{1} ( {\boldsymbol D}^{\boldsymbol \eta } )^{2} \frac{1}{1-\widetilde{A}_{\eta } } - { \Big[ \hspace{-1.3mm} \Big[ }\pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \Big( \frac{1}{1-\widetilde{A}_{\eta } } \Big) , \, B \, { \Big] \hspace{-1.3mm} \Big] }^{\eta }_{\widetilde{A}_{\eta }} = 0 . \end{aligned}$$
Let ${\boldsymbol d}$ be a coderivation constructed from a derivation $d$ of the dual $A_{\infty }$-products ${\boldsymbol D}^{\boldsymbol \eta }$, which implies that the $d$-derivation propery ${ [ \hspace{-0.6mm} [ }{\boldsymbol d} , {\boldsymbol D}^{\boldsymbol \eta } { ] \hspace{-0.6mm} ] }= 0$ holds. Then, we obtain ${ [ \hspace{-0.6mm} [ }{\boldsymbol D}_{\boldsymbol d} , {\boldsymbol \eta } { ] \hspace{-0.6mm} ] }= 0 $ with ${\boldsymbol D}_{\boldsymbol d} \equiv \widehat{\bf G}^{-1} \, {\boldsymbol d} \, \widehat{\bf G}$, which means that ${\boldsymbol D}_{\boldsymbol d}$ is “${\boldsymbol \eta }$-exact” and there exists a coderivation ${\boldsymbol \xi }_{\boldsymbol d}$ such that $$\begin{aligned}
{\boldsymbol D}_{\boldsymbol d}= \widehat{\bf G}^{-1} \, {\boldsymbol d} \, \widehat{\bf G} = ( - )^{{\boldsymbol d}} { [ \hspace{-0.6mm} [ }{\boldsymbol \eta } , {\boldsymbol \xi }_{\boldsymbol d} { ] \hspace{-0.6mm} ] }. \end{aligned}$$ Using this coderivation ${\boldsymbol \xi}_{\boldsymbol d}$, we can construct NS and R associated fields constructed from the derivation operator $d$. Note that the response of ${\boldsymbol d}$ acting on the group-like element of $\widetilde{A}_{\eta }=\widetilde{A}^{\rm NS}_{\eta } + \widetilde{A}^{\rm R}_{\eta }$ is given by $$\begin{aligned}
(-)^{{\boldsymbol d}} {\boldsymbol d} \, \frac{1}{1-\widetilde{A}_{\eta } } & = (-)^{{\boldsymbol d}} G G^{-1} \, {\boldsymbol d} \, G \frac{1}{1- \widetilde{\Psi } } = G \, {\boldsymbol \eta } \, {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Phi } } = {\boldsymbol D}^{{\boldsymbol \eta }} \, G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big)
{\nonumber\\}&= {\boldsymbol D}^{{\boldsymbol \eta }} \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi }} \Big) \otimes \frac{1}{1-\widetilde{A}_{\eta }} \Big) .
$$
An NS associatede field of $d$ is given by $$\begin{aligned}
\nonumber
\widetilde{A}^{\rm NS}_{d} & \equiv \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm NS}
$$ because one can directly check $$\begin{aligned}
(-)^{{\boldsymbol d}} {\boldsymbol d} \, \frac{1}{1-\widetilde{A}^{\rm NS}_{\eta } } & = (-)^{{\boldsymbol d}} {\boldsymbol d} \, \frac{1}{1-\widetilde{A}_{\eta } } \Big{|}^{\rm NS}
= {\boldsymbol D}^{{\boldsymbol \eta }} \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi }} \Big) \otimes \frac{1}{1-\widetilde{A}_{\eta }} \Big) \Big|^{\rm NS}
{\nonumber\\}\nonumber
& = {\boldsymbol D}^{{\boldsymbol \eta }} \Big( \frac{1}{1-\widetilde{A}^{\rm NS}_{\eta } } \otimes \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi }} \Big) \Big{|}^{\rm NS} \otimes \frac{1}{1-\widetilde{A}^{\rm NS}_{\eta }} \Big) . \end{aligned}$$ Picking up the relation on $\mathcal{H}$, or equivalently acting $\pi _{1}$ on this relation on $T(\mathcal{H})$, we obtain $$\begin{aligned}
\nonumber
(-)^{d} d \, \widetilde{A}^{\rm NS}_{\eta } & = \eta \, \widetilde{A}^{\rm NS}_{d} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm NS}_{d} { \big] \hspace{-1.1mm} \big] },\end{aligned}$$ which is the simplest case of $(-)^{d} d \widetilde{A}_{\eta } = \pi _{1} {\boldsymbol D}^{\boldsymbol \eta } \mathchar`- {\rm exact} \,\, {\rm term}$.
Similarly, an R associated field of $d$ is given by $$\begin{aligned}
\nonumber
\widetilde{A}^{\rm R}_{d} & \equiv \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm R} \end{aligned}$$ because one can directly check $$\begin{aligned}
(-)^{d} d \, \widetilde{A}^{\rm R}_{\eta } & = \pi _{1} (-)^{{\boldsymbol d}} {\boldsymbol d} \, \frac{1}{1-\widetilde{A}_{\eta } } \Big{|}^{\rm R} = \pi _{1} {\boldsymbol D}^{{\boldsymbol \eta }} \Big( \frac{1}{1-\widetilde{A}_{\eta } } \otimes \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi }} \Big) \otimes \frac{1}{1-\widetilde{A}_{\eta }} \Big) \Big|^{\rm R}
{\nonumber\\}& = \eta \, \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm R} - { \Big[ \hspace{-1.3mm} \Big[ }\widetilde{A}^{\rm NS}_{\eta } , \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm R} { \Big] \hspace{-1.3mm} \Big] }- { \Big[ \hspace{-1.3mm} \Big[ }\widetilde{A}^{\rm R}_{\eta } , \pi _{1} G \Big( {\boldsymbol \xi }_{{\boldsymbol d}} \frac{1}{1- \widetilde{\Psi } } \Big) \Big{|}^{\rm NS} { \Big] \hspace{-1.3mm} \Big] }{\nonumber\\}\nonumber
& = \eta \widetilde{A}^{\rm R}_{d} - { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm NS}_{\eta } , \widetilde{A}^{\rm R}_{d} { \big] \hspace{-1.1mm} \big] }- { \big[ \hspace{-1.1mm} \big[ }\widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm NS}_{d} { \big] \hspace{-1.1mm} \big] }. \end{aligned}$$ We obtain $(-)^{d} d \widetilde{A}^{\rm R}_{\eta } = D^{\rm NS}_{\eta } \widetilde{A}^{\rm R}_{d} + { [ \hspace{-0.6mm} [ }\widetilde{A}^{\rm R}_{\eta } , \widetilde{A}^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }$, namely, $(-)^{d} d \widetilde{A}_{\eta } = \pi {\boldsymbol D}^{\boldsymbol \eta } \mathchar`- {\rm exact} \,\, {\rm terms}$.
Equivalence of EKS and KO theories
----------------------------------
In section 2, we proposed the Wess-Zumino-Witten-like complete action $$\begin{aligned}
\nonumber
S_{\rm WZW} = \int_{0}^{1} dt \, \langle \mathcal{A}^{\ast }_{t} , Q \mathcal{A}_{\eta } + m_{2}|_{2} ( \mathcal{A}_{\eta } , \mathcal{A}_{\eta } ) \rangle , \end{aligned}$$ where ${\cal A}_{\eta } \equiv {\cal A}^{\rm NS}_{\eta } + {\cal A}^{\rm R}_{\eta }$ and ${\cal A}^{\ast }_{t} \equiv {\cal A}^{\rm NS}_{t} + \xi Y \partial _{t} ( P_{\eta } {\cal A}^{\rm R}_{\eta } ) $.
We found that one realization of this WZW-like complete action is given by setting $$\begin{aligned}
\label{KO}
\mathcal{A}^{\rm NS}_{\eta } : = \big( \eta e^{\Phi ^{\rm NS}} \big) e^{- \Phi ^{\rm NS} } \equiv A^{\rm NS}_{\eta } , \hspace{8mm} \mathcal{A}^{\rm R}_{\eta } : = F \Psi ^{\rm R} \equiv A^{\rm R}_{\eta } , \end{aligned}$$ which is just Kunitomo-Okawa’s action proposed in [@Kunitomo:2015usa]. Another realization of the action, which was proposed in section 3.1 and checked in section 3.2, is given by setting $$\begin{aligned}
\label{EKS}
\mathcal{A}^{\rm NS}_{\eta } : = \pi _{1} \widehat{\bf G} \frac{1}{1- \widetilde{\Psi } }\Big{|}^{\rm NS} \equiv \widetilde{A}^{\rm NS}_{\eta } , \hspace{5mm} \mathcal{A}^{\rm R}_{\eta } : = \pi _{1} \widehat{\bf G} \frac{1}{1- \widetilde{\Psi } }\Big{|}^{\rm R} \equiv \widetilde{A}^{\rm R}_{\eta } , \end{aligned}$$ which reproduces the Ramond equations of motion proposed in [@Erler:2015lya]. Note also that the kinetic terms of (\[S KO\]) and $(\ref{S tilde})$ have the same spectrum. As a result, we obtain the equivalence of two theories proposed in [@Kunitomo:2015usa] and [@Erler:2015lya], which are different parametrizations of (\[WZW-like complete action\]). See also [@Erler:2015rra; @GM2].
In other words, since both (\[KO\]) and (\[EKS\]) have the same WZW-like structure and gives the same WZW-like action (\[WZW-like complete action\]), we can identify $A_{\eta } = A^{\rm NS}_{\eta } + A^{\rm R}_{\eta }$ and $\widetilde{A}_{\eta }= \widetilde{A}^{\rm NS}_{\eta } + \widetilde{A}^{\rm R}_{\eta }$ in the same way as [@Erler:2015rra]. Then, the identification of pure-gauge-fields $$\begin{aligned}
\label{parametrizations}
A_{\eta } = \big( \eta e^{\Phi ^{\rm NS} } \big) e^{- \Phi ^{\rm NS} } + F \Psi ^{\rm R} \cong \pi _{1} \widehat{\bf G} \frac{1}{1- \widetilde{\Psi } }\Big{|}^{\rm NS} + \pi _{1} \widehat{\bf G} \frac{1}{1- \widetilde{\Psi } }\Big{|}^{\rm R} = \widetilde{A}_{\eta } \end{aligned}$$ trivially provides the equivalence of two actions (\[S KO\]) and (\[S tilde\]), namely the equivalence of two theories. Note that (\[parametrizations\]) gives the equivalence of two actions but does not [*directly*]{} give a field redefinition of two theories. A partial gauge fixing $\Phi ^{\rm NS} = \xi \Psi ^{\rm NS}$ is necessitated to relate $(\Psi ^{\rm NS} , \Psi ^{\rm R})$ and $(\widetilde{\Psi }^{\rm NS} , \widetilde{\Psi }^{\rm R})$. See also [@Erler:2015uba; @Erler:2015uoa; @Goto:2015hpa; @GM2]. Similarly, as demonstrated in [@Erler:2015uoa], if we start with $$\begin{aligned}
\label{large redef}
A_{t} & = \big( \partial _{t} e^{\Phi ^{\rm NS}} \big) e^{- \Phi ^{\rm NS}} + F \Xi \Big( \partial _{t} \Psi ^{\rm R} - { \big[ \hspace{-1.1mm} \big[ }\big( \partial _{t} e^{\Phi ^{\rm NS}} \big) e^{-\Phi ^{\rm NS}} , F \Psi ^{\rm R} { \big] \hspace{-1.1mm} \big] }\Big)
{\nonumber\\}& \cong \pi _{1} \widehat{\bf G} \Big( {\boldsymbol \xi}_{t} \frac{1}{1-\widetilde{\Psi }} \Big) \Big{|}^{\rm NS} + \pi _{1} \widehat{\bf G} \Big( {\boldsymbol \xi}_{t} \frac{1}{1-\widetilde{\Psi }} \Big) \Big{|}^{\rm R} = \widetilde{A}_{t} , \end{aligned}$$ we can read a field redefinition of $(\Phi ^{\rm NS} , \Psi ^{\rm R})$ and $(\widetilde{\Phi }^{\rm NS} , \widetilde{\Psi }^{\rm R})$ in the large Hilbert space for NS fields and in the restricted small space for R fields with a trivial up-lift $\widetilde{\Psi }^{\rm NS} = \eta \widetilde{\Phi }^{\rm NS}$. One can check that the same logic used for the NS sector in [@Erler:2015uoa] also goes in the case including the R sector because WZW-like relations exist as we explained. See appendix A.
Conclusion
==========
In this paper, we have clarified a Wess-Zumino-Wtten-like structure including Ramond fields and proposed one systematic way to construct gauge invariant actions, which we call WZW-like complete action. In this framework, once a WZW-like functional $\mathcal{A}_{\eta } = \mathcal{A}_{\eta } [\Psi ]$ of some dynamical string field $\Psi $ is constructed, one obtain one realization of our WZW-like complete action parametrized by $\Psi $. On the basis of this way, we have constructed an action $\widetilde{S}$ whose on-shell condition is equivalent to the Ramond equations of motion proposed in [@Erler:2015lya]. In particular, this action $\widetilde{S}$ and Kunitomo-Okawa’s action proposed in [@Kunitomo:2015usa] just give different parametrizations of the same WZW-like structure and action, which implies the equivalence of two theories [@Kunitomo:2015usa; @Erler:2015lya]. Let us conclude by discussing future directions.
It would be interesting to extend the result of [@Kunitomo:2015usa] to closed superstring field theories[@GKMO]. We expect that our idea of WZW-like structure and action also goes in heterotic and type II theories if the kinetic terms are given by the same form. Then, we need explicit expressions of ${\boldsymbol D}^{\boldsymbol \eta }$ and ${\boldsymbol l}$, where ${\boldsymbol D}^{\boldsymbol \eta }$ is a dual $L_{\infty }$ structure of the original $L_{\infty }$ products $\widetilde{\bf L}=\widehat{\bf G}^{-1} \, {\boldsymbol l} \, \widehat{\bf G}$ given in [@Erler:2015lya]. NS and NS-NS parts of these dual $A_{\infty }$/$L_{\infty }$ structures are discussed in [@GM2].
We would have to quantize the (WZW-like) complete action and clarify its relation with supermoduli of super-Riemann surfaces[@Verlinde:1987sd; @D'Hoker:1988ta; @Saroja:1992vw; @Belopolsky:1997jz; @Witten:2012bh] to obtain a better understandings of superstrings from recent developments in field theoretical approach. The Batalin-Vilkovisky formalism[@Batalin:1981jr; @Batalin:1984jr] is one helpful way to tackle these problems: A quantum master action is necessitated. As a first step, it is important to clarify whether we can obtain an $A_{\infty }$-morphism $\widehat{\bf G}$ which has the cyclic property consistent with the $XY$-projection. If it is possible, the resultant action would have an $A_{\infty }$ form and then the classical Batalin-Vilkovisky quantization is straightforward. A positive answer would be provided in upcoming work [@EOT2] for open superstring field theory without stubs. It would also be helpful to clarify more detailed relations between recent important developments.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author would like to thank Keiyu Goto, Hiroshi Kunitomo, and Yuji Okawa for comments. The author also would like to express his gratitude to his doctors, nurses, and all the staffs of University Hospital, Kyoto Prefectural University of Medicine, for medical treatments and care during his long hospitalization. This work was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Basic facts and some identities
===============================
We summarize important properties of the BPZ inner product and give proofs of some relations which we skipped in the text.
The BPZ inner product $\langle A , B \rangle $ in the large Hilbert space of any $A,B \in \mathcal{H}$ has the following properties with the BRST operator $Q$ and the Witten’s star product $m_{2}$: $$\begin{aligned}
\langle A , B \rangle & = (-)^{AB} \langle B , A \rangle ,
{\nonumber\\}\langle A , Q B \rangle & = - (-)^{A} \langle Q A , B \rangle ,
{\nonumber\\}\nonumber
\langle A , m_{2} ( B , C ) \rangle & = (-)^{A(B+C)} \langle B , m_{2} ( C ,A ) \rangle . \end{aligned}$$ Note also that with a projector onto $\eta $-exact states $P_{\eta }$ and $P_{\xi } = 1 - P_{\eta }$ and the zero mode of $\eta \equiv \eta _{0}$ of $\eta (z)$-current, the BPZ inner product satisfies $$\begin{aligned}
\nonumber
\langle P_{\eta} A , B \rangle &= \langle A , P_{\xi } B \rangle ,
{\nonumber\\}\nonumber
\langle A , \eta B \rangle &= (-)^{A} \langle \eta A , B \rangle . \end{aligned}$$ Similarly, for any states in the restricted small space $A^{\rm R}, B^{\rm R} \in \mathcal{H}_{R}$, the bilinear $\langle \xi Y A^{\rm R} ,B^{\rm R} \rangle $ has the following properties: $$\begin{aligned}
\langle \xi Y A^{\rm R} , Q B^{\rm R} \rangle & = (-)^{AB} \langle \xi Y B^{\rm R} , A^{\rm R} \rangle ,
{\nonumber\\}\nonumber
\langle \xi Y A ^{\rm R} , Q B^{\rm R} \rangle & = - (-)^{A} \langle \xi Y Q A^{\rm R} , B^{\rm R} \rangle . \end{aligned}$$
In the work of [@Kunitomo:2015usa], for any state $B\in \mathcal{H}$, the linear map $F$ is defined by $$\begin{aligned}
\nonumber
F B \equiv \sum_{n=0}^{\infty } \big( \Xi { \big[ \hspace{-1.1mm} \big[ }A^{\rm NS}_{\eta } , \hspace{3mm} { \big] \hspace{-1.1mm} \big] }\big) ^{n} B = \frac{1}{1 - \Xi (\eta - D^{\rm NS}_{\eta } )} B , \end{aligned}$$ where $D^{\rm NS}_{\eta } B = \eta B - { [ \hspace{-0.6mm} [ }A^{\rm NS}_{\eta } , B { ] \hspace{-0.6mm} ] }$ and $(D^{\rm NS}_{\eta })^{2} = 0$. Thus, its inverse is given by $F^{-1} = \eta \Xi + \Xi D^{\rm NS}_{\eta }$, which provides $\eta F^{-1} = F^{-1} D^{\rm NS}_{\eta }$ or equivalently, $F \eta F^{-1} = D^{\rm NS}_{\eta }$, and thus ${ ] \hspace{-0.6mm} ] }D^{\rm NS}_{\eta } , F \Xi { ] \hspace{-0.6mm} ] }= 1$. Then, we find that $A^{\rm R}_{\eta } \equiv F \Psi $ satisfies $D^{\rm NS}_{\eta } A^{\rm R}_{\eta } = 0$ as follows: $$\begin{aligned}
\nonumber
A^{\rm R}_{\eta } \equiv F \Psi ^{\rm R} = F \eta \xi \Psi ^{\rm R} = D^{\rm NS}_{\eta } F \Xi \Psi ^{\rm R} = D^{\rm NS}_{\eta } F \Xi A^{\rm R}_{\eta } . \end{aligned}$$ With $F^{-1} = \eta \Xi + \Xi D^{\rm NS}_{\eta }$ and ${ [ \hspace{-0.6mm} [ }d , \eta { ] \hspace{-0.6mm} ] }= 0$, a derivation $d$ acts on the state $F\Psi $ as $$\begin{aligned}
d ( F \Psi ) & = F ( d \Psi ) - F { [ \hspace{-0.6mm} [ }d , F^{-1} { ] \hspace{-0.6mm} ] }F \Psi
{\nonumber\\}& = F ( d \Psi ) - F { [ \hspace{-0.6mm} [ }d , \eta \Xi { ] \hspace{-0.6mm} ] }F \Psi - F { [ \hspace{-0.6mm} [ }d , \Xi D^{\rm NS}_{\eta } { ] \hspace{-0.6mm} ] }F \Psi
{\nonumber\\}& = F ( d \Psi ) - (-)^{d} F \big( \eta { [ \hspace{-0.6mm} [ }d , \Xi { ] \hspace{-0.6mm} ] }+ (-)^{d} { [ \hspace{-0.6mm} [ }d , \Xi { ] \hspace{-0.6mm} ] }D^{\rm NS}_{\eta } \big) F \Psi + F \Xi D^{\rm NS}_{\eta } { [ \hspace{-0.6mm} [ }A^{\rm NS}_{d} , F \Psi { ] \hspace{-0.6mm} ] }{\nonumber\\}\nonumber
& = D^{\rm NS}_{\eta } F \Xi \big( d \Psi - { [ \hspace{-0.6mm} [ }A^{\rm NS}_{d} , F \Psi { ] \hspace{-0.6mm} ] }- (-)^{d} \eta { [ \hspace{-0.6mm} [ }d , \Xi { ] \hspace{-0.6mm} ] }F \Psi \big) + { [ \hspace{-0.6mm} [ }A^{\rm NS}_{d} , F \Psi { ] \hspace{-0.6mm} ] }. \end{aligned}$$ We thus obtain $$\begin{aligned}
\nonumber
(-)^{d} d A^{\rm R}_{\eta } = D^{\rm NS}_{\eta } A^{\rm R}_{d} - { [ \hspace{-0.6mm} [ }A^{\rm R}_{\eta } , A^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }\end{aligned}$$ where $A^{\rm R}_{\eta } \equiv F \Psi $ is an R pure-gauge-like field and an R associated field $A^{\rm R}_{d}$ is defined by $$\begin{aligned}
\nonumber
A^{\rm R}_{d} & \equiv F \Xi \Big( (-)^{d} d \Psi + { [ \hspace{-0.6mm} [ }A^{\rm R}_{\eta } , A^{\rm NS}_{d} { ] \hspace{-0.6mm} ] }+ \eta { [ \hspace{-0.6mm} [ }d , \Xi { ] \hspace{-0.6mm} ] }A^{\rm R}_{\eta } \Big) . \end{aligned}$$
The original form of Kunitomo-Okawa’s complete action is $$\begin{aligned}
\nonumber
S = - \frac{1}{2} \langle \xi Y \Psi , Q \Psi \rangle - \int_{0}^{1} dt \, \langle A^{\rm NS}_{t} , Q A^{\rm NS}_{\eta } + m_{2} ( A^{\rm R}_{\eta } , A^{\rm R}_{\eta } ) \rangle , \end{aligned}$$ where $A^{\rm R}_{\eta } = F(t) \Psi ^{\rm R}$. Note that $F(t)$ satisfies $F(t=0)=0$ and $F(t=1) = F$. Using the cyclic property of the star product $m_{2}$, $A^{\rm R} = D^{\rm NS}_{\eta } F \Xi A^{\rm R}_{\eta }$, and ${ [ \hspace{-0.6mm} [ }D^{\rm NS}_{\eta } , F \Xi { ] \hspace{-0.6mm} ] }=1$, we find $$\begin{aligned}
- \int_{0}^{1} dt \, \langle A^{\rm NS}_{t} , m_{2} ( A^{\rm R}_{\eta } , A^{\rm R}_{\eta } ) \rangle
& = \frac{1}{2} \int_{0}^{1} dt \, \langle A^{\rm R}_{\eta } , { [ \hspace{-0.6mm} [ }A^{\rm NS}_{t} , A^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
= \frac{1}{2} \int_{0}^{1} dt \, \langle D^{\rm NS}_{\eta } F \Xi \Psi , { [ \hspace{-0.6mm} [ }A^{\rm NS}_{t} , A^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
{\nonumber\\}& = \frac{1}{2} \int_{0}^{1} dt \, \langle \Psi , F \Xi D^{\rm NS}_{\eta } { [ \hspace{-0.6mm} [ }A^{\rm NS}_{t} , A^{\rm R}_{\eta } { ] \hspace{-0.6mm} ] }\rangle
= \frac{1}{2} \int_{0}^{1} dt \, \langle \Psi , \partial _{t} ( F \Psi ) \rangle
{\nonumber\\}\nonumber
& = \frac{1}{2} \langle \Psi , F \Psi \rangle
= - \frac{1}{2} \langle \xi Y \Psi , \eta X F \Psi \rangle . \end{aligned}$$ Note also that $XY \Psi ^{\rm R}= \Psi ^{\rm }$ and $\eta \xi \Psi = \Psi ^{\rm R}$. Hence, we find that $$\begin{aligned}
\nonumber
- \frac{1}{2} \langle \xi Y \Psi , Q \Psi \rangle - \int_{0}^{1} dt \, \langle A^{\rm NS}_{t} , m _{2} ( F \Psi , F \Psi ) \rangle
& = - \frac{1}{2} \langle \xi Y \Psi , \, Q \Psi + \eta X F \Psi \rangle
= - \frac{1}{2} \langle \xi Y \Psi , Q F \Psi \rangle . \end{aligned}$$ As we explained in section 1.1, this is equal to (\[S KO\]).
We check that the identification $A_{t} = A^{\rm NS}_{t}+ A^{\rm R}_{t}$ and $\widetilde{A}_{t}=\widetilde{A}^{\rm NS}_{t} + \widetilde{A}^{\rm R}_{t}$ provide a field redefinition of $(\Phi ^{\rm NS} , \Psi ^{\rm R})$ and $(\widetilde{\Phi }^{\rm NS} , \widetilde{\Psi }^{\rm R})$ with $\widetilde{\Psi }^{\rm NS} = \eta \widetilde{\Phi }^{\rm NS}$. We start is $A_{t} - \widetilde{A}_{t} = 0$. Then the relation $\eta ( A_{t} - \widetilde{A}_{t} ) = 0$ automatically holds. Recall that we have WZW-like relation $\partial _{t} A_{\eta } = D_{\eta } A_{t}$ and $\partial _{t} \widetilde{A}_{\eta } = \widetilde{D}_{\eta } A_{t}$ where $D_{\eta } B = \eta B - m_{2}|_{0} ( A_{\eta } , B) - (-)^{A_{\eta } B} m_{2}|_{0} ( B , A_{\eta })$. Therefore, using these WZW-like relations and the identification $A_{t} = \widetilde{A}_{t}$, one can rewrite $\eta (A_{t} - \widetilde{A}_{t})=0$ as $$\begin{aligned}
\label{Eq}
\partial _{t} (A_{\eta } - \widetilde{A}_{\eta } ) = m_{2}|_{0} \big( A_{\eta } -\widetilde{A}_{\eta } , A_{t} \big) - m_{2}|_{0} \big( A_{t} , A_{\eta } -\widetilde{A} \big) . \end{aligned}$$ For brevity, we define ${\cal I}^{\rm NS} (t) \equiv A^{\rm NS}_{\eta } (t) - \widetilde{A}^{\rm NS}_{\eta } (t)$ and ${\cal I}^{\rm R} (t) \equiv A^{\rm R}_{\eta } (t) - \widetilde{A}^{\rm R}_{\eta } (t)$. Note that the $t=0$ values $A_{\eta } (t=0) = \widetilde{A}_{\eta } (t=0) = 0$ gives the initial conditions $\mathcal{I}^{\rm NS} (t=0)=0$ and $\mathcal{I}^{\rm R}(t=0)=0$. Then, the NS output state and R output state of (\[Eq\]) can be separated as $$\begin{aligned}
\label{Eq NS}
{\rm NS} \, : & \hspace{15mm} \frac{\partial }{\partial t} \, \mathcal{I}^{\rm NS} (t) = { \big[ \hspace{-1.1mm} \big[ }\mathcal{I}^{\rm NS} (t) , \, A^{\rm NS}_{t} (t) { \big] \hspace{-1.1mm} \big] },
\\ \label{Eq R}
{\rm R} \, : & \hspace{5mm} \frac{\partial }{\partial t} \, \mathcal{I}^{\rm R} (t) = { \big[ \hspace{-1.1mm} \big[ }\mathcal{I}^{\rm R} (t), \, A^{\rm NS}_{t} (t) { \big] \hspace{-1.1mm} \big] }+ { \big[ \hspace{-1.1mm} \big[ }\mathcal{I}^{\rm NS} (t) , \, A^{\rm R}_{t} (t) { \big] \hspace{-1.1mm} \big] }. \end{aligned}$$ The initial condition $\mathcal{I}^{\rm NS}(t=0) = 0$ provides the solution $\mathcal{I}^{\rm NS} (t) = 0$ of the differential equation (\[Eq NS\]), which means $A_{\eta }^{\rm NS} = \widetilde{A}^{\rm NS}_{\eta }$. Then, the R output equation (\[Eq R\]) reduces to $$\begin{aligned}
\nonumber
{\rm R} \, : & \hspace{5mm} \frac{\partial }{\partial t} \, \mathcal{I}^{\rm R} (t) = { \big[ \hspace{-1.1mm} \big[ }\mathcal{I}^{\rm R} (t), \, A^{\rm NS}_{t} (t) { \big] \hspace{-1.1mm} \big] }\end{aligned}$$ and the initial condition $\mathcal{I}^{\rm R}(t = 0) = 0$ also provides the solution $\mathcal{I}^{\rm R} (t) = 0$, which implies $A^{\rm R}_{\eta } = \widetilde{A}^{\rm R}_{\eta }$. As a result, under the identification $A_{t} \cong \widetilde{A}_{t}$, we obtain $A_{\eta } = \widetilde{A}_{\eta }$.
Under the identification $A_{t} \cong \widetilde{A}_{t}$, we obtained $A_{\eta } = \widetilde{A}_{\eta}$, which provides $$\begin{aligned}
\label{Rel}
\frac{1}{1 - A_{\eta }} \otimes A_{t} \otimes \frac{1}{1-A_{\eta }} = \widehat{\bf G} \frac{1}{1-\widetilde{\Psi } } \otimes {\boldsymbol \xi }_{t} \widetilde{\Psi } \otimes \frac{1}{1-\widetilde{\Psi }} , \end{aligned}$$ where $\widetilde{\Psi } = \eta \widetilde{\Phi }^{\rm NS} + \widetilde{\Psi }^{\rm R}$ and ${\boldsymbol \xi}_{t} = \xi \partial _{t}$. One can read the NS and R outputs of (\[Rel\]) as $$\begin{aligned}
{\rm NS} \, : & \hspace{5mm} \widetilde{\Phi }^{\rm NS} = \pi _{1} \, \int_{0}^{1} dt \, \Big( \widehat{\bf G}^{-1} \frac{1}{1 - A_{\eta }(t)} \otimes A_{t} (t) \otimes \frac{1}{1-A_{\eta } (t) } \Big) \Big{|}^{\rm NS} ,
\\
{\rm R} \, : & \hspace{5mm} \widetilde{\Psi }^{\rm R} = \pi _{1} \, {\boldsymbol \eta } \, \int_{0}^{1} dt \, \Big( \widehat{\bf G}^{-1} \frac{1}{1 - A_{\eta }(t) } \otimes A_{t} (t) \otimes \frac{1}{1-A_{\eta }(t)} \Big) \Big{|}^{\rm R} . \end{aligned}$$
[99]{}
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Work in progress.
T. Erler, Y. Okawa, and T. Takezaki, To appear.
[^1]: [email protected]
[^2]: Note that the starting NS action is that of $\mathbb{Z}_{2}$-reversed theory and has gauge invariance with $(\delta e^{\Phi })e^{-\Phi } = \eta \Omega - { [ \hspace{-0.6mm} [ }(\eta e^{\Phi }) e^{-\Phi } , \Omega { ] \hspace{-0.6mm} ] }+ Q \Lambda $, which is constructed from not a equation of bosonic pure gauge[@Berkovits:2004xh] but a equation of $\eta $-constraint. See also [@GM2] for these $\mathbb{Z}_{2}$-duals for open superstrings with stubs and closed superstrings.
[^3]: See [@Berkovits:2001im; @Michishita:2004by; @Kunitomo:2013mqa; @Kunitomo:2014hba; @Kunitomo:2014qla] for other fascinating approaches using Ramond fields in the large Hilbert space.
[^4]: These satisfy $P_{\eta } ^{2} = P_{\eta }$, $P_{\xi }^{2} = P_{\xi }$, $P_{\eta } P_{\xi } = P_{\xi } P_{\eta }=0$,and $P_{\eta } + P_{\xi } = 1$ on $\mathcal{H}$.
|
---
author:
- |
$^1$, V. Debattista$^3$, O. Agertz$^1$, L. Mayer$^{1}$, A. M. Brooks$^4$, F. Governato$^3$ and G. Lake$^1$\
$^1$Institute for Theoretical Physics, University of Zurich, Winterthurerstrasse 190 8047\
$^2$RCUK Fellow; Centre For Astrophysics, University of Central Lancashire, Preston, PR1 2HE\
$^3$Astronomy Dept., University of Washington, Box 351580, Seattle WA 98195-1580\
$^4$California Institute of Technology, M/C 130-33, Pasadena, CA 91125\
E-mail:
bibliography:
- 'refs.bib'
title: A dark matter disc in the Milky Way
---
Introduction {#sec:intro}
============
The case for dark matter in the Universe is based on a wide range of observational data, from galaxy rotation curves and gravitational lensing, to the Cosmic Microwave Background Radiation [@1996PhR...267..195J]; [@2006ApJ...648L.109C]. Of the many plausible dark matter candidates in extensions to the Standard Model, Weakly Interacting Massive Particles (WIMPs) stand out as well-motivated and detectable [@1996PhR...267..195J], giving rise to many experiments designed to detect WIMPs in the lab. Predicting the flux of dark matter particles through the Earth is key to the success of such experiments, both to motivate detector design, and for the interpretation of any future signal [@1996PhR...267..195J].
Previous work has modelled the phase space density distribution of dark matter at solar neighbourhood using cosmological simulations that model the dark matter alone (see e.g. [@2008arXiv0808.2981S], [@2008arXiv0809.0898S]). Here we make the first attempt to include the baryonic matter – the stars and gas that make up the Milky Way. The Milky Way stellar disc presently dominates the mass interior to the solar radius and likely did so also in the early Universe at redshift $z=1$, when the mean merger rate in a $\Lambda$CDM cosmology peaks [@2008MNRAS.389.1041R]. The stellar and gas disc is important because it biases the accretion of satellites, causing them to be dragged towards the disc plane where they are torn apart by tides. The material from these accreted satellites settles into a thick disc of stars and dark matter [@1989AJ.....98.1554L]. In this work, we quantify the expected properties of this dark disc. Its implications for direct detection experiments and the capture of WIMPs in the Sun and Earth are presented in the contribution from T. Bruch, this volume and in [@2008arXiv0804.2896B].
We use two different approaches. In the first approach (§\[sec:ap1\]), we use dark matter only simulations to estimate the expected merger history of a Milky Way mass galaxy, and then add a stellar disc to measure its effect. This work is presented in detail already in [@2008MNRAS.389.1041R]. In the second approach (§\[sec:ap2\]), we use cosmological hydrodynamic simulations of the Milky Way to hunt for dark discs. Both approaches are complementary in quantifying the expected properties of the dark disc. The former allows us to specify precisely the properties of the Milky Way disc at high redshift; the latter is fully self-consistent.
Approach \#1: adding a stellar disc to cosmological dark matter simulations {#sec:ap1}
===========================================================================
We used a cosmological dark matter only simulation already presented in [@2005MNRAS.364..367D] to estimate the frequency of satellite-disc encounters for a typical Milky Way galaxy; further details are given in [@2008MNRAS.389.1041R]. From the simulation volume, we extracted four Milky Way sized halos at a mass resolution of $m_p=5.7\times 10^5 M_\odot$. The subhalos inside each ‘Milky Way’ and at each redshift output were identified using the algorithm in [@2004MNRAS.351..399G] and then traced back in time to their progenitor halos as detailed in [@2008MNRAS.389.1041R].
From our sample of four Milky Way mass halos and assuming that mergers are isotropic (which we must do since we do not know how the disc should align with the dark matter halo), we find that a typical Milky Way sized halo will have 1($\pm 1$) subhalo merge within $\theta<20^\mathrm{o}$ of the disc plane with $\vmaxd > 80$km/s; $2(\pm 1)$ with $\vmaxd > 60$; and $5(\pm1)$ with $\vmaxd > 40$km/s. Away from the disc plane there will be twice as may mergers at the same mass [@2008MNRAS.389.1041R]. It is important to stress that these numbers come from the distribution of fully disrupted subhalos, not the surviving distribution that is significantly less damaging.
We then estimated the effect of a stellar disc on these mergers by running isolated disc-merger simulations. We set up our Milky Way (MW) model (disc+halo system) by adiabatically growing a disc inside a spherical halo, as detailed in [@2008MNRAS.389.1041R]. We chose three models for our satellite, but present just two here: LMC with $\vmaxd=60$km/s, and LLMC with $\vmaxd=80$km/s; these were set up as scaled versions of our MW model. We chose a wide range of initial inclination angles to the disc from $\theta = 10-60^\mathrm{o}$, one retrograde orbit, and range of pericentres and apocentres. The simulations were evolved using the collisionless tree-code, PkdGRAV [@2001PhDT........21S]. The final evolved systems were mass and momentum centred using the ‘shrinking sphere’ method described in [@2006MNRAS.tmp..153R], and rotated into their moment of inertia eigenframe with the $z$ axis perpendicular to the disc.
The results are shown in Figure \[fig:ddresults\]. The left panel shows the accreted stars (red) and dark matter (blue) at the end of a simulation where the LMC satellite merged at $\theta = 10^\mathrm{o}$ to the disc. Both the stars and the dark matter have settled into accreted discs. The middle panel shows the corresponding velocity distribution in $v_\phi$ (rotation velocity) at the solar neighbourhood (a cylinder $8<R<9$kpc, $|z|<0.35$kpc). The underlying dark matter halo is shown in green and is not rotating; the accreted stars and dark matter (red and blue) have kinematics similar to that of the underlying stellar disc (black). The right panel shows the dark matter disc to dark matter halo density ratio $\rhodd/\rhoh$ as a function of height above the disc plane for selected merger simulations, as marked. As the satellite impact angle $\theta$ is increased, the satellite contributes less material to a dark disc. For $\theta = 40^\mathrm{o}$, the density at the solar neighbourhood is nearly flat with $z$ and less than a tenth of the underlying halo density; there is correspondingly less rotation in this simulation. Summing over the expected number and mass of mergers, we find that the dark disc contributes $\sim 0.25 - 1$ times the non-rotating halo density at the solar position [@2008MNRAS.389.1041R]. It is important to stress that all satellites regardless of their initial inclination have some accreted material that is focused into the disc plane (see Figure \[fig:ddresults\]), right panel. As such, we expect that the accreted dark and stellar discs will comprise several accreted satellites; the most massive low-inclination mergers being the most important contributors.
The accreted stellar disc shares similar kinematics to the dark disc. Depending on assumptions about the mass to light ratio of accreted satellites, these accreted stars can make up $\sim 10 - 50$% of the Milky Way stellar thick disc [@2008MNRAS.389.1041R]. (The lower end of this range is more likely given the observed properties of satellite galaxies in the Universe today.) If future surveys of our Galaxy can disentangle accreted stars in the Milky Way thick disc from those that formed in-situ, then we will be able to infer the velocity distribution function of the dark disc from these stars.
Approach \#2: cosmological hydrodynamic simulations {#sec:ap2}
===================================================
We use three cosmological hydrodynamic simulations of Milky Way mass galaxies, two of which (,) have already been presented in [@2008arXiv0801.3845M] and [@2008ASPC..396..453G]. All three were run with the GASOLINE code [@2004NewA....9..137W] using the “blastwave feedback” described in [@2006MNRAS.373.1074S]. had cosmological parameters $(\Omega_{\rm m},\Omega_{\Lambda},\sigma_8,h)=(0.3,0.7,0.9,0.7)$; and used $(\Omega_{\rm m},\Omega_{\Lambda},\sigma_8,h)=(0.24,0.76,0.77,0.73)$. At redshift $z=0$ the typical particle masses for dark matter stars and gas were: $(M_{dm},M_*,M_{gas}) = (7.6,0.2,0.3)\times 10^5\,$M$_\odot$, with associated force softening: $(\epsilon_{dm},\epsilon_*,\epsilon_{gas}) = (0.3,0.3,0.3)\,$kpc. The analysis was performed as in §\[sec:ap1\].
\
Figure \[fig:cosmo\] shows the distribution of rotational velocities at the solar neighbourhood (top panels), and the orbital decay of the four most massive satellites (bottom panels) for , and . The right most panels (d) and (h) show the results for the galaxy simulated [*without*]{} any gas or stars – .
Notice that, with the exception of , in all cases the dark matter requires a double Gaussian fit to its local $v_\phi$ velocity distribution. Once the baryons (the stars and gas) are included in the simulation, there is a local dark matter disc that lags the rotation of the thin stellar disc by $\sim 50-150$km/s. The mass and rotation speed of the dark disc increase for the simulations that have more late mergers. has no significant mergers after redshift $z=2$ and has a less significant dark disc, with rotation lag $\sim150$km/s and dark disc to non-rotating dark halo density ratio $\rhodd/\rhoh = 0.23$ (obtained from the double Gaussian fit). and both have extreme dark discs with $\rhodd/\rhoh > 1$ and rotation lag $\simlt 60$km/s; they both have massive mergers at redshift $z<1$.
Conclusions {#sec:conclusions}
===========
Low inclination massive satellite mergers are expected in a $\Lambda$CDM cosmology. These lead to the formation of thick accreted dark and stellar discs. We used two different approaches to estimate the importance of the dark disc. Firstly, we used dark matter only simulations to estimate the expected merger history for a Milky Way mass galaxy, and then added a thin stellar disc to measure its effect. Secondly, we used three cosmological hydrodynamic simulations of Milky Way mass galaxies. In both cases, we found that a typical Milky Way mass galaxy will have a dark disc that contributes $\sim 0.25 - 1$ times the non-rotating halo density at the solar position. The dark disc has important implications for the direct detection of dark matter [@2008arXiv0804.2896B] and for the capture of WIMPs in the Sun and Earth (see contribution from T. Bruch, this volume).
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YITP-19-09
[Pawel Caputa$^1$, Shouvik Datta$^2$ and Vasudev Shyam$^3$]{}\
\
$^3$ *Perimeter Institute for Theoretical Physics\
31 N. Caroline St. Waterloo, ON, N2L 2Y5, Canada.*\
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Introduction
============
Our understanding of quantum gravity has been dramatically advanced by the AdS/CFT correspondence. In a sense, it provides a precise framework to tackle the gravity path integral by formulating it non-perturbatively in terms of a quantum field theory. As we still grapple with several challenges in black hole physics and cosmology, we require to develop newer tools for calculating various observables and engineer mechanisms to adapt holography to more general settings.
A question of fundamental importance is how can we formulate quantum gravity with some specified boundary conditions and can holography turn out to be useful in this context. A situation where we are posed with this problem appears is in the evolution of a closed universe, wherein the wavefunction of interest is calculated with fixed boundary conditions of an initial state [@Hartle:1983ai]. For specified spatial boundary conditions, there has been some progress from studies of the holographic renormalization group [@deBoer:1999tgo; @deHaro:2000vlm]. Here, attempts were made to interpret the radial cut-off as a Wilsonian cut-off in field theory whereby integrating out bulk geometry corresponds to integrating out high energy modes of the field theory [@Faulkner:2010jy; @Heemskerk:2010hk]. Identifying the radial cut-off in the bulk with a short-distance cut-off seems to work more naturally in this vein [@lecholrg].
Another procedure for implementing such a regularization is to deform the holographic CFT by some irrelevant operator, the scale associated to which is treated as the cut-off. The important question is then to pinpoint this operator in holographic CFTs. Along these lines, over the last couple of years a novel viewpoint has emerged via the $\ttb$ deformation of QFTs. Discovered first in two dimensions, the $\ttb$ operator induces a solvable, irrelevant, double-trace deformation of QFTs [@Zamolodchikov:2004ce; @Smirnov:2016lqw; @Cavaglia:2016oda]. The one-parameter family of theories parametrized by the $\ttb$ coupling has a number of special properties. Deforming an integrable theory by $\ttb$ preserves integrability. The deformed partition function obeys diffusion-like equations [@Cardy:2018sdv] and is modular invariant in a unique sense [@Datta:2018thy; @Aharony:2018bad]. The finite-size spectrum of this theory is exactly the same as Jackiw-Teitelbohm gravity coupled to the undeformed ‘matter’ [@Dubovsky:2018bmo].
The $\ttb$ deformation, for large-$c$ CFTs, has been proposed to be holographically dual to AdS$_3$ with a finite radial cut-off with Dirichlet boundary conditions [@McGough:2016lol]. Within the pure gravity sector, this geometric notion does reproduce some characteristics of the deformed CFT for a specific sign of the coupling [@Kraus:2018xrn]. Some further tests of this cut-off gravity/$\ttb$ relation include the finite-size spectrum, signal propagation velocities, stress tensor correlators [@Aharony:2018vux; @Kraus:2018xrn], entanglement entropy [@Donnelly:2018bef; @Chen:2018eqk; @Gorbenko:2018oov]. Interestingly, among other observations, [@McGough:2016lol] noticed that the flow equation for the trace of the energy momentum tensor in $T\bar{T}$ deformed theory can be re-written as the Hamiltonian constraint in bulk gravity theory on $AdS_3$ spacetime. A different version of $AdS_3$ holography in this context has been put forward in [@Giveon:2017myj; @Giveon:2017nie]. A Lorenz breaking cousin of this deformation and its holographic interpretation has been proposed in [@Guica:2017lia]. An application to de-Sitter holography was studied in [@Gorbenko:2018oov].
The holographic construction with a cut-off has also been generalized to higher dimensions in [@Taylor:2018xcy; @Hartman:2018tkw] (see also [@Chang:2018dge] for a supersymmetric generalization to higher dimensions). The guiding principle behind these works was to define holographic $TT$-deformed theories via a flow equation that originates from the Gauss-Codazzi equation the Hamiltonian constraint. At large $N$, this flow can indeed be seen as coming from the deforming operators that are quadratic in the energy-momentum tensor (see also [@Cardy:2018sdv] for generalization in the form of $\det T$). In [@Taylor:2018xcy; @Hartman:2018tkw] it was shown that such a procedure is consistent with finite cut-off holography through agreements of the quasi-local energy, speed of sound as well as simple correlators within the pure gravity sector. It was also shown in [@Shyam:2018sro] that for $d=4$ various results of holographic RG, such as the gradient form of the metric beta functions, are also captured by such irrelevant double trace deformations involving the stress tensor and other suitably defined deforming operators.
In this work, we study the sphere partition functions of $TT$ deformed CFTs in $d\geq 2$. The sphere partition function $Z_{\mathbb{S}^d}$ plays an important role in a wide variety of aspects. The QFT on the sphere is free from IR divergences and for several supersymmetric theories it has been computed exactly by localization techniques [@Marino:2011eh; @Fuji:2011km; @Pestun:2007rz; @Pestun:2016zxk]. These have led to many precision tests of holography and a better understanding of RG flows. For CFTs in even dimensions, it captures the anomalies. When anomalies are absent, for instance in 3 dimensions, $F=- \log |Z_{\mathbb{S}^3}|$ serves as an analogue to the central charge in counting the degrees of freedom; $F$ decreases along the RG flow from the UV to the IR [@Jafferis:2011zi; @Klebanov:2011gs]. Although the $F$-theorem is different in flavour from the even dimensional analogues (the $c$ and $a$ theorems), a unified formulation can be achieved by considering the sphere free energy. This quantity also has other uses. In odd dimensional CFTs, the finite piece of the sphere free energy $F_{d}$ measures the entanglement entropy across a spherical region $\mathbb{S}^{d-2}$ in flat spacetime $\mathbb{R}^{d-1,1}$.
The spectrum of $TT$ deformed CFTs can be computed from the flow equation. The flow equation relies on special factorization properties of the $TT$-operator [@Zamolodchikov:2004ce] which are expected to hold true for large $N$ theories in higher dimensions. For the case of the cylinder $\mathbb{S}^{d-1}\times \mathbb{R}^1$, this takes the form of the Burger’s equation in hydrodynamics. For the sphere on the other hand, owing to its maximal symmetry, the flow equation can be reduced to an algebraic equation and be solved exactly. This allows us to evaluate the sphere free energy/partition function from the field theory side in a very simple manner. In the cut-off AdS bulk, we calculate the on-shell action with the necessary counterterms [@Emparan:1999pm] and observe a precise agreement with the field theory analysis.
Since the holographic flow equation was deduced in [@Hartman:2018tkw] from re-writing the bulk Gauss-Codazzi equation in terms of the holographic stress tensor, one may wonder how universal or generic is this flow. Proving this equation starting from the definition of $TT$ operators in higher dimensions and on a curved background in ABJM or $\mathcal{N}=4$ SYM is beyond the scope of this work and remains an important future problem. However, we address the issue of how one can obtain such a universal flow equation from purely field theoretic considerations, namely starting already from a local Callan-Symanzik equation describing a CFT deformed only by an irrelevant $TT$ operator.
The Wheeler-DeWitt equation is a quantum constraint equation in a theory of quantum gravity that encodes the independence of the theory under choice of a foliation of space-time by co-dimension one hypersurfaces. Such a foliation is typically chosen in order to pass to the Hamiltonian formalism, as introduced in [@ADM]. The diffeomorphism invariance of the gravitational theory translates into the independence under the choice of foliation, and is thereby encoded in a constraint. In our context, the relevant foliation of the bulk is by successive finite radius cut-off surfaces. The $TT$ flow equation is then mapped to the semiclassical limit of the bulk Wheeler-DeWitt equation. In this work, we solve the mini-superspace Wheeler-DeWitt equation in the WKB approximation and find the Wheeler-DeWitt wavefunction that, indeed, up to holographic counterterms, matches our partition functions.
This paper is organized as follows. In section \[sec:holography\], we compute holographic stress-tensor and partition function in Euclidean anti-de Sitter geometries with spherical boundary at finite radial cut-off up to 6 dimensions. In section \[sec:field-theory\], we review the $TT$ deformation in field theory and consider the flow equation on the sphere and its solution. This allows us to obtain the sphere partition function and find an exact agreement with the gravity results, at large $N$. We provide field theoretical derivation of the flow equation starting from the local Callan–Symanzik equation and the regularization procedure of the $TT$ operator in section \[sec:FlowRegularization\]. In Section \[sec:Wheeler-DeWitt\], we discuss the (mini-superspace) Wheeler-DeWitt equation and its solution in the WKB approximation that captures our partition functions up to holographic counterterms. Finally, we conclude and pose some open problems in section \[sec:conclusions\]. Appendix \[sec:Gauss-Codazzi\] contains a review of the Gauss-Codazzi equation as a flow equation. Appendix \[sec:generateR\] has some details on the field theory derivation of the flow.
Finite cut-off holography {#sec:holography}
=========================
We begin by computing the holographic energy-momentum tensor as well as the sphere partition functions up to $d=6$ dimensions using AdS/CFT with a finite cut-off. In standard holographic computations, at large $N$ and strong ’t Hooft coupling, both quantities are related to the bulk gravity action evaluated on Euclidean AdS solution in one higher ($d+1$) dimension[^1].
More precisely, we consider regularized gravity action given by the Einstein-Hilbert (EH) and Gibbons-Hawking (GH) terms supplemented by local counterterms \[on-shell-init\] I\^[(d+1)]{}\_[on-shell]{}=-\_d\^[d+1]{}x(R-2)+\_d\^dx K+S\_[ct]{}, where the counterterm action, up to $d=6$, takes the form [@Emparan:1999pm] \[counterterm-action\] S\_[ct]{}=\_d\^dx, where $\kappa^2\equiv 8\pi G_N$, $\tilde{R}$ and $\tilde{R}_{ij}$ are the Ricci scalar and Ricci tensor of the cut-off surface. To keep track of different contributions, we introduce $c^{(1)}_d=1$ for $d\ge 2$, $c^{(2)}_d=1$ for $d\ge 3$ and $c^{(3)}_d=1$ is non-zero from $d\ge 5$.
We now consider a Euclidean AdS solution ds\^2=+r\^2d\^2\_d+\_[ij]{}(r,x)dx\^idx\^j, \[metric\] such that for a fixed value of $r=r_c$ the induced metric, $\gamma_{ij}(r_c,x)=r^2_c\gamma^b_{ij}(x)$, describes a sphere with radius $r_c$. Metric $\gamma^b_{ij}(x)$ of the unit sphere will later be identified with the metric of the boundary QFT theory.
The full on-shell action corresponding to this solution can be used to compute the energy-momentum tensor (Brown-York) and the holographic sphere partition function T\^d\_[ij]{}\[r\]-,Z\_[\^[d]{}]{}\[r\]-I\^[(d+1)]{}\_[on-shell]{}\[r\].\[PFT\] Note that both quantities explicitly depend on the radius $r$ at which we cut off spacetime. In standard holography, we take $r$ to infinity and the counterterm action yields finite answers (modulo logarithmic divergences that correspond to anomalies). However, in the context of finite cut-off holography, we keep the radial dependence finite – this can be holographically interpreted as a deformation by a generalization of the $T\bar{T}$ operator to arbitrary dimensions.
In what follows, we evaluate both quantities in from the on-shell action with a finite radial cut-off. From the symmetry of the problem, they are determined by a single function of the radius $\omega[r]$ that we extract from both computations, with exact agreement. We will later demonstrate that this function solves the algebraic flow equation that defines the deformed theory.
Holographic stress-tensors
--------------------------
The holographic stress-tensor [@Balasubramanian:1999re; @deHaro:2000vlm] is obtained by variation of the on-shell action with respect to the induced metric on the surface of constant $r=r_c$. We first compute the general variation and then show that, as constrained by the spherical symmetry, for our metric , energy momentum tensor is proportional to the metric. Performing the standard variations we obtain [@Emparan:1999pm; @Balasubramanian:1999re; @deHaro:2000vlm][^2] $$\begin{aligned}
\label{hol-stress}
T_{ij}=-\frac{1}{\kappa^2}&\left\{K_{ij}-K \gamma_{ij}-c^{(1)}_d\frac{d-1}{l}\gamma_{ij}+\frac{c^{(2)}_d\,l}{(d-2)}\tilde{G}_{ij}\right.\nn\\
&\quad \left.+\frac{c^{(3)}_dl^3}{(d-4)(d-2)^2}\left[2(\tilde{R}_{ikjl}-\frac{1}{4}\gamma_{ij}\tilde{R}_{kl})\tilde{R}^{kl}-\frac{d}{2(d-1)}\left(\tilde{R}_{ij}-\frac{1}{4}\tilde{R}\gamma_{ij}\right)\tilde{R}\right.\right.\nn\\
&\quad -\left.\left.\frac{1}{2(d-1)}\left(\gamma_{ij}\Box \tilde{R}+(d-2)\nabla_i\nabla_j\tilde{R}\right)+\Box \tilde{R}_{ij}\right]\right\},\end{aligned}$$ where $\tilde{G}_{ij}$, $\tilde{R}_{ikjl}$ are the Einstein and Riemann tensors and $\Box$ is the Laplace-Beltrami operator of the induced metric $\gamma_{ij}$. On the field theory side, the above holographic stress tensor will be used to construct the operator (or its expectation value) which deforms the CFT. For future reference, we note the contributions of additional terms for $d\geq 3$ in the counterterm action . Following the conventions of [@Hartman:2018tkw] we denote $$\begin{aligned}
\label{C-tensor-def}
{C}_{ij}=\,&\left\{{c^{(2)}_d}\tilde{G}_{ij}+c^{(3)}_db_d\left[2(\tilde{R}_{ikjl}-\frac{1}{4}\gamma_{ij}\tilde{R}_{kl})\tilde{R}^{kl}-\frac{d}{2(d-1)}\left(\tilde{R}_{ij}-\frac{1}{4}\tilde{R}\gamma_{ij}\right)\tilde{R}\right.\right.
\nn\\
&\quad -\left.\left.\frac{1}{2(d-1)}\left(\gamma_{ij}\Box \tilde{R}+(d-2)\nabla_i\nabla_j\tilde{R}\right)+\Box \tilde{R}_{ij}\right]\right\}.\end{aligned}$$ with $b_d=l^2/((d-4)(d-2))$.
Now we evaluate the stress-tensor for the metric with $r=r_c$, so that the induced metric[^3] becomes $\gamma_{ij}(r_c,x)$. Firstly, the extrinsic curvature terms on the constant $r=r_c$ surface become K\_[ij]{}-K \_[ij]{}= \_[ij]{}. The Ricci tensor at $r_c$ is also proportional to the metric $\tilde{R}_{ij}=\frac{d-1}{r^2_c}\gamma_{ij}$ such that the Einstein tensor for the sphere (Einstein manifold) is given by \_[ij]{}=-\_[ij]{}. Then, the contraction of the Riemann tensor with Ricci tensor is also proportional to the metric such that the second line of becomes 2(\_[ikjl]{}-\_[ij]{}\_[kl]{})\^[kl]{}-(\_[ij]{}-\_[ij]{})=\_[ij]{}. Finally, the last line vanishes for the sphere (constant curvature) and we get the holographic energy momentum-tensor for the $d$-dimensional sphere at $r=r_c$ in Euclidean $AdS_{d+1}$ \[omega-from-AdS\] T\^d\_[ij]{}\[r\_c\]=\_[ij]{}. Indeed, we see that it is proportional to the metric and we define the proportionality function as =. Clearly, we see that different counterterms in various dimensions contribute with polynomial terms whereas the EH and GH terms yield the square-root part. As we shall show in Section \[sec:field-theory\], this function can be obtained by solving the QFT flow equation that becomes an algebraic equation for $\omega[r_c]$.
Sphere partition functions
--------------------------
The next step involves evaluation of the regularized gravity actions and the holographic sphere partition functions with finite cut-off. We evaluate the action in AdS with a cut-off or wall at $r=r_c$ where we also take into account the counterterms .
Our metric has a constant negative curvature $R=-d(d+1)/l^2$ and is a solution of the vacuum Einstein equations with negative cosmological constant $\Lambda=-d(d-1)/(2l^2)$. With these ingredients and the formulae of the previous subsection, we can evaluate the on-shell action $$\begin{aligned}
I^{(d+1)}_{\rm on-shell}[r_c]=\frac{dl^{d}S_d}{\kappa^2l}&\left[\int^{r_c}_0\frac{r^d\,dr}{l^{d+1}\sqrt{1+\frac{r^2}{l^2}}}-\left(\frac{r_c}{l}\right)^{d-1}\sqrt{1+\frac{r^2_c}{l^2}}\right.\nn\\
&\quad + \frac{r^d_c}{l^d}\left.\left(c^{(1)}_d\frac{(d-1)}{d}+\frac{c^{(2)}_d(d-1)}{2(d-2)} \frac{l^2}{r^2_c}-\frac{c^{(3)}_d(d-1)}{8(d-4)}\frac{l^4}{r^4_c}\right)\right]\end{aligned}$$ The first term comes from the EH action, the second from the GH boundary term and second line from the counterterms. There is an overall factor of the sphere area, $S_d=(2\pi^{\frac{d+1}{2}})/\Gamma\left(\frac{d+1}{2}\right)$. Moreover, the first two terms the EH and GH terms, can be written under one integral as S\_[EH]{}+S\_[GH]{}=-\^q\_0 dq,\[osEHGH\] where we introduced $q=r^2_c$ and this expression will be important in the Wheeler-DeWitt analysis (Section \[sec:Wheeler-DeWitt\]). Performing this integral yields the hypergeometric function and writing the answer in terms of $r_c$ gives the full holographic sphere partition function (up to $d=6$) $$\begin{aligned}
\log Z_{\mathbb{S}^{d}}[r_c] =-\frac{dS_dr^d_c}{\kappa^2l}&\left[-\frac{l}{r_c}\,\,_2F_1\left(-\frac{1}{2},\frac{d-1}{2},\frac{d+1}{2},-\frac{r^2_c}{l^2}\right)\right.
\nn\\
& \quad
+\left.c^{(1)}_d\frac{(d-1)}{d}+\frac{c^{(2)}_d(d-1)}{2(d-2)}\frac{l^2}{r^2_c}-\frac{c^{(3)}_d(d-1)}{8(d-4)}\frac{l^4}{r^4_c}\right].\label{SPF} \end{aligned}$$ This is the main result of this section and in Section \[sec:Wheeler-DeWitt\] we will see how this expression is related to the solution of the Wheeler-DeWitt equation.
The on-shell action , also allows us to extract $\omega[r_c]$. Namely, in general dimensions, the derivative of the sphere partition function with respect to the radius is related to the expectation value of the trace of the energy-momentum tensor. Therefore, we have r\_c\_[r\_c]{} Z\_[\^[d]{}]{}\[r\_c\]=-d\^dxT\^i\_i=-r\^d\_cS\_dd, where we used that for the sphere $\vev{T_{ij}}=\omega[r_c] \gamma_{ij}$ and $\vev{T^i_i}=d\,\omega[r_c]$. Differentiating we obtain =. This is precisely the proportionality function derived in the previous subsection. In the next section, we show that it is the solution of the flow equations (with inclusion of anomalies) in all dimensions that we analyze.
Field theory analysis {#sec:field-theory}
=====================
$\ttb$ deformation in general dimensions
----------------------------------------
As alluded to in the introduction the $\ttb$ operator was initially introduced in 2$d$ by Zamolodchikov [@Zamolodchikov:2004ce]. This bi-local operator is defined as the following quadratic combination of the components of the stress-tensor $$\begin{aligned}
\label{tt-def}
\ttb(z,z') = T_{zz}(z) T_{\bz \bz} (z') - T_{z\bz}(z)T_{z\bz}(z').\end{aligned}$$ This definition is in flat Euclidean space ($z=x+it$). By using symmetries and conservation laws of the stress tensor, it can be shown that the expectation value of this operator is a constant. This fact motivates defining the operator at coincident points. Although there are divergences which do appear upon taking the coincident point limit, it can be shown that these appear as total derivative terms. The operator $\ttb$ therefore makes sense unambiguously within an integral. We can then deform a QFT by this operator as follows $$\begin{aligned}
\label{action-flow}
\frac{dS(\lambda)}{d\lambda} = \int d^2 x \, \ttb(x). \end{aligned}$$ It is crucial to observe that the stress tensor components appearing in the right hand side of the above equation are that of the action $S(\lambda)$ and, therefore, the deformation is in a sense recursive. This leads to modifications of the action/Lagrangian which are generically non-linear in the coupling $\lambda$, see [@Cavaglia:2016oda; @Bonelli:2018kik]. For deformations of CFTs, the $\ttb$ coupling is the only new dimensionful scale of the theory. If a single dimensionful scale is present, the following Ward identity for the effective action holds $$\begin{aligned}
\label{eff-Ward-id}
\lambda \frac{dW}{d\lambda} = -\frac{1}{2} \int d^2 x \, \vev{T_i^i} . \end{aligned}$$ Combining the equations and leads to the flow equation $$\begin{aligned}
\label{ttb-fac}
\vev{T_i^i} = -2 \lambda \vev{\ttb} = -2 \lambda \left(\vev{T_{zz}}\vev{T_{\bz \bz} }- \vev{T_{z\bz}}^2 \right).\end{aligned}$$ If the theory lives on a cylinder $\mathbb{R}\times \mathbb{S}^1$, the second equality of the above equation takes the same form as the inviscid Burger’s equation of hydrodynamics [@Smirnov:2016lqw; @Cavaglia:2016oda].
The above analysis can be generalized for curved spaces and also to higher dimensions. This has been carried out in [@Taylor:2018xcy; @Hartman:2018tkw]. The strategy there was to make use of the holographic stress tensor and higher dimensional analogues of and to build the deforming operator. Although the factorization property is not true in general for curved spaces and $d> 2$, it is still expected to hold for large $N$ theories. The deforming operator has the following structure $$\begin{aligned}
\label{x-def}
X_d = \left(T_{ij}+\frac{\alpha_d}{\lambda^{\frac{d-2}{d}}}C_{ij}\right)^2-\frac{1}{d-1}\left(T^{i}_i+\frac{\alpha_d}{\lambda^{\frac{d-2}{d}}}C^i_i\right)^2+\frac{1}{d }\frac{\alpha_d}{\lambda^{\frac{(d-2)}{d}}}\left(\frac{(d-2)}{2}R+C^i_i\right).\end{aligned}$$ The notation $(B_{ij})^2= B_{ij}B^{ij}$ has been used above. Here, $\alpha_d$ is a dimensionless parameter depending on the degrees of freedom of the theory – $\alpha_4 = N/(2^{7/2}\pi)$ for $\mathcal{N}=4$ super-Yang-Mills with an $SU(N)$ gauge group. The last term in vanishes for $d=3, 4$. The tensor $C_{ij}$ is the contribution to the holographic stress tensor from additional counterterms in $d\geq 3$, equation . For the field theory on a sphere becomes C\_[ij]{}=c\^[(2)]{}\_dG\_[ij]{}+c\^[(3)]{}\_d,\[CijQFT\] where $c^{(n)}_d$ are defined as in Section \[sec:holography\] (see below equation ) so the first term only appears from $3$ dimensions and the second from $5$ dimensions.\
For even dimensions, the appropriate anomaly terms are included as a part of the deforming operator. For $d=2$ the factors of $(d-2)$ in front of $R$ and in $\alpha_2$ cancel such that we recover the $T\bar{T}$ flow equation.
The above operator is quadratic in the stress tensor and should be viewed as the large $N$ approximation of a more general operator which could give rise to the dual quantum field theory for cut-off AdS. Specifically, the deforming operators across various dimensions are given by $$\begin{aligned}
X_2 &= \left( T_{ij} \right)^2 - \left( T^i_{i} \right)^2 + \frac{1}{2\lambda} \frac{c}{24\pi} R ,\\
X_3 &= \left( T_{ij} + \frac{\alpha_3}{\lambda^{1\over 3}} G_{ij} \right)^2 - \frac{1}{2}\left( T^i_{i} + \frac{\alpha_3}{\lambda^{1\over 3}} G^i_{i} \right)^2 ,\\
\label{4d-deform}
X_4 &= \left( T_{ij} + \frac{\alpha_4}{\lambda^{1\over 2}} G_{ij} \right)^2 - \frac{1}{3}\left( T^i_{i} + \frac{\alpha_4}{\lambda^{1\over 2}} G^i_{i} \right)^2 ,\\
X_5 &= \left( T_{ij} + \frac{\alpha_5}{\lambda^{3\over 5}} C_{ij} \right)^2 - \frac{1}{4}\left( T^i_{i} + \frac{\alpha_5}{\lambda^{3\over 5}}C^i_{i} \right)^2 +\frac{1}{5\lambda}\frac{\alpha_5}{\lambda^{\frac{3}{5}}} \left(\frac{3}{2}R + C_i^i\right), \\
X_6 &= \left( T_{ij} + \frac{\alpha_6}{\lambda^{2\over 3}} C_{ij} \right)^2 - \frac{1}{5}\left( T^i_{i} + \frac{\alpha_6}{\lambda^{2\over 3}} C^i_{i} \right)^2 + \frac{1}{6\lambda}\frac{\alpha_6}{\lambda^{\frac{2}{3}}} \left(2R +C_i^i\right).\end{aligned}$$ In $2d$, the relation $l^2 = \frac{c\lambda}{3\pi}$ has already been used to obtain the form above from . Also note that in $4d$, the squares appearing can be expanded and the terms corresponding to the anomaly can be manifestly separated $$\begin{aligned}
X_4 &= T_{ij}T^{ij} - \frac{1}{3}( T^i_{i})^2 + 2 \frac{\alpha_4}{\sqrt{\lambda}} \left( G_{ij}T^{ij} - \frac{1}{3} G_i^i T^i_i \right) + \frac{1}{4\lambda}\ \frac{C_T}{8\pi} \left(G_{ij}G^{ij} - \frac{1}{3} \left(G_{i}^i\right)^2\right) .\end{aligned}$$ Here we have used the relation between $\alpha_4$ and the central charge, $C_T=32 \pi \alpha_4^2$ (further details are provided below). This is the expression for the deforming operator in $4d$ which appears in [@Hartman:2018tkw]. Similarly, in 6 dimensions, using C\_[ij]{}=G\_[ij]{}+6(), we can write the operator as $$\begin{aligned}
X_6 =\ & T_{ij}T^{ij}-\frac{1}{5}( T^i_{i})^2 + 2 \frac{\alpha_6}{\lambda^{2/3}} \left( C_{ij}T^{ij} - \frac{1}{5} C_i^i T^i_i \right) \nonumber\\
&+\frac{144\alpha^3_6}{6\lambda}\left[R^{ij}R_{ikjl}R^{kl}-\frac{1}{2}RR_{kl}R^{kl}+\frac{3}{50}R^3\right]+\frac{1}{\lambda^{2/3}}O(R^4).\end{aligned}$$ The term with third order in curvature precisely matches the (negative of) the 6$d$ anomaly [@Henningson:1998gx] provided $
\alpha_6={N}/{24\pi}
$. Moreover, the terms quartic in curvature can be compactly written as O(R\^4)=(C\_[ij]{}-G\_[ij]{})\^2-(2R+C\^i\_i)\^2, and they come as important part of the operator needed for the correct solution of the flow equation.
The operator was arrived at by using the form of the holographic stress tensor [@Hartman:2018tkw] (see also Appendix \[sec:Gauss-Codazzi\]). In section \[sec:FlowRegularization\], we will also provide an independent procedure to derive $X_d$ by using a point-splitting procedure. However, for the rest of this section we assume that this is a correct flow equation in large $N$ holographic CFTs and employ in a concrete example.
The deformation on $\mathbb{S}^d$ {#ssec:remarks}
---------------------------------
We now consider the $TT$ deformation of a CFT on the unit sphere $\mathbb{S}^d$. Since the sphere is a maximally symmetric space, the stress-tensor expectation values are proportional to the metric[^4] $\vev{T_{ij}}= \omega_d \gamma_{ij}$. We can solve for $\omega_d$ by using the trace equation in higher dimensions $$\begin{aligned}
\label{trace-rel}
\vev{T_i^i}= - d {\lambda}\vev{X} .\end{aligned}$$ Inserting the explicit form of the operators, this equation becomes an algebraic equation for $\omega_d$ which can be compactly written as \[flow-alg\] d\_d&=&d. where $C_{ij}$ is defined in and the last term only depends on the curvature via f\_d(R)=(R+C\^i\_i)+d(C\_[ij]{}C\^[ij]{}-(C\^i\_i)\^2). The quadratic equation can be solved for $\omega_d$ in $d=2,3,4,5,6$ and we get a general formula \[om-general\] \_d\^[()]{}=(1-C\^i\_i), where the $-$ sign is taken in order to reproduce the anomalies in even dimensions as $\lambda\to0$.
In the “new" holographic dictionary, the $TT$ coupling, $\lambda$, is expressed by the bulk quantities via the relation [@McGough:2016lol; @Hartman:2018tkw] $$\begin{aligned}
\label{coupling-rel}
\lambda = {4 \pi G_N l \over d r^d_c}. \end{aligned}$$ We note that this relation implies that the $TT$ coupling is dimensionless. This is because there is an additional rescaling by the radius of the sphere, $r_c^d$.
Computing the counterterms and using , in all the examples up to $d\le 6$, the above field theory result agrees with the cut-off AdS computation of the stress tensor given $\omega[r_c]=r^{-d}_c\omega_d$ [^5]. We show this explicitly below.
$d=2$ {#d2 .unnumbered}
-----
The case for $d=2$ has been considered earlier in the context of entanglement entropy computations in [@Donnelly:2018bef]. We include it here for completeness. For $d=2$ equation is $$\begin{aligned}
\label{d2result}
\omega_2^{(\pm)}=\frac{1}{4\lambda}\left(1\pm\sqrt{1+\frac{c\lambda}{3\pi}}\right).\end{aligned}$$ The solution with a $-$ sign in front of the square-root agrees precisely with , with the identification for $d=2$ and the usual Brown-Henneaux relation $c = \frac{3l}{2G_N}$. The $\lambda\to 0$ limit of the $-$ branch above reproduces the $2d$ trace anomaly appropriately. The $+$ branch is ruled out since it does not reproduce the trace anomaly in the CFT limit.
$d>2$ {#d2-1 .unnumbered}
-----
For general $d$, the solution of the flow equation on $\mathbb{S}^d$ , is given by (with $\vev{T_{ij}}=\omega_d \gamma_{ij}$) $$\begin{aligned}
\label{field-theory-expression}
\hspace{-.3cm}\omega_d ^{(\pm)}
=\frac{d-1}{2d\lambda}\left[1+c_{d}^{(2)}\alpha_d\lambda^{\frac{2}{d}}d(d-2)\left(1-c_{d}^{(3)}\frac{\alpha_d\lambda^{\frac{2}{d}}d(d-2)}{2}\right)\pm\sqrt{1+2d(d-2)\alpha_d\lambda^{\frac{2}{d}}}\right]. \end{aligned}$$ There are two branches of the solution since the flow equation yields an algebraic equation quadratic in $\omega_d$.
Now for $3\leq d \leq 6$, the parameter $\alpha_d$ is related to gravitational quantities via the relation[^6] $$\begin{aligned}
\label{alpha-rel}
\alpha_d = \frac{1}{(2d)^{d-2 \over d} (d-2)} \left(l^{d-1} \over 8\pi G_N\right)^{2/d}.\end{aligned}$$ This quantity can be related to the rank of gauge groups of conventional CFT$_d$ duals of $AdS_{d+1}$ as follows $$\begin{aligned}
\alpha_3 = \frac{N_{\rm ABJM}}{6\, 2^{1/3}\pi^{2/3}}, \qquad
\alpha_4 = \frac{ N_{\rm SYM}}{2^{7/2}\pi}, \qquad
\alpha_6 = \frac{N_{(2,0)}}{24\pi},\end{aligned}$$ where, we used the relations for the ratio $l^{d-1}/G_N$ for ABJM, $\mathcal{N}=4$ super-Yang-Mills and the 6$d$ (2,0) theory respectively. Moreover, the following relation between $\alpha_d$, $l$ and $\lambda$ can be verified using and $$\begin{aligned}
\label{ads-radius}
\frac{l^2}{r^2_c} = 2d(d-2)\alpha_d\lambda^{2/d}. \end{aligned}$$ Once we use , $\omega^{(-)}_d$ is in precise agreement with bulk $\omega[r_c]$ in the bulk stress tensor .
The behavior of $\omega_d^{(+)}$ in the $\lambda\to 0$ limit is divergent and, similar to $2d$, this branch is ruled out since this does not reproduce the trace anomaly appropriately in the CFT limit. The situation here should be contrasted with that of the torus partition function, wherein non-perturbative ambiguities exist for the negative values of the coupling [@Aharony:2018bad]. In a sense, the CFT trace anomaly provides an additional constraint for partition functions on the sphere.
Finally, we have added appropriate counterterms to obtain the holographic stress tensor and while defining the $TT$ operator. Therefore, the $\lambda \to 0$ limit of the deformed $\mathbb{S}^d$ stress tensor , for $\omega^{(-)}_d$, is devoid of any divergences even in $d=5,6$.[^7] Explicitly, $\omega^{(-)}_d$ has the following forms in the undeformed CFT limit $$\begin{aligned}
\omega_{3,5}^{(-)} \approx 0, \qquad \omega_4^{(-)} \approx 12\alpha_4^2= \frac{3N_{\rm SYM}^2}{32\pi^2}, \qquad \omega_6^{(-)} \approx -2880\alpha_6^3 = - \frac{5N_{(2,0)}^3}{24\pi^3}. \end{aligned}$$ These values are perfectly consistent with trace anomalies of the undeformed holographic theory [@Henningson:1998gx].
$TT$ flow equation from the local Callan-Symanzik equation {#sec:FlowRegularization}
==========================================================
The flow equation we have been using so far was derived in [@Hartman:2018tkw] starting from the bulk Gauss-Codazzi equation, as explained in appendix \[sec:Gauss-Codazzi\], and is taken to be a definition of the dual theory on the boundary. In this section, we shed more light on this flow equation by utilising the Callan-Symanzik (CS) equation for a holographic CFT deformed only by a particular irrelevant operator constructed from the energy momentum tensor[^8]. In this section we will work up to $d=5$ and leave the technicalities of $d=6$ as a future problem.
$TT$ flow equation vs local CS equation
---------------------------------------
The flow equation at large $N$ that serves as the starting point for the analysis presented in the previous section is $$T^{i}_{i}+\frac{\alpha_{d}}{d\lambda^{\frac{d-2}{2}}}C^{i}_{i}=-d\lambda\left( \left(T_{ij}+\frac{\alpha_{d}}{\lambda^{\frac{d-2}{2}}}C_{ij}\right)^{2}-\frac{1}{d-1}\left(T^{i}_{i}+\frac{\alpha_{d}}{\lambda^{\frac{d-2}{2}}}C^{i}_{i}\right)^{2}\right)-\frac{(d-2)\alpha_{d}}{2\lambda^{\frac{d-2}{2}}}R. \label{rfe}$$ This can be made more compact by introducing the ‘bare’ energy momentum tensor $$\hat{T}^{ij}=T^{ij}+\frac{\alpha_{d}}{\lambda^{\frac{d-2}{2}}}C^{ij},$$ and now it reads $$\hat{T}^{i}_{i}=-d\lambda \left(\hat{T}^{ij}\hat{T}_{ij}-\frac{1}{d-1}(\hat{T}^{i}_{i})^{2}\right)-\frac{(d-2)\alpha_{d}}{d\lambda^{\frac{d-2}{2}}}R.\label{bfe}$$ In this section, we aim to obtain the above flow equation from a more intrinsically field theoretic starting point. Namely, the local Callan–Symanzik equation, which expresses the response of the field theory under a local change of scale. This is encoded in the expectation value of the trace of the energy momentum tensor.
First, we notice that on a flat background, the bare flow equation reduces to the one proposed in [@Taylor:2018xcy] $$\hat{T}^{i}_{i}|_{(\gamma_{ij}=\eta_{ij})}=-d\lambda\left(\hat{T}^{ij}\hat{T}_{ij}-\frac{1}{d-1}(\hat{T}^{i}_{i})^{2}\right).$$ On such a background, this equation can certainly be seen as coming from the relationship between the energy momentum tensor and the expectation value of a deforming operator[^9] $$\langle T^{i}_{i}\rangle|_{(\gamma_{ij}=\eta_{ij})}=-d\lambda \langle \mathcal{O}\rangle,$$ where $\mathcal{O}(x)$ is the irrelevant operator of interest, and the parameter $\lambda$ is the scale associated to the irrelevant deformation. This relationship is referred to as the local CS equation on flat space.
On curved spaces, this equation generalizes to $$\langle T^{i}_{i}\rangle=-d\lambda \langle \mathcal{O}(x)\rangle-\mathcal{A}(\gamma). \label{lrg}$$ For our purposes, $\mathcal{A}(\gamma)$ is the holographic anomaly which is present in even dimensions. This equation readily provides the correct flow equation in $d=2$. Here, in the large $c$ limit, we have $$\langle \mathcal{O}(x)\rangle|_{c\rightarrow \infty}=\lim_{y\rightarrow x}G_{ijkl}(x)\langle T^{ij}(x)T^{kl}(y)\rangle|_{c\rightarrow \infty} =\langle T^{ij} \rangle \langle T_{ij}\rangle -\langle T^{i}_{i}\rangle^{2},$$ where $G_{ijkl}=\gamma_{i(k}\gamma_{l)j}-\gamma_{ij}\gamma_{kl}$, and the anomaly takes the form $$\mathcal{A}(\gamma)=-\frac{c}{24\pi}R(\gamma).$$ So, in the end, the two dimensional $T\bar{T}$ deformed flow equation reads $$T^{i}_{i}=-2\lambda\left( T^{ij} T_{ij} -(T^{i}_{i})^{2} \right) -\frac{c}{24\pi}R(\gamma),$$ where the angle brackets are dropped in the large $c$ limit. From this derivation, we see that the coincidence between conformal anomaly and the Ricci scalar was crucially important.
This is no longer the case in $d=3,4,5$. In these dimensions, the anomaly in no longer provides for us the Ricci scalar term in . In fact, in $d=3$ and $d=5$ there is no conformal anomaly whilst in $d=4$ the anomaly is quadratic in the curvature. In order to obtain the Ricci scalar term in the flow equation, it must somehow be ‘generated’ from the definition of $\mathcal{O}(x)$. Furthermore, the anomaly in $d=4$ must somehow also be absorbed into the definition of this operator. These issues are addressed in what follows.
From local CS equation to the higher dimensional flow equation
--------------------------------------------------------------
Our aim, as described in the previous section, is to generate the Ricci scalar term in the equation , from the local CS equation .
In dimensions higher than 2, the deforming operator $\mathcal{O}(x)$ is defined as $$\mathcal{O}(x)=\lim_{y\rightarrow x}\frac{1}{4} \left(T_{ij}(x)-\frac{1}{d-1}T^{k}_{k}(x)g_{ij}(x)\right)T^{ij}(y).$$ It will help to introduce $$G_{ijkl}(x)=\left(\gamma_{i(k}(x)\gamma_{l)j}(x)-\frac{1}{d-1}\gamma_{ij}(x)\gamma_{kl}(x)\right),$$ so that $$T_{ij}(x)-\frac{1}{d-1}T^{k}_{k}(x)g_{ij}(x)=G_{ijkl}(x)T^{kl}(x).$$ From the definition of the energy momentum tensor, we have $$\langle\mathcal{O}(x)\rangle Z[\gamma]=\lim_{y\rightarrow x}G_{ijkl}(x)\left(\frac{1}{\sqrt{\gamma(x)}}\frac{\delta }{\delta \gamma_{ij}(x)}\left(\frac{1}{\sqrt{\gamma(y)}}\frac{\delta Z[\gamma]}{\delta \gamma_{kl}(y)}\right)\right).$$ In order to generate the $R$ term in , we choose more concrete means to implement the coincidence limit. This method is similar to the one of [@Ita:2017uvz] although the context is quite different. We choose to do this through the heat kernel $K(x,y;\epsilon)$, which satisfies the property $$\lim_{\epsilon\rightarrow 0}K(x,y;\epsilon)=\delta(x,y).$$ This property should be thought of as an initial condition for the heat equation $$\partial_{\epsilon}K(x,y;\epsilon)=(\nabla^{2}_{(x)}+\xi R_{(x)})K(x,y;\epsilon).$$ We can now implement the point splitting regularization as follows $$\lim_{y\rightarrow x}G_{ijkl}\langle T^{ij}(x)T^{kl}(y)\rangle Z[\gamma]=\lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}yK(x,y;\epsilon) G_{ijkl}(x)\frac{1}{\sqrt{\gamma}(x)}\frac{\delta}{\delta \gamma_{ij}(x)} \left(\frac{1}{\sqrt{\gamma}(y)}\frac{\delta Z[\gamma]}{\delta \gamma_{kl}(y)}\right).$$ We also exploit the fact that we can add to the effective action terms involving local functions of the metric $$Z[\gamma]\rightarrow e^{C[\gamma]}Z[\gamma],$$ where $C[\gamma]$ is chosen to be $$C[\gamma]=\alpha_{0}\left(\epsilon^{\frac{d}{2}-1}\int \textrm{d}^{d}x \sqrt{\gamma}+\frac{(d^{2}-3)\epsilon^{\frac{d}{2}}}{d(d-1)}\int \textrm{d}^{d}x \sqrt{\gamma}R\right).\label{impr}$$ Here, $\alpha_{0}$ is a constant given by $$\alpha_{0}=\frac{\alpha_{d}}{\lambda^{\frac{d+2}{2}}}\left(\frac{d-2}{2d^{2}\kappa(d)}\right),$$ where $$\kappa(d)=\frac{(d^{2}-3)(d(d(9d-11)-28)+42)}{12d (d-1)^{2}}.$$ With this choice of $\epsilon$ scaling in the improvement term $C[\gamma]$, one can show (as we do in appendix \[sec:generateR\]) that the deforming operator becomes $$\langle \mathcal{O}(x) \rangle= \lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y,\epsilon)G_{ijkl}(x)\langle T^{ij}(x) T^{kl}(y) \rangle+\alpha_{0}R(x)$$ which we then subject to the large $N$ limit to obtain $$\begin{aligned}
\langle \mathcal{O}(x)\rangle|_{N\rightarrow \infty}&=
\lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y;\epsilon)G_{ijkl}(x)\langle T^{ij}(x) \rangle \langle T^{kl}(y)\rangle+\alpha_{0}R(x) \nn \\
&=G_{ijkl}(x)\langle T^{ij}(x)\rangle \langle T^{kl}(x)\rangle+\alpha_{0}R(x).\end{aligned}$$ Here we have used the fact that the large $N$ factorized two point function does not suffer any coincidence divergences so the limit can be taken to turn the heat kernel into a delta function, and the $y$ integral can be performed. We can plug this back into the local CS equation , which now reads, at large $N$ $$T^{i}_{i}=-d\lambda G_{ijkl}T^{ij}T^{kl}-\frac{(d-2)\alpha_{d}}{2\lambda^{\frac{d-2}{2}}}R-\mathcal{A}(\gamma).$$ In $d=3$ and $d=5$, the anomaly $\mathcal{A}(\gamma)=0$. Here we immediately obtain provided we make the choice $\hat{T}^{ij}=T^{ij}$. In $d=4$, the holographic anomaly is given by $$\mathcal{A}=-\frac{\alpha^{2}_{4}}{\lambda^{2}}\left(G_{ij}G^{ij}-\frac{1}{3}(G^{i}_{i})^{2}\right),$$ where $G_{ij}=R_{ij}-\frac{1}{2}R g_{ij}$ is the Einstein tensor, and $a$ is the anomaly coefficient. This can be absorbed into an improvement of the energy momentum tensor, which is subsumed in the definition of the bare energy momentum tensor $$\hat{T}^{ij}= T^{ij}+\frac{\alpha_{4}}{\lambda}G^{ij}.$$ In other words, the equation $$T^{i}_{i}=-4\lambda(T^{ij}T_{ij}-\frac{1}{d-1}(T^{i}_{i})^{2})-\frac{\alpha_{4}}{\lambda}R-\mathcal{A}(\gamma)$$ becomes $$T^{i}_{i}+\frac{\alpha_{4}}{\lambda}G^{i}_{i}=-4\lambda \left(\left(T^{ij}+\frac{\alpha_{4}}{\lambda}G^{ij}\right)\left(T_{ij}+\frac{\alpha_{4}}{\lambda}G_{ij}\right)-\frac{1}{d-1}\left(T^{i}_{i}+\frac{\alpha_{4}}{\lambda}G^{i}_{i}\right)^{2}\right)-\frac{\alpha_{4}}{\lambda}R,$$ hence we get .
Limitations of this method
--------------------------
Despite the promise, we find that in $d=4$, this method allows us to readily obtain where as in $d=3,5$, we automatically obtain . The reason for this distinction is that the absorbing the anomaly into the improvement of the energy momentum tensor occurs only in $d=4$. In $d=3$ and $d=5$, the absence of the anomaly leaves us only with the bare flow equation. The inclusion of the counterterms, especially as involved as in $d=6$, should arise from a further improvement of the energy momentum tensor.
In other words, the counterterms are accounted for automatically in $d=4$ whereas must be thought of as an additional input in odd dimensions. Perhaps a different method or scheme would directly give us the renormalized flow equation no matter what dimension we are working in, starting from the local CS equation.
The Wheeler-DeWitt equation {#sec:Wheeler-DeWitt}
===========================
In this section, we comment on the role played by the Wheeler-DeWitt equation in deriving the deformed partition function. We shall see that the WKB solution of the (minisuperspace) Wheeler-DeWitt equation perfectly reproduces the bulk and boundary on-shell action without counterterms.
Let us briefly review the Wheeler-DeWitt equation that arises in the minisuperspace approximation (we closely follow [@Caputa:2018asc]). The minisuperspace ansatz for the Euclidean asymptotically $AdS$ metric is defined as ds\^2=\^2(r)dr\^2+a\^2(r)d\^2\_[d]{}, where $\mathbf{N}(r)$ is the lapse function and $a(r)$ is the scale factor.
We first evaluate the EH and GH actions on this metric and then, in the Euclidean gravity path integral, we redefine the lapse $\mathbf{N}\to \mathbf{N}a^{d-4}$ and introduce a variable[^10] $q=a^2$ such that the action takes the form (see [@Caputa:2018asc] and references therein) S\_[EH]{}+S\_[GH]{}=-dr , where $S_d$ is the sphere area.
To derive the Hamiltonian we compute the canonical momentum conjugate to $q(r)$ p==-q’, and a Legendre’s transform yields H==-. Inserting $p=\hbar\frac{d}{dq}$, we derive the Hamiltonian constraint, or the Wheeler-DeWitt equation for the wavefunction $\Psi[q]$ [@Caputa:2018asc] ==0. This equation can be solved exactly in terms of special functions for $d=2,3,4$ (in $d=3$ the solution is the Airy function that reproduces the ABJM partition function [@Marino:2011eh; @Fuji:2011km] with perturbative $1/N$ corrections). However, let us focus just on the semi-classical limit, $G_N\to 0$ (large $N$) but with fixed $q$. In this regime we can use the WKB approximation and the leading order solutions are $$\begin{aligned}
\Psi_{\rm WKB}(q) &\approx \exp \left[ \pm \left(d(d-1)S_d\over 2\kappa^2 l \hbar\right) \int_{0}^{q} \sqrt{ l^2 q^{d-3} + q^{d-2}}\, dq \right].\end{aligned}$$ Performing the integral, we can see that with $q=r^2_c$, the $-$ sign solution in $d$-dimensions becomes[^11] $$\begin{aligned}
\Psi_{\rm WKB}[r_c]
=\exp \left[ \frac{dS_dr^{d-1}_c}{\kappa^2 }\,\,_2F_1\left(-\frac{1}{2},\frac{d-1}{2},\frac{d+1}{2},-\frac{r^2_c}{l^2}\right)\right]
=e^{-\left(I^{\rm on-shell}_{GR}[r_c]-S_{ct}[r_c]\right)}\end{aligned}$$ where we have identified the exponent as the on-shell EH and GH actions (gravity on-shell action without counterterms) evaluated on our Euclidean $AdS$ metric with finite boundary cut-off I\^[on-shell]{}\_[GR]{}\[r\_c\]-S\_[ct]{}\[r\_c\]=S\_[EH]{}\[r\_c\]+S\_[GH]{}\[r\_c\], computed in . Analogous to [@Donnelly:2018bef], this bare partition function (translated to QFT) can be used to compute entanglement entropy and matched with the Ryu-Takayanagi prescription [@Ryu:2006bv] applied to a spacetime with finite cut-off. The details of this computation will be presented elsewhere [@CaputaHiranoWIP].
A few comments are in order at this point. Firstly, this concrete example for the sphere illustrates the known fact that the Wheeler-DeWitt wavefunction should be related to the holographic partition function [@deBoer:1999tgo; @deBoer:2000cz; @Papadimitriou:2004ap]. However, it is the minisuperspace approximation that turns this equation into a powerful tool. Secondly, the counterterms (for the full $TT$ partition function that is obtained from the flow equation) are included by additional canonical transformation as explained, for instance, in [@lecholrg]. Thirdly, in the large $N$ limit, it is the WKB solution of the Wheeler-DeWitt equation that can be matched with the on-shell action with finite cut-off. It is therefore an interesting future problem to compare the full solution of the Wheeler-DeWitt equation[^12] with the CFT deformations at finite $N$.
Finally, let us also recall that the connection between solutions to the radial Wheeler-DeWitt equation and the partition function of the $T\bar{T}$ deformed conformal field theories in two dimensions was first noticed in slightly different guise in [@Freidel:2008sh]. The idea there was to define the partition function for the deformed theory through an integral kernel as $$\label{couple}
Z_{\rm QFT}[e]=\int \mathcal{D}f \, e^{\frac{1}{\lambda}\int f^{+}\wedge f^{-}}Z_{\rm CFT}[e+f],$$ where $e^{I}_{i}$ is the dyad associated to the metric on the boundary $\gamma_{ij}$. It was then shown that this kernel, when applied to the Weyl Ward identity for the partition function $Z_{\rm CFT}[e]$, resulted in $Z_{QFT}[e]$ satisfying the Wheeler-DeWitt equation. From our discussion above, it follows that this object can be seen as the generating functional for the $T\bar{T}$ deformed theory not including the counter-terms. See [@Freidel:2008sh], [@McGough:2016lol] for more details.
It is intriguing to note that is very similar to the proposal involving coupling the CFT to Jackiw-Teitelboim gravity in [@Dubovsky:2018bmo]. It is also a very interesting open problem to find such integral kernels in higher dimensions.
Conclusions {#sec:conclusions}
===========
In this work we further explored generalized $TT$ deformations in large $N$ CFTs and holography with a finite cut-off. We focused on the deformations defined by the trace of the energy-momentum flow equation in holographic CFTs on the sphere. By computing the energy momentum tensor and sphere partition functions holographically (up to $d=6$), we saw that the crucial information is contained in the proportionality function, $\omega[r_c]$, of the stress tensor, $\vev{T_{ij}}=\omega[r_c]\gamma_{ij}$. In the field theory side, this function solves the (algebraic) $TT$ flow equation provided all the non-trivial ingredients of the holographic dictionary like precise anomalies on $\mathbb{S}^d$ as well as relation between the deformation coupling and the gravity parameters. This program can be generalized to other asymptotic geometries as well as black hole solutions and we leave this for future work.
Since the higher dimensional flow equation originates from the Gauss-Codazzi equation, or the Hamiltonian constraint in gravity, the above results may be seen as a consistency check of AdS/CFT. On the other hand, without the $T\bar{T}$ story, the relation between the radial direction and deformation by irrelevant operators would have remained elusive. This is why there is still a lot to be learned about this new ingredient of holography, especially in higher dimensions. In particular, the definition of $TT$ operators on curved manifolds or purely field theory origin of the flow equation remains challenging. In section \[sec:FlowRegularization\], we made some progress on the latter and showed how the field theory flow equation emerges from the regularization procedure in defining the $TT$ operator at large $N$.
We hope that our arguments can be sharpened so that they capture, in arbitrary dimensions, the $1/N$ corrections and additional matter content of the theories. Along these lines, a potentially promising direction to pursue would be to obtain the flow equation for a holographic theory away from large $N$. This is possible by first upgrading the parameter $\lambda$ to a local function of space, (a source) and then to apply the methods of the local renormalization group in the presence of irrelevant operator deformations as was studied recently in [@vanRees:2011ir; @Schwimmer:2019efk]. Then, setting the parameter to be constant would lead to a flow equation of the kind we are interested in.
The Wheeler-DeWitt equation is ubiquitous in quantum gravity and plays an important role in holographic RG [@deBoer:1999tgo; @deBoer:2000cz; @Papadimitriou:2004ap]. In our example we can see that its mini-superspace version can be employed to reproduce the holographic partition function with a finite cut-off. We may hope that the Wheeler-DeWitt equation can guide us in defining the $TT$ operator and identify its expectation value in the flow equation beyond large $N$. In particular, understanding the relation between the coupling of the $TT$ operator and the cut-off in quantum gravity remains a challenge.
Finally, it is important to explore physical quantities under the $TT$ deformation in various dimensions. In particular, how do correlation functions and transport coefficients ($\eta/s$) get modified. The quasi-normal modes get shifted upon putting a finite cut-off. This should in turn affect the retarded Green’s functions. Similarly, an interesting avenue to explore is how thermalization timescales get affected by $TT$. Since the deformation introduces new interaction terms in the Lagrangian and also leads to superluminal signal propagation, one might expect thermalization to occur faster. Last but not least, many recently developed quantum information theoretic quantities in holography correspond to bulk objects that are non-trivially modified by finite cut-off. Non-perturbative comparisons with deformed CFTs, perhaps even beyond the planar limit, may provide important lessons in this directions (see [@Akhavan:2018wla; @Hashemi:2019xeq]).
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank John Cardy, Thomas Dumitrescu, Monica Guica, Michael Gutperle, Shinji Hirano, Mukund Rangamani, Tadashi Takayanagi, Yunfeng Jiang, Per Kraus, Silviu Pufu and Edgar Shaghoulian for fruitful discussions. Some calculations in this work were performed with the aid of the collection of xAct Mathematica packages. PC is supported by the Simons Foundation through the “It from Qubit" collaboration and by JSPS Grant-in-Aid for Research Activity start-up 17H06787. SD would like to thank the participants and organizers of CHORD‘18 at the KITP for simulating discussions on related topics and UC Berkeley and IIT Kanpur for hospitality during the completion of this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. VS is supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
Gauss-Codazzi equation {#sec:Gauss-Codazzi}
======================
In this appendix we give a lightning review of the ideas presented in [@Taylor:2018xcy; @Hartman:2018tkw] for deriving a flow equation for the trace of the energy momentum tensor in large $N$ holographic CFTs in $d$-dimensions. The main objective of these works is to generalize the $T\bar{T}$ deformations from $2d$ CFTs that result in the finite cut-off in dual holographic geometry (as in [@McGough:2016lol]). As observed in [@McGough:2016lol], in two dimensions, the $T\bar{T}$ flow equation can be rewritten as the Gauss-Codazzi equation (Hamiltonian constraint) in gravity. Therefore, the logic to derive analogues in higher dimensions is to derive a flow starting from Gauss-Codazzi equation and postulate that it should be realized as a flow in holographic CFT at large $N$ deformed by the $TT$ operator.
In the Hamiltonian approach to holographic renormalization (see e.g. [@Papadimitriou:2004ap; @lecholrg]), it is convenient to write Einstein’s equations in terms of intrinsic ($\tilde{R}$) and extrinsic ($K$) curvatures of the hypersurfaces $\Sigma_r$ of constant radial direction $r$ with induced metric $\gamma_{ij}$. In this formalism, Einstein’s equations are equivalent to Gauss-Codazzi equations. In particular, in the case of pure gravity, their $(r,r)$ component for $d+1$-dimensional spacetime with $d$-dimensional $\Sigma_r$ is given by K\^2-K\_[ij]{}K\^[ij]{}=+.\[GC\] This equation is just the Hamiltonian constraint namely, with the canonical momentum conjugate to the boundary metric \_[ij]{}=(K\_[ij]{}-K\_[ij]{}),\^i\_i=(d-1)K, we have \^[ij]{}\_[ij]{}-(\^i\_i)\^2=, and the Gauss-Codazzi equation (after multiplying by $\kappa^2/\sqrt{\gamma}$) becomes +=0. This is the standard ADM Hamiltonian constraint $H=0$, introduced in [@ADM]. This becomes the Wheeler-DeWitt equation $H\Psi=0$ for the wavefunction $\psi$ after replacing canonical momenta with derivatives w.r.t the metric in the quantum theory.
Gauss-Codazzi as a holographic flow equation
--------------------------------------------
The Gauss-Codazzi equation is also equivalent to the flow equation for the expectation value of the trace of the holographic energy-momentum tensor. To see that, take a general form of the holographic stress tensor[^13] T\_[ij]{}=--a\_d C\_[ij]{} where $\kappa^2=8\pi G_N$ and $a_d=\frac{l}{(d-2)\kappa^2}$ is the known coefficient of the first counterterm above $d=2$. It is convenient to introduce a “bare" stress-tensor \_[ij]{}=T\_[ij]{}+a\_d C\_[ij]{},\[BareST\] such that \_[ij]{}=,\^i\_[i]{}=. From these relations, we have K\^2-K\_[ij]{}K\^[ij]{}=-\^i\_[i]{}-\^4+, and again, replacing the LHS with the Gauss-Codazzi equation yields the holographic “flow" equation for the bare stress-tensor \^i\_[i]{}=--.\[bareflow\] We can now also write this equation in terms of the “renormalized" stress tensor by inserting such that we get [@Hartman:2018tkw] T\^i\_[i]{}=-.\[FLGR\] Note that this is still purely phrased in terms of gravitational quantities but we can use the holographic dictionary to turn it into a flow in dual CFTs deformed by a generalized $TT$ operator [@Taylor:2018xcy; @Hartman:2018tkw].
A dual flow in deformed holographic CFTs
----------------------------------------
Formally, in holographic large $N$ CFTs, we can translate the flow equation to the boundary theory on a unit sphere $\mathbb{S}^d$ by introducing boundary quantities related by powers of the bulk radial cut-off $r_c$. The bulk quantities are translated into the boundary (with superscript $^b$) as [@Hartman:2018tkw] \_[ij]{}r\^[2]{}\_c\^b\_[ij]{},T\_[ij]{}r\^[2-d]{}\_cT\^b\_[ij]{} T\^[i]{}\_i=\^[ij]{}T\_[ij]{}r\^[-d]{}\_c(T\^b)\^i\_i,=r\^[-2]{}\_c\^b, such that the bare flow equation in QFT becomes (\^b)\^i\_i=-. If we want to interpret this as a QFT flow, we should have d\^dx(T\^b)\^i\_[i]{}=-dX, and we need in total two relations to replace $l$ and $G_N$ with boundary data. We can write =,,\[Param\] where $\alpha_d$ is a QFT parameter (see the main text). Using these, we can then write the bare QFT flow as (\^b)\^i\_i=-d, and similarly for the renormalized stress-tensor[^14] (T\^b)\^i\_i=-d. The expression can be expanded further (for simplicity we drop the superscript $b$) and we get the equation used in the main text T\^i\_i&=&-d. where (up to 6d) in field theory we have C\_[ij]{}=c\^[(2)]{}\_dG\_[ij]{}+c\^[(3)]{}\_d, where all the ingredients are of those of the unit metric on the sphere $\gamma_{ij}$.\
One last comment is that, naively, it appears that this formula is wrong for $d=2$ because it would kill the anomaly. However, for consistency we must have \_2=\_[d2]{}a\_d=\_[d2]{}=\_[d2]{}, where we used the Brown-Henneaux relation, and this precisely gives the anomaly piece when $C_{ij}=0$ in $d=2$.
Generating $R$ in the flow equation {#sec:generateR}
===================================
The specific $\epsilon$ scaling of the coefficients in the expression are chosen such that in the limit $\epsilon\rightarrow 0$, the following terms vanish[^15] $$\lim_{\epsilon \rightarrow 0}C[\gamma]=0=\lim_{\epsilon \rightarrow 0} \frac{\delta C[\gamma]}{\delta \gamma_{ij}}.$$ The second functional derivative however will remain finite, provided we smear it against the heat kernel. This means that if we distribute the limit, we have $$\begin{aligned}
&\lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y,\epsilon)G_{ijkl}(x)\frac{1}{\sqrt{\gamma}(x)}\frac{\delta }{\delta \gamma_{ij}(x)}\left(\frac{1}{\sqrt{\gamma}(y)}\frac{\delta(e^{C[\gamma]}Z[\gamma])}{\delta \gamma_{kl}(y)}\right)\nn \\
=&\lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y,\epsilon)G_{ijkl}(x)\frac{1}{\sqrt{\gamma}(x)}\frac{\delta }{\delta \gamma_{ij}(x)}\left(\frac{1}{\sqrt{\gamma}(y)}\frac{\delta Z[\gamma]}{\delta \gamma_{kl}(y)}\right)\nn \\
&+\left(\lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y,\epsilon)G_{ijkl}(x)\frac{1}{\sqrt{\gamma}(x)}\frac{\delta }{\delta \gamma_{ij}(x)}\left(\frac{1}{\sqrt{\gamma}(y)}\frac{\delta C[\gamma]}{\delta \gamma_{kl}(y)}\right)\right)Z[\gamma].\end{aligned}$$ We then take a closer look at the term on the second line of the RHS in the expression above $$\begin{aligned}
&\left(\lim_{\epsilon \rightarrow 0}\int \textrm{d}^{d}y K(x,y,\epsilon)G_{ijkl}(x)\frac{1}{\sqrt{\gamma}(x)}\frac{\delta }{\delta \gamma_{ij}(x)}\left(\frac{1}{\sqrt{\gamma}(y)}\frac{\delta C[\gamma]}{\delta \gamma_{kl}(y)}\right)\right)Z[\gamma],\nn \\
&= -\alpha_{0}\lim_{\epsilon\rightarrow 0}\left(\left(\frac{d(d^{2}-3)}{(d-1)}\right)\frac{\epsilon^{\frac{d}{2}-1}}{4}K(x,x;\epsilon)+\frac{2\epsilon^{\frac{d}{2}}}{d}\left(\nabla^{2}_{(x)}+\xi R_{(x)}\right)K(x,x;\epsilon)\right)Z[\gamma].\end{aligned}$$ Here $$\xi=-\left(\frac{3d^{3}-4d^{2}-9d+14}{2d(d-1)}\right).$$ Then, the heat equation implies that we can write $$\begin{aligned}
&\lim_{\epsilon\rightarrow 0}\alpha_{0}\left(\left(\frac{d(d^{2}-3)}{(d-1)}\right)\frac{\epsilon^{\frac{d}{2}-1}}{4}K(x,x;\epsilon)+\frac{2\epsilon^{\frac{d}{2}}}{d}\left(\nabla^{2}_{(x)}+\xi R_{(x)}\right)K(x,x;\epsilon)\right)\nn \\
&=\alpha_{0}\lim_{\epsilon\rightarrow 0}\left(\left(\frac{d(d^{2}-3)}{(d-1)}\right)\frac{\epsilon^{\frac{d}{2}-1}}{4}K(x,x;\epsilon)+\frac{2\epsilon^{\frac{d}{2}}}{d}\partial_{\epsilon}K(x,x;\epsilon)\right)=\alpha_{0}R(x).\end{aligned}$$
[10]{}
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[^1]: We ignore additional internal directions in this work.
[^2]: Note that the signs of the first two terms in differ from equation [@Hartman:2018tkw eq. (3.3)] since the extrinsic curvature is defined there with an opposite sign. The Gibbons-Hawking term in [@Hartman:2018tkw eq. (A.1)] also appears with a minus sign as opposed to our equation where it appears with a plus sign.
[^3]: In Section \[sec:holography\], $\gamma_{ij}$ refers to this induced metric and we suppress the explicit dependence on $(r_c,x)$. Later, in the field theory part, Section \[sec:field-theory\], we will work with boundary $\gamma_{ij}$ related by factor of $r^2_c$. We hope that notation should be clear depending on the context.
[^4]: Here $\gamma_{ij}$ refers to the metric on a unit sphere and all geometric quantities are computed using this metric.
[^5]: This comes form $T^{bulk}_{ij}=r^{2-d}_c T^{bdr}_{ij}$ and our definitions of $\omega$’s.
[^6]: This can be derived using the relation $a_d r_c^{d-2}=\alpha_d \lambda^{\frac{2-d}{d}}$ and $a_d=\frac{1}{8\pi G_N (d-2)}$ of [@Hartman:2018tkw]. Note that [@Hartman:2018tkw] works with $l=1$ and therefore powers of $l$ need to be appropriately reinstated.
[^7]: These additional counterterms have not been considered in [@Hartman:2018tkw].
[^8]: We would like to stress that, in this section, the Callan-Symanzik equation with only the $TT$ deformation is our starting point and we argue how the full flow equation in curved background emerges from the regularisation procedure of defining the $TT$ operator. We are *not* providing a prescription or an RG scheme that would justify the use of CS with only $TT$. We would like to thank Edgar Shaghoulian for correspondence and clarifications on this point.
[^9]: We assume no other deforming operators are present.
[^10]: The main advantage of the $q$ variable here is the canonical kinetic term.
[^11]: We also set $\hbar=1$ at the end.
[^12]: As shown in [@Caputa:2018asc], the full WDW wavefunction is an Airy function in 3$d$. In $2d$, the full solution can be written in terms of the $_1F_1$-hypergeometric function, whilst in 4$d$ it is the parabolic cylinder function or a Hermite polynomial upon variable transformations.
[^13]: We will drop the expectation values for simplicity of the notation
[^14]: Using $(\hat{T}^b)_{ij}\to (T^b)_{ij}+a_d r^{d-2}_c C_{ij}$. Note that when translating $C_{ij}$ into field theory, in different dimensiosn, its different components can have a different scaling with $r_c$. Namely, in our examples, in 4d we have $C_{ij}=G_{ij}=G^b_{ij}$ but the extra term in $6$ and $7$ dimensions has a scaling $r^{-2}_c$.
[^15]: Note that the order of limits here is to first take $\epsilon\rightarrow 0$ with $N$ fixed and then taking $N\rightarrow \infty$ at the end.
|
---
abstract: |
When studying safety properties of (formal) protocol models, it is customary to view the scheduler as an adversary: an entity trying to falsify the safety property. We show that in the context of security protocols, and in particular of anonymizing protocols, this gives the adversary too much power; for instance, the contents of encrypted messages and internal computations by the parties should be considered invisible to the adversary.
We restrict the class of schedulers to a class of admissible schedulers which better model adversarial behaviour. These admissible schedulers base their decision solely on the past behaviour of the system that is visible to the adversary.
Using this, we propose a definition of anonymity: for all admissible schedulers the identity of the users and the observations of the adversary are independent stochastic variables. We also develop a proof technique for typical cases that can be used to proof anonymity: a system is anonymous if it is possible to ‘exchange’ the behaviour of two users without the adversary ‘noticing’.
author:
- 'Flavio D. Garcia'
- Peter van Rossum
- Ana Sokolova
title: Probabilistic Anonymity and Admissible Schedulers
---
Introduction
============
Systems that include probabilities and nondeterminism are very convenient for modelling probabilistic (security) protocols. Nondeterminism is highly desirable feature for modelling implementation freedom, action of the environment, or incomplete knowledge about the system.\
It is often of use to analyze probabilistic properties of such systems as for example “in 30% of the cases sending a message is followed by receiving a message” or “the system terminates successfully with probability at least 0.9”. Probabilistic anonymity [@bp_2005_probabilistic] is also such a property. In order to be able to consider such probabilistic properties we must first eliminate the nondeterminism present in the models. This is usually done by entities called schedulers or adversaries. It is common in the analysis of probabilistic systems to say that a model with nondeterminism and probability satisfies a probabilistic property if and only if it satisfies it no matter in which way the nondeterminism was resolved, i.e., for *all possible schedulers*.\
On the other hand, in security protocols, adversaries or schedulers are malicious entities that try to break the security of the protocol. Therefore, allowing just any scheduler is inadmissible. We show that the well-known Chaum’s Dining Cryptographers (DC) protocol [@cha_1988_dining] is not anonymous if we allow for all possible schedulers. Since the protocol is well-known to be anonymous, this shows that for the treatment of probabilistic security properties, in particular probabilistic anonymity, the general approach to analyzing probabilistic systems does not directly fit.\
We propose a solution based on restricting the class of all schedulers to a smaller class of *admissible schedulers*. Then we say that a probabilistic security property holds for a given model, if the property holds after resolving the nondeterminism under *all admissible schedulers*.
Probabilistic Automata
======================
In this section we gather preliminary notions and results related to probabilistic automata [@SL94:concur; @Seg95:thesis]. Some of the formulations we borrow from [@Sok05:thesis] and [@Che06:thesis]. We shall model protocols with probabilistic automata. We start with a definition of probability distribution.
\[PrDisDef\] A function $\mu \colon S \to [0,1]$ is a discrete probability distribution, or distribution for short, on a set $S$ if $\sum_{x \in S}
\mu(x) = 1$. The set $\{x \in S|\ \mu(x) \gr 0\}$ is the support of $\mu$ and is denoted by $\operatorname{supp}(\mu)$. By $\mathcal{D}(S)$ we denote the set of all discrete probability distributions on the set $S$.
We use the simple probabilistic automata [@SL94:concur; @Seg95:thesis], or MDP’s [@Bellman_1957_markov] as models of our probabilistic processes. These models are very similar to the labelled transition systems, with the only difference that the target of each transition is a distribution over states instead of just a single next state.
\[ProbAutDef\] A probabilistic automaton is a triple ${\mbox{$\mathcal A$}}= \langle S , A , \alpha \rangle$ where:
- $S$ is a set of states.
- $A$ is a set of actions or action labels.
- $\alpha$ is a transition function $\alpha: S \to {\mathcal{P}}(A \times {{\operatorname{{\mathcal{D}}}}}S)$.
A terminating state of $\mathcal A$ is a state with no outgoing transition, i.e. with $\alpha(s) = \emptyset$. We might sometimes also specify an initial state $s_0 \in S$ of a probabilistic automaton ${\mbox{$\mathcal A$}}$. We write $s \stackrel{a}{\to} \mu$ for $(a,\mu) \in \alpha(s), \
s\in S$. Moreover, we write $s \stackrel{a,\mu}\leadsto t$ for $s, t \in S$ whenever $s \stackrel{a}{\to} \mu$ and $\mu(t) \gr 0$.
We will also need the notion of a fully probabilistic system.
\[FProbAutDef\] A fully probabilistic automaton is a triple ${\mbox{$\mathcal A$}}= \langle S , A , \alpha \rangle$ where:
- $S$ is a set of states.
- $A$ is a set of actions or action labels.
- $\alpha$ is a transition function $\alpha: S \to {{\operatorname{{\mathcal{D}}}}}(A \times S) + 1$.
Here $1 = \{*\}$ denotes termination, i.e., if $\alpha(s) = *$ then $s$ is a terminating state. It can also be understood as a zero-distribution i.e. $\alpha(s)(a,t) = 0$ for all $a \in A$ and $t \in S$. By $s_0 \in S$ we sometimes denote an initial state of ${\mbox{$\mathcal A$}}$. We write $s {\to} \mu$ for $\mu = \alpha(s), \ s\in S$. Moreover, we write $s \stackrel{a}\leadsto t$ for $s, t \in S$ whenever $s
{\to} \mu$ and $\mu(a,t) \gr 0$.
A major difference between the (simple) probabilistic automata and the fully probabilistic ones is that the former can express nondeterminism. In order to reason about probabilistic properties of a model with nondeterminism we first resolve the nondeterminism with help of schedulers or adversaries – this leaves us with a fully probabilistic model whose probabilistic behaviour we can analyze. We explain this in the sequel.\
\[PathsDef\] A path of a *probabilistic automaton* $\mathcal A$ is a sequence $$\pi = s_0 \stackrel{a_1,\mu_1}{\to} s_1 \stackrel{a_2,\mu_2}{\to} s_2 \dots$$ where $s_i \in S$, $a_i \in A$ and $s_i \stackrel{a_{i+1},\mu_{i+1}}{\leadsto} s_{i+1}$.
A path of a *fully probabilistic automaton* $\mathcal A$ is a sequence $$\pi = s_0 \stackrel{a_1}{\to} s_1 \stackrel{a_2}{\to} s_2 \dots$$ where again $s_i \in S$, $a_i \in A$ and $s_i \stackrel{a_{i+1}}{\leadsto} s_{i+1}$.
A path can be finite in which case it ends with a state. A path is complete if it is either infinite or finite ending in a terminating state. We let $\operatorname{last}(\pi)$ denote the last state of a finite path $\pi$, and for arbitrary path $\operatorname{first}(\pi)$ denotes its first state. A trace of a path is the sequence of actions in $A^{*} \cup A^{\infty}$ obtained by removing the states (and the distributions), hence above $\operatorname{trace}(\pi) = a_1a_2\ldots$. The length of a finite path $\pi$, denoted by $|\pi|$ is the number of actions in its trace. Let ${{\operatorname{{Paths}}}}(\mathcal A)$ denote the set of all paths, ${{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$ the set of all finite paths, and ${{\operatorname{{CPaths}}}}(\mathcal A)$ the set of all complete paths of an automaton $\mathcal A$.
Paths are ordered by the prefix relation, which we denote by $\leq$.
Let $\mathcal A$ be a (fully) probabilistic automaton and let $\pi_i$ for $i \geq 0$ be finite paths of $\mathcal A$ all starting in the same initial state $s_0$ and such that $\pi_i \leq \pi_j$ for $i \leq j$ and $|\pi_i| = i$, for all $i \geq 0$. Then by $\pi = \lim_{i \to
\infty}{\pi_i}$ we denote the infinite complete path with the property that $\pi_i \leq \pi$ for all $i\geq 0$.
\[ConeDef\] Let $\mathcal A$ be a (fully) probabilistic automaton and let $\pi \in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$ be given. The cone generated by $\pi$ is the set of paths $$C_\pi = \{ \pi'\in {{\operatorname{{CPaths}}}}(\mathcal A) \mid \pi \leq \pi'\}.$$
From now on we fix an initial state. Given a fully probabilistic automaton $\mathcal A$ with an initial state $s_0$, we can calculate the probability-value denoted by ${{\operatorname{\mathbf{P}}}}(\pi)$ of any finite path $\pi$ starting in $s_0$ as follows. $$\begin{aligned}
{{\operatorname{\mathbf{P}}}}(s_0) & = & 1\\
{{\operatorname{\mathbf{P}}}}(\pi \stackrel{a}{\to} s) & = & {{\operatorname{\mathbf{P}}}}(\pi)\cdot \mu(a,s) \quad \text{~where~} \operatorname{last}(\pi) \to \mu\end{aligned}$$
Let $\Omega_{\mathcal A} = {{\operatorname{{CPaths}}}}(\mathcal A)$ be the sample space, and let $\mathcal F_{\mathcal A}$ be the smallest $\sigma$-algebra generated by the cones. The following proposition (see [@Seg95:thesis; @Sok05:thesis]) states that ${{\operatorname{\mathbf{P}}}}$ induces a unique probability measure on $\mathcal F_{\mathcal A}$.
\[PMeasProp\] Let $\mathcal A$ be a fully probabilistic automaton and let ${{\operatorname{\mathbf{P}}}}$ denote the probability-value on paths. There exists a unique probability measure on $\mathcal F_{\mathcal A}$ also denoted by ${{\operatorname{\mathbf{P}}}}$ such that ${{\operatorname{\mathbf{P}}}}(C_\pi) = {{\operatorname{\mathbf{P}}}}(\pi)$ for every finite path $\pi$.
This way we are able to measure the probability of certain events described by sets of paths in an automaton with no nondeterminism. Since our models include nondeterminism, we will first resolve it by means of schedulers or adversaries. Before we define adversaries note that we can describe the set of all sub-probability distributions on a set $S$ by ${{\operatorname{{\mathcal{D}}}}}(S +1)$. These are functions whose sum of values on $S$ is not necessarily equal to 1, but it is bounded by 1.
A scheduler for a probabilistic automaton $\mathcal A$ is a function $$\xi \colon {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A) \to {{\operatorname{{\mathcal{D}}}}}(A \times {{\operatorname{{\mathcal{D}}}}}(S) + 1)$$ satisfying $\xi(\pi)(a, \mu) \gr 0$ implies $\operatorname{last}(\pi) \stackrel{a}{\to} \mu$, for each finite path $\pi$. By ${{\operatorname{{Sched}}}}(\mathcal A)$ we denote the set of all schedulers for $\mathcal A$.
Hence, a scheduler according to the previous definition imposes a probability distribution on the possible non-deterministic transitions in each state. Therefore it is randomized. It is history dependent since it takes into account the path (history) and not only the current state. It is partial since it gives a sub-probability distribution, i.e., it may halt the execution at any time.
A probabilistic automaton $\mathcal A = \langle S, A, \alpha\rangle$ together with a scheduler $\xi$ determine a fully probabilistic automaton $$\mathcal A_\xi = \langle {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A), A, \alpha_\xi \rangle.$$ Its set of states are the finite paths of $\mathcal A$, its initial state is the initial state of $\mathcal A$ (seen as a path with length 1), its actions are the same as those of $\mathcal A$, and its transition function $\alpha_\xi$ is defined as follows. For any $\pi \in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$, we have $\alpha_\xi(\pi) \in
{{\operatorname{{\mathcal{D}}}}}(A\times {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)) + 1$ as $$\alpha_\xi(\pi)(a,\pi') = \left\{\begin{array}{ll}
\xi(\pi)(a, \mu)\cdot \mu(s) & \,\,\, \pi' = \pi \stackrel{a,\mu}{\to} s\\
0 & \,\,\, \text{otherwise}
\end{array}\right.$$
Given a probabilistic automaton $\mathcal A$ and a scheduler $\xi$, we denote by ${{\operatorname{\mathbf{P}}}}_\xi$ the probability measure on sets of complete paths of the fully probabilistic automaton $\mathcal A_\xi$, as in Proposition \[PMeasProp\]. The corresponding $\sigma$-algebra generated by cones of finite paths of $\mathcal A_\xi$ we denote by $\Omega_\xi$. The elements of $\Omega_\xi$ are measurable sets.
By $\Omega$ we denote the $\sigma$-algebra generated by cones of finite paths of $\mathcal A$ (without fixing the scheduler!) and also call its elements measurable sets, without having a measure in mind. Actually, we will now show that any scheduler $\xi \in {{\operatorname{{Sched}}}}(\mathcal A)$ induces a measure ${{\operatorname{\mathbf{P}}}}_{(\xi)}$ on a certain $\sigma$-algebra $\Omega_{(\xi)}$ of paths in $\mathcal A$ such that $\Omega \subseteq
\Omega_{(\xi)}$. Hence, any element of $\Omega$ can be measured by any of these measures ${{\operatorname{\mathbf{P}}}}_{(\xi)}$. We proceed with the details.
Define a function $f: {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A_\xi) \to {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$ by $$\label{PathsFuncEq}
f(\hat{\pi}) = \operatorname{last}(\hat{\pi})$$ for any $\hat{\pi} \in \mathcal A_\xi$. The function $f$ is well-defined since states in $\mathcal A_\xi$ are the finite paths of $\mathcal
A$. Moreover, we have the following property.
\[PrefFLem\] For any $\hat\pi_1, \hat\pi_2 \in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A_\xi)$ we have $$\hat\pi_1 \leq \hat\pi_2\quad \iff \quad f(\hat\pi_1) \leq f(\hat\pi_2)$$ where the order on the left is the prefix order in ${{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A_\xi)$ and on the right the prefix order in ${{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$.
By the definition of $\mathcal A_\xi$ we have that for $\pi, \pi' \in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$ i.e. states of $\mathcal A_\xi$: $\pi
\stackrel{a}{\leadsto} \pi'$ if and only if $\xi(\pi) \gr 0$ and $\operatorname{last}(\pi) \stackrel{a,\mu}{ s}$ in $\mathcal A$ for some $\mu$ and $s$. In other words if $\pi \stackrel{a}{\to} \pi'$, then $\pi \leq \pi'$ and $|\pi'| = |\pi| + 1$ i.e. $\pi'$ extends $\pi$ in one step. Therefore, if we have a path $\pi_0 \stackrel{a_0}{\to} \pi_1 \stackrel{a_1}{\to} \pi_2 \stackrel{a_2}{\to}\cdots$ in $\mathcal A_\xi$ , then for all its states: if $i \leq j$, then $\pi_i \leq \pi_j$ and $|\pi_j| = |\pi_i| + (j-i)$. So if $\hat\pi_1, \hat\pi_2 \in
{{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A_\xi)$ are such that $\hat\pi_1 \leq \hat\pi_2$, then $\operatorname{last}(\hat\pi_1)$ is a state in $\hat\pi_2$ and therefore we at once get $\operatorname{last}(\hat\pi_1)\leq \operatorname{last}(\hat\pi_2)$. For the opposite implication, again from the definition we notice that if a path $\hat\pi
\in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$ contains a state $\pi \in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A)$, then it also contains all prefixes of $\pi$ as states. Hence, if $\operatorname{last}(\hat\pi_1) \leq \operatorname{last}(\hat\pi_2)$ for $\hat{\pi_1}, \hat{\pi_2} \in {{\operatorname{{Paths}^{\leq\omega}}}}(\mathcal A_\xi)$, then $\operatorname{last}(\hat\pi_1)$ is a state in $\hat\pi_2$ and also all its prefixes are. Since all paths start in the initial state (path), this implies that $\hat\pi_1 \leq \hat\pi_2$.
\[InjFCor\] The function $f$ defined by (\[PathsFuncEq\]) is injective.
By Lemma \[PrefFLem\] we can extend the function $f$ to $\hat f: {{\operatorname{{CPaths}}}}(\mathcal A_\xi) \to {{\operatorname{{CPaths}}}}(\mathcal A)$ by $$\hat f(\hat\pi) = \begin{cases}f(\hat\pi) & \pi \text{~is~finite}\\
\lim_{i \to infty} f(\hat\pi_i) & \hat\pi_i \leq \pi,\, |\hat\pi_i| = i\end{cases}$$ The properties from Lemma \[PrefFLem\] and Corollary \[InjFCor\] continue to hold for the extended function $\hat f$ as well. We will write $f$ for $\hat f$ as well.\
Recall that $\Omega_\xi$ denotes the $\sigma$-algebra on which the measure ${{\operatorname{\mathbf{P}}}}_\xi$ is defined. We now define a family of subsets $\Omega^{\xi}$ of ${{\operatorname{{CPaths}}}}({\mbox{$\mathcal A$}})$ by $$\label{OmegaXiEq}
\Omega^\xi = \{ \Pi\in {{\operatorname{{CPaths}}}}({\mbox{$\mathcal A$}}) \mid f^{-1}(\Pi) \in \Omega_\xi.$$
The following properties are instances of standard measure-theoretic results.
\[MeasurePropLem\] The family $\Omega^\xi$ is a $\sigma$-algebra on ${{\operatorname{{CPaths}}}}({\mbox{$\mathcal A$}})$ and by $${{\operatorname{\mathbf{P}}}}^\xi (\Pi) = {{\operatorname{\mathbf{P}}}}_\xi(f^{-1}(\Pi))$$ for $\Pi \in \Omega^\xi$ a measure on $\Omega^\xi$ is defined.
Recall that $\Omega$ denotes the $\sigma$-algebra on complete paths of ${\mbox{$\mathcal A$}}$ generated by the cones. We show that for any scheduler $\xi$, $\Omega \subseteq \Omega^\xi$. Hence, the measurable sets (elements of $\Omega$) are indeed measurable by the measure induced by any scheduler.
For any scheduler $\xi$, $\Omega \subseteq \Omega^\xi$.
Fix a scheduler $\xi$. Since $\Omega$ is generated by the cones it is enough to show that each cone is in $\Omega^\xi$. Let $C_{\pi_0,
\mathcal A}$ be a cone in ${{\operatorname{{CPaths}}}}({\mbox{$\mathcal A$}})$ generated by the finite path $\pi_0$, i.e. $$C_{\pi_0,\mathcal A} = \{\pi \in {{\operatorname{{CPaths}}}}({\mbox{$\mathcal A$}}) \mid \pi_0 \leq \pi\}.$$ We have $$\hat f^{-1}(C_{\pi_0,\mathcal A}) = \begin{cases}\emptyset & \pi_0 \not\in \hat f({{\operatorname{{CPaths}}}}({\mbox{$\mathcal A$}}))\\
C_{\hat\pi_0, \mathcal A_\xi} & \hat f(\hat\pi_0) = \pi_0\end{cases}.$$ by Lemma \[PrefFLem\]. Indeed, let $\pi_0 = \hat f(\hat\pi_0)$. Then $$\begin{aligned}
\hat f^{-1}(C_{\pi_0,\mathcal A}) & = & \{ \hat\pi \in {{\operatorname{{CPaths}}}}(\mathcal A_\xi) \mid \hat f(\hat\pi) \geq \pi_0\}\\
& = & \{ \hat\pi \in {{\operatorname{{CPaths}}}}(\mathcal A_\xi) \mid \hat f(\hat\pi) \geq \hat f(\hat\pi_0)\}\\
(\text{Lem.}~\ref{PrefFLem}) & = & \{ \hat\pi \in {{\operatorname{{CPaths}}}}(\mathcal A_\xi) \mid \hat\pi \geq \hat\pi_0\}\\
& = & C_{\hat\pi_0, \mathcal A_\xi}\end{aligned}$$
We next define two operations on probabilistic automata used for building composed models out of basic models: parallel composition and restriction. We compose probabilistic automata in parallel in the style of the process algebra ACP. That is, asynchronously with communication function given by a semigroup operation on the set of actions. This is the most general way of composing probabilistic automata in parallel (for an overview see [@SV04:voss]).
\[ParCompDef\] We fix an action set $A$ and a communication function $\cdot$ on $A$ which is a partial commutative semigroup. Given two probabilistic automata $\mathcal A_1 = \langle S_1, A, \alpha_1 \rangle$ and $\mathcal A_2 = \langle S_2, A, \alpha_2 \rangle$ with actions $A$, their parallel composition is the probabilistic automaton $\mathcal A_1 \parallel \mathcal A_2 = \langle S_1 \times S_2, A, \alpha\rangle$ with states pairs of states of the original automata denoted by $s_1 \parallel s_2$, the same actions, and transition function defined as follows. $s_1\parallel s_2 \stackrel{a}{\to} \mu$ if and only if one of the following holds
- $s_1 \stackrel{b}{\to} \mu_1$ and $s_2 \stackrel{c}{\to} \mu_2$ for some actions $b$ and $c$ such that $a = b\cdot c$ and\
$\mu = \mu_1 \cdot \mu_2$ meaning $\mu(t_1 \parallel t_2) = \mu_1(t_1)\cdot \mu_2(t_2)$.
- $s_1 \stackrel{a}{\to} \mu'$ where $\mu'(t_1) = \mu(t_1\parallel s_2)$ for all states $t_1$ of the first automaton.
- $s_2 \stackrel{a}{\to} \mu'$ where $\mu'(t_2) = \mu(s_1\parallel t_2)$ for all states $t_2$ of the second automaton.
Here, 1. represents a synchronous joint move of both of the automata, and 2. and 3. represent the possibilities of an asynchronous move of each of the automata. In case $s^0_1$ and $s^0_2$ are the initial states of $\mathcal A_1$ and $\mathcal A_2$, respectively, then the initial state of $\mathcal A_1 \parallel \mathcal A_2$ is $s^0_1 \parallel s^0_2$.
Often we will use input and output actions like $a?$ and $a!$, respectively, in the style of CCS.In such cases we assume that the communication is defined as hand-shaking $a? \cdot a! = \tau_a$ for $\tau_a$ a special invisible action.\
The operation of restriction is needed to prune out some branches of a probabilistic automaton that one need not consider. For example, we will commonly use restriction to get rid of parts of a probabilistic automaton that still wait on synchronization.
\[HidingDef\] Fix a subset $I \subseteq A$ of actions that are in the restricted set. Given an automaton $\mathcal A = \langle S, A, \alpha\rangle$, the automaton obtained from $\mathcal A$ by restricting the actions in $I$ is $\mathcal R_I(\mathcal A) = \langle S, A \setminus I,
\alpha'\rangle$ where the transitions of $\alpha'$ are defined as follows: $s \stackrel{a}{\to} \mu$ in $\mathcal R_I(\mathcal A)$ if and only if $s \stackrel{a}{\to} \mu$ in $\mathcal A$ and $a \not\in I$.
We now define bisimilarity - a behaviour equivalence on the states of a probabilistic automaton. For that we first need the notion of relation lifting.
Let $R$ be an equivalence relation on the set of states $S$ of a probabilistic automaton ${\mbox{$\mathcal A$}}$. Then $R$ lifts to a relation $\equiv_R$ on the set ${{\operatorname{{\mathcal{D}}}}}(S)$, as follows: $$\mu \equiv_R \nu \iff \sum_{s \in C} \mu(s) = \sum_{s\in C} \nu(s)$$ for any equivalence class $C \in S/R$.
Let $\cal A = \langle S, A, \alpha\rangle$ be a probabilistic automaton. An equivalence $R$ on its set of states $S$ is a bisimulation if and only if whenever $\langle s, t \rangle \in R$ we have\
if $s \stackrel{a}{\to} \mu_s$, then there exists $\mu_t$ such that $t \stackrel{a}{\to} \mu_t$ and $\mu_s \equiv_R \mu_t$.\
Two states $s, t \in S$ are bisimilar, notation $s \sim t$ if they are related by some bisimulation relation $R$.
Note that bisimilarity $\sim$ is the largest bisimulation on a given probabilistic automaton ${\mbox{$\mathcal A$}}$.
Anonymizing Protocols
=====================
Dining cryptographers {#sec:dining-crypt}
---------------------
The canonical example of an anonymizing protocol is Chaum’s Dining Cryptographers [@cha_1988_dining]. In Chaum’s introduction to this protocol, three cryptographers are sitting down to dine in a restaurant, when the waiter informs them that the bill has already been paid anonymously. They wonder whether one of them has paid the bill in advance, or whether the NSA has done so. Respecting each other’s right to privacy, the carry out the following protocol. Each pair of cryptographers flips a coin, invisible to the remaining cryptographer. Each cryptographer then reveals whether or not the two coins he say were equal or unequal. However, if a cryptographer is paying, he states the opposite. An even number of “equals” now indicates that the NSA is paying; an odd number that one of the cryptographers is paying.
Formally, Chaum states the result as follows. (Here we are restricting to the case with 3 cryptographers; Chaum’s version is more general.) Here, ${{\mathbb F}_2}$ is the field of two elements.
Let $K$ be a uniformly distributed stochastic variable over ${{\mathbb F}_2}^3$. Let $I$ be a stochastic variable over ${{\mathbb F}_2}^3$, taking only values in $\{ (1,0,0),\allowbreak (0,1,0),\allowbreak
(0,0,1),\allowbreak (0,0,0) \}$. Let $A$ be the stochastic variable over ${{\mathbb F}_2}^3$ given by $A = (I_1 + K_2 + K_3, K_1 + I_2 + K_3, K_1
+ K_2 + I_3)$. Assume that $K$ and $I$ are independent. Then $$\forall a \in {{\mathbb F}_2}^3\; \forall i \in \{1,2,3\}:
{{\mathbb P}}[ I = i ] > 0 \implies {{\mathbb P}}[A = a {\;|\;}I = i] = \tfrac14$$ and hence $$\forall a \in {{\mathbb F}_2}^3\; \forall i \in \{1,2,3\}:
{{\mathbb P}}[ I = i ] > 0 \implies {{\mathbb P}}[A = a {\;|\;}I = i] = {{\mathbb P}}[A = a].
\tag*{\qed}$$
In terms of the storyline, $K$ represents the coin flips, $I$ represents which cryptographer (if any) is paying, and $A$ represents the every cryptographer says.
We will now model this protocol as a probabilistic automaton. We will construct it as a parallel composition of seven components: the Master, who decides who will pay, the three cryptographers Crypt${}_i$, and the three coins Coin$_{i}$. The action $p_i!$ is used by the Master to indicate to Crypt$_{i}$ that he should pay; the action $n_i!$ to indicate that he shouldn’t. If no cryptographer is paying, the NSA is paying, which is not explicitly modelled here. The coin Coin${}_i$ is shared by Crypt${}_i$ and Crypt${}_{i-1}$ (taking the -1 modulo 3); the action $h_{i,j}!$ represents Coin${}_i$ signalling to Crypt${}_j$ that the coin was heads and similarly $t_{i,j}!$ signals tails. At the end, the cryptographers state whether or not the two coins they saw were equal or not by means of the actions $a_i!$ (agree) or $d_i$ (disagree). $$\PandocStartInclude{master.tex}\PandocEndInclude{input}{777}{21}
\qquad
\PandocStartInclude{coin.tex}\PandocEndInclude{input}{779}{19}
\qquad
\PandocStartInclude{crypt.tex}\PandocEndInclude{input}{781}{20}$$ Now DC is the parallel composition of Master, Coin${}_0$, Coin${}_1$, Coin${}_2$, Crypt${}_0$, Crypt${}_1$, and Crypt${}_2$ with all actions of the form $p_i$, $n_i$, $h_{i,j}$, and $t_{i,j}$ hidden.
Note that in Chaum’s version, there is no assumption on the probability distribution of $I$; in our version this is modelled by the fact that the Master makes a non-deterministic choice between the four options. Since we allow probabilistic schedulers, we later recover all possible probability distributions about who is paying, just as in the original version. Independence between the choice of the master and the coin flips ($I$ and $K$ in Chaum’s version) comes for free in the automata model: distinct probabilistic choices are always assumed to be independent.
In Section \[sec:PurelyProbabilisticSystems\] we formulate what it means for DC (or more general, for an anonymity automaton) to be anonymous.
Voting
------
At a very high level, a voting protocol can be seen as a blackbox that inputs the voters’ votes and outputs the result of the vote. For simplicity, assume the voters vote yes (1) or no (0), do not abstain, and that the numbers of voters is known. The result then is the number of yes-votes. $$\PandocStartInclude{voting.tex}\PandocEndInclude{input}{804}{19}$$ In such a setting, it is conceivable that an observer has some a-priori knowledge about which voters are more likely to vote yes and which voters are more likely to vote no. Furthermore, there definitely is a-posteriori knowledge, since the vote result is made public. For instance, in the degenerate case where all voters vote the same way, everybody’s vote is revealed. What we expect here from the voting protocol is not that the adversary has no knowledge about the votes (since he might already have a-priori knowledge), and also not that the adversary does not gain any knowledge from observing the protocol (since the vote result is revealed), but rather that observing the protocol does not augment the adversary’s knowledge beyond learning the vote result.
For the purely probabilistic case, this notion of anonymity is formalized in Section \[sec:PurelyProbabilisticSystems\].
Anonymity for Purely Probabilistic Systems {#sec:PurelyProbabilisticSystems}
==========================================
This section defines anonymity systems and proposes a definition for anonymity in its simplest configuration, i.e., for purely probabilistic systems. Purely probabilistic systems are simpler because there is no need for schedulers. Throughout the following sections, this definitions will be incrementally modified towards a more general setting.
Let $M = \langle S, {{\text{Act}}}, \alpha \rangle$ be a fully probabilistic automaton. An *anonymity system* is a triple $\langle M, I, \{ A_i
\}_{i \in I}, {{\text{Act}_O}}\rangle$ where
1. $I$ is the set of user identities,
2. $A_i$ is any measurable subset of ${{\operatorname{{CPaths}}}}(M)$ such that $A_i \cap A_j = \emptyset$ for $i
\not= j$.
3. ${{\text{Act}_O}}\subseteq {{\text{Act}}}$ is the set of observable actions.
4. ${Otrace}(\pi)$ is the sequence of elements in ${{\text{Act}_O}}$ obtained by removing form $\operatorname{trace}(\pi)$ the elements in ${{\text{Act}}}\setminus {{\text{Act}_O}}$.
Define $O$ as the set of observations, i.e., $O = \{\operatorname{trace}(\pi) {\;|\;}\pi \in {{\operatorname{{Paths}}}}(M)\}$. We also define $A = \bigcup_{i \in I} A_i$.
Intuitively, the $A_i$s are properties of the executions that the system is meant to hide. For example, in the case of the dining cryptographers $A_i$ would be “cryptographer $i$ payed”; in a voting scheme “voter $i$ voted for candidate $c$”, etc. Therefore, for the previous examples, the predicate $A$ would be “some of the cryptographers payed” or “the vote count” respectively.
Next, we propose a definition of anonymity for a purely probabilistic systems. We deviate from the definition proposed by Bhargava and Palamidessi [@bp_2005_probabilistic] for what we consider a more intuitive definition: We say that an anonymity system is anonymous if the probability of seeing a observation is independent of who performed the anonymous action ($A_i$), given that some anonymous action took place ($A$ happened). The formal definition follows.
A system $\langle M, I, \{ A_i \}_{i \in I}, {{\text{Act}_O}}\rangle$ is said to be anonymous if $$\begin{aligned}
\forall i \in I.\forall o \in O. {{\mathbb P}}[\pi \in A] > 0 \implies & {{\mathbb P}}[{Otrace}(\pi) = o \;\land\; \pi \in A_i {\;|\;}\pi \in A] =\\
& {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A]\, {{\mathbb P}}[\pi \in A_i{\;|\;}\pi \in A].\end{aligned}$$
In the above probabilities, $\pi$ is drawn from the probability space $\operatorname{Paths}(M)$.
The following lemma shows that this definition is equivalent to the one proposed in Bhargava and Palamidessi [@bp_2005_probabilistic].
A anonymity system is anonymous if and only if $$\begin{aligned}
\forall i,j \in I.\forall o \in O.({{\mathbb P}}[\pi \in A_i]>0 \land {{\mathbb P}}[\pi \in A_j]>0) \implies & {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_i] =\\
& {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_j] \;\end{aligned}$$
The only if part is trivial. For the if part we have $$\begin{aligned}
{{\mathbb P}}[ &{Otrace}(\pi) = o {\;|\;}\pi \in A]\, {{\mathbb P}}[\pi \in A_i {\;|\;}\pi \in A] \\
&= {{\mathbb P}}[\pi \in A_i {\;|\;}\pi \in A] \; \sum_{j \in I} {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_j \cap A] \; {{\mathbb P}}[\pi \in A_j {\;|\;}\pi \in A]\\
\intertext{\qquad\quad(since $A_i \cap A_j = \emptyset,i \not= j$)}
&= {{\mathbb P}}[\pi \in A_i {\;|\;}\pi \in A] \; \sum_{j \in I} {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_j] \; {{\mathbb P}}[\pi \in A_j {\;|\;}\pi \in A]\\
\intertext{\qquad\quad(by definition of $\pi \in A$)}
&= {{\mathbb P}}[\pi \in A_i {\;|\;}\pi \in A] \; {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_i] \;\sum_{j \in I} {{\mathbb P}}[\pi \in A_j {\;|\;}\pi \in A]\\
\intertext{\qquad\quad(by hypothesis)}
&= {{\mathbb P}}[\pi \in A_i {\;|\;}\pi \in A] \; {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_i] \;\frac{\sum_{j \in I} {{\mathbb P}}[\pi \in A_j]}{{{\mathbb P}}[\pi \in A]}\\
\intertext{\qquad\quad(since $A_j \subseteq A$)}
&= {{\mathbb P}}[\pi \in A_i {\;|\;}\pi \in A] \; {{\mathbb P}}[{Otrace}(\pi) = o {\;|\;}\pi \in A_i] \\
&= \frac{{{\mathbb P}}[\pi \in A_i]}{{{\mathbb P}}[\pi \in A]} \; \frac{{{\mathbb P}}[{Otrace}(\pi) = o \land \pi \in A_i]}{{{\mathbb P}}[\pi \in A_i]}\\
&= {{\mathbb P}}[{Otrace}(\pi) = o \land \pi \in A_i {\;|\;}\pi \in A]
\intertext{\qquad\quad(since $A_i \subseteq A$)}\end{aligned}$$ which concludes the proof.
Anonymity for Probabilistic Systems
===================================
We now try to extend the notion of anonymity to probabilistic automata that are not purely probabilistic, but that still contain some non-deterministic transitions.
One obvious try is to say that $M$ is anonymous if $M_\xi$ is anonymous for all schedulers $\xi$ of $M$. The following automaton $M$ and scheduler $\xi$ show that this definition would be problematic. $$\PandocStartInclude{choice1.tex}\PandocEndInclude{input}{898}{22}
\qquad
\PandocStartInclude{choice2.tex}\PandocEndInclude{input}{900}{22}$$ Here $a_1$ and $a_2$ are invisible actions; they represent which user performed the action that was to remain hidden. The actions $x_1$ and $x_2$ are observable. Intuitively, because the adversary cannot see the messages $a_1$ and $a_2$, she cannot learn which user actually performed the hidden action. On the right hand side $M_\xi$ is shown and the branches the scheduler does not take are indicated by dotted arrows. Now ${{\mathbb P}}_\xi[a_1 {\;|\;}x_1] = 1$, but ${{\mathbb P}}_\xi[a_1] = \frac{1}{2}$, showing that with this particular scheduler $M_\xi$ is not anonymous.
Note that this phenomenon can easily occur as a consequence of communication non-determinism. For instance, consider the following three automata and their parallel composition in which $c?$ and $c!$ are hidden. In this example the order of the messages $x_1$ and $x_2$ depends on a race-condition, but a scheduler can make it depend on whether $a_1$ or $a_2$ was taken. I.e., there exists a scheduler $\xi$ such that ${{\mathbb P}}_\xi[x_1 x_2 {\;|\;}a_1] = {{\mathbb P}}_\xi[x_2 x_1 {\;|\;}a_2] = 1$ and hence ${{\mathbb P}}_\xi[x_2 x_1 {\;|\;}a_1] = {{\mathbb P}}_\xi[x_1 x_2
{\;|\;}a_2] = 0$. $$\PandocStartInclude{comm1.tex}\PandocEndInclude{input}{916}{20}
\qquad\qquad
\PandocStartInclude{comm2.tex}\PandocEndInclude{input}{918}{20}$$ In fact, the Dining Cryptographers example from Section \[sec:dining-crypt\] suffers from exactly the same problem. The order in which the cryptographers say ${\operatorname{\it agree}}_i$ or ${\operatorname{\it disagree}}_i$ is determined by the scheduler and it is possible to have a scheduler that makes the paying cryptographer, if any, go last.
In [@bp_2005_probabilistic], a system $M$ is called anonymous if for all schedulers $\zeta$, $\xi$, for all observables $o$, and for all hidden actions $a_i$, $a_j$ such that ${{\mathbb P}}_\zeta[a_i] > 0$ and ${{\mathbb P}}_\xi[a_j] > 0$, ${{\mathbb P}}_\zeta[o {\;|\;}a_i] = {{\mathbb P}}_\xi[o {\;|\;}a_j]$. This definition, of course, has the same problems as above; in the Dining Cryptographers example in [@bp_2005_probabilistic] this is solved by fixing the order in which the the cryptographers say ${\operatorname{\it agree}}_i$ or ${\operatorname{\it disagree}}_i$. However, also a non-deterministic choice between two otherwise anonymous systems can become non-anonymous with this definition. For instance, let $P$ be some anonymous system. For simplicity, assume that $P$ is fully probabilistic (e.g., the Dining Cryptographers with a probabilistic master and a fixed scheduler) and let $P'$ be a variant of $P$ in which the visible actions have been renamed (e.g., the actions ${\operatorname{\it agree}}_i$ and ${\operatorname{\it disagree}}_i$ are renamed to ${\operatorname{\it equal}}_i$ and ${\operatorname{\it unequal}}_i$). Now consider the probabilistic automaton $M$ which non-deterministically chooses between $P$ and $P'$: $$\xymatrix{
& \circ \ar[dl] \ar[dr] \\
P\phantom{'} \drop\frm{-} && P' \drop\frm{-}
}$$ This automaton has only two schedulers: the one that chooses the left branch and then executes $P$ and the one that chooses the right branch and then executes $P'$. Let’s call these schedulers $l$ and $r$ respectively. Now pick any hidden action $a_i$ and observable $o$ such that ${{\mathbb P}}_l[o {\;|\;}a_i] > 0$. (e.g., $o = {\operatorname{\it agree}}_1 {\operatorname{\it disagree}}_2 {\operatorname{\it agree}}_3$ and $a_i = {\operatorname{\it pay}}_1$, for which ${{\mathbb P}}[o {\;|\;}{\operatorname{\it pay}}_1] =
\frac{1}{4})$. Then, nevertheless, ${{\mathbb P}}_r[o {\;|\;}a_i] = 0$, because the observation $o$ cannot occur in $P'$. So, even though intuitively this system should be anonymous, it is not so according to the definition in [@bp_2005_probabilistic].
Every time the problem is that the scheduler has access to information it shouldn’t have. When one specifies a protocol by giving a probabilistic automaton, an implementation of this protocol has to implement a scheduler as well. This is especially obvious if the non-determinism originates from communication. When we identify schedulers with adversaries, as is common, it becomes clear that the scheduler should not have access to too much information. In the next section we define a class of schedulers, called *admissible* schedulers that base their scheduling behavior on the information an adversary actually has access to: the observable history of the system.
Admissible Schedulers
=====================
As explained in the previous section, defining anonymity as a condition that should hold true for all possible schedulers is problematic. It is usual to quantify over all schedulers when showing theoretical properties of systems with both probabilities and non-determinism - for example we may say “no matter how the non-determinism is resolved, the probability of an event $X$ is at least $p$”. However, in the analysis of security protocols, for example with respect to anonymity, we would like to quantify over all possible “realistic” adversaries. These are not all possible schedulers as in our theoretic considerations since such a realistic adversary is not able to see all details of the probabilistic automaton under consideration. Hence, considering that the adversary is any scheduler enables the adversary to leak information where it normally could not. We call such schedulers interfering schedulers. This way protocols that are well-known to be anonymous turn out not to be anonymous. One such example is the dining cryptographers protocol explained above. We show that one gets a better definition of anonymity if one restricts the power of the schedulers, in a realistic way. In this section we define the type of schedulers with restricted power that we consider good enough for showing anonymity of certain protocols. We call these schedulers admissible.
Schedulers with restricted power have been treated in the literature. In general, as explained by Segala in [@Seg95:thesis], a scheduler with restricted power is given by defining two equivalences, one on the set of finite paths $\equiv_1$ and another one $\equiv_2$ on the set of possible transition, in this case ${{\operatorname{{\mathcal{D}}}}}(A\times S)$. Then a scheduler $\xi$ is oblivious relative to $\langle\equiv_1, \equiv_2\rangle$ if and only if for any two paths $\pi_1, \pi_2$ we have $$\pi_1 \equiv_1 \pi_2 \Longrightarrow \xi(\pi_1) \equiv_2 \xi(\pi_2).$$
Admissible schedulers based on bisimulation
-------------------------------------------
In this section we specify $\equiv_1$ and $\equiv_2$ and obtain a class of oblivious adversaries that suits the anonymity definition.
Defined $\equiv_1$ on the set of finite paths of an automaton $M$ as, $$\pi_1 \equiv_1 \pi_2 \iff \big(\operatorname{trace}(\pi_1) = \operatorname{trace}(\pi_2) \land \operatorname{last}(\pi_1) \sim \operatorname{last}(\pi_2)\big).$$ Recall that we defined $\equiv_R$ as the lifting of the equivalence relation $R$ on a set $S$ to an equivalence relation on ${{\operatorname{{\mathcal{D}}}}}(A\times
S)$. For $\equiv_2$ we take the equivalence $\equiv_{\sim}$ on ${{\operatorname{{\mathcal{D}}}}}(A\times S)$. This is well defined since bisimilarity is an equivalence. Hence, we obtain a class of oblivious schedulers relative $\langle\equiv_1, \equiv_{\sim}\rangle$. These schedulers we call admissible.
A scheduler is admissible if for any two finite paths $\pi_1$ and $\pi_2$ we have $$\big(\operatorname{trace}(\pi_1) = \operatorname{trace}(\pi_2) \land \operatorname{last}(\pi_1) \sim \operatorname{last}(\pi_2)\big)\Longrightarrow
\xi(\pi_1) \equiv_{\sim} \xi(\pi_2).$$
Intuitively, the definition of a admissible scheduler enforces that in cases when the schedular has observed the same history (given by the traces of the paths) and is in bisimilar states, it must schedule “the same” transitions up to bisimilarity.
Existence
---------
We now show that admissible schedulers do exist. In fact, we even show that admissible history-independent schedulers exist. A scheduler $\xi$ is history-independent if it is completely determined by its image of paths of length 0 i.e. if for any path $\pi$ it holds that $\xi(\pi) = \xi(\operatorname{last}(\pi))$.
There exists a admissible scheduler for every probabilistic automaton.
Take a probabilistic automaton $M$. We first show that there exists a map $\xi: S \to {{\operatorname{{\mathcal{D}}}}}(A\times S)\cup \{\bot\}$ with the property that $\xi(s) = \bot$ if and only if $s$ terminates and for all $s, t \in S$, if $s \sim t$, then $\xi(s) \equiv_\sim \xi(t)$.
Consider the set of partial maps $$\Xi = \left\{\xi: S \hookrightarrow {{\operatorname{{\mathcal{D}}}}}(A\times S)\cup \{\bot\} \left| \begin{array}{ll}
&\xi(s) = \bot \iff s \text{~terminates~},\\
& s \sim t \Longrightarrow \xi(s) \equiv_\sim \xi(t)\\
& \text{~for~} s, t \in dom(\xi)
\end{array}
\right\}\right..$$ This set is not empty since the unique partial map with empty domain belongs to it. We define an order $\leq$ on $\Xi$ in a standard way by $$\xi_1 \leq \xi_2 \iff \big( dom(\xi_1) \subseteq dom(\xi_2) \land \xi_2|_{dom(\xi_1)} = \xi_1 \big).$$
Consider a chain $(\xi_i)_{i \in I}$ in $\Xi$. Let $\xi = \cup_{i \in I} \xi_i$. This means that $dom(\xi) = \cup_{i\in I} dom(\xi_i)$ and if $x \in dom(\xi)$, then $\xi(x) = \xi_i(x)$ for $i \in I$ such that $x \in dom(\xi_i)$. Note that $\xi$ is well-defined since $(\xi_i)_{i
\in I}$ is a chain. Moreover, it is obvious that $\xi_i \leq \xi$ for all $i \in I$. We next check that $\xi \in \Xi$. Let $s, t \in
dom(\xi)$, such that $s \sim t$. Then $s \in dom(\xi_i)$ and $t \in dom(\xi_j)$ for some $i,j \in I$ and either $\xi_1 \leq \xi_2$ or $\xi_2 \leq \xi_1$. Assume $\xi_1 \leq \xi_2$. Then $s, t \in dom(\xi_j)$ and $\xi_j \in \Xi$ so we have that $\xi_j(s) \equiv_\sim
\xi_j(t)$ showing that $\xi(s) \equiv_\sim \xi(t)$ and we have established that $\xi \in \Xi$.
Hence, every ascending chain in $\Xi$ has an upper bound. By the Lemma of Zorn we conclude that $\Xi$ has a maximal element. Let $\sigma$ be such a maximal element in $\Xi$. We claim that $\sigma$ is a total map. Assume opposite, i.e., there exists $s \in S \setminus
dom(\sigma)$. If there exists a $t \in dom(\sigma)$ such that $s \sim t$ then we define a new partial scheduler $\sigma'$ as follows. If $\sigma(t) = \bot$ we put $\sigma'(s) = \bot$. If $\sigma(t) = \mu_t$, then, since $t \to \mu_t$ and $s \sim t$, there exists $\mu_s$ such that $s \to \mu_s$ and $\mu_t \equiv_\sim \mu_s$. In this case we put $\sigma'(s) = \mu_s$. Moreover, put $\sigma'(x) = \sigma(x)$ for $x
\in dom(\sigma)$. Then we have $\sigma'> \sigma$ and $\sigma' \in \Xi$ contradicting the maximality of $\sigma$. Hence $\sigma$ is a total map.
.
We are now ready to define anonymity for probabilistic systems, the formal definition follows.
A system $\langle M, I, \{ A_i \}_{i \in I}, {{\text{Act}_O}}\rangle$ is said to be anonymous if for all admissible schedulers $\xi$, for all $i \in
I$ and for all $o \in O$ $$\begin{aligned}
{{\mathbb P}}_\xi[\pi \in A] > 0 \implies & {{\mathbb P}}_\xi[{Otrace}(\pi) = o \;\land\; \pi \in A_i {\;|\;}\pi \in A] =\\
& {{\mathbb P}}_\xi[{Otrace}(\pi) = o {\;|\;}\pi \in A]\, {{\mathbb P}}_\xi[\pi \in A_i{\;|\;}\pi \in A].\end{aligned}$$
Anonymity Examples {#sec:examples}
==================
In the purely non-deterministic setting, anonymity of a system is often proved (or defined) as follows: take two users $A$ and $B$ and a trace in which user $A$ is “the culprit”. Now find a trace that looks the same to the adversary, but in which user $B$ is “the culprit” [@ho_2003_anonymity; @ghrp_2005_anonymity; @mvv_2004_anonymity; @HasuoK07a]. In fact, this new trace is often most easily obtained by switching the behavior of $A$ and $B$.
In this section, we make this technique explicit for anonymity in our setting, with mixed probability and non-determinism.
Let $M$ be a probabilistic automaton. A map $\alpha \colon S \to S$ is called an [*${{\text{Act}_O}}$-automorphism*]{} if $\alpha$ induces an automorphism of the automation $M_\tau$, which is a copy of $M$ with all actions not in ${{\text{Act}_O}}$ renamed to $\tau$.
The following result generalized the above-mentioned proof technique that is commonly used for a purely non-deterministic setting.
Consider an anonymity system $(M,I,{{\text{Act}_O}})$. Suppose that for every $i, j \in I$ there exists a ${{\text{Act}_O}}$-automorphism $\alpha \colon S \to S$ such that $\alpha(A_i) = A_j$. Then the system is anonymous.
Anonymity of the Dining Cryptographers {#anonymity-of-the-dining-cryptographers .unnumbered}
--------------------------------------
We can now apply the techniques from the previous section to the Dining Cryptographers. Concretely, we show that there exists a ${{\text{Act}_O}}$-automorphism exchanging the behaviour of the Crypt${}_1$ and Crypt${}_2$; by symmetry, the same holds for the other two combinations.
Consider the endomorphisms of Master and Coin${}_2$ indicated in the following figure. The states in the left copy that are not explicitly mapped (by a dotted arrow) to a state in the right copy are mapped to themselves. $$\hspace{3.3cm}\PandocStartInclude{mastermap.tex}\PandocEndInclude{input}{1117}{36}$$ $$\PandocStartInclude{coinmap.tex}\PandocEndInclude{input}{1121}{20}$$ Also consider the identity endomorphism on Crypt${}_i$ (for $i = 0, 1, 2$) and on Coin${}_i$ (for $i = 0, 1$). Taking the product of these seven endomorphisms, we obtain an endomorphism $\alpha$ of DC.
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---
abstract: 'We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet’s theorem in Diophantine approximation and prove very general results in this context.'
address: 'School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005, India'
author:
- Arijit Ganguly
- Anish Ghosh
title: 'Dirichlet’s theorem in function fields'
---
[^1]
Introduction
============
The set up
----------
Let $p$ be a prime and $q:= p^r$, where $r\in \mathbb{N}$, let $\mathbb{F}_q$ be the finite field of $q$ elements and consider the field of rational functions $\mathbb{F}_{q}(T)$. We define a function $|\cdot|: \mathbb{F}_{q}(T) \longrightarrow \mathbb{R}_{\geq 0}$ as follows. $$|0|:= 0\,\, \text{ and} \,\, \left|\frac{P}{Q}\right|:= e^{\displaystyle {\operatorname{deg}}P- {\operatorname{deg}}Q}
\text{ \,\,\,for all nonzero } P, Q\in \mathbb{F}_{q}[T]\,.$$ Clearly $|\cdot|$ is nontrivial, non-archimedian and a discrete absolute value in $\mathbb{F}_{q}(T)$. This absolute value gives rise to a metric in $\mathbb{F}_{q}(T)$.\
The completion field of $\mathbb{F}_{q}(T)$ with respect to this valuation is $\mathbb{F}_{q}((T^{-1}))$, the field of Laurent series over $\mathbb{F}_{q}$. The absolute value on $\mathbb{F}_{q}((T^{-1}))$, which we again denote by $|\cdot |$, is given as follows. Let $a \in \mathbb{F}_{q}((T^{-1}))$. For $a=0$, define $|a|=0$. If $a \neq 0$, then we can write $$a=\displaystyle \sum_{k\leq k_{0}} a_k T^{k}\,\,\mbox{where}\,\,\,\,k_0 \in \mathbb{Z},\,a_k\in \mathbb{F}_{q}\,\,\mbox{and}\,\, a_{k_0}\neq 0\,.$$ We define $k_0$ as the *degree* of $a$, which will be denoted by ${\operatorname{deg}}a$, and $|a|:= e^{{\operatorname{deg}}a}$. This clearly extends the absolute value $|\cdot|$ of $\mathbb{F}_{q}(T)$ to $\mathbb{F}_{q}((T^{-1}))$ and moreover, the extension remains non-archimedian and discrete like earlier. Let $\Lambda$ and $F$ denote $\mathbb{F}_{q}[T]$ and $\mathbb{F}_{q}((T^{-1}))$ respectively from now on. It is obvious that $\Lambda$ is discrete in $F$. For any $n\in \mathbb{N}$, $F^n$ is throughout assumed to be equipped with the *supremum norm* which is defined as follows $$||\mathbf{x}||:= \displaystyle \max_{1\leq i\leq n} |x_i|\text{ \,\,for all \,} \mathbf{x}=(x_1,x_2,...,x_n)\in F^{n}\,,$$
and with the topology induced by this norm. Clearly $\Lambda^n$ is discrete in $F^n$. Since the topology on $F^n$ considered here is the usual product topology on $F^n$, it follows that $F^n$ is locally compact as $F$ is locally compact. We shall also fix a Haar measure $\lambda$ on $F$.\
In this paper, we study analogues of Dirichlet’s theorem in Diophantine approximation and its improvability for vectors in $F^n$. An analogue of Dirchlet’s theorem for local fields of positive characteristic can be formulated as in the following:
\[thm:D\] Let $t$ be a nonnegative integer. For $\mathbf{y}:=(y_1,y_2,...,y_n)\in F^n$, there exist $q\in \Lambda \setminus \{0\}$ and $p\in \Lambda$ such that $$|y_1q_1+ y_2q_2+\cdot\cdot\cdot+y_nq_n-p|\,\textless \, \frac{1}{e^{nt}}\text{ \,\,and } \displaystyle
\max_{1\leq j\leq n} |q_j|\leq e^t\,.$$
The theorem above is clearly well known and Diophantine approximation in the context of local fields of positive characteristic has been extensively studied of late. We refer the reader to [@deM] for survey and [@AGP; @Kris; @Las1; @Las2] for more recent results. Indeed, the geometry of numbers, which can be used to prove Dirichlet’s theorem was developed in the context of function fields by Mahler [@Mahler] as early as the 1940’s. However, since we could not find a specific proof of the above in the literature, and in the interest of readability, we provide a proof in Section \[section:review\]. In fact we have proved there a stronger, namely a multiplicative statement (see Theorem \[thm:1.1\]). There are many interesting parallels and contrasts between the theory of Diophantine approximation over the real numbers and in positive characteristic. Many results hold in both settings, the main result of the paper being one such while there are some striking exceptions. For instance the theory of *badly approximable* numbers and vectors in positive characteristic offers several surprises: there is no analogue of Roth’s theorem, provided that the base field is finite, which we assume throughout this paper. We refer the reader to [@AB] for other results in this vein.
Improving Dirichlet’s theorem
-----------------------------
Following Kleinbock-Weiss [@KW1], the notion of “Dirichlet improvability" can now be introduced as follows. Let $0\,\textless\,\varepsilon \leq \frac{1}{e}$. A vector $\mathbf{y}:=(y_1,y_2,...,y_n)\in F^n$ is said to be *Dirichlet $\varepsilon$-improvable* if there is some $t_0\,\textgreater \,0$ such that for any choice of $n$ and nonnegative integers $t_1,t_2,...,t_n$ with $\max\{t,t_1,...,t_n\}\,\textgreater\,t_0$, where $t=t_1+
t_2+\,\cdot\,\cdot\,\cdot\,+t_n$, one can always find nonzero $(p,q_1,q_2,...,q_n)\in \Lambda \times \Lambda^n$ satisfying $$\displaystyle |y_1q_1+ y_2q_2+\cdot\cdot\cdot+y_nq_n-p|\,\textless \, \frac{\varepsilon}{e^t} \text{ and } |q_j|\,\textless\, \varepsilon e^{t_j}\text{ for } j=1,2,...,n\,.$$
Let ${\operatorname{DI}_{\varepsilon}}(n)$ denote the set of Dirichlet improvable vectors in $F^n$ or in ${{\mathbb R}}^n$, the context will make the field clear. Some remarks:\
1. In the definition above, we have invoked the more general, multiplicative analogue of Dirichlet’s thereorem, for which we provide a proof in Theorem \[thm:1.1\]. The results of this paper are valid in this, stronger, setting.\
2. This notion can be considered in greater generality, for systems of linear forms, as was done by Kleinbock and Weiss. We refer the reader to Definition \[defn:DI imp\].\
3. Dirichlet’s theorem can be formulated for global fields, i.e. one could consider number fields or finite extensions of positive characteristic fields. However, it is in general an open problem to determine the optimal constant in Dirichlet’s theorem in this setting, without which of course, the question of improvement does not arise. There are some cases where the constant can be determined. For example in [@GR], the theory of metric Diophantine approximation for certain *imaginary* quadratic extensions of function fields was developed. In these fields, an analogue of Dirichlet’s theorem with the same constant, i.e. $1$ holds, and it is plausible that the results of the present paper will work in that setting as well.\
We review briefly the state of the art on the question of improving Dirichlet’s theorem in the context of real numbers. Davenport and Schmidt [@DS1; @DS2] showed that the Lebesgue measure of ${\operatorname{DI}_{\varepsilon}}(n)$ is zero for every $\varepsilon < 1$. Starting with work of Mahler, the question of *Diophantine approximation on manifolds* has received considerable attention. In this subject, one asks if Diophantine properties which are typical with respect to Lebesgue measure are also typical with respect to the push forward of Lebesgue measure via smooth maps. The starting point to this theory was a conjecture due to Mahler which asked if almost every point on the curve $$\label{def:veronese}
(x, x^2, \dots, x^n)$$ is not very well approximable by rationals. Such maps (or measures) are referred to as *extremal*. This conjecture was resolved by V. G. Sprindzhuk who in turn stated two generalisations of Mahler’s conjecture which involved a *nondegenerate* collection of functions replacing the map above. We refer to the work of Kleinbock-Margulis [@KM] where Sprindzhuk’s conjectures are resolved, for the definitions. In a subsequent striking work, Kleinbock, Lindenstrauss and Weiss [@KLW] extended the results of [@KM] to a much wider class of measures, the so-called *friendly* measures. This class includes push-forwards of Lebesgue measure as well as many other self similar measures including the uniform measure on the middle-third Cantor set. As regards improving Dirichlet’s theorem for manifolds, in [@DS2], Davenport and Schmidt showed that for any $\varepsilon < 4^{-1/3}$ the set of $x \in {{\mathbb R}}$ for which $(x, x^2) \in {\operatorname{DI}_{\varepsilon}}(2)$ has zero Lebesgue measure. Further results in this vein were obtained by Baker and by Bugeaud in [@Ba1; @Ba2; @Bu]. In [@KW1], Kleinbock and Weiss proved several results in this direction and in particular showed the existence of $\varepsilon > 0$ such that for continuous, good and nonplanar maps $\mathbf{f}$ and Radon, Federer measures $\nu$, $\mathbf{f}_{*}(\nu)({\operatorname{DI}_{\varepsilon}}(n)) = 0$. We will define all these terms later in the paper. In particular, this generalises the work of Baker and Bugeaud. The result that is obtained in [@KW1] holds for $\varepsilon$ which are quite a bit smaller than $1$ and to prove an analogous result for every $\varepsilon < 1$ remains an outstanding open problem. In the case of curves, N. Shah has resolved this problem. See [@Shah1] and also [@Shah2; @Shah3] for related results.\
In another direction, Kleinbock and Tomanov [@KT; @KT-pre] established $S$-arithmetic analogues of Sprindzhuk’s conjectures. In positive characteristic, Sprindzhuk [@Spr1] established the analogues of Mahler’s conjecture, namely the extremality of the curve (\[def:veronese\]) over $F^n$ and also proved other interesting results, including a transference principle interpolating between simultaneous Diophantine approximation and systems of linear forms. The analogues of Sprindzhuk’s conjectures in positive characteristic were established by the second named author in [@G-pos]. However, the question of improving Dirichlet’s theorem in positive characteristic has been completely open as far as we are aware. In the present paper, we study the question of Dirichlet improvability of vectors, maps and measures in positive characteristic.\
Here is a special case of our main result, Theorem \[thm:1.2\].
\[thm:spcase\] Let $f_1,f_2,...,f_n$ be polynomials so that $1,f_1,f_2,...,f_n$ are linearly independent over $F$. Fix some open set $U$ of $F$ and consider the map $\mathbf{f}(x)=(f_1(x),f_2(x),...,f_n(x))$ defined for all $x\in U$. Then there exists $\varepsilon_0\,\textgreater \,0$ such that whenever $\varepsilon\,\textless\,\varepsilon_0$, $\mathbf{f}(x)$ is not Dirichlet $\varepsilon$-improvable for $\lambda$ almost all $x\in U$.
Theorem \[thm:1.2\], the main result of this paper is far more general and holds for good, non-planar maps and Radon, Federer measures. It may be regarded as a positive characteristic version of Theorem 1.5 of Kleinbock and Weiss [@KW1]. Since the statement of the general form of the Theorem is fairly technical, we have chosen to postpone it to later in the paper. The constant $\varepsilon$ can be estimated so the proof is “effective" in that sense. However, it is likely to be far from optimal. We compute $\varepsilon_0$ in the special case $n=2$ and $f_i(x)=x^i$ for $i=1,2$ (see Section \[section:example\]) as an example. Our proof proceeds along the lines of [@KW1] and the main tool is a quantitative non divergence result for certain maps in the space of unimodular lattices, which can be identified with the non compact quotient $\operatorname{SL}(n+1, F)/\operatorname{SL}(n+1, \Lambda)$.
Review of the classical theory {#section:review}
==============================
In this section, we provide a proof of Dirichlet’s theorem in positive characteristic for completeness, and to aid the reader. In what follows, for $k\in \mathbb{N}$, $\mathbb{Z}_{+}^{k}$ denotes the set of all $k$ tuples $(t_1,t_2,...,t_k)$ where each $t_i$ is a nonnegative integer . We prove the following:
\[thm:1.1\] Let $m,n \in \mathbb{N}$, $k=m+n$ and $$\mathfrak{a}^{+}:= \{\mathbf{t}:=(t_1,t_2,...,t_k)\in \mathbb{Z}_{+}^{k}\,:\,\displaystyle \sum_{i=1}^{m} t_{i} =\displaystyle \sum_{j=1}^{n}t_{m+j}\}\,.$$ Consider $m$ linear forms $Y_{1},Y_{2},...,Y_{m}$ over $F$ in $n$ variables. Then for any $\mathbf{t}\in \mathfrak{a}^+$, there exist solutions $\mathbf{q}=(q_1,q_2,...,q_n)\in \Lambda^{n} \setminus \{\mathbf{0}\}$ and $\mathbf{p}=(p_1,p_2,...,p_m)\in \Lambda^m$ of the following system of inequalities $$\label{eqn:1.1}\left \{ \begin{array}{rcl} |Y_{i}\mathbf{q}-p_i|\textless e^{-t_{i}} &\mbox{for} &i=1,2,...,m \\ |q_j|\leq e^{t_{m+j}}&\mbox{for} &j=1,2,...,n\,. \end{array} \right.$$
To prove this theorem, we first introduce the ‘polynomial part’ and ‘fractional part’ of a Laurent series. For any Laurent series $$a = \cdot\cdot\cdot+\frac{a_2}{T^2}+ \frac{a_1}{T}+(a_0+a_1T+a_2T^2+\cdot\cdot\cdot+a_kT^k)$$ in $F$, where $k \in \mathbb{Z},\,a_i\in \mathbb{F}_{q}$ and $a_{k}\neq 0$, let us define the *polynomial part* of $a$ as $$a_0+a_1T+a_2T^2+\cdot\cdot\cdot+a_k T^k$$ if $k\geq 0$, otherwise it is defined to be $0$; and the *fractional part* of $a$, denoted by $\langle a\rangle$, is defined as $$\alpha- \mbox{polynomial\,\,part}= \frac{a_1}{T}+\frac{a_2}{T^2}+\cdot\cdot\cdot\,.$$\
Now, let $a:=\frac{a_1}{T}+\frac{a_2}{T^2}+\cdot\cdot\cdot \in F$ and $\alpha:=\alpha_{0}+\alpha_{1}T+\alpha_{2}T^2+\cdot\cdot\cdot+\alpha_{k}T^k \in \Gamma\setminus \{0\}$ with degree $\leq k$, where $k\geq 0$ is an integer. Let us observe that, for any $s\in \mathbb{N}$, the coefficient of $\frac{1}{T^s}$ in $\alpha a$ is $$a_s \alpha_0+\cdot\cdot\cdot+a_{s+k}\alpha_{k}\,.$$ It follows that, for any $m\in \mathbb{N}$, $|\langle \alpha a\rangle |\textless \frac{1}{e^m}$ if and only if the system $A\textit{\textbf{x}}=\textbf{0}$ of linear equations over $\mathbb{F}_{q}$, where the coefficient matrix $$A:=\begin{bmatrix}a_1& a_2 & .&.&. & a_{k+1}\\a_2&
a_3 & .&.&. & a_{k+2}\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\a_m&a_{m+1} &.&.&.&a_{m+k}
\end{bmatrix}\,,$$ has $(\alpha_0,\alpha_1,...,\alpha_k)$ as a nontrivial solution.\
Continuing along the same line, let us now take two Laurent series $a=\frac{a_1}{T}+\frac{a_2}{T^2}+\cdot\cdot\cdot$, $b=\frac{b_1}{T}+\frac{b_2}{T^2}+\cdot\cdot\cdot,$ and two nonzero polynomials $\alpha:=\alpha_{0}+\alpha_{1}T+\alpha_{2}T^2+\cdot\cdot\cdot+\alpha_{k}T^k$, $\beta:=\beta_{0}+\beta_{1}T+\beta_{2}T^2+\cdot\cdot\cdot+\beta_{l}T^l$, with degree $\leq k,l$ respectively, where $k,l=0,1,2,...$. For any $s\in \mathbb{N}$, the coefficient of $\frac{1}{T^s}$ in $\alpha a+\beta b$ is easily seen to be $$a_s \alpha_0+\cdot\cdot\cdot+a_{s+k}\alpha_{k}+b_s \beta_0+\cdot\cdot\cdot+b_{s+l}\beta_{l}\,.$$ Therefore, for any $m\in \mathbb{N}$, $|\langle \alpha a+\beta b\rangle|\textless \frac{1}{e^m}$ if and only if $(\alpha_0,\alpha_1,...,\alpha_k,\beta_0,\beta_1,..., \beta_l)$ is a nontrivial solution of the following system $$\begin{bmatrix}\Huge A &\huge B\end{bmatrix}
\begin{bmatrix}\textit{\textbf{x}}\\ \textit{\textbf{y}}
\end{bmatrix}=\textbf{0}\,,$$ where $$A:= \begin{bmatrix}a_1& a_2 & .&.&. & a_{k+1}\\a_2&
a_3 & .&.&. & a_{k+2}\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\a_m&a_{m+1} &.&.&.&a_{m+k}
\end{bmatrix}\,\,\mbox{and}\,\, B:=\begin{bmatrix}b_1& b_2 & .&.&. & b_{l+1}\\b_2&
b_3 & .&.&. & b_{l+2}\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\b_m&b_{m+1} &.&.&.&b_{m+l}
\end{bmatrix}\,.$$ It is obvious that we can generalize this observation for any such $n$ Laurent series and nonzero polynomials.\
Now we are ready to start the proof of Theorem \[thm:1.1\]. Each $Y_i$, being a linear form over $F$ in $n$ variables, must be of the form $$y_{i1}x_1+y_{i2}x_2+\cdot \cdot \cdot +y_{in}x_n\,,$$ for some $y_{ij}\in \mathbb{F}_q$ , $j=1,2,...,n$. It suffices to consider the case $|y_{ij}|\,\textless\, 1$, i.e. the polynomial part of $y_{ij}$ is zero, for all $i=1,2,...,m$ and $j=1,2,...,n$.\
From the observations we made earlier, we see that each $y_{ij}$ gives rise to a matrix $M_{ij}$ having $t_i$ rows and $t_{m+j}+1$ columns and more importantly, the existence of solution of the system (\[eqn:1.1\]) is equivalent to the existence of nontrivial solutions of the following system of linear equations over $\mathbb{F}_q$ $$\begin{bmatrix}M_{11}& M_{12} & .&.&. & M_{1n}\\M_{21}&
M_{22} & .&.&. & M_{2n}\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\.&.&\,&\,&\,&.\\M_{m1}&M_{m2} &.&.&.&M_{mn}
\end{bmatrix}
\begin{bmatrix}\textit{\textbf{x}}_1\\ \textit{\textbf{x}}_2\\ .\\.\\.\\ \textit{\textbf{x}}_n
\end{bmatrix}=\textbf{0}\,.$$
Clearly the above coefficient matrix has $\displaystyle \sum_{i=1}^m t_i$ rows and $\displaystyle\sum_{j=1}^n (t_{m+j}+1)$ columns. As $ \sum_{i=1}^{m} t_{i} = \sum_{j=1}^{n}t_{m+j}$, we see that the matrix has more columns than rows and hence nontrivial solution exists. This completes our proof. $\Box$\
The main theorem
================
We shall now introduce the notion of “Dirichlet improvability” in a greater generality. Let $\mathfrak{a}^{+}$ be as given in Theorem \[thm:1.1\], $\mathcal{T}$ be an unbounded subset of $\mathfrak{a}^{+}$ and $0\,\textless \,\varepsilon\leq \frac{1}{e}$.
\[defn:DI imp\] For a system of linear forms $Y_{1},Y_{2},...,Y_{m}$ over $F$ in $n$ variables, we say that *DT can be $\varepsilon$-improved along $\mathcal{T}$*, or we use the notation $Y\in DI_{\varepsilon}(\mathcal{T})$, where $Y$ is the $m\times n$ matrix having $Y_i$ as the $i$ th row for each $i$, if there exists $t_0\,\textgreater \,0$ such that for every $\mathbf{t}:=(t_1,t_2,...,t_k)\in \mathcal{T}$ with $||\mathbf{t}||\,\textgreater\, t_0$ the following system admits nontrivial solutions $(\mathbf{p},\mathbf{q})\in \Lambda^m \times \Lambda^n$ : $$\label{eqn:1.4} \left \{\begin{array}{rcl} |Y_{i}\mathbf{q}-p_i|\,\textless \,\frac{\varepsilon}{e^{t_i}} &\mbox{for} &i=1,2,...,m
\\ |q_j|\,\textless\,\varepsilon e^{t_{m+j}}&\mbox{for} &j=1,2,...,n\,. \end{array} \right.$$ In particular, a vector $\mathbf{y}:=(y_1,y_2,...,y_n)\in F^n$ is said to be *Dirichlet $\varepsilon$-improvable along $\mathcal{T}$* if the corresponding row matrix $[y_1\,\,y_2\,\,\cdot\,\,\cdot\,\,\cdot\,\,y_n]\in DI_{\varepsilon}(\mathcal{T})$.
Exactly similar to that shown in [@KW1], here also we want to prove that if $\varepsilon\,\textgreater \,0$ is sufficiently small and an unbounded subset $\mathcal{T}$ of $\mathfrak{a}^{+}$ is chosen, the set of all Dirichlet $\varepsilon$-improvable vectors along $\mathcal{T}$ is negligible. The setup here is *multiplicative*, i.e. one studies Diophantine inequalities where Euclidean or supremum norm is replaced with the product of coordinates. The changed “norm" introduces several complications and the subject of multiplicative Diophantine approximation is generally considered more difficult than its euclidean counterpart.\
Before proceeding to our main theorem, we will recall the some terminology introduced in the papers of Kleinbock and Margulis, and Kleinbock, Lindenstrauss and Weiss and used in several subsequent works by many authors. The following is taken from §1 and 2 of [@KT].\
For the sake of generality, we assume $X$ is a Besicovitch metric space, $U\subseteq X$ is open, $\nu$ is a radon measure on $X$, $({\mathcal{F}},|\cdot|)$ is a valued field and $f: X\longrightarrow{\mathcal{F}}$ is a given function such that $|f|$ is measurable. For any $B\subseteq X$, we set $$||f||_{\nu,B} := \displaystyle \sup_{x\in B\cap \text{ supp }(\nu)} |f(x)|.$$
\[defn:C,alpha\] For $C,\alpha \textgreater\,0$, $f$ is said to be $(C,\alpha)-good$ on $U$ with respect to $\nu$ if for every ball $B\subseteq U$ with center in $\text{supp }(\nu)$, one has $$\nu(\{x\in B: |f(x)|\,\textless \varepsilon\})\leq C\left(\frac{\varepsilon}{||f||_{\nu,B}}\right)^{\alpha} \nu(B)\,.$$
The following properties are immediate from Definition \[defn:C,alpha\].
\[lem:C,alpha\] Let $X,U,\nu, {\mathcal{F}}, f, C,\alpha,$ be as given above. Then one has
(i) $f$ $(C,\alpha)-good$ $U$ with respect to $\nu \Longleftrightarrow \text{ so is } |f|$.
(ii) $f$ is $(C,\alpha)-good$ on $U$ with respect to $\nu$ $\Longrightarrow$ so is $c f$ for all $c \in {\mathcal{F}}$.
(iii) \[item:sup\] $\forall i\in I, f_i$ are $(C,\alpha)-good$ on $U$ with respect to $\nu$ and $\sup_{i\in I} |f_i|$ is measurable $\Longrightarrow$ so is $\sup_{i\in I} |f_i|$.
(iv) $f$ is $(C,\alpha)-good$ on $U$ with respect to $\nu$ and $g :V \longrightarrow \mathbb{R}$ is a continuous function such that $c_1\leq |\frac{f}{g}|\leq c_2$ for some $c_1,c_2 \,\textgreater \,0\Longrightarrow g$ is $(C(\frac{c_2}{c_1})^{\alpha},\alpha)$ good on $U$ with respect to $\nu$.
(v) Let $C_2 \,\textgreater \,1$ and $\alpha _2\,\textgreater\,0$. $f$ is $(C_1,\alpha_1)-good$ on $U$ with respect to $\nu$ and $C_1 \leq C_2, \alpha_2 \leq \alpha_1 \Longrightarrow f$ is $(C_2,\alpha_2)-good$ on $V$ with respect to $\nu$.
We say a map $\mathbf{f}=(f_1,f_2,...,f_n)$ from $U$ to ${\mathcal{F}}^n$, where $n\in \mathbb{N}$, is $(C,\alpha)-good$ on $U$ with respect to $\nu$, or simply $(\mathbf{f},\nu)$ is $(C,\alpha)-good$ on $U$, if every ${\mathcal{F}}$-linear combination of $1,f_1,...,f_n$ is $(C,\alpha)-good$ on $U$ with respect to $\nu$.
Let $\mathbf{f}=(f_1,f_2,...,f_n)$ be a map from $U$ to ${\mathcal{F}}^n$, where $n\in \mathbb{N}$. We say that $(\mathbf{f},\nu)$ is *nonplanar* if for any ball $B\subseteq U$ with center in $\text{supp }(\nu)$, the restrictions of the functions $1,f_1,...,f_n$ on $B\cap \text{ supp }(\nu)$ are linearly independent.
In other words, $\mathbf{f}(B\cap \text{ supp }(\nu))$ is not contained in any affine subspace of ${\mathcal{F}}^n$ for any ball $B\subseteq U$ with center in $\text{supp }(\nu)$.\
For $m\in \mathbb{N}$ and a ball $B=B(x;r)\subseteq X$, where $x\in X$ and $r\,\textgreater\,0$, we shall use the notation $3^mB$ to denote the ball $B(x;3^mr)$.
Let $D\,\textgreater\,0$. The measure $\nu$ is said to be $D-Federer$ on $U$ if for every ball $B$ with center in $\text{supp }(\nu)$ such that $3B\subseteq U$, one has $$\frac{\nu(3B)}{\nu(B)}\leq D\,.$$
We are now ready to state our main Theorem, which addresses improvements of Dirichlet’s theorem in the multiplicative setting for good, nonplanar maps and Federer measures over local fields of positive characteristic.
\[thm:1.2\] For any $d,n\in \mathbb{N}$ and $C,\alpha,D\,\textgreater\, 0$ there exists $\varepsilon_0=\varepsilon_0(n,C,\alpha,D)$ satisfying the following: whenever a radon measure $\nu$ on $F^d$, an open set $U$ of $F^d$ with $\nu(U)\,\textgreater \,0$ and $\nu$ is D-Federer on $U$, and a continuous map $\mathbf{f}:U\longrightarrow F^n$ such that $(\mathbf{f},\nu)$ is $(C,\alpha)-good$ and nonplanar is given then for any $\varepsilon < \varepsilon_{0}$,
$$\mathbf{f}_{*}\nu(DI_{\varepsilon}(\mathcal{T}))=0 \,\,\text{ for any unbounded \,}\mathcal{T}\subseteq \mathfrak{a}^{+}\,.$$
We shall use the so called “quantitative nondivergence”, a generalization of non-divergence of unipotent flows on homogeneous spaces, to prove our main theorem. Similarly to the approach adopted in [@KW1], we will first translate the property of a system of linear forms over $F$ being Dirichlet improvable into certain recurrent properties of flows on some homogeneous space in the following section.\
The correspondence {#section:2}
==================
Let $G:= \operatorname{SL}(k,F)$, $\Gamma := \operatorname{SL}(k, \Lambda)$ and $\pi$ be the quotient map $G\longrightarrow G/\Gamma$. $G$ acts on $G/\Gamma$ by left translations via the rule $g\pi(h)=\pi(gh)$ for $g,h\in G$. For $Y\in M_{m\times n}(F)$, define $$\tau (Y):= \begin{bmatrix}
I_m& Y\\0& I_n
\end{bmatrix}\,\,\mbox{and}\,\,\overline{\tau}:=\pi \circ \tau\,,$$ where $I_l$ stands for the $l\times l$ identity matrix, $l\in \mathbb{N}$. Since $\Gamma$ is the stabilizer of $\Lambda^k$ under the transitive action of $G$ on the set of unimodular lattices in $F^k$, which is denoted by $\mathcal{L}_k(F)$, we can identify $G/\Gamma\simeq \mathcal{L}_k(F)$. Thus $\overline{\tau}(Y)$ becomes identified with $$\{(Y\mathbf{q}-\mathbf{p},\mathbf{q})\,:\mathbf{p}\in \Lambda^m\,,\mathbf{q}\in \Lambda^n\}\,.$$
Now for $\varepsilon\textgreater 0$, let $K_{\varepsilon}$ denote the collection of all unimodular lattices in $F^k$ which contain nononzero vector of norm smaller than $\varepsilon$, that is, $$\label{eqn:cpt} K_{\varepsilon}:=\pi (\{g\in G\,:\,||g\mathbf{v}||\geq \varepsilon \,\,\forall
\,\mathbf{v}\in \Lambda^k\setminus \{\mathbf{0}\}\})\,.$$
Next, for $\mathbf{t}:=(t_1,t_2,...,t_k)\in \mathfrak{a}^{+}$, we associate the diagonal matrix $$g_{\mathbf{t}}:=\,\,\mbox{diag}\,(T^{t_1},.\,.\,.\,,T^{t_m},T^{-t_{m+1}},.\,.\,.\,,T^{-t_{k}})\in G\,.$$ Let us come to the relevance of defining the above objects. An immediate observation shows that, for given $\mathbf{t}\in \mathfrak{a}^{+}$, the system (\[eqn:1.4\]) has nonzero polynomial solutions if and only if $$g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\,.$$ Thus we have
\[prop: 2.1\] Let $0< \varepsilon \leq \frac{1}{e}$ and unbounded $\mathcal{T}\subseteq \mathfrak{a}^+$ be given. Then for any $Y\in M_{m\times n}(F)$, $$Y\in DI_{\varepsilon}(\mathcal{T})\Longleftrightarrow g_{\mathbf{t}}\overline{\tau}(Y)\notin
K_{\varepsilon} \,\,\forall \mathbf{t}\in
\mathcal{T}\,\,\mbox{with} \,\,||\mathbf{t}||\gg 1\,,$$ or equivalently one has, $$DI_{\varepsilon}(\mathcal{T})= \displaystyle \bigcup_{n=1}^{\infty} \,\,\bigcap_{\mathbf{t}\in \mathcal{T}\,,\,||\mathbf{t}||\textgreater n}
\{Y\in M_{m\times n}(F):\,g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\}\,.$$
Hence, in view of the above proposition, it is clear that if in addition a radon measure $\nu$ on $F^d$, an open set $U$ of $F^d$ and a map $F:U\longrightarrow M_{m\times n}(F)$ are given then to prove $F_{*}\nu(DI_{\varepsilon}(\mathcal{T}))=\nu (F^{-1}(DI_{\varepsilon}(\mathcal{T}))=0$, it is enough to show $$\label{eqn:2.1} \nu \left(F^{-1} \left( \displaystyle \bigcap_{\mathbf{t}\in \mathcal{T}\,,\,||\mathbf{t}||\textgreater n}
\{Y\in M_{m\times n}(F):\,g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\}\right)\right) = 0$$ for all $n\in \mathbb{N}$. Suppose now that we have some $c \in (0,1)$ with the property that for any ball $B\subseteq U$ centered in $\mbox{supp}\,(\nu)$, there exists $s\,\textgreater\, 0$ such that $$\label{eqn:2.2}\displaystyle
\nu( B \cap F^{-1}(\{Y\,:\,g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\}))= \nu
(\{\textbf{x}\in B\,:\,g_{\mathbf{t}}\overline{\tau}(F(\textbf{x}))\notin K_{\varepsilon}\})\leq c\nu (B)$$ holds for any $\mathbf{t}\in \mathfrak{a}^+$ with $||\mathbf{t}||\geq s$. Then it is easy to see that, for any $n\in \mathbb{N}$ and any ball $B\subseteq U$ centered in $\mbox{supp}\,(\nu)$, $$\label{eqn:2.3} \begin{array}{rcl}\displaystyle \frac{\nu \left(B \cap F^{-1} \left ( \displaystyle \bigcap_{\mathbf{t}\in \mathcal{T}\,,\,||\mathbf{t}||\textgreater n}
\{Y\in M_{m\times n}(F):\,g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\}\right)\right)}{\nu(B)}\\=
\displaystyle \frac{\nu \left(\displaystyle \bigcap_{\mathbf{t}\in \mathcal{T}\,,\,||\mathbf{t}||\textgreater n}
B\cap F^{-1}(\{Y\in M_{m\times n}(F):\,g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\})\right)}{\nu(B)}& \leq
\displaystyle \frac{c\,\nu(B)}{\nu(B)}=c\textless 1\,.\end{array}$$
It follows that, for any given $n\in \mathbb{N}$, no $\mathbf{x} \in U\cap\mbox{supp}\,(\nu)$ is a point of density of the set $$F^{-1} \left(\displaystyle \bigcap_{\mathbf{t}\in \mathcal{T}\,,\,||\mathbf{t}||\textgreater n}
\{Y\in M_{m\times n}(F):\,g_{\mathbf{t}}\overline{\tau}(Y)\notin K_{\varepsilon}\}\right)\,,$$ as (\[eqn:2.3\]) holds true for any ball $B$ with $\mathbf{x}\in B\subseteq U$. Thus (\[eqn:2.1\]) will be achieved in view of Theorem \[thm:3.1\].
The proof of Theorem \[thm:1.2\]
================================
As $F^d$ is locally compact, hausdorff and second countable, every open set is the union of some countable collection of compact subsets. Hence to prove the Theorem \[thm:1.2\], once correct $\varepsilon_0=\varepsilon_0(n,C,\alpha,D)$ is found, it suffices to show that for all $\mathbf{y}\in U\cap\text{ supp }(\nu)$, there exists a ball $\mathfrak{B}\subseteq U$ containing $\mathbf{y}$ such that $$\label{eqn:4.1} \nu(\mathfrak{B}\cap \mathbf{f}^{-1}(DI_{\varepsilon}(\mathcal{T})))=
\nu(\{\mathbf{x}\in \mathfrak{B}\,:\, \mathbf{f}(\mathbf{x})\in DI_{\varepsilon}(\mathcal{T})\})=0$$ for all $\varepsilon\,\textless\, \varepsilon_0$. From our discussion of Section \[section:2\], we see that (\[eqn:4.1\]) is guaranteed as soon as we can show the existence of some $c\in (0,1)$ which satisfies the following: whenever a ball $B$ with center in $\mbox{supp}\,(\nu)$ is contained in $\mathfrak{B}$ then, $$\label{eqn:suff}
\mbox{there exists}\,\, s\,\textgreater \,0 \,\,\mbox{such that for all}\,\,\mathbf{t}\in \mathfrak{a}^+\,\
\mbox{with}\,\,||\mathbf{t}||\geq s, (\ref{eqn:2.2})\,\,\mbox{holds}\,.$$ Now the following proposition shows our way.
\[main prop\] For any $d,n\in \mathbb{N}$ and any $C,\alpha, D\,\textgreater \,0$ there exists $\tilde C=\tilde C(n,C, D)$ with the following property:\
whenever a ball $ B$ centered in $\mbox{supp}\,(\nu)$, a radon measure $\nu$ on $F^d$ which is D-Federer on $\tilde B:= 3^{n+1} B$ and a continuous map $\mathbf{f}:\tilde {B} \longrightarrow F^n$ are given so that
(i) any $F$-linear combination of $1,f_1,\,.\,.\,.\,,f_n$ is $(C,\alpha)-good$ on $\tilde B$ with respect to $\nu$ and,
(ii) the restrictions of $1,f_1,\,.\,.\,.\,,f_n$ to $B\cap \mbox{supp}\,(\nu)$ are linearly independent over $F$;
then we can find some $s\,\textgreater\, 0$ such that for all $\mathbf{t}\in \mathfrak{a}^+$ with $||\mathbf{t}||\geq s$ and any $\varepsilon\leq \frac{1}{e}$, one has $$\label{eqn:prop}
\nu(\{\mathbf{x}\in B\,:\,g_{\mathbf{t}}\overline{\tau}(\mathbf{f}(\mathbf{x}))\notin K_{\varepsilon}\})\leq \tilde C \varepsilon ^{\alpha} \nu(B)\,.$$
Theorem \[thm:1.2\] follows easily from the Proposition \[main prop\]. In fact, we first choose $ 0\,\textless\, \varepsilon_0\leq \frac{1}{e}$ so that $\displaystyle \tilde C \varepsilon_{0} ^{\alpha}\,\textless \,1$. Clearly this $\varepsilon_0$ depends only on $(n, C,\alpha, D)$. Let $\mathbf{y}\in U\cap\mbox{supp}\,(\nu)$. Choose a ball $\mathfrak{B}$ such that $\mathbf{y}\in \mathfrak{B} \subseteq \tilde {\mathfrak{B}}:= 3^{n+1} \mathfrak{B} \subseteq U$. Now pick any ball $B\subseteq \mathfrak{B}$ having center in $\mbox{supp}\,(\nu)$ and consider the corresponding $\tilde B$. Since $(\mathbf{f},\nu)$ is $(C,\alpha)-good$ and nonplanar, the conditions (i) and (ii) of Proposition \[main prop\] hold here immediately. Hence, if we set $c= \tilde C \varepsilon_{0} ^{\alpha}$, the assertion (\[eqn:suff\]) is immediate from Proposition \[main prop\] whenever $0\,\textless\, \varepsilon \,\textless\, \varepsilon_0$. Thus the proof of Theorem \[thm:1.2\] is complete. $\Box$\
We now need to prove Proposition \[main prop\]. We shall show this as a consequence of a more general result, namely the ‘Quantitative nondivergence theorem’. All these will be discussed in Section \[section:5\].
Quantitative nondivergence and the proof of\
Proposition \[main prop\] {#section:5}
============================================
We shall first recall the ‘Quantitative nondivergence theorem’ in the most generality, as it is developed in §6 of [@KT]. Finally, we shall prove Proposition \[main prop\] from this.
Quantitative nondivergence
--------------------------
We start this subsection by assuming that $\mathcal{D}$ is an integral domain, $K$ is the field of quotients of $\mathcal{D}$ and $\mathcal{R}$ is a commutative ring containing $K$ as a subring.\
Let $m\in \mathbb{N}$. If $\Delta$ is a $\mathcal{D}$-submodule of $\mathcal{R}^m$, let us denote by $K\Delta$ (respectively $\mathcal{R}\Delta$) its $K$- (respectively $\mathcal{R}$) linear span inside $\mathcal{R}^m$. We use the notation ${\operatorname{rank}}(\Delta)$ to denote the rank of $\Delta$ which is defined as $${\operatorname{rank}}(\Delta):= \displaystyle \dim_{K} (K\Delta)\,.$$ For example ${\operatorname{rank}}({\mathcal{D}}^m)=m$. If $\Theta$ is a $\mathcal{D}$-submodule of $\mathcal{R}^m$ and $\Delta$ is a submodule of $\Theta$, we say that $\Delta$ *is primitive in* $\Theta$ if any submodule of $\Theta$ containing $\Delta$ and having rank equal to ${\operatorname{rank}}(\Delta)$ is equal to $\Delta$. We see that the set of all nonzero primitive submodules of a fixed $\mathcal{D}$-submodule $\Theta$ of $\mathcal{R}^m$ is a partially ordered set with respect to set inclusion and its length is equal to ${\operatorname{rank}}(\Theta)$. When $\Theta = \mathcal{D}^m$, we can even characterize the primitive submodules of $\mathcal{D}^m$ from the following observation: $$\Delta\,\,\text{ is primitive }\,\,\Longleftrightarrow \Delta= K\Delta \cap \mathcal{D}^m \Longleftrightarrow \Delta= \mathcal{R}\Delta \cap \mathcal{D}^m\,.$$ This also shows that for any submodule $\Delta'$ of $\mathcal{D}^m$ there exists a unique primitive submodule $\Delta\supseteq \Delta'$ such that ${\operatorname{rank}}(\Delta)= {\operatorname{rank}}(\Delta')$, namely $\Delta := K\Delta'\cap \mathcal{D}^m$.\
Let $\mathcal{R}$ have a topological ring structure in addition. We consider the topological group $\operatorname{GL}(m,\mathcal{R})$ of $m\times m$ invertible matrices with entires in $\mathcal{R}$. It is obvious that any $g\in \operatorname{GL}(m,\mathcal{R})$ maps $\mathcal{D}$-submodules of $\mathcal{R}^m$ to $\mathcal{D}$-submodules of $\mathcal{R}^m$ preserving their rank and inclusion relation. Let $${\mathfrak{M}}({\mathcal{R}},{\mathcal{D}},m):= \{g\Delta\,:\, g\in \operatorname{GL}(m,{\mathcal{R}}),\,\,\Delta \,\,\mbox{is a submodule of of}\,\,{\mathcal{D}}^m\}\,.$$ We also denote the set of all nonzero primitive submodules of ${\mathcal{D}}^m$, which is a poset of length $m$ with respect to inclusion relation as we have already seen, by ${\mathfrak{P}}({\mathcal{D}},m)$.\
For a given function $||\cdot||: {\mathfrak{M}}({\mathcal{R}},{\mathcal{D}},m) \longrightarrow \mathbb{R}_{\geq 0}$, one says that $||\cdot||$ is *norm-like* if the following three conditions hold:
1. \[item:N1\] For any $\Delta,\Delta'\in {\mathfrak{M}}({\mathcal{R}},{\mathcal{D}},m)$ with $\Delta'\subseteq \Delta$ and ${\operatorname{rank}}(\Delta')= {\operatorname{rank}}(\Delta)$, we always have $||\Delta'||\geq ||\Delta||$;
2. \[item:N2\] there exists $C_{||\cdot||} \textgreater 0$ such that $||\Delta + {\mathcal{D}}\gamma||\leq C_{||\cdot||} ||\Delta||\,||{\mathcal{D}}\gamma||$ holds for any $\Delta \in {\mathfrak{M}}({\mathcal{R}},{\mathcal{D}},m)$ and any $\gamma\notin {\mathcal{R}}\Delta$; and
3. \[item:N3\] the function $\operatorname{GL}(m,{\mathcal{R}})\longrightarrow \mathbb{R}_{\geq 0},\,g\mapsto ||g\Delta||$ is continuous for every submodule $\Delta$ of ${\mathcal{D}}^m$.
With the notations and terminologies defined so above, it is now time to state the ‘Quantitative nondivergence theorem’.
\[thm:qn\] Let $B\subseteq X$ be a ball in a Besicovitch metric space $X$ and $h:\tilde{B} \longrightarrow \operatorname{GL}(m,{\mathcal{R}})$, where $\tilde{B}:=3^m B$, be a continuous map. Suppose $\nu$ is a radon measure on $X$ which is D-Federer on $\tilde B$. Assume that a norm-like function $||\cdot||$ is given on ${\mathfrak{M}}({\mathcal{R}},{\mathcal{D}},m)$. Assume further that for some $C,\alpha \,\textgreater \,0$ and $\rho \in \displaystyle (0, 1/C_{||\cdot||}]$, the following conditions hold:
1. \[item:C1\] for every $\Delta \in {\mathfrak{P}}({\mathcal{D}},m)$, the function $x\mapsto ||h(x)\Delta||$ is $(C,\alpha)-good$ on $\tilde{B}$ w.r.t $\nu$;
2. \[item:C2\]for every $\Delta \in {\mathfrak{P}}({\mathcal{D}},m)$, $\displaystyle \sup_{x\,\,\in B\cap \,\mbox{supp}\,(\nu)}||h(x)\Delta||\geq \rho$; and
3. \[item:C3\]$\forall x\in \tilde{B} \cap \mbox{supp}\,(\nu)$, $\#\{\Delta \in {\mathfrak{P}}({\mathcal{D}},m)\,:\,||h(x)\Delta||\,\textless \,\rho\} \,\textless \,\infty$.
Then for any positive $\varepsilon \leq \rho$, one has $$\label{eqn:qn}
\displaystyle \nu\left(\left\{x\in B\,:\,||h(x)\gamma||\,\textless\, \varepsilon\,\,\mbox{for some}
\,\,\gamma \in {\mathcal{D}}^m\setminus \{\mathbf{0}\}\right\}\right)\leq mC(N_X D^2)^m \left(\frac{\varepsilon}{\rho}\right)^{\alpha} \nu(B)\,,$$ where $N_X$ is the ‘Besicovitch constant’.
For the proof, see ([@KT], §6, Theorem).
The proof of Proposition \[main prop\]
--------------------------------------
From the definition of $K_{\varepsilon}$, as in $(\ref{eqn:cpt})$, it is obvious that for $\mathbf{t}\in \mathfrak{a}^+$ and $\mathbf{x}\in B$, $$g_{\mathbf{t}}\overline{\tau}(\mathbf{f}(\mathbf{x}))\notin K_{\varepsilon} \Longleftrightarrow
||(g_{\mathbf{t}}\tau(\mathbf{f}(\mathbf{x})))\mathbf{v}||\textless \varepsilon \,\,\mbox{for some}\,\,\mathbf{v}\in \Lambda^{n+1}
\setminus \{\mathbf{0}\}\,.$$ This inspires us to use Theorem \[thm:qn\] in the setting $$\begin{array}{rcl}{\mathcal{D}}=\Lambda, {\mathcal{R}}=F,
X=F^d,m=n+1;\\ \nu, B, C, \alpha \,\,\mbox{and}\,\,D\,\,\mbox{as in Proposition \ref{main prop}};\\
h(\mathbf{x})= g_{\mathbf{t}}\tau(\mathbf{f}(\mathbf{x}))\,\,\forall\mathbf{x}\in \tilde{B}\,;\,\end{array}$$ and $||\cdot||$ as the following:\
Since $\Lambda$ is a PID, any submodule of the $\Lambda$ module $\Lambda^{n+1}$, being submodule of a free module of rank $n+1$, is free of rank $\leq n+1$. Thus any nonzero $\Delta \in {\mathfrak{M}}(F,\Lambda,n+1)$ has a $\Lambda$ basis, say $\{\mathbf{v}_1,\,.\,. \,.\,,\mathbf{v}_j\}$, where $1\leq j\leq n+1$. We consider the $j$-vector $\mathbf{w}:=\mathbf{v}_1\wedge \cdot\cdot\cdot \wedge \mathbf{v}_j\in \bigwedge^j (F^{n+1})$. Recall that the $j$-vectors $e_{i_1}\wedge e_{i_2}\wedge \cdot\cdot\cdot\wedge e_{i_j}$ with integers $1\leq i_1\textless i_2 \textless \cdot\cdot\cdot\textless i_j\leq n+1$ form a basis of $\bigwedge^j (F^{n+1})$ and thus $\bigwedge^j (F^{n+1})$ can be identified with $F^{\binom{n+1}{j}}$. Therefore one can naturally talk about the *supremum norm* on $\bigwedge^j (F^{n+1})$ using this identification. We define $$||\Delta||:= \text{ supremum norm of }\mathbf{w}\,.$$ It is a routine verification that this definition does not depend on the choice of the ordered basis of $\Delta$. If $\Delta=\{\mathbf{0}\}$, we define $||\Delta||=1$.\
In order to prove that the just defined $||\cdot||$ is indeed norm-like, we need to verify the conditions (N\[item:N1\])-(N\[item:N3\]). (N\[item:N1\]) and (N\[item:N3\]) follow easily from the basic properties of exterior product, while (N\[item:N2\]) can be proved by a verbatim repetition of the proof of Lemma 5.1 of [@KM] as follows.\
We claim that $C_{||\cdot||}$ can be taken as $1$. If $\Delta=\{\bf{0}\}$ then it is immediate. Otherwise let $\{\mathbf{v}_1,\,.\,. \,.\,,\mathbf{v}_j\}$ be a basis of $\Delta$. Clearly $\{\mathbf{v}_1,\,.\,. \,.\,,\mathbf{v}_j,\gamma\}$ is a basis of $\Delta+\Lambda \gamma$. Now writing $\mathbf{v}_1\wedge \cdot\cdot\cdot \wedge \mathbf{v}_j= \displaystyle\sum_{\tiny \begin{array}{rcl}I\subseteq \{1,2,...,n+1\},\\ \# I=j\end{array}}w_I e_I$ and $\gamma=\displaystyle \sum_{i=1}^{n+1} w_ie_i$ (in usual notations) and using the ultrametric property, we see that $$\begin{array}{rcl}
||\Delta+\Lambda \gamma||=\left|\left|\displaystyle\sum_{{\tiny \begin{array}{rcl}I\subseteq \{1,2,...,n+1\},\\ \# I=j\end{array}}}w_I e_I\wedge \sum_{i=1}^{n+1} w_ie_i\,\right|\right|
\leq
\displaystyle \max_{1\leq i\leq n+1} \left|\left|\displaystyle\sum_{{\tiny \begin{array}{rcl}I\subseteq \{1,2,...,n+1\},\\ \# I=j\end{array}}}w_I w_i (e_I\wedge e_i)\,\right|\right|
\\ \leq \displaystyle \max_{1\leq i\leq n+1} \max_{{\tiny \begin{array}{rcl}I\subseteq \{1,2,...,n+1\},\\\# I=j\end{array}}}|w_Iw_i|\\ \leq
\displaystyle \max_{{\tiny \begin{array}{rcl}I\subseteq \{1,2,...,n+1\},\\ \#I=j\end{array}}}|w_I| \displaystyle \max_{1\leq i\leq n+1}|w_i|\\=
||\Delta||\,||\Lambda \gamma||\,.
\end{array}$$
Now we have to check the conditions (C\[item:C1\]), (C\[item:C2\]) and (C\[item:C3\]) of Theorem \[thm:qn\]. From the discreteness of $\bigwedge^j (\Lambda^{n+1})$ in $\bigwedge^j (F^{n+1})$ for all $j=1,2,...,n+1$, (C\[item:C3\]) is immediate. To investigate the validity of others, we have to do the explicit computation exactly in the similar manner to that of §3.3 in [@KW1].\
$\bullet$ **Checking (C\[item:C1\]):** Here, for the sake of convenience in computation, it is customary to bring a few minor changes in some of the notations we have been using so far. For the rest of this section, we write $\{\mathbf{e}_0,\mathbf{e}_1,...,\mathbf{e}_n\}$ the standard basis of $F^{n+1}$ and for $$\label{eqn:I} I=\{i_1,...,i_j\}\subseteq \{0,...,n\}\text{ where \,}i_i\,\textless \,i_2\,\textless \cdot\cdot\cdot\textless \,i_j\,,$$ we let $\mathbf{e}_I$ denote $\mathbf{e}_{i_1}\wedge \cdot \cdot \cdot \wedge \mathbf{e}_{i_j}$. Similarly, it will be convenient to put any $\mathbf{t}\in {\mathfrak{a}}^+$ as $$\mathbf{t}=(t_0,t_1,...,t_n)\,\,\,\mbox{where}\,\,t_0=\displaystyle \sum_{i=1}^n t_i\,.$$ Let us observe that for any $\mathbf{y}\in F^n$, $\tau(\mathbf{y})$ fixes $\mathbf{e}_0$ and sends any other $\mathbf{e}_i$ to $\mathbf{e}_i+y_i\mathbf{e}_0$. Thus for any $I$ as in (\[eqn:I\]), we have $$\label{eqn:5.1}
\tau(\mathbf{y})\mathbf{e}_I= \left \{\begin{array}{rcl}
\mathbf{e}_I & \mbox{if}\,\,0\in I\\
\mathbf{e}_I + \sum_{i\in I} \pm y_i \mathbf{e}_{I\cup \{0\}\setminus \{i\}} & \mbox{otherwise}\,.
\end{array}
\right.$$ Likewise, we can also see that for any $I$ as in (\[eqn:I\]), $$\label{eqn:5.2}
g_{\mathbf{t}}\mathbf{e}_I= \left \{\begin{array}{rcl}
\displaystyle T^{t_0- \sum_{i\in I\setminus \{0\}}t_i} \,\mathbf{e}_I & \mbox{if}\,\,0\in I\\
T^{-\sum_{i\in I} t_i}\,\mathbf{e}_I & \mbox{otherwise}\,.
\end{array}
\right.$$
Suppose $\Delta \in {\mathfrak{P}}(\Lambda,n+1)$ and $\{\mathbf{v}_1,\,.\,. \,.\,,\mathbf{v}_j\}$ is a basis of $\Delta$ and let $$\mathbf{w}:= \mathbf{v}_1\wedge \cdot\cdot\cdot \wedge \mathbf{v}_j= \displaystyle\sum_{\tiny \begin{array}{rcl}I
\subseteq \{0,...,n\},\\ \# I=j\end{array}}w_I \mathbf{e}_I\,;\,\,\,\,w_I\in \Lambda\,.$$ From (\[eqn:5.1\]) and (\[eqn:5.2\]), it follows that for any $\mathbf{x}\in \tilde B$, one has $$h(\mathbf{x})\mathbf{w}= \displaystyle\sum_{\tiny \begin{array}{rcl}I\subseteq \{0,...,n\},\\ \# I=j\end{array}} h_I(\mathbf{x})\mathbf{e}_I\,$$ where $$\label{eqn:coeff} h_I(\mathbf{x}):= \left \{ \begin{array}{rcl}\displaystyle T^{- \sum_{i\in I} t_i}w_I & \mbox{if} \,\, 0\notin I \\
\displaystyle T^{\sum_{i\notin I} t_i}(w_I + \sum_{i\notin I} \pm w_{I \cup \{i\} \setminus \{0\}} f_i(\mathbf{x})) &\mbox{otherwise} .
\end{array} \right.$$
In particular, the coordinate maps $h_I$ of the map $\mathbf{x}\mapsto h(\mathbf{x})\mathbf{w},\,\mathbf{x}\in \tilde B$ are $F$-linear combinations of $1,f_1,\,.\,.\,.\,,f_n$ and hence, by (i) of Proposition \[main prop\], all of them are $(C,\alpha)-good$ on $\tilde B$ with respect to $\nu$. Therefore, from (\[item:sup\]) of Lemma \[lem:C,alpha\], it follows that the function $$\displaystyle \mathbf{x}\mapsto ||h(\mathbf{x})\Delta||= ||h(\mathbf{x})\mathbf{w}||=
\max_{I}|h_I(\mathbf{x})|$$ is $(C,\alpha)-good$ on $\tilde{B}$ with respect to $\nu$. Thus (C\[item:C1\]) is established.\
$\bullet$ **Checking (C\[item:C2\]):** Let $\Delta \in {\mathfrak{P}}(\Lambda,n+1)$, $\{\mathbf{v}_1,\,.\,. \,.\,,\mathbf{v}_j\}$ be a basis of $\Delta$ and let $$\mathbf{w}:= \mathbf{v}_1\wedge \cdot\cdot\cdot \wedge \mathbf{v}_j= \displaystyle\sum_{\tiny \begin{array}{rcl}I
\subseteq \{0,...,n\},\\ \# I=j\end{array}}w_I \mathbf{e}_I\,;\,\,\,\,w_I\in \Lambda\,.$$ *Case 1:* Assume $w_I=0$ whenever $0\notin I$. Then there must be some $J\subseteq \{0,...,n\}$ containing $0$ such that $w_J\neq 0$ as all $w_I$ can not be zero. Pick any $\mathbf{t}\in {\mathfrak{a}}^+$. Now from (\[eqn:coeff\]), we see that $$|h_J(\mathbf{x})|=\displaystyle |T^{\sum_{i\notin J} t_i}w_J|\geq1\text{ \,\,for any }\mathbf{x}\in \tilde B\,.$$ Therefore in this case, we have $$\label{eqn:C1}\begin{array}{rcl} \displaystyle \sup_{\tiny \mathbf{x}\in B\,\cap \,\mbox{supp}\,(\nu)}||h(\mathbf{x})\Delta||
=\sup_{\tiny \mathbf{x}\in B\,\cap \,\mbox{supp}\,(\nu)}||h(\mathbf{x})\mathbf{w}|| = \displaystyle
\sup_{\tiny \mathbf{x}\in B\,\cap \,\mbox{supp}\,(\nu)} \max_{I} |h_I(\mathbf{x})| \\ \displaystyle \geq \sup_{\tiny \mathbf{x}\,\,\in B\,\cap \,\mbox{supp}\,(\nu)} |h_J(\mathbf{x})|\geq 1
\\ \forall \,\mathbf{t}\in {\mathfrak{a}}^+\,.\end{array}$$ *Case 2:* Suppose $w_I\neq 0$ for some $I\subseteq \{1,...,n\}$. Choose $l\in\{1,...,n\}$ such that $t_l= \max_{1\leq i\leq n}t_i$. If $l\in I$, set $J= I\cup \{0\}\setminus \{l\}$. Clearly $J$ contains $0$ but does not contain $l$. In view of (\[eqn:coeff\]), the coefficient of $f_l$ in the expression of $h_J$ is easily seen to be $\pm T^{\sum_{i\notin J} t_i}w_I$ and its absolute value is
$$\label{eqn:lbc}\displaystyle |T^{\sum_{i\notin J} t_i}w_I|
\geq e^{\sum_{i\notin J} t_i}\geq e^{t_l}\geq e^{t_0 /n}=e^{||\mathbf{t}||/n}\,.$$
If $l \notin I$, choose any $i\in I$ and let $J= I\cup \{0\}\setminus \{i\}$. Like before, $\pm T^{\sum_{i\notin J} t_i}w_I$ turns out as the coefficient of $f_i$ in $h_J$ so that we obviously get the analogue of (\[eqn:lbc\]). Thus in this case, there always exists $J$ such that\
$$\label{eqn:coeff*} \mbox{at least one
of the coefficients of}\,\,\, f_1,f_2,...\,,f_n \,\,\mbox{in}\,\, h_J\,\,\mbox{has absolute value}\geq e^{||\mathbf{t}||/n}\,.$$
Now, from the assumption (ii) of Proposition \[main prop\], it follows that there exists $\delta\, \textgreater\, 0$ such that $\sup_{\tiny \mathbf{x}\in B\,\cap\, \mbox{supp}\,(\nu)} |c_0+c_1f_1(\mathbf{x})+\cdot \cdot \cdot +c_nf_n(\mathbf{x})|\geq \delta$ for any $c_0,c_1,...,c_n\in F$ with $\max_{\tiny 0\leq i\leq n} |c_i|\geq 1$. We choose $M\in \mathbb{N}$ such that $\delta e^M\geq 1$.\
Let $||\mathbf{t}||\geq nM$. Then, because of (\[eqn:coeff\*\]), one surely has at least one of the coefficients of $f_1,f_2,...\,,f_n$ in $\frac{1}{T^M}\,h_J$ has absolute value at least $1$ and thus
$$\sup_{\tiny \mathbf{x}\,\cap \,\mbox{supp}\,(\nu)} \left|\frac{1}{T^M}\,h_J (\mathbf{x})\right|\geq \delta\,.$$ This gives, $$\sup_{\tiny \mathbf{x}\,\cap \,\mbox{supp}\,(\nu)} \left|h_J (\mathbf{x})\right|\geq \delta e^M\geq 1\,.$$ So, even here, we can see that $$\label{eqn:C2}
\begin{array}{rcl}\displaystyle \sup_{\tiny \mathbf{x}\,\,\in B\,\cap \,\mbox{supp}\,(\nu)}||h(\mathbf{x})\Delta||
= \sup_{\tiny \mathbf{x}\,\,\in B\,\cap \,\mbox{supp}\,(\nu)}||h(\mathbf{x})\mathbf{w}||=
\sup_{\tiny \mathbf{x}\,\,\in B\,\cap \,\mbox{supp}\,(\nu)} \max_{I}
|h_I(\mathbf{x})|\\ \displaystyle \geq \sup_{\tiny \mathbf{x}\,\,\in B\,\cap \,\mbox{supp}\,(\nu)} |h_J(\mathbf{x})|\geq 1 \\ \forall\, ||\mathbf{t}||\geq nM\,.
\end{array}$$
Letting $\rho=1$, $(C2)$ is thus immediate from (\[eqn:C1\]) and (\[eqn:C2\]) whenever $||\mathbf{t}||\geq nM$.\
Finally, $\tilde C$ and $s$ are taken as $(n+1)CD^{2(n+1)}$ and $nM$ repectively, and one applies Theorem \[thm:qn\] to show (\[eqn:prop\]). $\Box$.
Explicit constants: an example {#section:example}
==============================
In this section, we talk about a simple application of our Theorem (\[thm:1.2\]) to a concrete example, with special attention on the explicit constant $\varepsilon_0$. For us, here $d=1, n=2$ and $\nu$ is the unique Haar measure on $F$ that satisfies $\nu(B[0;1])=1$. It is not difficult to show that $\nu$ is $e^2$-Federer. Let $$f: B(0,1)\longrightarrow F^2\,,x\mapsto (x,x^2)\,.$$ We claim that $f$ is $(2,1/2)-good$, i.e. in other words, so is any $\phi\in F[x]$ having degree $\leq 2$. To see this, we shall apply the same technique which used in the proof of proposition 3.2 of [@KM].\
Let $\varepsilon\,\textgreater \,0$ and $\mathfrak{B}\subseteq B(0;1)$. We have to show $$\label{eqn:2,1/2}
\nu(\{x\in \mathfrak{B}\,:\, |\phi(x)|\,\textless\, \varepsilon\})\leq 2 \left(\frac{\varepsilon}{||\phi||_{\mathfrak{B}}}\right)^{1/2} \nu(\mathfrak{B})\,.$$ For convenience, put $S:= \{x\in \mathfrak{B}\,:\, |\phi(x)|\,\textless \,\varepsilon\}$. If $\nu(S)$, i.e. the LHS of (\[eqn:2,1/2\]), is $0$ then there is nothing to prove. Otherwise, we will show that $$m\leq 2 \left(\frac{\varepsilon}{||\phi||_{\mathfrak{B}}}\right)^{1/2} \nu(\mathfrak{B})$$ of equivalently, $$\label{eqn:2,1/2, suff}
|\phi(x)|\leq \varepsilon \left(\frac{\nu({\mathfrak{B}})}{m/2}\right)^2 \text{\,\,\,for all } x\in {\mathfrak{B}}\,,$$ whenever $0\,\textless \,m\,\textless \,\nu(S)$.\
From the continuity of $\phi$, we see that for each $x\in S$, there is a ball $B_x$ with center at $x$ and radius $ \textless \, \frac{m}{2}$ such that $B_x\subseteq S$. Now from the Besicovitch nature of $F$, one can extract a countable subcover $\{B_1,B_2,...\}$ consisting of mutually disjoint open balls from the cover $\{B_x\,:x\in S\}$ of $S$. Clearly $\nu(B_i)\leq \frac{m}{2}$ for each $i$. Thus in view of their size, it follows that the subcover has at least three balls. Let us denote their centers as $x_1, x_2\text{ and } x_3$. Then the centers $x_i\in S$ and they must satisfy $$\label{eqn:center}
|x_i- x_j|\geq \frac{m}{2}\text{\,\,\, for all }i,j= 1,2,3\,;\, i\neq j\,.$$ It is now time to employ the ‘Lagrange’s interpolation formula’ to complete the proof. By this formula, we can write $\phi$ as $$\label{eqn:lag}\begin{array}{rcl}\displaystyle \phi(x)=\phi(x_1)\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}+\phi(x_2)\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}\\ \displaystyle +\phi(x_3)
\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\,.\end{array}$$ As ${\mathfrak{B}}$ is a ball, certainly there exist $a\text{ and }m\in \mathbb{N}$ such that ${\mathfrak{B}}=B[a;\frac{1}{e^m}]$. Therefore ${\operatorname{diameter}}({\mathfrak{B}})=\frac{1}{e^m}=\nu({\mathfrak{B}})$. In view of this, (\[eqn:center\]) and (\[eqn:lag\]), it follows at once that $$|\phi(x)|\leq \varepsilon\frac{({\operatorname{diameter}}({\mathfrak{B}}))^2}{m^2/4}= \varepsilon \left(\frac{\nu({\mathfrak{B}})}{m/2}\right)^2 \text{\,\,\,for all } x\in {\mathfrak{B}}\,;$$ and that shows (\[eqn:2,1/2, suff\]).\
Thus all the conditions of the hypothesis of Theorem \[thm:1.2\] hold here and so that existence of desired $\varepsilon_0\, \textgreater\, 0$ is confirmed. We are interested to compute it. In the proof of Theorem \[thm:1.2\], we have also observed that our $\varepsilon_0$ can be taken as any positive quantity which is $\displaystyle \textless \,\frac{1}{\tilde C^{1/\alpha}}$; where $\tilde C$ was set, as we did in the proof of Proposition \[main prop\], as $(n+1)CD^{2(n+1)}$. Therefore in our example, we obtain that $$\varepsilon_0 \textless\, \frac{1}{\tilde C^2}= \frac{1}{(3\times 2\times (e^2)^ {2\times 3})^2}=\frac{1}{36\,e^{24}}\,.$$
Appendix: The density theorem {#section:density}
=============================
Fix $d\in \mathbb{N}$ and a radon measure $\nu$ on $F^d$. Let $\Omega$ be a measurable subset of $F^d$ and $\mathbf{x}\in \mbox{supp}\,(\nu)$. We say that $\mathbf{x}$ is a *point of density* of $\Omega$ if $$\displaystyle \lim_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\ \mathbf{x}\in B\end{array}}\frac{\nu(B\cap \Omega)}{\nu(B)}=1\,.$$ The above definition is nothing but the counterpart of the classical notion ‘point of Lebsegue density’ in our setting. Likewise, it is thus natural to expect the following:
\[thm:3.1\] Suppose $\Omega$ is a measurable subset of $F^d$. Then :
1. Almost every $\mathbf{x}\in \Omega$ is a point of density of $\Omega$.
2. Almost every $\mathbf{x}\notin \Omega$ is not a point of density of $\Omega$.
Now to prove Theorem \[thm:3.1\], we shall take up the same strategy of the proof for euclidean spaces. Namely, we develop our version of the ‘Lebsegue differentiation theorem’ first and Theorem \[thm:3.1\] follows then as a consequence. To begin with, we need to introduce ‘locally integrable’ functions as follows:\
A measurable function $f$ on $F^d$ is said to be *locally integrable* if for every ball $B\subseteq F^d$, the function $f\chi_{B} \in L^1(F^d,\nu)$.\
With this terminology, let us state our ‘differentiation theorem’:
\[thm:3.2\] Let $f$ be a locally integrable function on $F^d$. Then $$\label{eqn:3.1}\displaystyle \lim_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\ \mathbf{x}\in B\end{array}}
\frac{1}{\nu(B)} \int_{B} f\,\,d\nu= f(\mathbf{x})\,\,\mbox{for almost all}\,\,\mathbf{x}\in \mbox{supp}\,(\nu)\,.$$
An application of the above theorem to the characteristic function of $\Omega$ immediately yields Theorem \[thm:3.1\].\
To prove Theorem \[thm:3.2\], let us observe that $f$ can be assumed to be integrable without any loss in generality. To see this, suppose that the conclusion of the theorem is established for integrable functions. Now for any $n \in \mathbb{N}$, we apply (\[eqn:3.1\]) to the integrable function $f\chi_{B(\mathbf{0};n)}$ and obtain that the set of all $\mathbf{x}\in \mbox{supp}\,(\nu)\cap B(\mathbf{0};n)$ for which $$\label{eqn:3.2}\displaystyle \lim_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\ \mathbf{x}\in B\end{array}}
\frac{1}{\nu(B)} \int_{B} f\,\,d\nu \neq f(\mathbf{x})$$ has zero measure. Since any $\mathbf{x}\in \mbox{supp}\,(\nu)$ that satisfies (\[eqn:3.2\]) is contained in some $B(\mathbf{0};n)$, we are done. Thus we shall always let $f\in L^1(F^d,\nu)$ in the rest of this section.\
It suffices to show that, for all $\alpha\, \textgreater \,0$, $$E_{\alpha} := \left \{\mathbf{x}\in \mbox{supp}\,(\nu)\,:\,\displaystyle {\varlimsup}_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\
\mathbf{x}\in B\end{array}}\left|\frac{1}{\nu(B)}\int_{B} f\,\,d\nu\,-f(\mathbf{x})\right|\,\textgreater \,2\alpha \right\}$$ is null with respect to $\nu$. To achieve this, we need to introduce the ‘Hardy-Littlewood maximal function’ and make use of its main property. The relevant definition goes as follows.\
Let $f\in L^1(F^d,\nu)$. The *maximal function* of $f$, denoted by $f^*$, is defined as $$f^{*}(\mathbf{x})= \displaystyle \sup_{\mathbf{x}\in B}\, \frac{1}{\nu(B)}\int_B |f|\,\,d\nu\,\,\,for\,\,\mathbf{x}\in\mbox{supp}\,(\nu)\,\,
\mbox{and}\,\, \infty \,\,\,\mbox{otherwise} \,.$$ It is easy to see that, $\forall \alpha \in \mathbb{R}$, the set $\{\mathbf{x}\in F^d\,:\,f^{*}(\mathbf{x})\,\textgreater \,\alpha \}$ is open, because if $\mathbf{x}\in \mbox{supp}\,(\nu)$ and $f^{*}(\mathbf{x})\,\textgreater \,\alpha$ then there exists a ball $B$ containing $\mathbf{x}$ for which $$\frac{1}{\nu(B)}\int_B |f|\,\,d\nu \,\textgreater\, \alpha\,.$$ Clearly for any $\overline{\mathbf{x}} \in B \cap \mbox{supp}\,(\nu)$, one has $$f^*(\overline{\mathbf{x}})\geq \frac{1}{\nu(B)}\int_B |f|\,\,d\nu\, \textgreater \,\alpha\,.$$ Hence $f^*$ is measurable. The main property of this maximal function is given by the following theorem.
\[thm:3.3\] Let $f$ be integrable. Then for all $\alpha \,\textgreater \,0$ $$\label{eqn:3.3}
\nu (\{\mathbf{x}\in F^d\,:\, f^*(\mathbf{x})\,\textgreater\, \alpha\}) \leq \frac{1}{\alpha}\,||f||_1\,,$$ where $||f||_1=\int_{F^d} |f|\,\,d\nu$. As a consequence, $f^*(\mathbf{x})\,\textless \,\infty$ for almost all $\mathbf{x}$.
*Proof* : Since $\nu$ is radon and the set $\mathcal{A}_{\alpha} := \{\mathbf{x}\in F^d\,:\, f^*(\mathbf{x})\,\textgreater \,\alpha\}$ is open, as seen earlier, one has $$\nu(\mathcal{A}_{\alpha})=\displaystyle \sup_{\tiny \begin{array}{rcl}K\subseteq \mathcal{A}_{\alpha}\\K\,\mbox{ compact}\end{array}} \nu(K)\,.$$ It is thus enough to show that, for any compact subset $K$ of $\mathcal{A}_{\alpha}$, $$\nu(K)\,\textless\, \frac{1}{\alpha}\,||f||_1\,.$$ For each $\mathbf{x}\in K \cap \mbox{supp}\,(\nu)$, we have a ball $B_{\mathbf{x}}$ satisfying $$\frac{1}{\nu(B_{\mathbf{x}})}\int_{B_{\mathbf{x}}} |f|\,\,d\nu\, \textgreater \,\alpha\,,$$ i.e. $$\label{eqn:3.4}\frac{1}{\alpha}\int_{B_{\mathbf{x}}} |f|\,\,d\nu \,\textgreater \,\nu(B_{\mathbf{x}})\,.$$ The compact set $ K \cap \,\mbox{supp}\,(\nu)$ can be covered by finitely many such balls, say $B_1\,\,B_2\,,\,.\,.\,.\,,\,B_r$. Without any loss in generality, we can assume that the collection $\{B_i\}_{i=1}^r$ of balls is mutually disjoint.\
Now, in view of (\[eqn:3.4\]), we find that $$\begin{array}{rcl}\nu (K)= \nu (K \cap \,\mbox{supp}\,(\nu))\leq \displaystyle \sum_{i=1}^{r} \nu(B_{i})\,
\textless \,\displaystyle \sum_{i=1}^{r} \frac{1}{\alpha}\int_{B_{i}} |f|\,\,d\nu =
\displaystyle \frac{1}{\alpha} \sum_{i=1}^{r} \int_{B_{i}} |f|\,\,d\nu \\ = \displaystyle \frac{1}{\alpha} \int_{\cup_{i=1}^r B_i} |f|\,\,d\nu
\\ \leq \displaystyle \frac{1}{\alpha}\int_{F^d} |f|\,\,d\nu\\= \displaystyle \frac{1}{\alpha} ||f||_1\,.\,\,\,\,\,\,\Box \end{array}$$ We shall also need the following lemma
\[lem:3.1\] Let $g$ be an integrable function on $F^d$ and $\mathbf{x}\in \mbox{supp}\,(\nu)$ is a point of continuity of $g$. Then $$\displaystyle \lim_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\ \mathbf{x}\in B\end{array}}
\frac{1}{\nu(B)} \int_{B} g\,\,d\nu= g(\mathbf{x})\,.$$
*Proof* : Let $\varepsilon\, \textgreater \,0$. From the continuity of $g$ at the point $\mathbf{x}$, we get $r\,\textgreater \,0$ such that $|g(\mathbf{y})-g(\mathbf{x})|\,\textless\, \varepsilon$ for each $\mathbf{y}\in B(\mathbf{x};r)$. We set $$\delta := \frac{1}{2} \,\nu(B(\mathbf{x};r))\,.$$ Pick any ball $B$ containing $\mathbf{x}$ with $\nu(B)\,\textless\, \delta$. Clearly $B\subseteq B(\mathbf{x};r)$, because otherwise $B(\mathbf{x};r) \subsetneqq B$ will hold, as $\mathbf{x}\in B\cap B(\mathbf{x};r)$. But then we will have $\delta\, \textless\, \nu (B(\mathbf{x};r))\leq \nu(B)$ which is impossible. Now, it is easy to see that $$\left|\frac{1}{\nu(B)} \int_{B} g\,\,d\nu- g(\mathbf{x})\right|\leq \frac{1}{\nu(B)} \left|\int_{B} (g-g(\mathbf{x}))\,d\nu \right|\leq\frac{1}{\nu(B)} \int_{B} \left|g-g(\mathbf{x})\right|\,d\nu\leq
\frac{\varepsilon \nu(B)}{\nu(B)}= \varepsilon\,.\,\,\,\,\Box$$\
Let us prove Theorem \[thm:3.2\] now. Assume $\varepsilon\,\textgreater\, 0$. Since $F^d$ is locally compact and hausdorff, $C_c(F^d)$ is dense in $L^1(F^d,\nu)$. So we can choose a continuous and compactly supported function $g$ such that $$\label{eqn:3.5}
||f-g||_1 \,\textless \,\varepsilon\,.$$
For any $\mathbf{x}\in \mbox{supp}\,(\nu)$, writing $\frac{1}{\nu(B)}\int_{B} f\,\,d\nu\,-f(\mathbf{x})$ as $$\frac{1}{\nu(B)}\int_{B} (f-g)\,\,d\nu+ \frac{1}{\nu(B)} \int_{B} g\,\,d\nu\,- g(\mathbf{x})+(g(\mathbf{x})-f(\mathbf{x}))\,,$$ we can see that $$\label{eqn:3.6}
\begin{array}{rcl}\displaystyle {\varlimsup}_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\
\mathbf{x}\in B\end{array}}\left|\frac{1}{\nu(B)}\int_{B} f\,\,d\nu\,-f(\mathbf{x})\right|\leq \displaystyle {\varlimsup}_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\
\mathbf{x}\in B\end{array}}\frac{1}{\nu(B)}\int_{B} |f-g|\,\,d\nu \\ + \displaystyle {\varlimsup}_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\
\mathbf{x}\in B\end{array}}\frac{1}{\nu(B)}\left| \int_{B} g\,\,d\nu\,- g(\mathbf{x})\right|\\ + |g(\mathbf{x})-f(\mathbf{x})|\,.\end{array}$$ Let us observe that first term on the RHS of (\[eqn:3.6\]) is $\leq (f-g)^*(\mathbf{x})$ and the middle term vanishes in view of Lemma \[lem:3.1\]. Thus we have
$$\displaystyle {\varlimsup}_{\tiny \begin{array}{rcl}\nu (B)\rightarrow 0\\
\mathbf{x}\in B\end{array}}\left|\frac{1}{\nu(B)}\int_{B} f\,\,d\nu\,-f(\mathbf{x})\right|\leq (f-g)^*(\mathbf{x}) +
|g(\mathbf{x})-f(\mathbf{x})|\,.$$
This shows that $E_{\alpha}\subseteq F_{\alpha}\,\cup \,G_{\alpha}$, where $$F_{\alpha}:= \{\mathbf{x}\in F^d\,:\, (f-g)^*(\mathbf{x})\,\textgreater\, \alpha\}\,\,\mbox{and}\,\,G_{\alpha}:=\{\mathbf{x}\in F^d\,:\,
|g(\mathbf{x})-f(\mathbf{x})|\,\textgreater \,\alpha\}\,.$$ We shall now estimate $\nu(F_{\alpha})$ and $\nu(G_{\alpha})$ one by one. On the one hand, as $f-g$ is integrable, it is immediate that $$\label{eqn:3.7}
\nu(G_{\alpha})\leq \frac{1}{\alpha}\,||f-g||_1\,.$$ On the other hand, Theorem \[thm:3.3\] provides $$\label{eqn:3.8}
\nu(F_{\alpha})\leq \frac{1}{\alpha}\,||f-g||_1\,.$$ Finally from (\[eqn:3.7\]), (\[eqn:3.8\]) and (\[eqn:3.5\]), it follows that $$\nu(E_{\alpha})\leq \nu(F_{\alpha})+\nu(G_{\alpha})\leq 2\,\frac{1}{\alpha}\,
||f-g||_1\leq \frac{2}{\alpha}\,\varepsilon\,.\,\,\,\,\Box$$
[99]{}
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[^1]: Ghosh is supported by an ISF-UGC grant.
|
---
address:
- ', , '
- ', , '
author:
- 'Niklas Georg\*'
- Ulrich Römer
title: 'Conformally Mapped Polynomial Chaos Expansions for Maxwell’s Source Problem with Random Input Data'
---
Introduction {#sec1}
============
Due to recent developments in [@xiu2009], studying random parameter variations within the numerical simulation of fields and waves comes into reach. The present study is motivated from the design of optical components and plasmonic structures, where relatively large variabilities of nano-scale geometrical parameters can be observed, see, e.g. [@preiner2008]. In this work, we focus on the forward problem, i.e., the propagation of uncertainties from the model inputs to the outputs, in order to compute statistics and sensitivities for physical . We rely on surrogate modeling [@lemaitre2010] to reduce the computational complexity of sampling the underlying Maxwell solver. Although motivated from a forward model perspective, the surrogate construction could equally be used in an inverse problem context. Examples of surrogate modeling in electromagnetics can be found for instance in [@austin2013; @Georg_2019aa; @Corno2019] where microwave circuits and accelerator cavities are considered.
expansions [@xiu2002] are powerful tools for forward uncertainty propagation. They are based on an orthogonal polynomial basis with respect to the underlying probability distribution of the input parameters, to achieve good convergence properties. However, applying may still be challenging, the computational cost to handle large parameter uncertainties and parametric sensitivities for instance may be quite high. To remedy this issue, conformal maps can be utilized in order to improve the convergence of polynomial-based methods. The acceleration of quadrature methods by the use of conformal maps, has been considered in [@hale2008; @trefethen2013; @jantsch2018sparse]. In [@georg2018], conformal maps were combined with a stochastic collocation method, indicating significant gains in the accuracy of the corresponding surrogate model. In this work, we propose a new orthogonal basis by combining and conformal maps. We note that, the proposed basis is constructed such that it fulfills the same orthogonality properties as . Accordingly, advantages of methods are preserved, e.g., stochastic moments and Sobol coefficients can be directly computed from the expansion coefficients. It should also be noted that various approaches employing Polynomial Chaos expansions with basis rotation have been reported recently, see [@tsilifis2017reduced; @papaioannou2019pls]. Although, these works equally rely on mapped Polynomial Chaos approximations, the transformations are linear (affine) and not based on conformal mappings. Also, the emphasis there is on high dimensional approximation instead of convergence acceleration.
[.6]{} 
[.38]{} 
The proposed numerical scheme is applied to quantify uncertainties via surrogate models for Maxwell’s equations in the frequency domain. In particular, we consider the source problem on periodic domains with a plane wave excitation and uncertainties in the material interface geometry. Such model equations can describe, for instance, the coupling into plasmon modes with subwavelength diffraction gratings, which is illustrated in Fig. \[fig:mimPlasmonMode\]. Although illustrated by means of this particular application example, we note that the employed methodologies apply in a much broader context.
This paper is structured as follows: Section \[sec:maxwell\] contains a brief description of Maxwell’s source problem. The uncertainty quantification part can be found in Section \[sec:uq\], where we briefly recall standard before discussing the proposed extension based on conformal mappings. Section \[sec:application\] reports numerical results for an analytical RLC circuit and the aforementioned optical grating coupler, before conclusions are drawn.
Maxwell’s Source Problem {#sec:maxwell}
========================
We consider Maxwell’s source problem for periodic structures excited by an incident plane wave. For further details on this subject, we refer to [@georg2018; @jin2015]. We start with the time-harmonic curl-curl equation $$\nabla \times \left(\mu_{\mathrm{r}}^{-1} \nabla \times \ensuremath{\mathbf{E}}\right)- \omega^2 \varepsilon\mu_0 \ensuremath{\mathbf{E}} = 0 \qquad \text{in }\ensuremath{D}, \label{eq:curlcurl}$$ for the electric field phasor $\ensuremath{\mathbf{E}}$ in the computational domain $D$, where $\omega$ denotes the angular frequency, $\varepsilon$ the complex permittivity and $\mu_{\mathrm r}, \mu_0$ denote the relative and vacuum permeability, respectively. Note that assumes absence of charges and source currents in $D$. Based on Floquet’s theorem [@jin2015 Chapter 13], the computational domain $\ensuremath{D}$ can be reduced to a unit cell of the periodic structure, as we assume a periodic excitation. Such a unit cell is depicted in Fig. \[fig:unit\_cell\]. Due to the oblique angle of the incident wave, the excitation has a different periodicity than the geometry and, hence, periodic phase-shift boundary conditions need to be imposed on the respective boundaries. To truncate the structure in the non-periodic direction, a Floquet absorbing boundary condition and a boundary condition are applied. This leads to the boundary value problem $$\begin{aligned}
\nabla \times \left(\mu_{\mathrm{r}}^{-1} \nabla \times \ensuremath{\mathbf{E}}\right)- \omega^2 \varepsilon\mu_0 \ensuremath{\mathbf{E}} &= 0 &&\text{in }D\\
\ensuremath{\mathbf{E}}\vert_{\ensuremath{{\Gamma_{x^+}}}} e^{j\ensuremath{k^{\mathrm{inc}}_x} \ensuremath{{d_x}}}&= \ensuremath{\mathbf{E}}\vert_{\ensuremath{{\Gamma_{x^-}}}} \ \ &&\text{on}~{\ensuremath{{\Gamma_{x^+}}} \cup \ensuremath{{\Gamma_{x^-}}}}\\
\ensuremath{\mathbf{E}}\vert_{\ensuremath{{\Gamma_{y^+}}}} e^{j\ensuremath{k^{\mathrm{inc}}_y} \ensuremath{d_y}}&= \ensuremath{\mathbf{E}}\vert_{\ensuremath{{\Gamma_{y^-}}}} \ \ &&\text{on}~{\ensuremath{{\Gamma_{y^+}}}\cup \ensuremath{{\Gamma_{y^-}}}}\\
\ensuremath{\mathbf{n}} \times \ensuremath{\mathbf{E}} &= 0 && \text{on }\ensuremath{{\Gamma_{z^-}}} \\
(\mu_\mathrm{r}^{-1} \nabla \times \ensuremath{\mathbf{E}})\times \ensuremath{\mathbf{n}} + \mathcal{F}(\ensuremath{\mathbf{E}} ) &= \mathcal{G}(\ensuremath{\mathbf{E}}^{\mathrm{inc}}) \ \ &&\text{on } \ensuremath{{\Gamma_{z^+}}},\end{aligned}$$ where we refer to [@georg2018 Appendix A] for a derivation and definition of the functionals $\mathcal F(\cdot), \mathcal G(\cdot)$.
We assume in the following that the complex permittivity $\varepsilon$ depends smoothly on a parameter vector $\ensuremath{\mathbf{y}}\in \Xi \subset \mathbb R^N$. These parameters can then be used to model variations in the refractive indices or extinction coefficients of the (different) materials in $D$, as well as changes in the geometry of the material interfaces inside the domain $D$. Following a standard Galerkin procedure, cf. [@georg2018], we then obtain a model in the form $$\label{eq:pde_weak}
\text{find }\ensuremath{\mathbf{e}}(\ensuremath{\mathbf{y}}) \in V \text{ s.t.} \quad a_\ensuremath{\mathbf{y}}(\ensuremath{\mathbf{e}}(\ensuremath{\mathbf{y}}),\ensuremath{\mathbf{v}}) = l_\ensuremath{\mathbf{y}}(\ensuremath{\mathbf{v}}) \quad \forall \ensuremath{\mathbf{v}} \in V,$$ where $a_{\mathbf y}(\cdot, \cdot)$ is a continuous sesquilinear form, $l_{\mathbf y}(\cdot)$ is a continuous (anti)linear form and $V$ denotes a discrete subspace of ${\ensuremath{\mathbf{H}\left(\text{curl};\ensuremath{D}\right)}}$ [@monk2003], enforcing periodic phase-shift conditions on the traces at the periodic boundaries and homogeneous Dirichlet conditions at $\Gamma_{z^-}$. To achieve a curl-conforming discretization of , we employ N[é]{}d[é]{}lec’s elements of the first kind [@nedelec1980] and 2nd order on a tetrahedral mesh of $\ensuremath{D}$. As we consider the fundamental reflection coefficient $\mathcal Q(\mathbf e(\mathbf y))$, i.e. a scattering parameter, which can be computed as an affine-linear functional of the electric field $\ensuremath{\mathbf{e}}$ in post-processing [@georg2018]. For brevity, we replace $\mathcal Q(\mathbf e(\mathbf y))$ by $\mathcal Q(\mathbf y)$ in the following.
Uncertainty Quantification {#sec:uq}
==========================
To account for uncertainty, we model the input parameters $\mathbf y$ as independent with joint probability density function $\rho$ and image set $\Xi\subset \mathbb R^N$, where we assume in this section for brevity of notation that $\Xi$ is given as the hypercube $[-1,1]^N$. Note that different image sets $\Xi$ or stochastic dependence could also be considered, e.g. by a Rosenblatt transformation [@lebrun2009rosenblatt]. Additionally, we assume that the map $\mathcal Q: \Xi \rightarrow \mathbb C$ is holomorphic. Note that this assumption can often be justified for boundary value problems with random influences, see, e.g., [@hiptmair2018]. Holomorphy of the solution of Maxwell’s source problem with respect to general shape parametrizations was established in [@aylwin2019].
As discussed in the following, in this work we propose a method for surrogate modeling, where the basis functions are mapped polynomials based on [@xiu2002] combined with a conformal mapping. To compute the corresponding coefficients we rely on pseudo-spectral projection based on mapped quadrature rules [@hale2008].
Generalized Polynomial Chaos
----------------------------
For convenience of the reader, we briefly recall the standard polynomial chaos expansions, going back to Wiener [@wiener1938]. Considering Gaussian random variables, any $\mathcal Q(\mathbf y)$ with bounded variance, can be accurately represented using Hermite polynomials as basis functions. Employing the Askey-scheme [@xiu2002], for different probability distributions $\rho$, basis functions $\Psi_m:\Xi \rightarrow \mathbb R$ which are orthonormal w.r.t. the probability density $\rho$, i.e., $$\mathbb E[\Psi_i\Psi_j] := \int_\Xi \Psi_i(\mathbf y)\Psi_j(\mathbf y) \rho(\mathbf y) \,\mathrm d\mathbf y =\delta_{ij}, \label{eq:orth}$$ can be obtained. We note that can also be constructed for arbitrary densities $\rho$ [@soize2004]. The approximation is then given as $$\label{eq:mtermsapprox}
\mathcal Q_M^{\mathrm{PC}}\left(\mathbf{y}\right)
= \sum_{m=0}^{M} s_m \Psi_m\left(\mathbf{y}\right),$$ where the $s_m \in \mathbb{C}$ denote the coefficients. In practice, in order to obtain a computable expression, the sum in has to be truncated to $M<\infty$ and limited polynomial degrees are considered. The coefficients $s_m$ can then be determined in various ways, e.g. by regression or stochastic collocation, see [@xiu2010] for an overview. Here we consider projection, i.e., $$s_m = \mathbb E[\mathcal Q\Psi_m] = \int_\Xi \mathcal Q(\mathbf y) \Psi_m(\mathbf y) \rho(\mathbf y)\,\mathrm{d}\mathbf y. \label{eq:gpc_proj}$$ The integral in is usually not readily computable and is hence often approximated by numerical quadrature. Due to orthogonality of the basis, stochastic moments as well as variance-based sensitivity indices can then be calculated directly from the coefficients $s_m$ without further approximations, see [@xiu2010]. These methods show spectral convergence, e.g., in the norm $||u||_{L^2_\rho}:=\sqrt{\mathbb E[u^2]}$ [@xiu2002]. In particular, if the map $\mathbf y\mapsto \mathcal Q(\mathbf y)$ is analytic, exponential convergence can be expected, as discussed in the following. Note that, for simplicity, we first consider the univariate case, i.e., $N=1$, while generalizations to the multivariate case $N>1$ will be discussed later.
We assume that $\mathcal Q_\mathrm{1D}:[-1,1]\rightarrow\mathbb C$ can be analytically extended onto an open Bernstein ellipse $E_r\subset \mathbb C$. A Bernstein ellipse $E_r$ is an ellipse with foci at $\pm1$ and the size $r$ is given by the sum of the length of semi-major and semi-minor axis. This is illustrated in Fig. \[fig:Bernstein\]. Following [@trefethen2013], the error of the polynomial best approximation $\mathcal Q_{M}^\mathrm{PC^*}$ with degree $M$ can be estimated as $$\begin{aligned}
\|\mathcal Q_\mathrm{1D}-\mathcal Q_{M}^\mathrm{PC^*}\|_\infty &\le \frac {C_\text{B} r^{-M}}{r-1}, \label{eq:conv_est}\end{aligned}$$ where $\|\cdot\|_\infty$ denotes the supremum-norm on $[-1,1]$ and the constant $C_\text{B}>0$ depends on the uniform bound of $\mathcal Q_\mathrm{1D}$ in $E_r$. Note that convergence in the supremum-norm implies convergence in the $||\cdot||_{L^2_\rho}$ norm as well, as $$||\mathcal Q_\mathrm{1D}-\mathcal Q_{M}^\mathrm{PC^*}||_{L^2_\rho} =\Bigl( \int_{[-1,1]} \bigl(\mathcal Q_\mathrm{1D}-\mathcal Q_{M}^\mathrm{PC^*}\bigr)^2\rho_{\mathrm{1D}} \,\mathrm d y\Bigr)^{\frac 1 2} \le \|\sqrt{\rho_{\mathrm{1D}}}\|_\infty \,\|\mathcal Q_\mathrm{1D}-\mathcal Q_{M}^\mathrm{PC^*}\|_\infty \Bigl(\int_{[-1,1]} 1\,\mathrm dy \Bigr)^{\frac 1 2}=\sqrt{2} \,\|\sqrt{\rho_{\mathrm{1D}}}\|_\infty \,\|\mathcal Q_\mathrm{1D}-\mathcal Q_{M}^\mathrm{PC^*}\|_\infty.$$ We further note that the additional aliasing error introduced by the discrete projection does not harm the convergence order for well-resolved smooth function, cf. [@xiu2002 Chapter 3.6].
Conformally Mapped Generalized Polynomial Chaos
-----------------------------------------------
Equation shows that the convergence is connected to the region of analyticity, in particular the convergence order $r$ depends on the size of the largest Bernstein ellipse not containing any poles of the continuation of $\mathcal Q_\mathrm{1D}$ (in the complex plane). However, established procedures [@babuska2007] inferring the regularity of parametric problems based on a sensitivity analysis, do not lead to elliptical regions, but rather prove analyticity in an $\epsilon$-neighborhood of the unit interval. In this case, a conformal map $g$ can be employed, which maps Bernstein ellipses to *straighter* regions and thus, enlarges the domain of analyticity, as illustrated in Fig. \[fig:mapping\]. To this end, there are various mappings which could be employed, cf. [@hale2009]. Here, we focus for simplicity on the so-called $9$-th order sausage mapping $$g(s) = \frac 1 {53089} (40320s+6720s^3+3024s^5+1800s^7 +1225s^9)\label{eq:sausage_map}$$ introduced in [@hale2008], which represents a normalized Taylor approximation of the inverse sine function. Note that $g$ maps the unit interval to itself, i.e., $$g([-1,1])=[-1,1] ~\text{and}~g(\pm 1)=\pm1.\label{eq:unit_interval}$$
[0.45]{} 
[0.5]{} 
Conformal maps were employed in [@hale2008] to derive new numerical quadrature formulas, and have also recently been considered in the context of stochastic collocation methods [@jantsch2018sparse; @georg2018]. In this work, we address the combination of conformal maps and polynomial chaos expansions. Based on the assumption that $h:=\mathcal Q_\mathrm{1D}\circ g$ has a larger Bernstein ellipse than $\mathcal Q_\mathrm{1D}$, and is hence better suited to be approximated with polynomials, we propose a new orthogonal basis $$\Phi_m \coloneqq \tilde \Psi_m\circ g^{-1},\quad m=0,\ldots,M \label{eq:map_basis}$$ where $\tilde \Psi_m$ are orthonormal polynomials w.r.t. the transformed density $$\tilde \rho_{\mathrm{1D}} (s) := g'(s) \rho_{\mathrm{1D}}(g(s)).
\label{eq:transformed_pdf}$$ We emphasize that $\{\Phi_m\}_{m=0}^M$ forms an orthonormal basis w.r.t. the input probability distribution $\rho$. This can be shown by a change of variables $y=g(s)$ $$\begin{aligned}
\mathbb E[\Phi_i \Phi_j] &= \int_{-1}^1 (\tilde \Psi_i\circ g^{-1})(y)(\tilde \Psi_j\circ g^{-1})(y) \rho_{\mathrm{1D}}(y)\,\mathrm{d}y \\
&=\int_{-1}^1 \tilde \Psi_i(s) \tilde \Psi_j(s) \underbrace{\rho_{\mathrm{1D}}(g(s))g'(s)}_{\tilde \rho_{\mathrm{1D}}(s)}\,\mathrm{d}s= \delta_{ij}, \end{aligned}$$ where the last line holds by construction of the polynomials $\tilde \Psi_m$. Due to the orthogonality, the corresponding coefficients $s_m$ of the mapped approximation $${Q}_M(y) = \sum_{m=0}^M s_m \Phi_m(y) \label{eq:map_approx}$$ can then be determined by the projection $$s_m = \mathbb E[\Phi_m Q_\mathrm{1D}] = \int_{-1}^1 \Phi_m(y) Q_\mathrm{1D}(y) \rho_{\mathrm{1D}}(y)\,\mathrm{d}y. \label{eq:mapped_proj}$$ Note that, by abuse of notation, we use the same symbol $s_m$ for the coefficients and the mapped coefficients. The mapped polynomial best approximation $ \mathcal Q_M^*$ converges as $$\begin{aligned}
\|\mathcal Q_\mathrm{1D}- {\mathcal Q}_M^*\|_\infty &= \|\mathcal Q_\mathrm{1D}\circ g-{\mathcal Q}_M^*\circ g\|_\infty =\|(h-h_M^{\mathrm{PC}^*}) \|_\infty \le \frac{\tilde C_B \tilde r^{-M}}{\tilde r-1},\end{aligned}$$ where $h_M^{\mathrm{PC}^*}$ denotes the polynomial best approximation of $h$ and $\tilde r$ the size of a Bernstein ellipse $E_{\tilde r}$ on which an analytic continuation of $h$ exists.
In particular, the convergence order of the mapped approximation $\mathcal Q_M$ is given by the size of the largest Bernstein ellipse $E_{\tilde r_{\mathrm{max}}}$ which is fully mapped into the region of analyticity of $\mathcal Q_\mathrm{1D}(y)$. Note that $\tilde r_{\mathrm{max}}>r_{\mathrm{max}}$ for any positive $\epsilon<0.75$ [@georg2018], and hence, a convergence improvement is to be expected in those cases, i.e., for functions analytic in such $\epsilon$-neighborhoods. It should be mentioned nevertheless that this procedure does not always yield improved convergence rates. One can easily imagine poles located such that a Bernstein ellipse may lead to a larger region of analyticity than a strip-like geometry. In the examples considered in this work, however, convergence acceleration could indeed be obtained.
To numerically compute , we derive mapped quadrature rules, cf. [@hale2008; @hale2009]. As pointed out in [@trefethen2013] for instance, Gaussian quadrature is derived from polynomial approximations and, hence, the convergence order also depends on the size of the Bernstein ellipse corresponding to the regularity of the integrand, see e.g. [@hale2008 Theorem 1]. Therefore, relying again the assumption that $\mathcal Q_\mathrm{1D}\circ g$ has a larger Bernstein ellipse, we apply a change of variables $y=g(s)$ in $$s_m = \mathbb E[\Phi_m \mathcal Q_\mathrm{1D}] = \int_{-1}^1 \Phi_m(y) \mathcal Q_\mathrm{1D}(y) \rho_{\mathrm{1D}}(y)\,\mathrm{d}y =
\int_{-1}^1 \Phi_m(g(s)) \mathcal Q_\mathrm{1D}(g(s)) \underbrace{\rho_{\mathrm{1D}}(g(s)) g'(s)}_{\tilde \rho_{\mathrm{1D}}}\,\mathrm{d}s. \label{eq:mapped_proj_trans}$$ The mapped quadrature scheme is then obtained by application of Gaussian quadrature w.r.t. the transformed density $\tilde \rho_{\mathrm{1D}}$, i.e. quadrature nodes $\{\tilde y^{(i)}\}_{i=0}^{M_\mathrm{quad}}$ and correspondings weights $\{\tilde w^{(i)}\}_{i=0}^{M_\mathrm{quad}}$, to the transformed integrand in $$s_m \approx \sum_{i=0}^{M_\text{quad}} \Phi_m(g(\tilde y^{(i)})) \mathcal Q_\mathrm{1D}(g( \tilde y^{(i)})) \tilde w^{(i)} = \sum_{i=0}^{M_\text{quad}} \Phi_m(\hat y^{(i)}) \mathcal Q_\mathrm{1D}(\hat y^{(i)})\hat w^{(i)}.$$ Note that the mapped quadrature nodes are obtained as $\hat y^{(i)} := g(\tilde y^{(i)})$, while the mapped weights are given as $\hat w^{(i)} := \tilde w^{(i)}$. Due to , it is ensured that the mapped quadrature nodes $\hat y^{(i)}$ do not require the evaluation of the analytic continuation of $\mathcal Q_\mathrm{1D}$ in the complex plane, which is, in practice, not always possible. A convergence improvement is expected based on the assumption that the transformed integrand in has a larger Bernstein ellipse. For further details on mapped quadrature schemes, we refer to [@hale2008]. However, we note the (minor) differences that in this work we employ Gaussian quadrature w.r.t. the transformed density $\tilde \rho_{\mathrm{1D}}$ to derive the mapped quadrature scheme, while [@hale2008] only considers unweighted Gaussian quadrature and, thereby, takes $g'(s)$ as part of the integrand (instead of the weight).
We proceed with a discussion of the multivariate case $N>1$. To this end, we introduce the multivariate mapping $\mathbf g(\mathbf s) = [g_1(s_1), \ldots, g_N(s_N)]$. In this work, we employ, for simplicity, the same mapping for all parameters, i.e. $g_1=\ldots=g_N=g$. However, different choices would be possible as well. We also note that, for the trivial mapping $\mathbf g_\mathrm{triv}: \mathbf s\mapsto \mathbf s$ standard polynomial chaos expansions would be recovered. For each parameter $y_i$ with univariate $\rho_i$, we define the transformed $\tilde \rho_i(y_i) := \rho_i(g_i(y_i)) g_i'(y_i)$. The corresponding transformed joint is then given by $\tilde \rho(\mathbf y) = \tilde \rho_1(y_1) \ldots \tilde \rho_N(y_N)$. In the following, we denote by $\{\tilde \Psi_{\mathbf{m}}\}_{\mathbf m}$ an orthonormal polynomial basis w.r.t. to the transformed density $\tilde \rho$, i.e. $$\mathbb E_{\tilde\rho}[\tilde \Psi_{\mathbf i}\tilde \Psi_{\mathbf j}]:=\int_{\Xi}\tilde \Psi_{\mathbf i}(\mathbf y)\tilde \Psi_{\mathbf j}(\mathbf y)\tilde\rho(\mathbf y) \,\mathrm d\mathbf y = \delta_{i_1j_1}\ldots\delta_{i_Nj_N}, \label{eq:orth_ND}$$ where we introduced the multi-index $\mathbf m = (m_1,\ldots, m_N)$ holding the univariate polynomial degrees, such that $\tilde \Psi_{\mathbf m}$ is a tensor-product polynomial of order $m_j$ in dimension $j=1,\ldots, N$. The respective mapped polynomials are then obtained as $$\Phi_{\mathbf m}(\mathbf y) := (\tilde \Psi_{\mathbf m} \circ \mathbf g^{-1})(\mathbf y).$$ The coefficients of the multivariate mapped approximation $${\mathcal Q}_p(\mathbf y) := \sum_{\| \mathbf m\|_\infty \le p} s_{\mathbf m} \Phi_{\mathbf m}(\mathbf y), \label{eq:map_approx_ND}$$ where we consider for simplicity a tensor-product construction of maximum degree $p$, can then again be obtained by projection $$s_{\mathbf m} = \mathbb E[ \Phi_{\mathbf m} \mathcal Q] = \int_\Xi \Phi_{\mathbf m}(\mathbf y) \mathcal Q(\mathbf y) \rho(\mathbf y)\,\mathrm{d}\mathbf y. \label{eq:multivariate_projection}$$ To evaluate the multi-dimensional integral in , we employ mapped Gaussian quadrature. In this case the mapped nodes and weights are given by $\hat{\mathbf y}^{(i)} := \mathbf g(\tilde{\ensuremath{\mathbf{y}}}^{(i)})$ and $\hat w^{(i)}:=\tilde w^{(i)}$, respectively, where, in turn, $\tilde{\ensuremath{\mathbf{y}}}^{(i)}$ and $\tilde w^{(i)}$ are the nodes and weights of a Gaussian quadrature w.r.t. $\tilde \rho$.
Finally, we emphasize that, since the mapped representation uses an orthogonal basis, the coefficients $s_{\mathbf m}$ can be used to directly compute stochastic moments as well as variance-based sensitivity indices. For instance, the mean value is given by $$\mathbb E[{\mathcal Q}_p] = \int_{\Xi } \Bigl(\sum_{\| \mathbf m\|_\infty \le p} s_{\mathbf m} \Phi_{\mathbf m}(\mathbf y)\Bigr) \rho(\mathbf y)\,\mathrm d\mathbf y = s_{\mathbf 0},$$ where we employed, that the mapped basis function $\Phi_{\mathbf 0}$ is constant on $\Xi$, as well as the orthonormality condition . Accordingly the variance is given by $$\mathbb V[{\mathcal Q}_p]=\mathbb E[{\mathcal Q}_p^2] -\mathbb E[{\mathcal Q}_p]^2 = \sum_{0<\|\mathbf m\|_\infty\le p} s_{\mathbf m}^2.$$ Additionally, Sobol sensitivity indices [@Sobol2001], based on a decomposition of the variance, can also be directly derived from the coefficients. Regarding the estimation of Sobol indices, we will focus on the so-called main-effect (1st order) and total-effect (total order) indices. We define the multi-index sets $\Lambda_n^{\text{main}}, \Lambda_n^{\text{total}} \subset \Lambda^\text{TP}_{p}:=\{\mathbf m\,|\, 0\le \|\mathbf m \|_\infty \le p\}$, $n=1,2,\dots,N$, such that $$\begin{aligned}
\Lambda_n^\text{main} &= \{\mathbf{m} \in \Lambda_{p}^\text{TP} \; : \; m_n \neq 0 \hspace{0.5em} \text{and} \hspace{0.5em} m_j = 0, n \neq j\},\\ \Lambda_{n}^{\text{total}} &= \{\mathbf{m} \in \Lambda^\text{TP}_{p} \; : \; m_n \neq 0\}.\end{aligned}$$ We then define the partial variances $\mathbb V_n^{\text{main}}\left[{\mathcal Q}_p\right]$ and $\mathbb V_n^{\text{total}}\left[{\mathcal Q}_p\right]$, such that $$\begin{aligned}
\mathbb V_n^{\text{main}}\left[{\mathcal Q}_p\right] = \sum_{\mathbf{m} \in \Lambda_n^{\text{main}}} s_{\mathbf{m}}^2, &&
\mathbb V_n^{\text{total}}\left[{\mathcal Q}_p\right]= \sum_{\mathbf{m} \in \Lambda_n^{\text{total}}} s_{\mathbf{m}}^2.\end{aligned}$$ Then, the main-effect and total-effect Sobol indices, $ S_n^{\text{main}}$ and $S_n^{\text{total}}$, respectively, are given as $$\begin{aligned}
S_n^{\text{main}}[{\mathcal Q}_p] = \frac{\mathbb V_n^{\text{main}}\left[{\mathcal Q}_p\right]}{\mathbb V\left[{\mathcal Q}_p\right]}, &&
S_n^{\text{total}} = \frac{\mathbb V_n^{\text{total}}\left[{\mathcal Q}_p\right]}{\mathbb V\left[{\mathcal Q}_p\right]}. \label{eq:sobol_indices}\end{aligned}$$
Application {#sec:application}
===========
We apply the methods presented in the last section to two model problems. We first consider an academic example of an stochastic RLC circuit, since there is a closed-form solution available which allows us to illustrate the main ideas of the proposed approach in detail. We then consider the optical grating coupler [@preiner2008], which is a non-trivial benchmark example from nanoplasmonics.
RLC circuit
-----------
[.49]{} 
[.49]{} 
We consider the model of an RLC circuit, as illustrated in Fig. \[fig:RLC\_circuit\]. Assuming harmonic time dependency, the electric current $i$ is given by $$\Bigl(-L\omega^2 +j\omega R + \frac 1{C} \Bigr)i = j\omega u_\text{e} \label{eq:rlc_ana_sol}$$ We consider, arbitrarily chosen, an angular frequency $\omega=10^4\,$$\mathrm{s}^{-1},$ exciting voltage $u_\mathrm e = \SI{1}{V},$ capacitance $C=\SI{10}{\micro\farad},$ and a (rather small) resistance of $R=\SI{1}{\ohm}$. Additionally, we consider a variable inductance $L(y)=\SI{1}{mH}+\SI{0.25}{mH}\cdot y$. The parameter $y$ is then modeled as a uniformly distributed random variable with probability density function $\rho=\mathcal U (-1,1)$, such that a stochastic model is obtained. As $\mathcal Q$, we consider the amplitude of the current $\mathcal Q:=|i|$. Fig. \[fig:i\_over\_l\] shows the parametric dependency of the $|i|$ with respect to $y$, which is analytic for $y\in[-1,1]$. However, the continuation in the complex plane has poles at $$y =\pm i\frac R {\omega \cdot \SI{0.25}{mH}}. \label{eq:poles}$$ This complex conjugate pole pair limits the size of the largest Bernstein ellipse, where $\mathcal Q(y)$ is analytic, which is illustrated in Fig. \[fig:circuit\_ellipses\] for different values of $R$.
In each case, we compute approximations of increasing order for $\mathcal Q(y)$ using the `Chaospy` toolbox [@feinberg2015]. In particular, the coefficients of an $M-$th order approximations are computed by pseudo-spectral projection using Gaussian quadrature of order $M+1$. The accuracy of the surrogate models is then quantified in the empirical $L^2_\rho$ norm. In particular, we apply cross-validation using $N^\text{cv}=1000$ random parameter realizations $y_{\mathrm{cv}}^{(i)}$ drawn according to the probability density $\rho$, to compute the error $$E^\text{cv}= \frac {1}{N^\text{cv}} \sum_{i=1}^{N^\text{cv}} |\mathcal Q^\mathrm{PC}_M(y_{\mathrm{cv}}^{(i)})-\mathcal Q(y_{\mathrm{cv}}^{(i)})|^2. \label{eq:E_cv}$$ Additionally, we compute the error in the first-stochastic moment, i.e. the mean value of the approximation given by the first polynomial coefficient $s_0$. The reference solutions for the expected values are obtained by Gaussian quadrature of order $200$ up to machine accuracy. The convergence of the corresponding surrogate model w.r.t. the polynomial order $M$, in terms of cross-validation and mean value accuracy, are presented in Fig. \[fig:Conv\_gpc\_l2\] and Fig. \[fig:Conv\_gpc\_mean\], respectively. The plots confirm numerically, showing a decreasing convergence order for decreasing values of $R$ corresponding to decreasing sizes of the associated Bernstein ellipses. Note that, according to a similar behaviour as for decreasing damping can be expected for increasing amplitudes of the considered input variation.
[.33]{} 
[.33]{} 
[.33]{} 
Next, we apply the conformally mapped expansions proposed in the last section. The implementation is done in `Python` based on `Chaospy` [@feinberg2015]. Fig. \[fig:uniform\_densities\] shows the transformed density for a uniform input distribution $\rho$. Fig. \[fig:uniform\_basisfuns\] depicts some exemplary basis functions of and mapped . Note that the basis functions are in this case Legendre polynomials, while the mapped basis functions, given by , are no polynomials. We then study the convergence of the corresponding surrogate models, where mapped quadrature of order $M+1$ is used to compute the mapped expansions of order $M$. Fig. \[fig:uniform\_cv\_err\], Fig. \[fig:uniform\_mean\_err\] and Fig. \[fig:uniform\_std\_err\] demonstrate the improved convergence order of the mapped approach, in terms of the cross-validation error, as well as the accuracy of the computed mean value and the computed standard deviation.
[.43]{} 
[.55]{}
[.3]{} 
[.3]{} 
[.3]{} 
Optical Grating Coupler
-----------------------
![Numerical model of considered optical grating coupler. Excitation by incident plane wave at upper boundary.[]{data-label="fig:design_coupler"}](images/fig6)
We now consider the model of an optical grating coupler [@preiner2008], which was introduced in the beginning, see Fig. \[fig:mimPlasmonMode\]. The structure’s design [@cstTutorial] is shown in Fig. \[fig:design\_coupler\]. A plane wave at optical frequency hits the surface of the grating coupler. The incident wave couples with a plasmon mode, which propagates along the metallic surface. It is found that the resonance has a significant shift (in energy) as a function of the grating depth [@preiner2008] and therefore, it is of great interest to evaluate the influence of nano-technological manufacturing imperfections.
We use <span style="font-variant:small-caps;">FEniCS</span> [@alnaes2015] for the discretization and implement a design element approach [@braibant1984] for the geometry parametrization. The numerical model is described in greater detail in [@georg2018]. Note that we only consider periodic variations, modeling a systemic offset in the fabrication process, and do not address local uncertainties leading to different unit cells. Readers interested in the latter case are referred to [@schmitt2019optimization]. The fundamental scattering parameter is considered as $\mathcal Q\in \mathbb C$. We consider three sensitive geometrical parameters as uncertain, in particular the thicknesses of the upper gold layer $t_1=\SI{12}{nm}+\Delta y_1$, the thickness of the dielectric layer $t_2=\SI{14}{nm}+\Delta y_2$ and the grating depth $T=\SI{20}{nm} + \Delta y_3$, as illustrated in Fig. \[fig:design\_coupler\]. We model those parameters as independent beta distributed in the range of $\pm \Delta=\pm\SI{2}{nm}$. The corresponding shape parameters are chosen such that a normal approximation is approximated. The corresponding probability distribution $\rho_i$ of the $y_i,\,i=1,\ldots,3$ is shown in Fig. \[fig:beta\_densities\], together with the transformed density $\tilde \rho_i$. The univariate polynomials which are Jacobi polynomials in this case, as well as the mapped polynomials are illustrated in Fig. \[fig:beta\_basisfuns\].
[.43]{} ![ for stochastic RLC circuit with beta distributed input parameter.[]{data-label="fig:beta_gpc"}](images/fig7a "fig:")
[.55]{} ![ for stochastic RLC circuit with beta distributed input parameter.[]{data-label="fig:beta_gpc"}](images/fig7b "fig:")
### Decay of Fourier Coefficients
We first study the decay of polynomial coefficients to numerically investigate the smoothness of the mapping from the input parameters to the complex S-parameter $\mathcal Q$ and justify the use of (mapped) polynomial approximations. It has been shown, see e.g. [@nobile2009 Lemma 2] where Legendre polynomials are considered, that if this mapping is smooth, the Fourier coefficients $s_\ensuremath{\mathbf{m}}$ of an $N-$variate approximation decay exponentially, i.e. $$\begin{aligned}
|s_\ensuremath{\mathbf{m}}|^2 \le C e^{-\sum_{n=1}^N g_n m_n},\end{aligned}$$ where $C$ and $g_n, ~n = 1,\ldots,N$ are positive constants independent of $\ensuremath{\mathbf{m}}$ and we have assumed that the polynomials are normalized. We consider the maximum of the absolute value of the Fourier coefficients $s_\ensuremath{\mathbf{m}}$ with fixed maximum-degree $w = ||\ensuremath{\mathbf{m}}||_\infty := \max_i m_i$ $$\begin{aligned}
\max_{||\ensuremath{\mathbf{m}}||_\infty=w} |s_\ensuremath{\mathbf{m}}|^2
\le \max_{||\ensuremath{\mathbf{m}}||_\infty=w} Ce^{-\sum_{n=1}^N g_n m_n}
= Ce^{-\min_{||\ensuremath{\mathbf{m}}||_\infty =w}\sum_{n=1}^N g_n m_n}
\le Ce^{-(\min_n g_n) w}. \end{aligned}$$ It can be seen that the maximum Fourier coefficient is expected to decay exponentially with an increasing maximum-degree $w$.
![Decay of Fourier coefficients of multivariate (mapped) approximation.[]{data-label="fig:fouriercoeffs"}](images/fig8)
We construct a multivariate approximation with a tensor-product basis of order $m_{\max} = 15$. The multivariate integrals of the pseudo-spectral projection are then computed by a Gauss quadrature of order 17. All coefficients $s_\ensuremath{\mathbf{m}}$ are plotted in Fig. \[fig:fouriercoeffs\] in red color, where an exponential decay can indeed be observed. This can be seen as a numerical indicator for smoothness of the approximated mapping $\mathcal Q(\ensuremath{\mathbf{y}})$. Additionally, we also construct a mapped approximation of same order and plot the corresponding coefficients in black color. It can be observed that the mapped coefficients exhibit a faster convergence and hence the mapped approach can be expected to show, again, an improved convergence.
### Uncertainty Quantification {#uncertainty-quantification}
[.3]{} 
[.3]{} 
[.3]{} 
Next, we consider approximations of the magnitude of the S-parameter $|\mathcal Q(\mathbf y)|$ using (mapped) tensor-product expansions of increasing order $M$, where pseudo-spectral projections of order $M+1$ is employed to compute the coefficients. Fig. \[fig:conv\_coupler\_l2\] compares and the proposed mapped counterpart in terms of the $L^2_\rho$-error , in particular, again, by cross-validation with $10^3$ random parameter realizations. It can be observed that the mapped approach converges about $30\%$ faster w.r.t. the order $M$ than . However, the respective computational gain grows, in this case, exponentially w.r.t. the number of inputs and, hence, the required number of model evaluation to reach a prescribed accuracy reduces roughly by a factor of $2$. Similar findings hold for the stochastic moments, in particular, we present the convergence of the mean value in Fig. \[fig:conv\_coupler\_mean\] and the computed standard deviation in Fig. \[fig:conv\_coupler\_std\]. In this case, the reference solutions are obtained by Gaussian quadrature of order 30.
Finally, the most accurate surrogate model, i.e. the mapped expansion of order $14$, is used to compute the mean value $\mathbb{E}[|\mathcal Q|]\approx0.786$ and the standard deviation $\sqrt{\mathbb{V}[|\mathcal Q|]}\approx 0.077$ of the . Additionally, Sobol indices are computed and presented in Fig. \[fig:coupler\_sobol\]. The thickness of the dielectric layer $t_2$ is identified as the most influential parameter. We note that there is a significant difference between the main- and total-effect indices. In particular, the sum of the first order indices is only $34\%$, while the remaining $66\%$ can be attributed to strong coupling effects among the parameters.
![Sensitivity of input parameters.[]{data-label="fig:coupler_sobol"}](images/fig10)
Conclusions
===========
In this paper an efficient surrogate modeling technique for quantifying uncertainties in the material and geometry of high-frequency and optical devices was presented. The proposed method is based on to achieve spectral convergence. Through a combination with conformal maps we were able to enlarge the region of analyticity. This lead to an improved convergence rate, which was numerically demonstrated for two benchmark problems. In particular, the approach showed significant gains in either accuracy or computational cost, without adding any relevant extra computational effort. Due to orthogonality of the proposed basis, stochastic moments as well as Sobol indices can be directly obtained from the coefficients. It is worth noting that this technique can also be combined with other techniques for convergence acceleration such as adjoint-error correction, sparse-grids and (adjoint-based) adaptivity for the multivariate case [@georg2018].
Acknowledgements {#acknowledgements .unnumbered}
----------------
This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – RO4937/1-1. The work of Niklas Georg is also partially funded by the *Excellence Initiative* of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.
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---
abstract: 'In this work we reexamine the LDA+U method of Anisimov and coworkers in the framework of a plane-wave pseudopotential approach. A simplified rotational-invariant formulation is adopted. The calculation of the Hubbard $U$ entering the expression of the functional is discussed and a linear response approach is proposed that is internally consistent with the chosen definition for the occupation matrix of the relevant localized orbitals. In this way we obtain a scheme whose functionality should not depend strongly on the particular implementation of the model in ab-initio calculations. We demonstrate the accuracy of the method, computing structural and electronic properties of a few systems including transition and rare-earth correlated metals, transition metal monoxides and iron-silicate.'
author:
- 'Matteo Cococcioni[^1] and Stefano de Gironcoli'
title: A linear response approach to the calculation of the effective interaction parameters in the LDA+U method
---
Introduction
============
The description and understanding of electronic properties of strongly correlated materials is a very important and long standing problem for ab-initio calculations. Widely used approximations for the exchange and correlation energy in density functional theory (DFT), mainly based on parametrization of (nearly) homogeneous electron gas, miss important features of their physical behavior. For instance both local spin-density approximation (LSDA) and spin-polarized generalized gradient approximations ($\sigma-$GGA), in their several flavors, fail in predicting the insulating behavior of many simple transition metal oxides (TMO), not only by severely underestimating their electronic band gap but, in most cases, producing a qualitatively wrong metallic ground state.
TMOs have represented for long time the most notable failure of DFT. When the high-T$_{{\rm c}}$ superconductors entered the scene (their parent materials are also strongly correlated systems) the quest for new approaches that could describe accurately these systems by first principles received new impulse, and in the last fifteen years many methods were proposed in this direction. Among these, LDA+U approach, first introduced by Anisimov and coworkers [@AZA91; @ani2; @SDA94], has allowed to study a large variety of strongly correlated compounds with considerable improvement with respect to LSDA or $\sigma-$GGA results. The successes of the method have led to further developments during the last decade which have produced very sophisticated theoretical approaches[@DMFT] and efficient numerical techniques.
The formal expression of LDA+U energy functional is adapted from model hamiltonians (Hubbard model in particular) that represent the “natural” theoretical framework to deal with strongly correlated materials. As in these models, a small number of localized orbitals is selected and the electronic correlation associated to them is treated in a special way. The obtained results strongly depend on the definition of the localized orbitals and on the choice of the interaction parameters used in the calculation, that should be determined in an internally consistent with. This is not always done and a widespread but, in our opinion, unsatisfactory attitude is to determine the value of the electronic couplings by seeking a good agreement of the calculated properties with the experimental results in a semiempirical way.
In this work a critical reexamination of the LDA+U approach is proposed, which starting from the formulation of Anisimov and coworkers [@AZA91; @ani2; @SDA94], and its further improvements [@LAZ95; @ani5; @Pickett98], develops a simpler approximation. This is, in our opinion, the “minimal” extension of the usual approximate DFT (LDA or GGA) schemes needed when atomic-like features are persistent in the solid environment.
In the central part of this work we describe a method, based on a linear response approach, to calculate in an internally consistent way—without aprioristic assumption about screening and/or basis set employed in the calculation—the interaction parameters entering the LDA+U functional used. In this context our plane-wave pseudopotential (PWPP) implementation of the LDA+U approach is presented and discussed in some details. We stress however that the proposed method is basis-set independent.
Our methodology is then applied to the study of the electronic properties of some real materials, chosen as representative of “normal” (bulk iron) and correlated (bulk cerium) metals, as well as a few examples of strongly correlated systems (iron oxide, nickel oxide and fayalite).
Standard LDA+U implementation:
==============================
In order to account explicitly for the on-site Coulomb interaction responsible for the correlation gap in Mott insulators and not treated faithfully within LDA, Anisimov and coworkers [@AZA91; @ani2; @SDA94] correct the standard functional adding an on-site Hubbard-like interaction, $E_{Hub}$: $$\begin{aligned}
\label{simpleu}
E_{LDA+U}[n({\bf r})] &=& E_{LDA}[n({\bf r})] + \nonumber \\
&&E_{Hub}[\{n^{I\sigma}_{m}\}] -
E_{dc}[\{n^{I\sigma}\}]\end{aligned}$$ where $n({\bf r})$ is the electronic density, and $n^{I\sigma}_{m}$ are the atomic-orbital occupations for the atom $I$ experiencing the “Hubbard” term. The last term in the above equation is then subtracted in order to avoid double counting of the interactions contained both in $E_{Hub}$ and, in some average way, in $E_{LDA}$. In this term the total, spin-projected, occupation of the localized manifold is used: $n^{I\sigma}=\sum_{m}n^{I\sigma}_{m}$.
In its original definition the functional defined in Eq. \[simpleu\] was not invariant under rotation of the atomic-orbital basis set used to define the occupancies $n^{I\sigma}_{m}$. A rotationally invariant formulation has then been introduced [@LAZ95; @ani5] where the orbital dependence of $E_{Hub}$ is borrowed from atomic Hartree-Fock with renormalized slater integrals: $$\begin{aligned}
\label{ub1}
E_{Hub}[\{n^{I}_{mm'}\}] &=&
\frac{1}{2}\sum_{\{m\},\sigma,I}\{ {\langle}m,m''|V_{ee}|m',m'''{\rangle}n^{I\sigma}_{mm'}n^{I-\sigma}_{m''m'''} \nonumber \\
&&+ ({\langle}m,m''|V_{ee}|m',m'''{\rangle}\nonumber \\
&&- {\langle}m,m''|V_{ee}|m''',m'{\rangle}) n^{I\sigma}_{mm'}n^{I\sigma}_{m''m'''} \}\end{aligned}$$ with $${\langle}m,m''|V_{ee}|m',m'''{\rangle}= \sum_{k=0}^{2l} a_{k}(m,m',m'',m''') F^{k}$$ where $l$ is the angular moment of the localized ($d$ or $f$) electrons and $$a_{k}(m,m',m'',m''') = \frac{4\pi}{2k+1}\sum_{q=-k}^{k}
{\langle}lm|Y_{kq}|lm'{\rangle}{\langle}lm''|Y^{*}_{kq}|lm'''{\rangle}.$$
The double-counting term $E_{dc}$ is given by: $$\begin{aligned}
\label{ub2}
E_{dc}[\{n^{I}\}] &=&\sum_{I} \frac{U}{2}n^{I}(n^{I}-1) \nonumber \\
&-&\sum_{I}\frac{J}{2}
[n^{I\uparrow}(n^{I\uparrow}-1)+n^{I\downarrow} (n^{I\downarrow}-1)] .\end{aligned}$$ The radial Slater integrals $F^{k}$ are the parameters of the model ($F^0$,$F^2$ and $F^4$ for $d$ electrons, while also $F^6$ must be specified for $f$ states) and are usually re-expressed in terms of only two parameters, $U$ and $J$, describing screened on-site Coulomb and exchange interaction, $$\begin{aligned}
\label{ueff}
U &=& \frac{1}{(2l+1)^{2}} \sum_{m,m'}
{\langle}m,m'|Vee|m,m'{\rangle}= F^{0} \\
J &=& \frac{1}{2l(2l+1)} \sum_{m \ne m',m'}
{\langle}m,m'|Vee|m',m{\rangle}\nonumber = \frac{F^{2}+F^{4}}{14}, \nonumber\end{aligned}$$ by assuming atomic values for $F^{4}/F^{2}$ and $F^{6}/F^{4}$ ratios.
To obtain $U$ and $J$, Anisimov and coworkers [@AnGun91; @SDA94] propose to perform LMTO calculations in supercells in which the occupation of the localized orbitals of one atom is constrained. The localized orbitals of all atoms in the supercell are decoupled from the remainder of the basis set. This makes the treatment of the local orbitals an atomic-like problem—making it easy to fix their occupation numbers—and allows to use Janak theorem [@Janak] to identify the shift in the corresponding eigenvalue with the second-order derivative of the LDA total energy with respect to orbital occupation. It has however the effect of leaving a rather artificial system to perform the screening, in particular when it is not completely intra-atomic. In elemental metallic Iron, for instance, Anisimov and Gunnarsson [@AnGun91] showed that only half of the screening charge is contained in the Wigner-Seitz cell. This fact, in addition to a sizable error due to the Atomic Sphere Approximation used [@AnGun91], could be at the origin of the severe overestimation of the computed on-site coulomb interaction with respect to estimates based on comparison of spectroscopic data and model calculations[@ExpFeU1; @ExpFeU2].
Basis set independent formulation of LDA+U method
=================================================
Some aspects of currently used LDA+U formulation, and in particular of the determination of the parameters entering the model, have been so far tied to the LMTO approach. This is not a very pleasant situation and some efforts have been done recently [@Pickett98; @Bengone00] to reformulate the method for different basis sets. Here we want to elaborate further on these attempts and provide an internally consistent, basis-set independent, method for the calculation of the needed parameters.
Localized orbital occupations
-----------------------------
In order to fully define how the approach works the first thing to do is to select the degrees of freedom on which “Hubbard $U$" will operate and define the corresponding occupation matrix, $n^{I\sigma}_{mm'}$. Although it is usually straightforward to identify in a given system the atomic levels to be treated in a special way (the $d$ electrons in transition metals and the $f$ ones in the rare earths and actinides series) there is no unique or rigorous way to define occupation of localized atomic levels in a multi-atom system. Equally legitimate choices for $n^{I\sigma}_{mm'}$ are $i)$ projections on normalized atomic orbitals, or $ii)$ projections on Wannier functions whenever the relevant orbitals give raise to isolated band manifolds, or $iii)$ Mulliken population or $iv)$ integrated values in (spherical) regions around the atoms of the angular-momentum-decomposed charge densities. Taking into account the arbitrariness in the definition of $n^{I\sigma}_{mm'}$ no particular significance should be attached to any of them (or other that could be introduced) and the usefulness and reliability of an approximate DFT+U method (aDFT+U), and of its more recent and involved evolutions like the aDFT+DMFT method, should be judged from its ability to provide a correct physical picture of the systems under study irrespective of the details of the formulation, once all ingredients entering the calculation are determined consistently.
All above mentioned definitions for the occupation matrices can be put in the generic form $$\label{eq:nijdef}
n^{I\sigma}_{mm'} = \sum_{{\rm \bf k},v}f^\sigma_{{\rm \bf k}v}
{\langle}\psi^{\sigma}_{{\rm \bf k}v}| P^I_{mm'}|\psi^{\sigma}_{{\rm \bf k}v}{\rangle}$$ where $\psi^{\sigma}_{{\rm \bf k}v}$ is the valence electronic wavefunction corresponding to the state (${\rm \bf k}v$) with spin $\sigma$ of the system and $f^\sigma_{{\rm \bf k}v}$ is the corresponding occupation number. The $P^I_{mm'}$’s are generalized projection operators on the localized-electron manifold that satisfy the following properties: $$\begin{aligned}
\nonumber & \sum_{m'} P^I_{mm'} P^I_{m'm''} = P^I_{mm''};
\quad P^I_{mm'} = (P^I_{m'm})^\dagger ; &\\
& P^I_{mm'}P^I_{m''m'''} = 0 \quad {\rm when} \quad m'\not=m'' . &\end{aligned}$$ In particular $P^I = \sum_m P^I_{mm} $ is the projector on the complete manifold of localized states associated with atom at site $I$ and therefore $$\label{eq:nI}
n_I = \sum_{\sigma} \sum_{{\rm \bf k},v}f^\sigma_{{\rm \bf k}v}
{\langle}\psi^{\sigma}_{{\rm \bf k}v}| P^I|\psi^{\sigma}_{{\rm \bf k}v}{\rangle}=\sum_{\sigma,m} n^{I\sigma}_{mm}$$ is the total localized-states occupation for site $I$. Orthogonality of projectors on different sites is [*not*]{} assumed.
In the applications discussed in this work we will define localized-level occupation matrices projecting on atomic pseudo-wavefunctions. The needed projector operators are therefore simply $$P^I_{mm'} = |\varphi^I_m{\rangle}{\langle}\varphi^I_{m'}|$$ where $|\varphi^{I}_m{\rangle}$ is the valence atomic orbital with angular momentum component $|lm{\rangle}$ of the atom sitting at site $I$ (the same wavefunctions are used for both spins). Since we will be using ultrasoft pseudopotentials to describe valence-core interaction, all scalar products between crystal and atomic pseudo-wavefunctions are intended to include the usual S matrix describing orthogonality in presence of charge augmentation [@USPP].
As already mentioned, other choices could be used as well and different definitions for the occupation matrices will require, in general, different values of the parameter entering the aDFT+U functional, as it has been pointed out recently also by Pickett et al. [@Pickett98] where, for instance, the value of Hubbard $U$ in FeO shifts from 4.6 to 7.8 eV when atomic d-orbitals for Fe$^{2+}$ ionic configuration are used instead of those of the neutral atom. In an early study [@McMahan88] the U parameter in La$_2$CuO$_4$ varies from 6.8 to 7.7 eV upon variation of the atomic sphere radius employed in the LMTO calculation. As pointed out in these works it is not fruitful to compare numerical values of U obtained by different methods but rather comparison should be made between results of complete calculations.
A simplified rotationally invariant scheme and the meaning of U
---------------------------------------------------------------
In order to simplify our analysis and gaining a more transparent physical interpretation of the “+U” correction to standard aDFT functionals we concentrate on the main effect associated to on-site Coulomb repulsion. We thus neglect the important but somehow secondary effects associated to non sphericity of the electronic interaction and the proper treatment of magnetic interaction, that in the currently used rotational invariant method is dealt with assuming a screened Hartree-Fock form. [@LAZ95].
We are therefore going to assume in the following that parameter $J$ describing these effects can be set to zero, or alternatively that its effects can be mimicked redefining the U parameter as $U_{eff}= U - J $, a practice that have been sometime used in the literature [@Dudarev98]. The Hubbard correction to the energy functional, Eqs. \[ub1\] and \[ub2\], greatly simplifies and reads: $$\begin{aligned}
\label{our1}
E_{U}[\{n^{I\sigma}_{mm'}\}] &=& E_{Hub}[\{n^{I}_{mm'}\}] -
E_{dc}[\{n^{I}\}] \nonumber \\
&=&\frac{U}{2}\sum_{I}\sum_{m,\sigma}
\{n_{mm}^{I\sigma}-\sum_{m'}n_{mm'}^{I\sigma}n_{m'm}^{I\sigma}\}
\nonumber \\
&=&\frac{U}{2}\sum_{I,\sigma}
Tr[{\bf n}^{I\sigma}(1-{\bf n}^{I\sigma})].\end{aligned}$$
Choosing for the localized orbitals the representation that diagonalizes the occupation matrices $$\begin{aligned}
\label{diago}
{\bf n}^{I\sigma} {\bf v}^{I\sigma}_i =
\lambda^{I\sigma}_i {\bf v}^{I\sigma}_i\end{aligned}$$ with $0 \le \lambda^{I\sigma}_i \le 1$, the energy correction becomes $$\begin{aligned}
\label{our2}
E_{U}[\{n^{I\sigma}_{mm'}\}]
&=&\frac{U}{2}\sum_{I,\sigma} \sum_{i} \lambda^{I\sigma}_i ( 1 -\lambda^{I\sigma}_i ).\end{aligned}$$ from where it appears clearly that the energy correction introduces a penalty, tuned by the value of the U parameter, for partial occupation of the localized orbitals and thus favors disproportionation in fully occupied ($\lambda \approx 1$) or completely empty ($\lambda \approx
0$) orbitals. This is the basic physical effect built in the aDFT+U functional and its meaning can be traced back to known deficiencies of LDA or GGA for atomic systems.
An atom in contact with a reservoir of electrons can exchange integer numbers of particles with its environment. The intermediate situation with fractional number of electrons in this open atomic system is described not by a pure state wave function, but rather by a statistical mixture so that, for instance, the total energy of a system with $N+\omega$ electrons (where N is an integer and $0 \le \omega \le 1$) is given by: $$\label{efrac}
E_{n} = (1-\omega)E_{N} + \omega E_{N+1}$$ where $E_{N}$ and $E_{N+1}$ are the energies of the system corresponding to states with $N$ and $N+1$ particles respectively, while $\omega$ represents the statistical weight of the state with $N+1$ electrons. The total energy of this open atomic system is thus represented by a series of straight-line segments joining states corresponding to integer occupations of the atomic orbitals as depicted in fig. \[parab\]. The slope of the energy vs electron-number curve is instead piece-wise constant, with discontinuity for integer number of electrons, and corresponds to the electron affinity (ionization potential) of the N (N+1) electron system.
![\[parab\] Sketch of the total energy profile as a function of number of electrons in a generic atomic system in contact with a reservoir. The bottom curve is simply the difference between the other two (the LDA energy and the “exact” result for an open system).](FIG_1.ps){width="9.0truecm"}
Exact DFT correctly reproduce this behavior [@perd2; @perd0], which is instead not well described by the LDA or GGA approach, which produces total energy with unphysical curvatures for non integer occupation and spurious minima in correspondence of fractional occupation of the orbital of the atomic system. This leads to serious problems when one consider the dissociation limit of hetero-polar molecules or an open-shell atom in front of a metallic surface [@perd2; @perd0], and is at the heart of the LDA/GGA failure in the description of strongly correlated systems[@AZA91]. The unphysical curvature is associated basically to the incorrect treatment by LDA or GGA of the self-interaction of the partially occupied Kohn-Sham orbital that gives a non-linear contribution to the total energy with respect to orbital occupation (with mainly a quadratic term coming from the Hartree energy not canceled properly in the exchange-correlation term).
Nevertheless, it is well known [@jones1] that total energy differences between different states can be reproduced quite accurately by the LDA (or GGA) approach, if the occupation of the orbitals is [*constrained*]{} to assume integer values. As an alternative, we can recover the physical situation (an approximately piece-wise linear total energy curve) by adding a correction to the LDA total energy which vanishes for integer number of electrons and eliminates the curvature of the LDA energy profile in every interval with fractional occupation (bottom curve of fig. \[parab\]). But this is exactly the kind of correction that is provided by eq. \[our1\] if the numerical value of the parameter $U$ is set equal to the curvature of the LDA (GGA) energy profile.
This clarifies the meaning of the interaction parameter $U$ as the (unphysical) curvature of the LDA energy as a function of $N$ which is associated with the spurious self-interaction of the fractional electron injected into the system. From this analysis it is clear that the numerical value of $U$ will depend in general not only, as noted in the preceding section, on the definition adopted for the occupation matrices but also on the particular approximate exchange-correlation functional to be corrected, and should [*vanish*]{} if the exact DFT functional were used.
The situation is of course more complicated in solids where fractional occupations of the atomic orbitals can occur due to hybridization of the localized atomic-like orbitals with the crystal environment and the [*unphysical*]{} part of the curvature has to be extracted from the total LDA/GGA energy, which contains also hybridization effects. In the next section this problem is discussed and a linear response approach to evaluate Hubbard $U$ is proposed.
Internally consistent calculation of U
--------------------------------------
Following previous seminal works [@McMahan88; @HSC89; @AnGun91] we compute $U$ by means of constrained-density-functional calculations [@ConstrDFT84]. What we need is the total energy as a function of the localized-level occupations of the “Hubbard” sites:
$$\label{eq:Eofn}
E [\{q_{I}\}] = \min_{n({\bf r}),\alpha_I} \left \{
E [n({\bf r})] + \sum_I \alpha_I ( n_{I} - q_{I} )\right \} ,$$
where the constraints on the site occupations, $n_{I}$’s from Eq. \[eq:nI\], are applied employing the Lagrange multipliers, $\alpha_I$’s. From this dependence we can compute numerically the curvature of the total energy with respect to the variation, around the unconstrained values $\{n^{(0)}_I\}$ , of the occupation of one isolated site. A supercell approach is adopted in which occupation of one representative site in a sufficiently large supercell is changed leaving unchanged all other site occupations. This curvature contains the energy cost associated to the localization of an electron on the chosen site including all screening effects from the crystal environment, but it is not yet the Hubbard $U$ we want to compute. In fact, had we computed the same quantity from the total energy of the non-interacting Kohn-Sham problem associated to the same system, $$E^{KS} [\{q_I\}] = \min_{n({\bf r}),\alpha_I} \left\{ E^{KS} [n({\bf r})] +
\sum_I \alpha^{KS}_I ( n_I - q_I ) \right \},$$ we would have obtained a non vanishing results as well because by varying the site occupation a rehybridization of the localized orbitals with the other degrees of freedom is induced that gives rise to a non-linear change in the energy of the system. This curvature coming from rehybridization, originating from the non-interacting band structure but present also in the interacting case, has clearly nothing to do with the Hubbard $U$ of the interacting system and should be subtracted from the total curvature: $$\label{dedn1}
U = \frac{\partial^{2}E[\{q_I\}]}{\partial q_I^{2}} -
\frac{\partial^{2}E^{KS}[\{q_I\}]}{\partial q_I^{2}} .$$
In Ref. [@AnGun91] Anisimov and Gunnarsson, in order to avoid dealing with the above mentioned non-interacting curvature, exploited the peculiarities of the LMTO method, used in their calculation, and decoupled the chosen localized orbitals from the remainder of the crystal by suppressing in the LMTO hamiltonian the corresponding hopping terms. This reduced the problem to the one of an isolated atom embedded in an artificially disconnected charge background. Thanks to Janak theorem [@Janak] the second order derivative of the total energy in Eq. \[eq:Eofn\] can then be recast as a first order derivative of the localized-level eigenvalue. In our approach the role played in Refs. [@AnGun91; @SDA94] by the eigenvalue of the artificially isolated atom is taken by the Lagrange multiplier, used to enforce level occupation[@ConstrDFT84]: $$\begin{aligned}
\label{deriv1}
&&\frac{\partial E[\{q_{J}\}]}{\partial q_{I}} = - \alpha_{I}, \quad
\frac{\partial^{2}E[\{q_{J}\}]}{\partial q_{I}^{2}} = - \frac{\partial\alpha_{I}}{\partial q_{I}},\\
\nonumber
&&\frac{\partial E^{KS}[\{q_{J}\}]}{\partial q_{I}} = - \alpha^{KS}_{I}, \quad
\frac{\partial^{2}E^{KS}[\{q_{J}\}]}{\partial q_{I}^{2}} = - \frac{\partial\alpha^{KS}_{I}}{\partial q_{I}}.\end{aligned}$$ At variance with the original method of Refs. [@AnGun91; @SDA94], in our approach we need to compute and subtract the band-structure contribution, $-{\partial\alpha^{KS}_{I}}/{\partial q_{I}}$, from the total curvature, but, in return, Hubbard $U$ is computed in exactly the same system to which it is going to be applied and the screening from the environment is more realistically included. The present method was inspired by the linear response scheme proposed by Pickett and coworkers [@Pickett98] where however the role of the non-interacting curvature was not appreciated.
In actual calculations constraining the localized orbital occupations is not very practical and it is easier to pass, via a Legendre transform, to a representation where the independent variables are the $\alpha_I$’s $$\begin{aligned}
\label{eq:Eofa}
E [\{\alpha_I\}] & =& \min_{n({\bf r})} \left \{ E [n({\bf r})]
+ \sum_I \alpha_I \; n_{I} \right\}, \\
\nonumber E^{KS}[\{\alpha^{KS}_I\}] & =& \min_{n({\bf r})} \left \{
E^{KS} [n({\bf r})] + \sum_I \alpha^{KS}_I n_{I} \right\}.\end{aligned}$$ Variation of these functionals with respect to wavefunctions shows that the effect of the $\alpha_I$’s is to add to the single particle potential a term, $\Delta V = \sum_I \alpha_I P^{I} $ (or $ \Delta V
= \sum_I \alpha^{KS}_I P^{I} $ for the non-interacting case), where localized potential shifts of strength $\alpha_I$ ($\alpha^{KS}_I$) are applied to the localized levels associated to site $I$.
It is useful to introduce the (interacting and non-interacting) density response functions of the system with respect to these localized perturbations: $$\begin{aligned}
\label{chidef}
\chi_{IJ}
& = & \frac{\partial ^2E}{\partial \alpha_I \partial \alpha_J}
= \frac{\partial n_I}{\partial \alpha_J},
\\
\chi^0_{IJ}
& = & \frac{\partial ^2E^{KS}}{\partial \alpha^{KS}_I \partial \alpha^{KS}_J}
= \frac{\partial n_I}{\partial \alpha^{KS}_J}.
\nonumber\end{aligned}$$ Using this response-function language, the effective interaction parameter $U$ associated to site $I$ can be recast as: $$\label{ueff1}
U = + \frac{\partial \alpha^{KS}_I}{\partial q_I} -
\frac{\partial \alpha^I}{\partial q_I} = \left( \chi_{0}^{-1} -
\chi^{-1} \right)_{II}$$ that is reminiscent of the well known random-phase approximation [@RPA] in linear response theory giving the interacting density response in terms of the non-interacting one and the Coulomb kernel. A similar result is obtained within DFT linear response [@epsDFT] where the interaction kernel also contains an exchange-correlation part.
The response functions, Eq. \[chidef\], needed in Eq. \[ueff1\] are computed taking numerical derivatives. We perform a well converged LDA calculation for the unconstrained system ($\alpha_I=0$ for all sites in the supercell) and—starting from its self-consistent potential—we add small (positive and negative) potential shifts on each non equivalent “Hubbard” site $J$ and compute the variation of the occupations, $n_I$’s, for all sites in the supercell in two ways: $i)$ letting the Kohn-Sham potential of the system readjust self-consistently to optimally screen the localized perturbation, $\Delta V=\alpha_J P_J$, and $ii)$ without allowing this screening. This latter result is nothing but the variation computed from the first iteration in the self-consistent cycle leading eventually to the former (screened) results. The site-occupation derivatives calculated according to $i)$ and $ii)$ give the matrices $\chi_{IJ}$ and $\chi^0_{IJ}$ respectively.
Further considerations
----------------------
Before moving to examine some specific examples in the next section, let’s end the present one discussing a few additional technical points.
As mentioned earlier, Hubbard $U$ is computed, ideally, from variation of the site occupation of a single site in an infinite crystal and in practice adopting a supercell approach where periodically repeated sites are perturbed coherently. In order to speed up the convergence of the computed $U$ with supercell size it may result useful to enforce explicitly charge neutrality for the perturbation, that is to be introduced in the response functions, thus enhancing its local character and reduce the interaction with its periodic images. In this procedure we introduce in the response functions, $\chi$ and $\chi^0$, —in addition to the degrees of freedom associated to the localized sites—also a “delocalized background” representing all other degrees of freedom in the system. This translates in one more column and row in the response matrices, whose elements are determined imposing overall charge neutrality of the perturbed system for all localized perturbations, ($\sum_I \chi_{IJ} = 0, \quad \sum_I \chi^0_{IJ}=0, \quad \forall J$) and absence of any charge density variation upon perturbing the system with a constant potential ($\sum_J \chi_{IJ} = 0, \quad \sum_J \chi^0_{IJ}=0, \quad \forall I$). From a mathematical point of view both $\chi$ and $\chi_{0}$ acquire a null eigenvalue, corresponding to a constant potential shift, and the needed inversions in Eq. \[ueff1\] must be taken with care. It can be shown that their singularities cancel out when computing the difference $\chi_{0}^{-1} - \chi^{-1}$ and the final result is well defined. We stress that in the limit of infinitely large supercell the coupling with the background gives no contribution to the computed $U$, but we found that this limit is approached more rapidly when this additional degrees of freedom is included.
In the same spirit we found that the spatial locality of the response matrices can be rather different from the one of their inverse and a supercell sufficient to decouple the periodically repeated response may be too small to describe correctly the inverse in eq. \[ueff1\]. As a practical procedure, therefore, after evaluating the response function matrices in a given supercell, we extrapolate the result to much larger supercells assuming that the most important matrix elements in $\chi_{0}$ and $\chi$ involve the atoms in the few nearest coordination-shells accessible in the original supercell. The corresponding matrix elements of the larger supercell are filled with the values extracted from the smaller one while all other, more distant, interactions are neglected. Again, when a sufficiently large supercell to extract the matrix elements of the response functions is considered, the effect of this extrapolation vanishes, but, as we will see in the following, this scheme capture a large fraction of the system-size dependence of the calculated $U$ and it may allow to reach more rapidly the converged result.
As a final remark we notice that the electronic structure of a system described within the LDA+U approach may largely differ from the one obtained within the LDA used to compute $U$. In a more refined approach one might seek internal consistency between the band structure used in the calculation of $U$ and the one obtained using it. We have not addressed this issue here, but one can imagine performing the same type of analysis leading to the $U$ determination for a functional already containing an LDA+U correction. The computed $U$ would in that case be a correction to be added to the original $U$ and internal consistency would be reached when the correction vanishes.
Examples
========
Metals: Iron and Cerium
-----------------------
In their seminal paper Anisimov and Gunnarsson [@AnGun91] computed the effective on site Coulomb interaction between the localized electrons in metallic Fe and Ce. For Ce the calculated Coulomb interaction was about 6 eV in good agreement with empirical and experimental estimates ranging from 5 to 7 eV [@ConstrDFT84; @Herbst78; @ExpCeU], while the result for Fe (also about 6 eV) was surprisingly high since $U$ was expected to be in the range of 1-2 eV for elemental transition metals, with the exception of Ni [@ExpFeU1; @ExpFeU2]. Let us apply the present approach to these two system, starting with Iron.
In its ground state elemental Iron has a ferromagnetic (FM) spin arrangement and a body-centered cubic (BCC) structure. Gradient corrected exchange-correlation functional are needed in order to stabilize the experimental structure as compared with non-magnetic face-centered cubic (FCC) structure preferred by LDA. The Perdew-Burke-Ernzherof (PBE) [@perd1] GGA functional was employed here. Iron ions were represented by ultrasoft pseudopotential and kinetic energy cutoffs of 35 Ry and 420 Ry were adopted for wavefunction and charge density Fourier expansion. Brillouin Zone integrations where performed using 8$\times$8$\times$8 Monkhorst and Pack special point grids [@AB] using Methfessel and Paxton smearing technique [@met1] with a smearing width of 0.005 Ry in order to smooth the Fermi distribution.
The calculation of the effective Hubbard $U$ followed the procedure outlined in preceding section: a supercell was selected containing a number of inequivalent Iron atoms; then, after a well converged self-consistent calculation, we applied to one of these atoms small, positive and negative, potential shifts, $\Delta
V= \alpha P_{d}$ (with $\alpha = \pm $0.2-0.5 eV), where $P_d$ is the projector on the localized $d$ electron of the selected atom. From the variation of the $d$-level occupations of all Iron atoms in the cell one column of $\chi$ and $\chi^0$ response functions was extracted and all other matrix elements were reconstructed by symmetry, including the background as explained previously. Hubbard $U$ was then calculated from Eq. \[ueff1\].
In order to describe response for an isolated perturbation four supercells were considered: [*i)*]{} a simple cubic (SC) cell containing two inequivalent iron atoms, the perturbed atom and one of its nearest neighbors; [*ii)*]{} a 2$\times$2$\times$2 BCC supercell containing 8 inequivalent Iron atoms, 4 in the nearest-neighbor shell of the perturbed atom and 3 belonging to the second shell of neighbors; [*iii)*]{} a 2$\times$2$\times$2 SC cell containing 16 atoms, including also some third nearest-neighbor atom and [*iv)*]{} a 4$\times$4$\times$4 BBC supercell containing 64 inequivalent Iron atoms; we used this largest cell just to extrapolate the results from the smaller ones.
The convergence properties of the effective $U$ of bulk iron with the size of the used supercell are shown in fig. \[unei\].
![Calculated Hubbard $U$ in metallic Iron for different supercells. Lines connect results from the cell-extrapolation procedure described in the text and different symbols correspond to inclusion of screening contributions up to the indicated shell of neighbors of the perturbed atom.[]{data-label="unei"}](FIG_2.ps){width="9.5truecm"}
![Lattice spacing dependence of the calculated Hubbard $U$ parameter for Iron.[]{data-label="uls"}](FIG_3.ps){width="9.5truecm"}
The Hubbard $U$ obtained from the SC 2-atom cell, once inserted in the 64-atom supercell, captures most of the effective interaction; second nearest neighbors shell brings some significant corrections to the final extrapolated result, while third nearest neighbor shell has a smaller effect. We believe that contributions from further neighbor rapidly vanish and that an accurate value of $U$ can be extracted from the SC supercell containing 16 atoms. The extrapolation from this cell to larger cells brings only minor variations which are within the finite numerical accuracy that we estimate within a fraction of an eV. From this analysis our estimate for the Hubbard $U$ in elemental Iron at the experimental lattice parameter is therefore 2.2 $\pm$ 0.2 eV.
This results is in very good agreement with the experimental estimates [@ExpFeU1; @ExpFeU2], but disagrees with Anisimov and Gunnarsson result [@AnGun91]. We can only recall here that many technical details differ in the two approaches. In particular [*i)*]{} in the original approach the perturbed atom is disconnected from the rest of the crystal by removing all hopping terms, thus leaving a rather unphysical environment to perform the screening, while in our approach the actual system is allowed to screen the perturbation, [*ii)*]{} the Atomic Sphere Approximation (ASA) was employed in the original LMTO calculation while no shape approximation is made in our case.
In order to further test our approach on this element we investigate the dependence of the Hubbard parameter on crystal structure. The dependence of the calculated interaction parameter on the lattice spacing of the unit cell is shown in fig. \[uls\] where a marked increase of the Hubbard $U$ can be observed when the lattice parameter is squeezed below its experimental value. Despite this may appear counterintuitive, as correlation effects are expected to become less important when atoms gets closer, one should actually compare the increasing value of $U$ with the much steeper increase of bandwidth when reducing the interatomic distance. Upon increase of the lattice parameter the Hubbard parameter should approach the atomic limit that can be estimated from all-electron atomic calculations where the local neutrality of the metallic system is maintained: $U = E(d^8s^0) + E(d^6s^2) - 2 \times E (d^7s^1) = 2.1 $ eV, in reasonable agreement with the results of fig. \[uls\].
Using the calculated volume dependent Hubbard $U$ parameter we have studied the effect of the LDA+U approximation on the structural properties of Iron.
$a_{0}$ (a.u.) $B_{0}$ (Mbar) $\mu_{0}$ ($\mu_{B}$)
-------------- ---------------- ---------------- -----------------------
Expt. 5.42 1.68 2.22
LSDA 5.22 2.33 2.10
$\sigma$-GGA 5.42 1.45 2.46
LDA+U 5.53 2.12 2.60
LDA+U (AMF) 5.34 1.53 2.00
: Comparison between the calculated lattice constant ($a_{0}$), bulk modulus ($B_{0}$) and magnetic moment ($\mu_{0}$) within several approximate DFT schemes and experimental results quoted from [@exp1]. []{data-label="ulats"}
Results are reported in table \[ulats\] where they are compared with results obtained within LSDA and $\sigma$-GGA(PBE) approximation and with experimental data. From these data it appears that, although simple $\sigma$-GGA(PBE) approximation appears to be superior in this case, LDA+U provides a reasonable description of the data, of the same quality as LSDA. In weakly correlated metals it has been suggested [@amf] that a formulation of LDA+U in terms of occupancy fluctuations around the uniform occupancy of the localized level could be more appropriate than the standard one. This “around mean field” (AMF) LDA+U approach has been revisited recently [@amf1; @amf2] and an “optimally mixed” scheme has also been proposed [@amf2]. We don’t want to enter in this discussion here, but we mention that by following the AMF recipe the description of structural and magnetic properties of metallic Iron improves as it is evident from table \[ulats\].
![Band structure of bulk iron obtained within the AMF LDA+U approach. Green lines are for minority spin states, black ones for majority spin levels. Photoemission results from [@turner1] are also reported for comparison.[]{data-label="febamf"}](FIG_4.ps){width="9.0truecm"}
Using the calculated value of $U$ we have obtained the electronic structure of Iron at the experimental lattice spacing. The theoretical band structure obtained using the AMF version of LDA+U is reported in fig. \[febamf\] together with some experimental results [@turner1]. The overall agreement is rather good for this scheme. However, when using the standard LDA+U scheme a somehow worse agreement with experimental data was obtained, mainly due to a rigid downward shift of the majority spin bands of about 1 eV. This is an indication that LDA+U approximation may still require some fine tuning in order to describe accurately both strongly and weakly correlated systems [@amf2].
Let us proceed to examine the Cerium case. Elemental cerium presents a very interesting phase diagram with a peculiar isostructural $\alpha-\gamma$ phase transition between a low volume ($\alpha$) and a high volume ($\gamma$) phase, both FCC. This phase transition has attracted much experimental and theoretical interest and in the last 20 years [@cerium_varie], many interpretations have been put forward to explain its occurrence. It is clear now that standard LDA or GGA approximations do not describe the transition and it appears that a treatment of the correlation at the DMFT level might be required [@cerium_DMFT], however a full understanding of the nature of the transition is still under debate [@cerium_last_prl]. Here, we do not want to address this delicate topics but we simply want to follow Anisimov and Gunnarsson [@AnGun91] by computing the Hubbard $U$ parameter for elemental cerium in the high volume $\gamma$ phase.
The interaction of valence-electrons with Ce nuclei and its core electrons was described by a non-local ultrasoft pseudopotential [@USPP] generated in the $5s^25p^65d^14f^1$ electronic configuration. Kinetic cutoffs of 30 Ry and 240 Ry were adopted for wavefunction and charge density Fourier expansion. The LSDA approximation was adopted for the exchange and correlation functional. Brillouin Zone integrations where performed using 8$\times$8$\times$8 Monkhorst and Pack special point grids [@AB] using Methfessel and Paxton smearing technique [@met1] with a smearing width of 0.05 Ry.
![Calculated Hubbard $U$ in metallic Cerium for different supercells. Lines connect results from the cell-extrapolation procedure and different symbols correspond to inclusion of screening contributions up to the indicated shell of neighbors of the perturbed atom.[]{data-label="fig:CeU"}](FIG_5.ps){width="8.0truecm"}
To obtain the response to an isolated perturbation we have perturbed a Cerium atom in three different cells: [*i)*]{} the fundamental face-centered cubic (FCC) cell containing just one inequivalent atom, [*ii)*]{} a simple-cubic (SC) cell containing 4 atoms (giving access to the first nearest-neighbor response) and [*iii)*]{} a 2$\times$2$\times$2 FCC cell (8 inequivalent atoms) including also the response of second-nearest neighbor atoms. The result of these calculations and their extrapolation to very large SC cells is reported in Fig. \[fig:CeU\] where it can be seen that the converged value for $U$ approaches 4.5 eV.
The screening in metallic cerium is extremely localized, as can be seen from the fact that inclusion of the first-nearest neighbor response is all is needed to reach converged results. This is at variance with what we found in metallic Iron where third nearest-neighbor response was still significant (see Fig. \[unei\]). The calculated value is not far from the value (5-7 eV) expected from empirical and experimental estimates [@ConstrDFT84; @Herbst78; @ExpCeU], especially if we consider that the parameter $U$ we compute plays the role of $U-J$ in the simplified rotational invariant LDA+U scheme adopted [@Dudarev98].
As a check, we performed all-electron atomic calculations for $Ce^+$ ions where localized $4f$ electrons were promoted to more delocalized $6s$ or $5d$ states and obtained $U = E(f^3s^0) + E(f^1s^2) - 2 \times E (f^2s^1)
= 4.4 $ eV, or $U = E(f^2s^0d^1) + E(f^0s^2d^1) - 2 \times E(f^1s^1d^1)
= 6.4$ eV, depending on the selected atomic configurations. This confirms the correct order of magnitude of our calculated value in the metal.
The present formulation is therefore able to provide reasonable values for the on-site Coulomb parameter both in Iron and Cerium, at variance with the original scheme of ref. [@AnGun91] where only the latter was satisfactorily described. We believe that a proper description of the interatomic screening, rather unphysical in the original scheme where atoms were artificially disconnected from the environment, is important to obtain a correct value for Hubbard $U$ parameter, especially in Iron where this response is more long-ranged.
Transition metal monoxides: FeO and NiO
---------------------------------------
The use of the LDA+U method for studying FeO is mainly motivated by the attempt to reproduce the observed insulating behavior. In fact, as for other transition metal oxides (TMO), standard DFT methods, as LDA or GGA, produce an unphysical metallic character due to the fact that crystal field and electronic structure effects are not sufficient in this case to open a gap in the three-fold minority-spin $t_{2g}$ levels that host one electron per Fe$^{2+}$ atom. As already addressed in quite abundant literature on TMO (and FeO in particular), a better description of the electronic correlations is necessary to obtain the observed insulating behavior and the structural properties of this compound at low pressure [@isa1; @zon1; @zon2; @maz1]. The application of our approach to this material will thus allow us to check its validity by comparison of our results with the ones from experiments and other theoretical works.
The unit cell of this compound is of rock-salt type, with a rhombohedral symmetry introduced by a type II antiferromagnetic (AF) order (see fig. \[cell\]) which sets in along the \[111\] direction below a Neél temperature of 198 K, at ambient pressure.
![\[cell\]The unit cell of FeO: blue spheres represent Oxygen ions, red ones are Fe ions, with arrows showing the orientation of their magnetic moments. Ferromagnetic (111) planes of iron ions alternate with opposite spins producing type II antiferromagnetic order and rhombohedral symmetry.](FIG_6.ps){width="8.0truecm"}
The calculations on this materials were all performed in the antiferromagnetic phase starting from the cubic (undistorted) unit cell of fig. \[cell\] with the experimental lattice spacing. We used a 40 Ry energy cut off for the electronic wavefunctions (400 Ry for the charge density due to the use of ultrasoft pseudopotentials [@USPP] both for Fe and O) and a small smearing width of 0.005 Ry which required a 4$\times$4$\times$4 k-points mesh.
To compute the Hubbard effective interactions, we performed GGA calculations with potential shifts on one Hubbard site in larger and larger unit cells, that we named C1, C4, and C16, containing 2, 8, and 32 iron ions respectively, and extrapolated their results up to a supercell containing 256 magnetic ions (called C128). The result for the undistorted cubic cell at the experimental lattice spacing is reported in fig. \[ufeoconv\]. We can observe that the effective interaction obtained from C4 is already very well converged, when extrapolated to the largest cell, with respect to inclusion of screening from additional shells of neighborers,
![\[ufeoconv\] Convergence of Hubbard $U$ parameter of FeO with the number of iron included in the supercell used in the extrapolation. Lines connect results including the screening contributions extracted from the indicated cell.](FIG_7.ps){width="9.0truecm"}
The final result for the Hubbard $U$ is 4.3 eV which is smaller than most of the values obtained (or simply assumed) in other works [@zon1; @zon2; @maz1]. If we use this value in a LDA+U calculation we can obtain the observed insulating behavior as shown in the band structure plot of fig. \[feob\] where a comparison is made with GGA (metallic) results.
![\[feob\] The band structure of FeO in the undistorted (cubic) AF configuration at the experimental lattice spacing obtained within GGA (top panel) and LDA+U using the computed Hubbard $U$ of 4.3 eV (bottom panel). The zero of the energy is set at the top of the valence band.](FIG_8.ps){width="8.5truecm"}
![\[feodosu\] Projected density of states of FeO in the undistorted (cubic) AF configuration at the experimental lattice spacing obtained within GGA (top panel) and LDA+U using the computed Hubbard $U$ of 4.3 eV (bottom panel).](FIG_9.ps){width="8.5truecm"}
A gap opens around the Fermi level whose minimal width is about 2 eV. The band gap is direct and located at the $\Gamma$ point. The corresponding transition, of $3d$(Fe)-$2p$(O)$\rightarrow$$4s$(Fe) character, should be quite weak due to the vanishing weight of Iron $s$ states at the bottom of the valence band (fig. \[feodosu\], bottom picture). We can expect that a stronger absorption line will appear instead around 2.6 eV due to the transition, of $3d$(Fe)-$2p$(O) $\rightarrow$ $3d$(Fe) character, among two pronounced peaks of the density of states around the Fermi level. This picture is in very good agreement with experiments (and other theoretical results [@maz1; @wei1]) where a first weak absorption is reported between 0.5 and 2 eV and a stronger line appears around 2.4 eV [@jmmm]. The large mixing between majority-spin Iron $3d$ states and the Oxygen $2p$ manifold over a wide region of energy and the finite contribution of the Oxygen states at the top of the valence band—a feature not present within $\sigma$-GGA (see top panel in fig. \[feodosu\])— are also in good agreement with experiments, which indicate for FeO a moderate charge transfer character of the insulating state.
Despite our $U$ is smaller than the ones used in literature, we find a good agreement of our results about the electronic structure of the system with experiments and other theoretical works. These findings confirm the validity of our internally consistent method to compute $U$. We now want to extend its application to the study of structural properties. This is indeed a very important test because a good ab-initio method should be able to describe the true ground state of a system and provide a complete description of both electronic and structural properties. Furthermore the plane-wave implementation we use allows a straightforward calculation of Hellmann-Feynman forces and stresses, thus giving easily access to equilibrium crystal structure.
As observed in experiments [@yagi1], the cubic rock salt structure of FeO shown in fig. \[cell\] becomes unstable under a pressure of 16 GPa (at room temperature) toward a rhombohedral distortion. In the distorted phase the unit cell is elongated along the \[111\] direction with a consequent shrinking of the interionic distances on the (111) planes. This transition is driven by the onset of the AFII magnetic order [@yagi1] (the Neél temperature reaches room value at about 16 GPa) which imposes a rhombohedral symmetry even in the cubic phase. Upon increasing pressure above the threshold value the distortion of the unit cell is observed to increase producing more elongated structures [@yagi1].
![\[angp\] The pressure dependence of the rhombohedral angle in FeO for the various approximations described in the text is compared with experimental results. These latter results were extracted extrapolating the data for the non stoichiometric compound Fe$_{1-x}$O up to the stoichiometric composition [@yagi1; @will1].](FIG_10.ps){width="8.0truecm"}
We have computed the Hubbard $U$ on a grid of possible values for the rhombohedral distortion and cell parameter and then from the corresponding total energy calculations we determined the rhombohedral distortion and the enthalpy of the system as a function of the pressure up to 250 Kbar.
As evident from fig. \[angp\], while GGA overestimates the rhombohedral distortion and his pressure dependence, LDA+U method—in the standard electronic configuration examined so far–overcorrects the GGA results and introduces even larger errors with respect to experimental results. In fact not only we obtain a distortion with the wrong sign (of compressive character along the \[111\] direction), but also the wrong pressure dependence. The reason for this failure can be traced back to the different occupation of the orbitals around the gap/Fermi level in the two cases. Even in the undistorted cell, the rhombohedral symmetry, induced by the antiferromagnetic order, lifts the degeneracy of the minority spin $t_{2g}$ states of iron and split them in one state of $A_{1g}$ character—which is essentially the m=0 ($z^{2}$) state along the \[111\] quantization axis—and two states of $e_g$ symmetry localized on the iron (111) planes. Within GGA, the Iron minority-spin $3d$ electrons partially occupy the two equivalent $e_g$ orbitals giving rise to two half filled bands and a (wrong) metallic state which is delocalized on the (111) plane. The system gains energy by filling the lowest half of the $e_g$ states and tends to elongate in the \[111\] direction, shrinking in the plane, because this increases the overlap of the $e_g$ states and their bandwidth. Within LDA+U, fractional occupation of orbitals is energetically disfavored and the system would like to have completely filled or empty $3d$ states. In the standard unit-cell considered so far in the literature—and used by us in the calculation above—this can be accomplished only by filling the non-degenerate $A_{1g}$ level, corresponding to wavefunctions elongated along \[111\], and pushing upward in energy the in-plane $e_g$ states, leaving them empty. As a consequence, the system tends to pull apart the ions on the same (111) plane, so that the bandwidth of the state in the plane is reduced, and increases instead the inter-plane overlap of the $A_{1g}$ states. This simple picture gives an explanation of the fact that GGA overestimates the elongation of the unit cell in the \[111\] direction, as well as the (wrong) compressive behavior of the standard LDA+U solution. We are thus left with the paradoxical situation that a correct pressure dependence of the structural properties can be obtained from the wrong band structure and viceversa.
![\[dist\] Lattice distortion in the (111) iron planes used to induce symmetry breaking in the electronic configuration of FeO.](FIG_11.ps){width="8.0truecm"}
We have found that it is possible to solve this paradox by allowing the possibility that the system partially occupies, as within GGA, the $e_g$ levels, thus maintaining the driving force for the right rhombohedral deformation, and still opens a gap, as in standard LDA+U, by some orbital ordering that breaks the equivalence of the iron ions in the (111) plane. This possibility has been sometimes proposed in literature [@maz1; @cohen] but has never been clearly addressed.
From a simple tight-binding picture one finds that the optimal broken symmetry phase would be the one where occupied $e_g$ orbitals have the highest possible hopping term with unoccupied $e_g$ orbitals in nearest-neighbor atoms in the plane, in order to maximize the kinetic energy gain coming from delocalization, and the lowest possible hopping term with neighboring occupied $e_g$ orbitals, in order to minimize bandwidth that tends to destroy the insulating state. In bipartite lattice this is simply achieved by making occupied orbitals in nearest-neighbor sites orthogonal but, in the triangular lattice, formed by iron atoms in (111) planes, this is not exactly possible, the system is topologically frustrated and some compromise is necessary.
It is generally believed [@HEISENBERG] that Heisenberg model in the triangular lattice, to which our system resemble in some sense, displays a three-sublattice 120$^\circ$ Néel long-range order. We thus imposed a symmetry breaking to the system where three nearest-neighbor atoms in the (111) plane were made inequivalent by slightly displacing them from the ideal positions in the way shown in fig. \[dist\]. This induced the desired symmetry breaking of the electronic structure and opened a gap that was robust and persisted when the atoms were brought back into the ideal positions. We found, quite satisfactorily, that the new broken symmetry phase (BSP) corresponds to a lower energy minimum than the “standard” LDA+U solution and that therefore it is, to say the least, a more consistent description of the ground state of FeO. The one depicted in fig. \[dist\] is, of course, only one of three equivalent distortions we could have imposed to the electronic structure of the system and three symmetry related BSPs could be defined. In the actual system an effective equivalence of the ions in the (111) planes is probably restored by a (dynamical) switching among equivalent states but considering the atoms as strictly equivalent, as in the standard solution, leads to incorrect results.
![\[newdos\] The projected density of states of FeO as obtained in the “standard” LDA+U ground state (top panel) and in the proposed broken symmetry phase (bottom panel). On the right of each DOS is a picture of the corresponding occupied Fe-$3d$ minority states. ](FIG_12.ps){width="10.0truecm"}
The comparison of the projected density of state in the “standard” LDA+U solution and in the novel BSP phase is shown in fig. \[newdos\] where also a pictorial representation of the occupied minority-spin orbitals in the two cases is shown. As we can observed, no remarkable qualitative difference in the DOS appears apart from the different ordering of the $d$ states around the gap. In fact the minority-spin $d$ electron is now accommodated on a state lying on the (111) plane (shown on the right panel) while the one with $A_{1g}$ ($z^{2}$) character has been pushed above the energy gap. The gap width and the charge transfer character of the system do not change significantly and are still in very good agreement with the experiments.
We repeated the structural calculations (according to the same procedure described above) in the BSP, and obtained the LDA+U (BSP) curve reported in fig. \[angp\]. The agreement with experiments is much improved with respect to both GGA and LDA+U “standard” ground states. The mechanism leading to the pressure behavior in BSP case is basically the same already producing the correct evolution of distortion in the GGA calculations. When the unit cell elongates along the cubic diagonal the iron ions in the (111) plane get closer and the hopping between nearest-neighbor orbitals increased with a consequent lowering of the electronic kinetic energy.
We therefore conclude that LDA+U, not only improves the description of the structural and electronic properties with respect to GGA, but that a close examination of both electronic and structural properties is in this case necessary in order to describe the correct ground state of the system.
Another classical example of TMO we want to study in order to test the present implementation of LDA+U is Nickel Oxide. It is a very well studied material and there is a good number of theoretical [@Bengone00] and experimental works, including some photoemission experiments [@zx1; @kule1], our results can be compared with. At variance with FeO, no compositional instability is observed for NiO so that the stoichiometric compound is easy to study and is much better characterized than iron oxide. It has cubic structure with the same AF spin arrangements of rhombohedral symmetry as FeO, but does not show tendencies toward geometrical distortions of any kind and is therefore easier to study.
In this case we did not perform any structural relaxation and calculated the value of $U$ at the experimental lattice spacing for the cubic unit cell imposing the rhombohedral AF magnetic order which is the ground state spin arrangement for this compound. The GGA approximation (in the PBE prescription) was used in the calculation. US pseudopotentials for Nickel and Oxygen (the same as in FeO) were used with the same energy cutoffs (of 40 and 400 Ry respectively) for both the electronic wavefunctions and the charge density as for FeO and also the same 4$\times$4$\times$4 k-point grid for reciprocal space integrations.
In the calculation of the Hubbard $U$ of NiO we did not studied the convergence properties of $U$ with system size as we did in FeO but, assuming a similar convergence also in this case, we performed a constrained calculation only in the C4 cell and then extrapolated the obtained result to the C128 supercell. The calculated value of the $U$ parameter is 4.6 eV. This value is smaller than literature values for the same parameter that are rather in the range of 7-8 eV [@AZA91], however it has been recently pointed out [@Dudarev98; @Bengone00] that in obtaining these values self-screening of $d$ electrons is neglected and that better agreement with experimental results is obtained using an effective Hubbard $U$ of the order of 5-6 eV.
The magnetic moment of the Ni ions is correctly described within the present GGA+U approach which gives a value of 1.7 $\mu_{B}$ well within the experimental range of values ranging from 1.64 and 1.9 $\mu_{B}$ [@Alperin62; @Cheetham83], better than the value of 1.55 $\mu_{B}$ obtained within GGA.
![\[nioband\] The band structure of NiO in the undistorted (cubic) AF configuration at the experimental lattice spacing obtained within GGA (top panel) and with the computed Hubbard $U$ of 4.6 eV (bottom panel). The zero of the energy is set at the top of the valence band. Experimental data from ref. [@zx1] (empty symbols) and [@kule1] (solid symbols) are also reported.](FIG_13.ps){width="8.5truecm"}
![\[niodos\] Projected density of states of NiO in the undistorted AF configuration at the experimental lattice spacing obtained with $U$ = 4.6 eV.](FIG_14.ps){width="8.7truecm"}
In fig. \[nioband\] and fig. \[niodos\] the band structure and atomic-state projected density of states of NiO obtained with this value of $U$ is shown, along with the results of standard GGA, and compared with the photoemission data in the $\Gamma$X direction extracted from ref. [@zx1; @kule1].
Despite the agreement with the experimental band-dispersion is not excellent—the valence band width is somehow overestimated by both GGA and GGA+U calculations—, GGA+U band structure reproduces well some features of the photoemmission spectrum for this compound and gives a much larger band gap than the one obtained within GGA approximation. A very important feature to be noticed in the density of states reported in fig.\[niodos\] is the fact that GGA+U modifies qualitatively the nature of the states at the top of the valence band, and hence the nature of the band gap: in GGA approximation the top of valence band is dominated by Nickel $d$-states while in the GGA+U calculation the Oxygen $p$-states give the most important contribution. In both approaches the bottom of the conduction band is mainly Nickel $d$-like and therefore the predicted band gap is primarily of charge-transfer type within GGA+U, in agreement with experimental and theoretical evidence [@Lee91; @wei1; @Sawatzky84], while it is wrongly described as of Mott-Hubbard type according GGA approximation.
Our GGA+U value for the optical gap is $\approx$ 2.7 eV around the T point, smaller that commonly accepted experimental values that range from 3.7 to 4.3 eV [@Powell70; @Adler70; @McNatt69; @Hufner86]. More recently however, a re-examination [@Hufner92] of the best available optical absorption data [@Powell70] pointed out that optical absorption in NiO starts at photon energy as low as 3.1 eV, not far from our theoretical result. Indeed, Bengone and coworkers [@Bengone00] reported recently an LDA+U calculation in NiO where different empirical values of $U$ were employed. When $U=5$ eV was used—a value close to our present first-principles result—, they obtained an optical gap of 2.8 eV, very close to our results, [*and*]{} an excellent agreement between the calculated and experimental [@Powell70] optical absorption spectra. The same calculation with the literature value of $U=8$ eV, gave a larger value for the optical gap but a very poor agreement with the experimental absorption spectrum.
Minerals: Fayalite
------------------
As a final example we want to apply the present methodology to Fayalite, the iron-rich end member of (Mg,Fe)$_2$SiO$_4$ olivine (orthorhombic structure), one of the most abundant minerals in Earth’s upper mantle. Recently [@NoiFayalite] we showed that, although good structural and magnetic properties could be obtained for this mineral within LDA or GGA, its electronic properties were incorrectly described as metallic, confirming the correlated origin of the observed insulating behavior.
![\[fig:str\]The unit cell of Fayalite. Large dark ions are Fe, small dark ions are O, light ions are Si.](FIG_15.ps){width="9.0truecm"}
From x-rays diffraction studies it is known that Fayalite has an orthorhombic cell, whose experimental lattice parameters are (in atomic units) a = 19.79, b = 11.50, c = 9.11. The unit cell (depicted in fig.\[fig:str\]) contains four formula units, 28 atoms: 8 iron, 4 silicon, and 16 oxygen atoms. Silicon ions are tetrahedrally coordinated to oxygens, whereas iron ions occupy the centers of distorted oxygen octahedra. The point group symmetry of the non magnetic crystal is mmm (D$_{2h}$ in the Schoenflies notation) and the space group is Pnma. The magnetization of iron reduces the original symmetry and only half of the symmetry operations survive. The general expression for the internal structural degrees of freedom is given in table \[tab:wyckoff\] in the Wyckoff notation [@Wyckoff].
Ion Class Coordinates
----------------- ------- ---------------------------------
Fe1 4a (0,0,0), (1/2,0,1/2)
(0,1/2,0), (1/2,1/2,1/2)
Fe2, Si, O1, O2 4c $\pm$(u,1/4,v),
$\pm$(u+1/2,1/4,1/2-v)
O3 8d $\pm$(x,y,z), $\pm$(x,1/2-y,z),
$\pm$(x+1/2,1/2-y,1/2-z),
$\pm$(x+1/2,y,1/2-z)
: Definition of the Wyckoff structural parameters appropriate for Fayalite structure
\[tab:wyckoff\]
Iron sites can be divided into two classes (see fig. \[fig:str\] and tab. \[tab:wyckoff\]): Fe1 centers which are structured in chains running parallel to the $b$, \[010\], side of the orthorhombic cell, and Fe2 sites which belong to mirror planes for the non magnetic crystal structure perpendicular to the $b$ side and cutting it at 1/4 and 3/4 of its length. The main structural units are the iron centered oxygen octahedra which are distorted from the cubic symmetry and tilted with respect to each other both along the chains and on nearest Fe2 sites. Fayalite is known to be an antiferromagnetic (AF) compound with slightly non collinear arrangement of spin on Fe1 iron site (this non collinearity will not be addressed here). Magnetic moments along the central and the edge Fe1 chains are antiferromagnetically oriented and from our previous work [@NoiFayalite] the most stable spin configuration is the one in which the magnetization of Fe2 ion is parallel to the one of the closest Fe1 iron. This magnetic structure is consistent with an iron-iron magnetic interaction via a superexchange mechanism through oxygen $p$ orbitals.
![\[fayuconv\]Convergence of Hubbard parameters of Fayalite with the number of iron included in the supercell used in the extrapolation. U1 is the value obtained for Fe1 ions, U2 the one for Fe2.](FIG_16.ps){width="9.0truecm"}
The calculation of $U$ was performed for the experimental geometry, in the above mentioned spin configuration. As the primitive unit cell of fayalite is already quite large, we performed the constrained calculation only in this cell and used larger supercells only to extrapolate the results. We considered three supercells in addition to the primitive one: [*i)*]{} a cell duplicated in the $[0,1,0]$ chain direction (a 1$\times$2$\times$1 supercell), containing 16 iron atoms; [*ii)*]{} a cell, containing 64 iron ions, obtained by duplicating the primitive structure in all directions (a 2$\times$2$\times$2 supercell) and [*iii)*]{} a 4$\times$4$\times$2 supercell (256 iron ions). Other computational details where similar to those used in our previous work [@NoiFayalite]. As GGA approximation provided a slightly better description of the system than LDA, we assumed this functional as the starting point to be improved; the same pseudopotentials used in ref.[@NoiFayalite] for Fe, O and Si were adopted here; somehow larger energy cutoff for the electronic wave functions and charge density (36 and 288 Ry respectively) and a small smearing width of 0.005 Ry were used. A 2$\times$4$\times$4 Monkhorst-Pack grid of k-points in the primitive cell was found sufficient for the BZ integration.
The results of the $U$ calculation for the two different families of iron sites (Fe1 and Fe2) are reported in fig. \[fayuconv\] where the rapid convergence with respect supercell dimension can be seen. The final results for the on-site Coulomb parameters are $U_1=4.9$ eV for Fe1 ions and $U_2=4.6$ eV for Fe2, which are in fairly good agreement with the approximate (average) value of 4.5 eV obtained in ref.[@NoiFayalite] from a rather crude estimate.
![\[FayBANDS\] The band structure of Fayalite obtained within the present LDA+U approach. The zero of the energy is set to the top of the valence band. Complete degeneracy among spin up and spin down states is present.](FIG_17.ps){width="10.0truecm"}
The GGA+U band structure of Fayalite is shown in fig. \[FayBANDS\] while in fig. \[FayDOS\] some atomic-projected density of states are reported. At variance with the GGA results reported in ref.[@NoiFayalite] a band gap of about 3 eV now separates the valence manifold from the conduction one, in reasonable agreement with the experimental result of about 2 eV [@fayexp1] at zero pressure.
![\[FayDOS\] Some atomic-projected density of states of Fayalite obtained within the present LDA+U approach. Contributions from majority- and minority-spin $3d$ states of one of the Fe1 iron ions and from the total $2p$ manifold of one oxygen ion are shown.](FIG_18.ps){width="9.0truecm"}
The minority spin $t_{2g}$ manifold of iron ions, that within GGA cross the Fermi energy, is split into two subgroups by the gap opening. The conduction-band states are shrunk to a narrow energy range and moved above the bottom of the iron $s$-states band which remains almost unaffected; the lower-energy minority-spin $d$-states, instead, merge in the group of states below the Fermi level where they mix strongly with states originating from Oxygen $p$ orbitals: the two sets of states, well separated in the GGA results, collapse into a unique block. The most evident consequence of the gap opening consists in a pronounced shrinking of the $d$ states of iron which become flatter than in the GGA case. This is evident on the top of the valence band, but also for states well below this energy level, which thus reveal a more pronounced atomic-like behavior. Beside the gap opening between the two groups of the minority-spin states, a strong mixing occurs among the oxygen $p$ states and the iron $d$ levels over a rather large region extending down to 8 eV below the top of the valence band. A finite contribution of the oxygen states is present close to the top of the valence manifold showing that the gap is mainly of Mott-Hubbard type with a partial charge-transfer character.
We have then relaxed the geometric structure of the system (both internal and cell degrees of freedom) assuming no dependence of $U_1$ and $U_2$ on the atomic configuration. The resulting structural parameters ($a=20.18$, $b=11.75$, $c=9.29$ atomic unit) as well as the internal coordinates reported in table \[tab:faystrut\] are in very good agreement with the experimental results, even better than the already satisfactory agreement obtained in ref. [@NoiFayalite] within GGA.
Ion u v x y z
--------- ----- ------- ------- ------- ------- -------
Exp.
Fe2 0.780 0.515
Si 0.598 0.071
O1 0.593 0.731
O2 0.953 0.292
O3 0.164 0.038 0.289
$GGA+U$
Fe2 0.779 0.515
Si 0.597 0.072
O1 0.593 0.735
O2 0.951 0.289
O3 0.165 0.036 0.286
: Comparison of the experimental and LDA+U calculated values for the Wyckoff structural parameters of Fayalite as defined in table \[tab:wyckoff\]
\[tab:faystrut\]
Although we did not studied other spin-configurations, magnetic properties seam to improve slightly in the GGA+U approximation. The magnetic moment on each iron (both Fe1 and Fe2) was found to be 3.9 $\mu_{B}$, in closer agreement with the spin-only value (4 $\mu_{B}$) of the experimental result (4.4 $\mu_{B}$) than the one obtained by GGA only (3.8 $\mu_{B}$). This improvement is probably due to the enhanced atomic-like character of iron $d$ states, which is consequence of the gap opening.
In conclusion, the GGA+U provides a quite good description of structural, magnetic [*and*]{} electronic properties of fayalite, reproducing the observed insulating behavior with a reasonable value for its fundamental band gap.
Summary
=======
In this work we have reexamined the LDA+U approximation to DFT and a simplified rotational-invariant form of the functional was adopted. We then developed a method, based on a linear response approach, to calculate in an internally consistent way the interaction parameters entering the LDA+U functional, without making aprioristic assumption about screening and/or basis set employed in the calculation. Our methodology was then successfully tested on a few systems representative of normal and correlated metals, simple transition metal oxides and iron silicates. In all cases we obtained rather accurate results indicating that our scheme allows us to study both electronic and structural properties of strongly correlated material on equal footing, without resorting to any empirical parameter adjustment.
This work has been supported by the MIUR under the PRIN program and by the INFM in the framework of the [*Iniziativa Trasversale Calcolo Parallelo*]{}.
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[^1]: present address: Massachusetts Institute of Technology, 77 Massachusetts avenue, Cambridge MA, 02139 USA.
|
---
abstract: 'The S–transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a $B$–valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra $B$. Also, a new proof of the additivity of the R–transform in this setting is given.'
address:
- |
- Mathematisches Institut\
Westfälische Wilhelms–Universität Münster\
Einsteinstr. 62\
48149 Münster\
Germany
- |
- Department of Mathematics\
Texas A&M University\
College Station TX 77843–3368, USA
author:
- 'Kenneth J. Dykema'
date: '11 January, 2005'
title: 'On the S–transform over a Banach algebra'
---
[^1]
Introduction and statement of the main result
=============================================
Let $B$ be a unital complex Banach algebra. (In this paper, all Banach algebras will be over the complex numbers.) A $B$–valued Banach noncommutative probability space is a pair $(A,E)$ where $A$ is a unital Banach algebra containing an isometrically embedded copy of $B$ as a unital subalgebra and where $E:A\to B$ is a bounded projection satisfying the conditional expectation property $$E(b_1ab_2)=b_1E(a)b_2\qquad(a\in A,\,b_1,b_2\in B).$$ In the free probability theory of Voiculescu, see [@V85] and [@VDN], elements $x$ and $y$ of $A$ are said to be free if their mixed moments $E(b_1a_1\cdots b_na_n)$, where $a_j\in\{x,y\}$ and $b_j\in B$, are determined in a specific way from the moments of $x$ and of $y$. Of particular interest, for example to garner spectral data, are the symmetric moments $$\label{eq:bxy}
E(bxybxy\cdots bxy)$$ of the product $xy$, for $b\in B$.
In the case $B={{\mathbf C}}$, Voiculescu [@V87] invented the S–transform of an element $x\in A$ satisfying $E(x)\ne0$. The S–transform can be used to find the generating function for the symmetric moments of $xy$ in terms of those for $x$ and $y$ individually, when $x$ and $y$ are free and when $E(x)\ne0$ and $E(y)\ne0$. In particular, Voiculescu showed that the S–transform is multiplicative: $$\label{eq:Smult}
S_{xy}=S_xS_y$$ when $x$ and $y$ are free.
In [@V95], Voiculescu gave a definition of an S–transform in the context of an arbitrary noncommutative probability space. However, this definition was quite complicated and involved differential equations.
Recently, Aagaard [@Aa] took the straightforward extension of Voiculescu’s definition [@V87] of the scalar–valued S–transform to the Banach algebra situation and generalized Voiculescu’s result to the case when $B$ is a commutative unital Banach algebra and $E(x)$ and $E(y)$ are invertible elements of $B$.
In this paper, we treat the case when $B$ is an arbitrary unital Banach algebra. We make an improvement in Aagaard’s definition of the S–transform. For us, $S_x$ is a $B$–valued analytic function defined in a neighborhood of $0$ in $B$. We write $S_{xy}$ in terms of $S_x$ and $S_y$ (again assuming $E(x)$ and $E(y)$ are invertible). Instead of simple multiplicativity , we have in general a twisted multiplicativity, as stated in our main theorem immediately below, which reduces to when $B$ is commutative.
\[thm:main\] Let $B$ be a unital complex Banach algebra and let $(A,E)$ be a $B$–valued Banach noncommutative probability space. Let $x,y\in A$ be free in $(A,E)$ and assume both $E(x)$ and $E(y)$ are invertible elements of $B$. Then $$\label{eq:main}
S_{xy}(b)=S_y(b)S_x(S_y(b)^{-1}bS_y(b)).$$
Our definition of the S–transform and our proof of Theorem \[thm:main\] rely on the theory of analytic functions between Banach spaces – see for example Chapters III and XXVI of [@HP] and papers cited there.
In [@H], Haagerup gave two new proofs of the multiplicativity of the S–transform in the case $B={{\mathbf C}}$. Our proof of Theorem \[thm:main\] is very much inspired by one of Haagerup’s proofs, namely Theorem 2.3 of [@H], which uses creation and annihilation operators in the full Fock space. In particular, we consider a $B$–valued Banach algebra analogue of the full Fock space and we construct random variables having arbitrary moments up to a given finite order, using analogues of the creation and annihilation operators. These are reminiscent of, though slightly different from, Voiculescu’s constructions in [@V95].
In §\[sec:def\] below, we define the S–transform $S_a$ (assuming the expectation of $a$ is invertible). Then, considering Taylor expansions about zero, we show that the $n$th order term in the expansion for $S_a$ depends only on the moments up to $n$th order of $a$. In §\[sec:tmult\], we construct operators analogous to the creation and annihilation operators on full Fock space, and we use these to prove the main result, Theorem \[thm:main\]. In §\[sec:R-t\], we offer a new proof of additivity of the R–tranfrom over a Banach space, using the operators and techniques introduced in the preceding sections.
[*Acknowledgements.*]{} The author wishes to thank Joachim Cuntz and the Mathematics Institute of the Westfälische Wilhelms–Universität Münster for their generous hospitality during the author’s year–long visit, when this research was conducted.
The S–transform in a Banach noncommutative probability space {#sec:def}
============================================================
Let $B$ be a unital Banach algebra. For $n\ge1$ we will let ${{\mathcal{B}}}_n(B)$ denote the set of all bounded $n$–multilinear maps $$\alpha_n:\underset{n\text{ times}}{\underbrace{B{\times\cdots\times}B}}\to B,$$ where multilinearity means over ${{\mathbf C}}$ and a multilinear map $\alpha_n$ is bounded if $$\|\alpha_n\|:=\sup\{\|\alpha_n(b_1,\ldots,b_n)\|\mid b_j\in B,\,\|b_1\|,\ldots,\|b_n\|\le1\}<\infty.$$ We say $\alpha_n$ is [*symmetric*]{} if it is invariant under arbitrary permutations of its $n$ arguments.
From the theory of analytic functions between complex Banach spaces, any $B$–valued analytic function $F$ defined on a neighborhood of zero in $B$ has an expansion $$\label{eq:Fexp}
F(b)=F(0)+\sum_{n=1}^\infty F_n(b,\ldots,b),$$ for some symmetric multilinear functions $F_n\in{{\mathcal{B}}}_n(B)$, with $\limsup_{n\to\infty}\|F_n\|^{1/n}<\infty$; see, for example, Theorem 3.17.1 of [@HP] and its proof. Here $F_1$ is just the Fréchet derivative of $F$ at $0$ and the multilinear function $F_n$ appearing in is $1/n!$ times the $n$th variation of $F$, i.e. $n!F_n(h_1,\ldots,h_n)$ is the $n$–fold Fréchet derivative taken with respect to increments $h_1,\ldots,h_n$. For convenience we will write $F_0$ for $F(0)$. We will refer to as the [*power series expansion*]{} of $F(b)$ around $0$ and to $F_n(b,\ldots,b)$ as the $n$th term in this power series expansion. Note that the full symmetric multilinear function $F_n$ can be recovered from knowing its diagonal $b\mapsto F_n(b,\ldots,b)$; for example, $n!F_n(b_1,\ldots,b_n)$ is the obvious partial derivative of $$F_n(t_1b_1+\cdots+t_nb_n,\ldots,t_1b_1+\cdots+t_nb_n)$$ at $(0,\ldots,0)$, where $t_1,\ldots,t_n$ are real variables.
Let $(A,E)$ be a Banach noncommutative probability space over $B$, let $a\in A$ and suppose $E(a)$ is an invertible element of $B$. Consider the function $$\label{eq:Psi}
\Psi_a(b)=E((1-ba)^{-1})-1=\sum_{n=1}^\infty E((ba)^n),$$ defined for $\|b\|<\|a\|^{-1}$. Then $\Psi_a$ is Fréchet differentiable on its domain, i.e. is analytic there. We also have $$\label{eq:PsiPhi}
\Psi_a(b)=b\Phi_a(b),$$ where $$\label{eq:Phi}
\Phi_a(b)=E(a(1-ba)^{-1});$$ clearly $\Phi_a$ is analytic on the domain of $\Psi_a$. The Fréchet differential of $\Psi_a$ at $b=0$ is easily found to be the bounded linear map $$\label{eq:hE}
h\mapsto hE(a)$$ from $B$ to itself. By hypothesis, this linear map has bounded inverse $h\mapsto hE(a)^{-1}$. By the usual Banach space inverse function theorem, there are neighborhoods $U$ and $V$ of zero in $B$ such that $U$ lies in the domain of $\Psi_a$ and the restriction of $\Psi_a$ to $U$ is a homeomorphism onto $V$. Moreover, letting $\Psi_a{^{\langle-1\rangle}}$ denote the inverse with respect to composition of the restriction of $\Psi_a$ to $U$, the function $\Psi_a{^{\langle-1\rangle}}$ is Fréchet differentiable on its domain and is, therefore, analytic there.
\[lem:Ha\] Assuming $E(a)$ is invertible, there is an open neighborhood of $0$ in $B$ and unique analytic $B$–valued function $H_a$ defined there such that $\Psi_a{^{\langle-1\rangle}}(b)=bH_a(b)$.
Uniqueness of $H_a$ is clear by uniqueness of power series expansions about zero. Let us show existence. Using , we seek $H_a$ such that $bH_a(b)\Phi_a(bH_a(b))=b$, and it will suffice to find $H_a$ such that $$\label{eq:HPhi}
H_a(b)\Phi_a(bH_a(b))=1.$$ The existence of $H_a$ follows from an easy application of the implicit function theorem for functions between Banach spaces, which is a result of Hildebrandt and Graves [@HG] (see also the discussion on p. 655 of [@G]). Indeed, $H_a(0)=E(a)^{-1}$ is a solution of at $b=0$ and the Fréchet differential of the function $x\mapsto x\Phi_a(bx)$ at $b=0$ is the map , which has bounded inverse.
Let $a\in A$ and assume $E(a)$ is invertible. The [*S–transform*]{} of $a$ is the $B$–valued analytic function $$\label{eq:S}
S_a(b)=(1+b)H_a(b),$$ which defined in some neighborhood of $0$ in $B$, where $H_a$ is the function from Lemma \[lem:Ha\].
Note that $S_a(0)=E(a)^{-1}$.
We may write $$\label{eq:AaS}
S_a(b)=(1+b)b^{-1}\Psi_a{^{\langle-1\rangle}}(b),$$ which is the same formula given by Voiculescu [@V87] and used by Aagaard [@Aa]. In the case $B={{\mathbf C}}$, the definition yields, of course, the same function as Voiculescu’s S–transform. Moreover, the only difference between the definition and the one appearing in [@Aa] is that we have used the implicit function theorem to show that makes sense for all $b$ in a neighborhood of zero.
If $F$, $G$ and $H$ are $B$–valued analytic functions defined on neighborhoods of $0$ in $B$, then the product $FG$ is analytic and, if $H(0)=0$, also the composition $F\circ H$ is analytic in some neighborhood of $0$ in $B$. Straightforward asymptotic analysis yields the following formulas for the diagonals of the multilinear functions appearing in the power series expansions of $FG$ and $F\circ H$.
\[lem:FG\] We have for $n\ge0$ $$\label{eq:FG}
(FG)_n(b,\ldots,b)=\sum_{k=0}^nF_k(b,\ldots,b)G_{n-k}(b,\ldots,b)$$ and for $n\ge1$ $$\label{eq:FoH}
(F\circ H)_n(b,\ldots,b)=
\sum_{k=1}^n\sum_{\substack{p_1,\ldots,p_k\ge1 \\ p_1+\cdots+p_k=n}}
F_k(H_{p_1}(b,\ldots,b),\ldots,H_{p_k}(b,\ldots,b)).$$
\[lem:Finv\] Let $F$ be analytic in a neighborhood of $0$. If $F(0)$ is an invertible element of $B$, then $G(b)=F(b)^{-1}$ defines a function that is analytic in a neighborhood of $0$, and the $n$th term of its power series expansion is $G_0=F_0^{-1}$ and, for $n\ge1$, $$\label{eq:Frecip}
G_n(b,\ldots,b)=-F_0^{-1}\sum_{k=1}^nF_k(b,\ldots,b)G_{n-k}(b,\ldots,b).$$ On the other hand, if $F(0)=0$ and if $F_1$ has a bounded inverse, then $F$ has an inverse with respect to composition, denoted $F{^{\langle-1\rangle}}$, that is analytic in a neighborhood of $0$. Taking $H=F{^{\langle-1\rangle}}$, we have $H_1=(F_1){^{\langle-1\rangle}}$ and, for $n\ge2$, $$\label{eq:Finv}
H_n(b,\ldots,b)=
-(F_1){^{\langle-1\rangle}}\bigg(\sum_{k=2}^n\sum_{\substack{p_1,\ldots,p_k\ge1 \\ p_1+\cdots+p_k=n}}
F_k(H_{p_1}(b,\ldots,b),\ldots,
H_{p_k}(b,\ldots,b))\bigg).$$
Assuming $F(0)$ is invertible, that $G(b)=F(b)^{-1}$ is analytic is clear, and we have $(FG)_0=1$ and $(FG)_n=0$ for $n\ge1$. Now the expression results from solving for $G_n$.
If $F(0)=0$ and the Fréchet derivative $F_1$ of $F$ at $0$ has bounded inverse, then by the inverse function theorem for Banach spaces, $F$ has an inverse with respect to composition $F{^{\langle-1\rangle}}$ that is analytic in a neighborhood of $0$. Taking $H=F{^{\langle-1\rangle}}$, we have $(F\circ H)_1={{\operatorname{id}}}_B$ and $(F\circ H)_n=0$ for all $n\ge2$. Solving in for $H_n$ yields the expression .
Consider an element $a\in A$ as at the begining of this section. We say the [*$n$th moment function*]{} of $a$ is the multilinear function $\mu_{a,n}\in{{\mathcal{B}}}_n(B)$ given by $$\mu_{a,n}(b_1,\ldots,b_n)=E(b_1ab_2a\cdots b_na).$$
\[prop:nth\] Assume $E(a)$ is an invertible element of $B$. Then the $n$th term $(S_{a})_n(b,\ldots,b)$ in the power series expansion of the S–transform $S_a$ of $a$ about zero depends only on the first $n$ moment functions $\mu_{a,1},\,\mu_{a,2},\,\ldots,\,\mu_{a,n}$ of $a$.
The symmetric $n$–multilinear function $(\Psi_a)_n$ appearing in the power series expansion of $\Psi_a$ is the symmetrization of $\mu_{a,n}$. Using Lemma \[lem:Finv\], we see that the $n$th term $(\Psi_a{^{\langle-1\rangle}})_n(b,\ldots,b)$ in the power series expansion of $\Psi_a{^{\langle-1\rangle}}(b)$ around $0$ depends only on $\mu_{a,1},\ldots,\mu_{a,n}$. But $$\begin{gathered}
(\Psi_a{^{\langle-1\rangle}})_n(b,\ldots,b)=b\,(H_a)_n(b,\ldots,b) \\
(S_{a})_n(b,\ldots,b)=(1+b)\,(H_a)_n(b,\ldots,b)\end{gathered}$$ and the result is proved.
Twisted multiplicativity of the S–transform {#sec:tmult}
===========================================
Let $B$ be a unital Banach algebra over ${{\mathbf C}}$ and let $I$ be a set. Let $D=\ell^1(I,B)$ be the Banach space of all functions $d:I\to B$ such that $\|d\|:=\sum_{i\in I}\|d(i)\|<\infty$. For $i\in I$, $\delta_i\in D$ will denote the function taking value $1$ at $i$ and $0$ at all other elements of $I$. We have the obvious left action of $B$ on $D$ by $(bd)(i)=b\,d(i)$, and the resulting algebra homomorphism $B\to{{\mathcal{B}}}(D)$ is isometric. (Whenever $X$ is a Banach space, we denote by ${{\mathcal{B}}}(X)$ the Banach algebra of all bounded linear operators from $X$ to itself.) For $k\ge1$, let $D^{{\hat{\otimes}}k}=D{\hat{\otimes}}\cdots{\hat{\otimes}}D$ be the $k$–fold Banach space projective tensor product of $D$ with itself (over the complex field). Consider the Banach space $$\label{eq:F}
{{\mathcal{F}}}=B\Omega\oplus\bigoplus_{k=1}^\infty D^{{\hat{\otimes}}k}{\hat{\otimes}}B,$$ where also ${\hat{\otimes}}B$ is the Banach space projective tensor product and where the we take the direct sum with respect to the $\ell^1$–norm. Here, $B\Omega$ signifies just a copy of $B$ and $\Omega$ denotes the identity element of this copy of $B$, consdered as a vector in ${{\mathcal{F}}}$. Let $\lambda:B\to{{\mathcal{B}}}({{\mathcal{F}}})$ be the map defined by $$\begin{gathered}
\lambda(b)(b_0\Omega)=(bb_0)\Omega \\
\lambda(b)(d_1{\otimes\cdots\otimes}d_k\otimes b_0)=(bd_1)\otimes d_2{\otimes\cdots\otimes}d_k\otimes b_0\end{gathered}$$ for $k\in{{\mathbf N}}$, $d_1,\ldots,d_k\in D$ and $b_0\in B$. Then $\lambda$ is an isometric algebra homomorphism. We will often omit to write $\lambda$, and just think of $B$ as included in ${{\mathcal{B}}}({{\mathcal{F}}})$ by this left action.
For specificity, we took the $\ell^1$ norms in the definitions of $D$ and ${{\mathcal{F}}}$, but we actually have considerable flexibility. For $D$ we need only a Banach space completion of the set of all functions $d:I\to B$ vanishing at all but finitely many elements in $I$ with the property $\|b\delta_i\|=\|b\|$, and similarly for ${{\mathcal{F}}}$. Moreover, we could replace the projective tensor norm ${\hat{\otimes}}B$ in with any tensor norm so that $\|d\otimes B\|=\|d\|\,\|b\|$ for all $d\in D^{{\hat{\otimes}}k}$ and $b\in B$.
Let $P:{{\mathcal{F}}}\to B$ be the projection onto the summand $B\Omega=B$ that sends all summands $D^{{\hat{\otimes}}k}{\hat{\otimes}}B$ to zero and let ${{\mathcal{E}}}:{{\mathcal{B}}}({{\mathcal{F}}})\to B$ be ${{\mathcal{E}}}(X)=P(X\Omega)$. Then ${{\mathcal{E}}}$ has norm $1$ and satisfies ${{\mathcal{E}}}\circ\lambda={{\operatorname{id}}}_B$. Let $\rho:B\to{{\mathcal{B}}}({{\mathcal{F}}})$ be the map defined by $$\begin{gathered}
\rho(b)(b_0\Omega)=(b_0b)\Omega \\
\rho(b)(d_1{\otimes\cdots\otimes}d_k\otimes b_0)= d_1{\otimes\cdots\otimes}d_k\otimes (b_0b).\end{gathered}$$ Then $\rho$ is an isometric algebra isomorphism from the opposite algebra $B^{{\rm op}}$ into ${{\mathcal{B}}}({{\mathcal{F}}})$. Let ${{\mathcal{B}}}({{\mathcal{F}}})\cap\rho(B)'$ denote the set of all bounded operators on ${{\mathcal{F}}}$ that commute with $\rho(b)$ for all $b\in B$. Note that $\lambda(B)\subseteq{{\mathcal{B}}}({{\mathcal{F}}})\cap\rho(B)'$.
The restriction of ${{\mathcal{E}}}$ to ${{\mathcal{B}}}({{\mathcal{F}}})\cap\rho(B)'$ satisfies the conditional expectation property $${{\mathcal{E}}}(b_1Xb_2)=b_1{{\mathcal{E}}}(X)b_2\qquad(X\in{{\mathcal{B}}}({{\mathcal{F}}})\cap\rho(B)',\,b_1,b_2\in B).$$
We have $$\begin{aligned}
{{\mathcal{E}}}(b_1Xb_2)
&=P(\lambda(b_1)X\lambda(b_2)\Omega)
=P(\lambda(b_1)X\rho(b_2)\Omega) \\
&=P(\rho(b_2)\lambda(b_1)X\Omega)
=P(\lambda(b_1)X\Omega)b_2
=b_1P(X\Omega)b_2
=b_1{{\mathcal{E}}}(X)b_2.\end{aligned}$$
For $i\in I$, let $L_i\in{{\mathcal{B}}}({{\mathcal{F}}})$ be defined by $$\begin{aligned}
L_i(b_0\Omega)&=\delta_i\otimes b_0 \\
L_i(d_1{\otimes\cdots\otimes}d_k\otimes b_0)&=\delta_i\otimes d_1{\otimes\cdots\otimes}d_k\otimes b_0.\end{aligned}$$ Thus, $$b_1\delta_{i_1}\otimes b_2\delta_{i_2}{\otimes\cdots\otimes}b_k\delta_{i_k}\otimes b_0
=b_1L_{i_1}b_2L_{i_2}\cdots b_kL_{i_k}b_0\Omega.$$ Recall that ${{\mathcal{B}}}_n(B)$ denotes the set of all bounded multilinear functions from the $n$–fold product of $B$ to $B$. We will also let ${{\mathcal{B}}}_0(B)=B$. If $i\in I$, $n\in{{\mathbf N}}$ and $\alpha_n\in{{\mathcal{B}}}_n(B)$, define $V_{i,n}(\alpha_n)$ and $W_{i,n}(\alpha_n)$ in ${{\mathcal{B}}}({{\mathcal{F}}})$ by $$\begin{aligned}
V_{i,n}(\alpha_n)(b_0\Omega)&=0 \\[1ex]
V_{i,n}(\alpha_n)(d_1{\otimes\cdots\otimes}d_k\otimes b_0)&=
\begin{cases}
0,&k<n \\
\alpha_n(d_1(i),\ldots,d_n(i))b_0\Omega,&k=n \\
\alpha_n(d_1(i),\ldots,d_n(i))d_{n+1}{\otimes\cdots\otimes}d_k\otimes b_0,&k>n
\end{cases}\end{aligned}$$ and $$\begin{aligned}
W_{i,n}(\alpha_n)&(b_0\Omega)=0 \displaybreak[2] \\[1ex]
W_{i,n}(\alpha_n)&(d_1{\otimes\cdots\otimes}d_k\otimes b_0)= \\
&=\begin{cases}
0,&k<n \\
\alpha_n(d_1(i),\ldots,d_n(i))\delta_i\otimes b_0,&k=n \\
\alpha_n(d_1(i),\ldots,d_n(i))\delta_i\otimes d_{n+1}{\otimes\cdots\otimes}d_k\otimes b_0,&k>n.
\end{cases}\end{aligned}$$ Finally, taking $n=0$ and $\alpha_0\in B$, let $$V_{i,0}(\alpha_0)=\alpha_0\qquad W_{i,0}(\alpha_0)=\alpha_0L_i.$$ These formulas are guaranteed to define bounded operators on ${{\mathcal{F}}}$, because we took the projective tensor product in $D^{{\hat{\otimes}}k}$. The expression $V_{i,n}(\alpha_n)$, $n\ge1$, is a sort of $n$–fold annihilation operator, while $W_{i,n}(\alpha_n)$ is $n$–fold annihilation combined with single creation, and, of course, $W_{i,0}$ is a single creation operator. Note that in all cases we have $V_{i,n}(\alpha_n),\,W_{i,n}(\alpha_n)\in{{\mathcal{B}}}({{\mathcal{F}}})\cap\rho(B)'$.
The relations gathered in the following lemma are easily verified.
\[lem:rel\] Let $n,m\in{{\mathbf N}}$ and $\alpha_n\in{{\mathcal{B}}}_n(B)$, $\beta_m\in{{\mathcal{B}}}_{m}(B)$ and take $b\in B$. Then
1. $$V_{i,n}(\alpha_n)\lambda(b)=V_{i,n}({{\tilde\alpha}}_n)\qquad W_{i,n}(\alpha_n)\lambda(b)=W_{i,n}({{\tilde\alpha}}_n),$$ where $${{\tilde\alpha}}_n(b_1,\ldots,b_n)=\alpha_n(bb_1,b_2,\ldots,b_n);$$
2. if $n=1$, then $$V_{i,1}(\alpha_1)L_i=\lambda(\alpha_1(1)),\qquad W_{i,1}(\alpha_1)L_i=\lambda(\alpha_1(1))L_i$$ and for $n\ge2$ we have $$V_{i,n}(\alpha_n)L_i=V_{i,n-1}({{\tilde\alpha}}_{n-1}),\qquad W_{i,n}(\alpha_n)L_i=W_{i,n-1}({{\tilde\alpha}}_{n-1}),$$ where here $${{\tilde\alpha}}_{n-1}(b_1,\ldots,b_{n-1})=\alpha_n(1,b_1,\ldots,b_{n-1});$$
3. we have $$V_{i,n}(\alpha_n)V_{i,m}(\beta_m)=V_{i,n+m}(\gamma_{n+m}),
\qquad W_{i,n}(\alpha_n)V_{i,m}(\beta_m)=W_{i,n+m}(\gamma_{n+m}),$$ where $$\gamma_{n+m}(b_1,\ldots,b_{m+n})=
\alpha_n(\beta_m(b_1,\ldots,b_{m})b_{m+1},b_{m+2},\ldots,b_{m+n});$$
4. $$\begin{aligned}
V_{i,n}(\alpha_n)W_{i,m}(\beta_m)&=V_{i,n+m-1}(\gamma_{n+m-1}), \\
W_{i,n}(\alpha_n)W_{i,m}(\beta_m)&=W_{i,n+m-1}(\gamma_{n+m-1}),\end{aligned}$$
where $$\gamma_{n+m-1}(b_1,\ldots,b_{m+n-1})=
\alpha_n(\beta_m(b_1,\ldots,b_{m}),b_{m+1},b_{m+2},\ldots,b_{m+n-1});$$
5. $$\lambda(b)V_{i,n}(\alpha_n)=V_{i,n}(b\alpha_n),$$
6. if $i'\ne i$ and $n\ge1$, then $$V_{i,n}(\alpha_n)L_{i'}=0=W_{i,n}(\alpha_n)L_{i'}.$$
\[prop:free\] For $i\in I$ let ${{\mathfrak A}}_i\subseteq{{\mathcal{B}}}({{\mathcal{F}}})\cap\rho(B)'$ be the subalgebra generated by $$\lambda(B)\cup\{L_i\}\cup\{V_{i,n}(\alpha_n)\mid n\in{{\mathbf N}},\,\alpha_n\in{{\mathcal{B}}}_n(B)\}
\cup\{W_{i,n}(\alpha_n)\mid n\in{{\mathbf N}},\,\alpha_n\in{{\mathcal{B}}}_n(B)\}.$$ Then the family $({{\mathfrak A}}_i)_{i\in I}$ is free with respect to ${{\mathcal{E}}}$.
Using Lemma \[lem:rel\], we see that every element of ${{\mathfrak A}}_i$ can be written as a sum of finitely many terms of the following forms:
1. $\lambda(b)$
2. $\lambda(b_0)L_i\lambda(b_1)\cdots L_i\lambda(b_n)$
3. $V_{i,n}(\alpha_n)$
4. $\lambda(b_0)L_i\lambda(b_1)L_i\cdots \lambda(b_{k})L_iV_{i,n}(\alpha_n)$
5. $\lambda(b)W_{i,n}(\alpha_n)$
6. $\lambda(b_0)L_i\lambda(b_1)L_i\cdots \lambda(b_{k-1})L_i\lambda(b_k)W_{i,n}(\alpha_n)$.
Now all terms of the forms (ii)–(vi) lie in $\ker{{\mathcal{E}}}$, while ${{\mathcal{E}}}(\lambda(b))=b$. Therefore, ${{\mathfrak A}}_i\cap\ker{{\mathcal{E}}}$ is the set of all finite sums of terms of the forms (ii)–(vi).
Let $p\in{{\mathbf N}}$ with $p\ge2$ and take $i_1,\ldots,i_p\in I$ with $i_1\ne i_2,\,i_2\ne i_3,\ldots,i_{p-1}\ne i_p$. Suppose $a_j\in {{\mathfrak A}}_{i_j}\cap{{\mathcal{E}}}$ ($1\le j\le p$) and let us show ${{\mathcal{E}}}(a_1\cdots a_p)=0$. From Lemma \[lem:rel\] part (vi), we see $a_1a_2\cdots a_p=0$ unless either $\forall j$ $a_j$ is of the form (ii) or $\forall j$ $a_j$ is of the form (iii) or (v). But $V_{i,n}(\alpha_n)\Omega=0=W_{i,n}(\alpha_n)\Omega$ when $n\ge1$, so if $a_p$ is of the form (iii) or (v), then ${{\mathcal{E}}}(a_1\cdots a_p)=0$. We are left to consider the case when $a_1\ldots a_p$ can be written as $$\begin{gathered}
(\lambda(b_0)L_{i_1}\lambda(b_1^{(1)})L_{i_1}\lambda(b_2^{(1)})\cdots L_{i_1}\lambda(b_{k(1)}^{(1)}))
(L_{i_2}\lambda(b_1^{(2)})\cdots L_{i_2}\lambda(b_{k(2)}^{(2)}))\cdots \\
\cdots(L_{i_p}\lambda(b_1^{(p)})\cdots L_{i_p}\lambda(b_{k(p)}^{(p)})),\end{gathered}$$ where all $k(j)\ge1$. But in this case, clearly ${{\mathcal{E}}}(a_1\cdots a_p)=0$.
\[lem:Nthmoment\] Let $N\in{{\mathbf N}}$ and for every $n\in\{0,1,\ldots,N\}$ let $\alpha_n\in{{\mathcal{B}}}_n(B)$. Fix $i\in I$ and let $$\begin{aligned}
X&=\sum_{n=0}^{N-1}(V_{i,n}(\alpha_n)+W_{i,n}(\alpha_n)) \\[1ex]
Y&=X+V_{i,N}(\alpha_N)+W_{i,N}(\alpha_N).\end{aligned}$$ Then for any $b_0,\ldots,b_N\in B$, we have $${{\mathcal{E}}}(b_0Yb_1Y\cdots b_NY)=b_0\alpha_N(b_1\alpha_0,b_2\alpha_0,\ldots,b_N\alpha_0)
+{{\mathcal{E}}}(b_0Xb_1X\cdots b_NX).$$
To evaluate ${{\mathcal{E}}}(b_0Yb_1Y\cdots b_NY)$, first write $$Y=\sum_{n=0}^N(V_{i,n}(\alpha_n)+W_{i,n}(\alpha_n))$$ and distribute. Now using the creation and annihilation properties of the $W_{i,n}(\alpha_n)$ and $V_{i,n}(\alpha_n)$ operators, we see that the only term involving $\alpha_N$ to contribute a possibly nonzero quantity to ${{\mathcal{E}}}(b_0Yb_1Y\cdots b_NY)$ is $${{\mathcal{E}}}(b_0V_{i,N}(\alpha_N)b_1W_{i,0}(\alpha_0)\cdots b_NW_{i,0}(\alpha_0)),$$ whose value is $b_0\alpha_N(b_1\alpha_0,b_2\alpha_0,\ldots,b_N\alpha_0)$. The other terms involve only $\alpha_0,\ldots,\alpha_{N-1}$ and their sum is ${{\mathcal{E}}}(b_0Xb_1X\cdots b_NX)$.
\[prop:Smom\] Let $(A,E)$ be a $B$–valued Banach noncommutative probability space and let $a\in A$, $N\in{{\mathbf N}}$. Suppose $E(a)$ is an invertible element of $B$. Let $\alpha_0=E(a)$. Then there are $\alpha_1,\ldots,\alpha_N$, with $\alpha_n\in{{\mathcal{B}}}_n(B)$, such that if $$X=\sum_{n=0}^N(V_{i,n}(\alpha_n)+W_{i,n}(\alpha_n))\in{{\mathcal{B}}}({{\mathcal{F}}}),$$ then $$\label{eq:EcE}
{{\mathcal{E}}}(b_0Xb_1X\cdots b_kX)=E(b_0ab_1a\cdots b_ka)$$ for all $k\in\{1,\ldots,N\}$ and all $b_0,\ldots,b_N\in B$.
Using Lemma \[lem:Nthmoment\], The maps $\alpha_k$ can be chosen recursively in $k$ so that holds.
For the remainder of this section, we take $I=\{1,2\}$.
\[lem:X\] Let $\alpha_0\in B$ be invertible. Let $N\in{{\mathbf N}}$ and choose $\alpha_n\in{{\mathcal{B}}}_n(B)$ for $n\in\{1,\ldots,N\}$, and let $$F(b)=\alpha_0+\sum_{n=1}^N\alpha_n(b,\ldots,b).$$ Note that $F(b)$ is invertible for $\|b\|$ sufficiently small. Let $$\label{eq:X}
X=\sum_{n=0}^N(V_{1,n}(\alpha_n)+W_{1,n}(\alpha_n))\in{{\mathcal{B}}}({{\mathcal{F}}}).$$ Then the S–transform of $X$ is $S_X(b)=F(b)^{-1}$.
For $b\in B$, $\|b\|<1$, let $$\omega_b=\Omega+\sum_{k=1}^\infty(b\delta_1)^{\otimes k}\otimes1\in{{\mathcal{F}}}$$ We have $V_{1,0}(\alpha_0)\omega_b=\alpha_0\omega_b$ and, for $n\ge1$, $$V_{1,n}(\alpha_n)\omega_b=\alpha_n(b,\ldots,b)\Omega+\sum_{k=n+1}^\infty
\alpha_n(b,\ldots,b)(b\delta_1)^{\otimes(k-n)}\otimes1
=\alpha_n(b,\ldots,b)\omega_b.$$ Moreover, $W_{1,0}(\alpha_0)\omega_b=\alpha_0L_1\omega_b$ and, for $n\ge1$, $$\begin{aligned}
W_{1,n}(\alpha_n)\omega_b&=\alpha_n(b,\ldots,b)\delta_1\otimes1
+\sum_{k=n+1}^\infty\alpha_n(b,\ldots,b)\delta_1\otimes(b\delta_1)^{\otimes(k-n)}\otimes1 \\
&=\alpha_n(b,\ldots,b)L_1\omega_b.\end{aligned}$$ Thus, $$X\omega_b=F(b)(1+L_1)\omega_b.$$ For $\|b\|$ sufficiently small, we get $$\begin{gathered}
F(b)^{-1}X\omega_b=\omega_b+L_1\omega_b \\
bF(b)^{-1}X\omega_b=b\omega_b+(\omega_b-\Omega) \\
\Omega=(1+b)\omega_b-bF(b)^{-1}X\omega_b \\
\Omega=(1-bF(b)^{-1}X(1+b)^{-1})(1+b)\omega_b \\
(1-bF(b)^{-1}X(1+b)^{-1})^{-1}\Omega=(1+b)\omega_b \\
{{\mathcal{E}}}((1-bF(b)^{-1}X(1+b)^{-1})^{-1})\begin{aligned}[t]
=&P((1+b)\omega_b) \\
=&1+b.\end{aligned}\end{gathered}$$ Conjugating with $(1+b)$ yields $$1+b={{\mathcal{E}}}((1-(1+b)^{-1}bF(b)^{-1}X)^{-1})=1+\Psi_X((1+b)^{-1}bF(b)^{-1}).$$ Hence, $$\Psi_X{^{\langle-1\rangle}}(b)=(1+b)^{-1}bF(b)^{-1}$$ and $S_X(b)=F(b)^{-1}$.
\[lem:XY\] Let $\alpha_0,\ldots,\alpha_n$, $F$ and $X$ be as in Lemma \[lem:X\]. Let $\beta_0\in B$ be invertible and let $\beta_n\in{{\mathcal{B}}}_n(B)$ for $n\in\{1,\ldots,N\}$. Let $$G(b)=\beta_0+\sum_{n=1}^N\beta_n(b,\ldots,b)$$ and let $$\label{eq:Y}
Y=\sum_{n=0}^N(V_{2,n}(\alpha_n)+W_{2,n}(\alpha_n))\in{{\mathcal{B}}}({{\mathcal{F}}}).$$ Then the S–transform of $XY$ is $$\label{eq:SXY}
S_{XY}(b)=G(b)^{-1}F(G(b)bG(b)^{-1})^{-1}=S_Y(b)S_X(S_Y(b)^{-1}bS_Y(b)).$$
From Lemma \[lem:X\], we have $S_Y(b)=G(b)^{-1}$ and $S_X(b)=F(b)^{-1}$, so the final equality in is true. For $b\in B$ let $$Z_b=bL_2+bG(b)^{-1}L_1G(b)+bG(b)^{-1}L_1G(b)L_2\in{{\mathcal{B}}}({{\mathcal{F}}})$$ and insist that $\|b\|$ be so small that $\|Z_b\|<1$. Let $$\sigma_b=(1-Z_b)^{-1}\Omega=\Omega+\sum_{k=1}^\infty Z_b^k\Omega.$$ Using Lemma \[lem:rel\], we find for $n,k\ge0$, $$V_{2,n}(\beta_n)Z_b^k=
\begin{cases}
V_{2,n-k}({{\tilde\beta}}_{n-k}),&k<n, \\
\beta_n(b,\ldots,b),&k=n, \\
\beta_n(b,\ldots,b)Z_b^{k-n},&k>n
\end{cases}$$ and $$W_{2,n}(\beta_n)Z_b^k=
\begin{cases}
W_{2,n-k}({{\tilde\beta}}_{n-k}),&k<n, \\
\beta_n(b,\ldots,b)L_2,&k=n, \\
\beta_n(b,\ldots,b)L_2Z_b^{k-n},&k>n,
\end{cases}$$ where $${{\tilde\beta}}_{n-k}(b_1,\ldots,b_{n-k})=\beta_n(\underset{k}{\underbrace{b,\ldots,b}},b_1,\ldots,b_{n-k}).$$ Therefore, $$V_{2,n}(\beta_n)Z_b^k\Omega=
\begin{cases}
0,&k<n, \\
\beta_n(b,\ldots,b)\Omega,&k=n, \\
\beta_n(b,\ldots,b)Z_b^{k-n}\Omega,&k>n
\end{cases}$$ and $$W_{2,n}(\beta_n)Z_b^k\Omega=
\begin{cases}
0,&k<n, \\
\beta_n(b,\ldots,b)L_2\Omega,&k=n, \\
\beta_n(b,\ldots,b)L_2Z_b^{k-n}\Omega,&k>n
\end{cases}$$ and we get $$Y\sigma_b=G(b)(1+L_2)\sigma_b.$$ Letting $b'=G(b)bG(b)^{-1}$, we similarly find for $n,k\ge0$, $$V_{1,n}(\alpha_n)G(b)Z_b^k=
\begin{cases}
V_{1,n-k}({{\tilde\alpha}}_{n-k})G(b),&k<n, \\
\alpha_n(b',\ldots,b')G(b)(1+L_2),&k=n, \\
\alpha_n(b',\ldots,b')G(b)(1+L_2)Z_b^{k-n},&k>n
\end{cases}$$ and $$W_{1,n}(\alpha_n)G(b)Z_b^k=
\begin{cases}
W_{1,n-k}({{\tilde\alpha}}_{n-k})G(b),&k<n, \\
\alpha_n(b',\ldots,b')L_1G(b)(1+L_2),&k=n, \\
\alpha_n(b',\ldots,b')L_1G(b)(1+L_2)Z_b^{k-n},&k>n,
\end{cases}$$ where $${{\tilde\alpha}}_{n-k}(b_1,\ldots,b_{n-k})=\alpha_n(\underset{k}{\underbrace{b',\ldots,b'}},b_1,\ldots,b_{n-k}).$$ Therefore, we get $$XY\sigma_b=F(b')(1+L_1)G(b)(1+L_2)\sigma_b.$$ Thus, for $\|b\|$ sufficiently small we get $$\begin{gathered}
F(b')^{-1}XY=(1+L_1)G(b)(1+L_2)\sigma_b \\
F(b')^{-1}XY=G(b)\sigma_b+(G(b)L_2+L_1G(b)+L_1G(b)L_2)\sigma_b \\
bG(b)^{-1}F(b')^{-1}XY\sigma_b=b\sigma_b+Z_b\sigma_b \\
bG(b)^{-1}F(b')^{-1}XY\sigma_b=b\sigma_b+(\sigma_b-\Omega) \\
\Omega=((1+b)-bG(b)^{-1}F(b')^{-1}XY)\sigma_b \\
\Omega=(1-bG(b)^{-1}F(b')^{-1}XY(1+b)^{-1})(1+b)\sigma_b \\
(1-bG(b)^{-1}F(b')^{-1}XY(1+b)^{-1})^{-1}\Omega=(1+b)\sigma_b \\
{{\mathcal{E}}}((1-bG(b)^{-1}F(b')^{-1}XY(1+b)^{-1})^{-1})\begin{aligned}[t]
=&P((1+b)\sigma_b) \\
=&1+b.\end{aligned}\end{gathered}$$ Conjugating with $(1+b)$ yields $$\Psi_{XY}((1+b)^{-1}bG(b)^{-1}F(b')^{-1})
={{\mathcal{E}}}((1-(1+b)^{-1}bG(b)^{-1}F(b')^{-1}XY)^{-1})-1=b.$$ Hence, $$\Psi_{XY}{^{\langle-1\rangle}}(b)=(1+b)^{-1}bG(b)^{-1}F(b')^{-1}$$ and holds.
The formula asserts the equality of the germs of two analytic $B$–valued functions. This is equivalent to asserting the equality of the $n$th terms in their respective power series expansions around zero, for every $n\ge0$. By Lemmas \[lem:FG\] and \[lem:Finv\], the $n$th term, call it RHS$_n$, in the expansion for the right hand side of depends only on the $0$th through the $n$th terms of the power series expansions for $S_x(b)$ and $S_y(b)$. Hence, by Proposition \[prop:nth\], RHS$_n$ depends only on the moment functions $\mu_{x,1},\,\ldots,\,\mu_{x,n}$ and $\mu_{y,1},\,\ldots,\,\mu_{y,n}$. On the other hand, again by Proposition \[prop:nth\], the $n$th term in the power series expansion for the left hand side of , call it LHS$_n$, depends only on $\mu_{xy,1},\,\ldots,\,\mu_{xy,n}$. But by freeness of $x$ and $y$, for each $k\ge1$ the moment function $\mu_{xy,k}$ depends only on $\mu_{x,1},\,\ldots,\,\mu_{x,k}$ and $\mu_{y,1},\,\ldots,\,\mu_{y,k}$. Thus, both LHS$_n$ and RHS$_n$ depend only on $\mu_{x,1},\,\ldots,\,\mu_{x,n}$ and $\mu_{y,1},\,\ldots,\,\mu_{y,n}$.
Hence, in order to prove at the level of the $n$th terms in the power series expansion, it will suffice to prove for some free pair $X$ and $Y$ of elements in a Banach noncommutative probability space over $B$, whose first $n$ moment functions agree with those of $x$ and $y$, respectively. However, by Propositions \[prop:free\] and \[prop:Smom\], such $X$ and $Y$ can be chosen of the forms and . By Lemma \[lem:XY\], the equality holds for these operators.
A proof of the additivity of the R–transform over a Banach algebra {#sec:R-t}
==================================================================
The R–transform over a general unital algebra $B$ has been well understood since Voiculescu’s work [@V95] (and see also Speicher’s approach in [@Sp]). However, for completeness, in this section we offer a new proof, using the techniques and constructions of the previous two sections, of the additivity of the R–transform for free random variables in a Banach noncommutative probability space. This proof is, of course, analogous to Haagerup’s proof of Theorem 2.2 of [@H] in the scalar–valued case.
Let $(A,E)$ be a Banach noncommutative probability space over $B$ and let $a\in A$. Consider the function $${{\mathcal{C}}}_a(b)=E((1-ba)^{-1}b)=\sum_{n=0}^\infty E((ba)^nb),$$ defined and analytic for $\|b\|<\|a\|^{-1}$. We have ${{\mathcal{C}}}_a(b)=b+b\Phi_a(b)b$, where $\Phi_a$ is as in . Since the Fréchet differential of ${{\mathcal{C}}}_a$ at $b=0$ is the identity map, ${{\mathcal{C}}}_a$ is invertible with respect to composition in a neighborhood of zero.
\[prop:R\] There is a unique $B$–valued analytic function $R_a$, defined in a neighborhood of $0$ in $B$, such that $$\label{eq:CR}
{{\mathcal{C}}}_a{^{\langle-1\rangle}}(b)=(1+bR_a(b))^{-1}b=b(1+R_a(b)b)^{-1}.$$
Again, uniqueness is clear by the power series expansions.
The right–most equality in holds for any analytic function $R_a$. We seek a function $R_a$ such that $${{\mathcal{C}}}_a((1+bR_a(b))^{-1}b)=b.$$ But $${{\mathcal{C}}}_a((1+bR_a(b))^{-1}b)=\begin{aligned}[t]
&(1+bR_a(b))^{-1}b \\
&+(1+bR_a(b))^{-1}b\,\Phi_a\big((1+bR_a(b))^{-1}b\big)\,(1+bR_a(b))^{-1}b,\end{aligned}$$ so it will suffice to find $R_a$ so that any of the following hold: $$\begin{gathered}
\notag
(1+bR_a(b))^{-1}+(1+bR_a(b))^{-1}b\,\Phi_a\big((1+bR_a(b))^{-1}b\big)\,(1+bR_a(b))^{-1}=1, \\
\notag
1+b\,\Phi_a\big((1+bR_a(b))^{-1}b\big)\,(1+bR_a(b))^{-1}=1+bR_a(b), \\
\notag
b\,\Phi_a\big((1+bR_a(b))^{-1}b\big)\,(1+bR_a(b))^{-1}=bR_a(b), \\
\Phi_a\big((1+bR_a(b))^{-1}b\big)\,(1+bR_a(b))^{-1}=R_a(b). \label{eq:Ra}\end{gathered}$$ However, $R_a(0)=E(a)$ is a solution of at $b=0$, and the Fréchet differential of the function $x\mapsto \Phi_a((1+bx)^{-1}b)(1+bx)^{-1}-x$ at $b=0$ is the negative of the identity map, hence is invertible. The implicit function theorem of Hildebrandt and Graves [@HG] (see also the discussion on p. 655 of [@G]) guarantees the existence of $R_a$.
The [*R–transform*]{} of $a$ is defined to be the analytic function $R_a$ from Proposition \[prop:R\].
Analogously to Proposition \[prop:nth\], we have the following.
\[prop:Rn+1\] The $n$th term $(R_a)_n(b,\ldots,b)$ in the power series expansion for $R_a$ about zero depends only on the first $n+1$ moment functions $\mu_{a,1},\ldots,\mu_{a,n+1}$ of $a$.
Here is the analogue to Lemma \[lem:Nthmoment\], which can be proved similarly.
Let $N\in{{\mathbf N}}$ and for every $n\in\{0,1,\ldots,N\}$ let $\alpha_n\in{{\mathcal{B}}}_n(B)$. Fix $i\in I$ and let $$\begin{aligned}
X&=L_i+\sum_{n=0}^{N-1}V_{i,n}(\alpha_n) \\[1ex]
Y&=X+V_{i,N}(\alpha_N).\end{aligned}$$ Then for any $b_0,\ldots,b_N\in B$, we have $${{\mathcal{E}}}(b_0Yb_1Y\cdots b_NY)=b_0\alpha_N(b_1,b_2,\ldots,b_N)
+{{\mathcal{E}}}(b_0Xb_1X\cdots b_NX).$$
We immediately get the following analogue of Proposition \[prop:Smom\].
\[prop:Rmom\] Let $(A,E)$ be a $B$–valued Banach noncommutative probability space and let $a\in A$, $N\in{{\mathbf N}}$. Then there are $\alpha_0,\alpha_1,\ldots,\alpha_N$, with $\alpha_n\in{{\mathcal{B}}}_n(B)$, such that if $$X=L_i+\sum_{n=0}^NV_{i,n}(\alpha_n)\in{{\mathcal{B}}}({{\mathcal{F}}}),$$ then $${{\mathcal{E}}}(b_0Xb_1X\cdots b_kX)=E(b_0ab_1a\cdots b_ka)$$ for all $k\in\{1,\ldots,N\}$ and all $b_0,\ldots,b_N\in B$.
Now we have the following analogues of Lemmas \[lem:X\] and \[lem:XY\].
\[lem:R-tX\] Let $N\in{{\mathbf N}}$ and choose $\alpha_n\in{{\mathcal{B}}}_n(B)$ for $n\in\{0,1,\ldots,N\}$, and let $$F(b)=\alpha_0+\sum_{n=1}^N\alpha_n(b,\ldots,b).$$ Let $$X=L_1+\sum_{n=0}^NV_{1,n}(\alpha_n)\in{{\mathcal{B}}}({{\mathcal{F}}}).$$ Then the R–transform of $X$ is $R_X(b)=F(b)$.
With $\omega_b$ defined as in the proof of Lemma \[lem:X\], we have $$\begin{gathered}
X\omega_b=L_1\omega_b+F(b)\omega_b \\
bX\omega_b=(\omega_b-\Omega)+bF(b)\omega_b \\
(1+bF(b)-bX)\omega_b=\Omega \\
(1-bX(1+bF(b))^{-1})^{-1}\Omega=(1+bF(b))\omega_b \\
{{\mathcal{E}}}((1-bX(1+bF(b))^{-1})^{-1})\begin{aligned}[t]&=P((1+bF(b))\omega_b) \\
&=1+bF(b).\end{aligned}\end{gathered}$$ Conjugating yields $${{\mathcal{E}}}((1-(1+bF(b))^{-1}bX)^{-1})=1+bF(b),$$ so $${{\mathcal{C}}}_X((1+bF(b))^{-1}b)={{\mathcal{E}}}((1-(1+bF(b))^{-1}bX)^{-1})(1+bF(b))^{-1}b=b.$$ Thus, $${{\mathcal{C}}}_X{^{\langle-1\rangle}}(b)=(1+bF(b))^{-1}b$$ and $R_a(b)=F(b)$.
\[lem:R-tXY\] Let $\alpha_0,\ldots,\alpha_n$, $F$ and $X$ be as in Lemma \[lem:R-tX\]. Let $\beta_n\in{{\mathcal{B}}}_n(B)$ for $n\in\{0,1,\ldots,N\}$. Let $$G(b)=\beta_0+\sum_{n=1}^N\beta_n(b,\ldots,b)$$ and let $$Y=L_2+\sum_{n=0}^NV_{2,n}(\alpha_n)\in{{\mathcal{B}}}({{\mathcal{F}}}).$$ Then the R–transform of $X+Y$ is $$R_{X+Y}(b)=F(b)+G(b)=R_X(b)+R_Y(b).$$
For $b\in B$ with $\|b\|<1/2$, let $$\sigma_b=(1-b(L_1+L_2))^{-1}\Omega
=\Omega+\sum_{k=1}^\infty(b\delta_1+b\delta_2)^{\otimes k}\otimes 1\in{{\mathcal{F}}}.$$ Then $$\begin{gathered}
(X+Y)\sigma_b=(L_1+L_2)\sigma_b+(F(b)+G(b))\sigma_b \\
b(X+Y)\sigma_b=(\sigma_b-\Omega)+b\,(F(b)+G(b))\sigma_b.\end{gathered}$$ Now arguing as in the proof of Lemma \[lem:R-tX\] above yields $R_{X+Y}(b)=F(b)+G(b)$.
Finally, we get a proof, which is analogous to our proof of Theorem \[thm:main\], of the additivity of the R–transform in a Banach noncommutative probability space.
Let $B$ be a unital complex Banach algebra and let $(A,E)$ be a $B$–valued Banach noncommutative probability space. Let $x,y\in A$ be free in $(A,E)$. Then $$R_{x+y}(b)=R_x(b)+R_y(b).$$
By Proposition \[prop:Rn+1\], it will suffice to show that given $n\in{{\mathbf N}}$ we have $R_{X+Y}=R_X+R_Y$ for some free pair $X$ and $Y$ of elements in a Banach noncommutative probability space over $B$ whose first $n$ moment functions agree with those of $x$ and $y$, respetively. Precisely this fact follows from Proposition \[prop:Rmom\], Proposition \[prop:free\] and Lemmas \[lem:R-tX\] and \[lem:R-tXY\].
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[^1]: Supported in part by NSF grant DMS–0300336 and by the Alexander von Humboldt Foundation.
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---
abstract: 'We establish a new model of coupling between a cosmic dark fluid and electrodynamic systems, based on an analogy with effects of electric and magnetic striction, piezo-electricity and piezo-magnetism, pyro-electricity and pyro-magnetism, which appear in classical electrodynamics of continuous media. Extended master equations for electromagnetic and gravitational fields are derived using Lagrange formalism. A cosmological application of the model is considered, and it is shown that a striction-type interaction between the dark energy (the main constituent of the dark fluid) and electrodynamic system provides the universe history to include the so-called unlighted epochs, during which electromagnetic waves can not propagate and thus can not scan the universe interior.'
author:
- 'Alexander B. Balakin'
- 'Nadejda N. Dolbilova'
title: |
Electrodynamic phenomena induced by a dark fluid:\
Analogs of pyromagnetic, piezoelectric, and striction effects
---
Introduction
============
Dark fluid composed of a dark energy and a dark matter is considered nowadays as a key constitutive element of modern cosmological models (see, e.g., [@DE1; @DE2; @DE3; @DM1; @DM2; @DM3; @DMDE; @DEcosmo; @DE2011; @DEmodified; @DF1; @DF2; @DF3; @DF4; @DF99]). Both the dark energy and dark matter are assumed to consist of electrically neutral particles and thus the dark fluid does not interact with an electromagnetic field [*directly*]{}. That is why, we would like, first of all, to explain the terminological context, which allows us to speak about electrodynamic phenomena induced by the dark fluid. Let us imagine a hierarchical cosmological system, in which the dark fluid (energetically dominating substrate with a modern contribution about $95\%$) is considered to be the guiding element, and an electrodynamic subsystem (as a part of baryonic matter with its modern contribution about $5\%$) to be the subordinate element. In this context the electrodynamic subsystem plays the role of a [*marker*]{}, which signalizes about the variations in the state of dark fluid, the energetic reservoir, into which this marker is immersed. We are interested to answer the question: what mechanisms might be responsible for a (possible) transmission of information about the dark fluid state to the electrodynamic system. Since electrodynamic systems form the basis for the most important channel of information about the universe structure, one could try to reconstruct features of the dark fluid evolution by tracking down specific fine details of the spectrum of observed electromagnetic waves, of their phase and group velocities.
The most known [*marker-effect*]{} of such type is the polarization rotation of electromagnetic waves travelling through the axionic dark matter [@PR1; @PR2; @PR3; @PR5]. Let us remind a few details of this phenomenon. From the physical point of view, the corresponding mechanism is connected with magneto-electric cross-effect [@Dell; @HehlObukhov], which is generated in the medium by the pseudoscalar field associated with dark matter axions. From the mathematical point of view, this mechanism is described by inserting a special term into the Lagrangian, $\frac14 \phi F^*_{mn}F^{mn}$, which is linear in the pseudoscalar (axion) field $\phi$ and is proportional to the pseudo-invariant of the electromagnetic field quadratic in the Maxwell tensor [@ax1]. The model of this axion-photon coupling was extended for the non-stationary state of the dark matter (see, e.g., [@BBT1; @BBT2]), and for the states, for which nonminimal effects linear in the space-time curvature are significant (see, e.g., [@BNi]).
Availability of the example of the coupling of photons with the dark matter axions encourages us to search for marker-effects related to the interaction of electrodynamic system with the dark energy, the main constituent of the dark fluid. We assume that electrodynamic systems can be influenced by the pressure of the dark energy in analogy with mechanical stresses, which are known to control the response in electric and magnetic materials in industry and technique. To be more precise, we can search for dark fluid analogies with the following classical effects. First, we mean the analogy with the classical piezo-electric effect (the appearance of an electric polarization in the medium influenced by mechanical stress and vice-versa), and with the classical piezo-magnetic effect (appearance of a magnetization under stress)(see, e.g., [@Nye; @SSh; @LL; @Mauginbook]). Second, we would like to consider an analogy with the inverse electrostriction effect (a combination of external pressure and electric field generates the electric polarization in the medium), and an analogy with the inverse magnetostriction effect (a combination of external pressure and magnetic field generates the magnetization in the medium), as well as, an analogy with the magneto-electric cross-effect displayed by the external stress. Based on results of classical electrodynamics of continuous media, we can expect that piezo-effects will be visualized, when the dark fluid is anisotropic (e.g., in the early universe). The striction effects due to their symmetries are expected to be available in the isotropic universe also. In addition to piezo- and striction- effects induced by the dark fluid pressure we can expect the appearance of marker-signals similar to pyro-electric and pyro-magnetic responses of the medium, in which the temperature changes with time [@Nye; @SSh; @LL] (pyro-effect also is hidden, when the dark fluid is isotropic). One of the important characteristic of the dark fluid is its macroscopic velocity four-vector and covariant derivative of this four-vector. When we focus on the influence of the dark fluid non-uniform motion on the properties of electrodynamic system, we, in fact, search for analogs of dynamo-optical phenomena [@LL]; we hope to consider these phenomena in detail in the next paper. In principle, we could consider an analogy with the so-called thermo-electric and thermo-magnetic effects, induced by heat-fluxes in the medium, but this sector of physical modeling is out of scope of this paper. Also, in this paper we do not consider magneto-electric cross-effects induced by the combination of the dark energy pressure and of the axionic dark matter. Effects of this type are worthy of special consideration.
We have to emphasize that mathematical theory of pyro-, piezo- and striction- effects is developed in detail for classical electrodynamics of continuous media, and below we consider a general relativistic extension of that theory for the case of dark fluid action on the electrodynamic system. In this sense, we take the mathematical scheme of the description of such interactions, which is well-tested, has clear interpretation and is based on the Lagrange formalism, and then we construct its general relativistic analog, using this scheme in the context of dark fluid electrodynamics.
This paper is organized as follows. In Section II we remind the terminology and introduce the Lagrangian and master equations for the model of electromagnetically inactive dark fluid. Section III contains detailed description of the extended model: in Section III.A we extend the Lagrangian by the terms, which describe interactions of the pyro-, piezo- and striction- types between dark energy and electrodynamic system; in Section III.B we derive extended electrodynamic equations and discuss the structure of tensor coefficients describing pyro- (III.B.1), piezo- (III.B.2) and striction-(III.B.3) coefficients associated with the coupling to dark energy; in Section III.C we obtain the extended gravity field equations in general form. In Section III.D we write the equation for a axion field attributed to the dark matter. In Section IV we reduce the derived master equations to the case, when the medium is spatially isotropic: Section IV.A contains details of reduced electrodynamic equations; in Section IV.B we collect details of modified gravity field equations; in Section IV.C we consider an example of extended model with hidden magnetic and/or electric anisotropy. In Section V we consider a cosmological application of the established model to the problem of description of the so-called unlighted epochs in the universe history, and their relations to the striction-type interactions of electrodynamic systems with the cosmic dark energy. In Section VI we summarize the results. Appendix includes working formulas for the extended variation procedure.
Electromagnetically inactive dark fluid
=======================================
In order to remind the standard elements of the theory and to introduce new details, let us start with the model, in the framework of which the electromagnetic field interacts with the standard matter only, and the dark fluid is coupled with the electrodynamic system by the gravitational field only.
The Lagrangian
--------------
Let us remind that the standard Einstein-Maxwell model is described by the action functional $$S_{0} {=} \int d^4 x \sqrt{{{-}}g} \left[\frac{R}{2\kappa} {+}
L_{({\rm DF})} {+} \frac{1}{4} C_{(0)}^{ikmn} F_{ik}F_{mn} {+}
L_{({\rm m})} \right], \label{0actmin}$$ which is quadratic in the Maxwell tensor $F_{ik}$. Here $g$ is the determinant of the metric tensor $g_{ik}$, the term $R$ is the Ricci scalar, $\kappa {=} \frac{8 \pi G}{c^4}$ is the Einstein constant, $L_{({\rm DF})}$ is the Lagrangian of the dark fluid, $L_{({\rm m})}$ is the Lagrangian of a standard matter. The tensor $C^{ikmn}_{(0)}$ describes standard linear electromagnetic response of the medium formed by the standard matter; in vacuum the corresponding term takes the form $\frac{1}{4} F^{mn}F_{mn}$. As usual, we assume that $L_{({\rm m})}$ does not include the Maxwell tensor $F_{mn}$, nevertheless, it can depend on the potential four-vector $A_i$, if the medium is conductive.
Standard electrodynamic equations
---------------------------------
The Maxwell tensor is represented in terms of a four-vector potential $A_i$ as $$F_{ik} = \nabla_i A_{k} - \nabla_k A_{i} \,, \label{maxtensor}$$ and thus satisfies the condition $$\nabla_{k} F^{*ik} =0 \,, \label{Emaxstar}$$ where $F^{*ik} \equiv \frac{1}{2} \epsilon^{ikpq}F_{pq}$ is the tensor dual to $F_{pq}$, the term $\epsilon^{ikpq} \equiv
\frac{1}{\sqrt{-g}} E^{ikpq}$ is the Levi-Civita tensor, $E^{ikpq}$ is the absolutely skew-symmetric Levi-Civita symbol with $E^{0123}=1$. The variation of the action functional (\[0actmin\]) with respect to the four-vector potential $A_i$ gives the following electrodynamic equations $$\nabla_{k} H^{ik} = - \frac{4\pi}{c} I^i \,, \quad H^{ik} =
C_{(0)}^{ikmn}F_{mn} \,. \label{induc1}$$ Here $H^{ik}$ is the excitation tensor [@HehlObukhov], and the four-vector $I^i$ defined as $$I^i = \frac{1}{4\pi} \frac{\delta L_{({\rm m})}}{\delta A_i} \,,
\quad \nabla_i I^i = 0 \,, \label{E8}$$ describes the electric current. In this paper we consider the medium to be non-conducting, i.e., $I^i {=}0$.
Gravity field equations
-----------------------
Variation of the action functional (\[0actmin\]) with respect to metric gives the equations of the gravitational field, which can be written in the following form $$R_{ik}-\frac{1}{2}Rg_{ik} = \kappa \left[ T^{(0)}_{ik} + T^{({\rm
DE})}_{ik} + T^{({\rm DM})}_{ik} + T^{({\rm m})}_{ik}\right] \,. \label{EineqMIN}$$ Here $T^{(0)}_{ik}$ is the effective symmetric traceless stress-energy tensor of electromagnetic field in a continuous medium $$T^{(0)}_{ik} \equiv \left[\frac{1}{4}g_{ik} F_{mn} {-} \frac12
\left(g_{im}F_{kn}{+}g_{km}F_{in} \right)\right]
C^{mnpq}_{(0)}F_{pq}\,, \label{TEM}$$ (see, e.g., [@2007A; @2007B] for details). The Lagrangian of the electromagnetically inactive dark fluid is presented as a sum $L_{({\rm DF})} \to L_{({\rm DE})} {+}L_{({\rm DM})}$, and the corresponding stress-energy tensors enter the right-hand side of equations (\[EineqMIN\]) also as the sum. The stress-energy tensor of the dark energy is defined as $$T^{({\rm DE})}_{ik} = - \frac{2}{\sqrt{-g}} \frac{\delta
\left[\sqrt{-g} \ L_{({\rm DE})}\right]}{\delta g^{ik}} \,.
\label{TAX}$$ Let us define the unit four-vector $U^i$ of the macroscopic velocity as an eigen-vector of the stress-energy tensor of the dark energy, i.e., let us assume that $T^{({\rm DE})}_{ls} U^s {=}
W U_l$, $U^iU_i{=}1$ (the so-called Landau-Lifshitz definition). Then this tensor can be algebraically represented in the following form $$T^{({\rm DE})}_{ik} \equiv W U_i U_k + {\cal P}_{ik}\,,
\label{fluid}$$ where the eigen-value $W$ is interpreted as the energy density of the dark energy, and ${\cal P}_{ik}$ is its pressure tensor. These quantities can be written as follows: $$W = U^l T^{({\rm DE})}_{ls} U^s \,, \quad {\cal
P}_{ik} = \Delta^l_i T^{({\rm DE})}_{ls} \Delta_k^s \,,
\label{1fluid}$$ where $\Delta^l_i \equiv \delta^l_i {-}U^lU_i$ is the projector. The tensors $T^{({\rm DM})}_{ik}$ and $T^{({\rm m})}_{ik}$, which describe the contributions of the dark matter and standard matter, respectively, can be obtained by the formulas similar to (\[TAX\]), however, their algebraic decompositions are more sophisticated, since the macroscopic velocity four-vector $U^i$ is already fixed as an eigen-vector of the dark energy stress-energy tensor. For instance, $T^{({\rm m})}_{ik}$ is of the form $$T^{({\rm m})}_{ik} \equiv W^{({\rm m})} U_i U_k + U_i I_k^{({\rm
m})} + U_k I_i^{({\rm m})} + P^{({\rm m})}_{ik}\,, \label{Tmatter}$$ and includes the heat-flux four-vector $I_k^{({\rm m})} \equiv
\Delta^l_k T^{({\rm m})}_{ls} U^s$ of the matter in addition to the matter energy-density $W^{({\rm m})}$ and the matter pressure tensor $P^{({\rm m})}_{ik}$. When we describe the dark matter, one should change the symbol $({\rm m})$ by $({\rm DM})$ in the formula (\[Tmatter\]).
Electromagnetically active dark fluid
=====================================
Extended Lagrangian
-------------------
Let us extend the action functional to include the terms describing the interaction between the dark fluid and electrodynamic system. We assume the extension to have the following form $$S {=} \int d^4 x \sqrt{{-}g} \left\{\frac{R}{2\kappa} {+} L_{({\rm
DE})} {+} \frac{1}{4} C_{(0)}^{ikmn} F_{ik}F_{mn} {+} L_{({\rm m})}
{+} \right.$$ $$\left.
{+} \frac12 \Psi^2_0 \left[- \nabla_k \phi \nabla^k \phi + {\cal V}(\phi^2) \right] {+} \frac14 \phi F^*_{mn} F^{mn} {+}
\right.$$ $$\left.
{+} \frac12 \left( \pi^{ik} {+} \frac12 \lambda^{ikmn} F_{mn} \right)F_{ik} DW {+}
\right.$$ $$\left.
{+} \frac12 \left({\cal D}^{ikpq} {+} \frac{1}{2} \
Q^{ikmnpq} F_{mn} \right) F_{ik} \ {\cal P}_{pq} \right\}\,, \label{actmin}$$ which again is up to second order in the Maxwell tensor $F_{ik}$, but in addition to the second order terms also the terms linear in $F_{mn}$ appeared. Here and below we use the symbol $D$ for the convective derivative $D \equiv U^i \nabla_i$. We specified the Lagrangian of the dark matter $L_{({\rm DM})}$ as the one for a pseudoscalar (axion) field $\phi$; in this context the quantity $\frac{1}{\Psi_0}$ is a coupling constant of the axion-photon interaction, and ${\cal V}(\phi^2)$ is the potential of the pseudoscalar field. In this terms, the cross-invariant $\frac14
\phi F^*_{mn} F^{mn}$ describes the coupling between the electromagnetic and pseudoscalar fields, i.e., the interaction between axionic dark matter and electromagnetic field [@ax1]. In other words, we deal here with the example of electromagnetically active dark fluid, and this type of activity is connected with the dark matter part of the dark fluid.
Now we consider the dark fluid activity related to the interaction of the electromagnetic field with dark energy constituent of the dark fluid. Based on the analogy with electrodynamics of continuous media we can consider the first term linear in the Maxwell tensor, $\frac12 \pi^{ik}F_{ik} DW$, as describing the analog of the pyro effects (pyro-electric and/or pyro-magnetic). Of course, in classical electrodynamics of continuous media one deals with the convective derivative of the temperature $DT$, when one speaks about pyro - effects, nevertheless, assuming that $DW{=}\frac{dW}{dT} DT$ we keep this terminology for the dark fluid also and indicate the tensor $\pi^{ik}$ as the tensor of pyro-coefficients. The second term linear in the Maxwell tensor, $\frac12{\cal D}^{ikpq}F_{ik} {\cal P}_{pq}$, includes the pressure tensor of the dark energy ${\cal P}_{pq}$ and thus describes analogs of piezo-effects (piezo-electric and/or piezo-magnetic). The corresponding piezo-coefficients are encoded in the tensor ${\cal D}^{ikpq}$. The term quadratic in $F_{mn}$ and linear in $DW$ describes the part of the linear electromagnetic response, which depends on the rate of evolution of the energy density of the dark fluid. The last term in (\[actmin\]) is quadratic in the Maxwell tensor and linear in the pressure tensor, thus describing the response associated with electro- and magneto-striction, induced by the dark energy; the tensor $Q^{ikmnpq}$ introduces coefficients of electro- and magneto-striction.
Extended electrodynamic equations
---------------------------------
The variation of the action functional (\[actmin\]) with respect to the four-vector potential $A_i$ gives the following electrodynamic equations $$\nabla_{k} H^{ik} = - \frac{4 \pi}{c} I^i \,, \quad H^{ik} = {\cal
H}^{ik} {+} \phi F^{*ik} {+} {\cal C}^{ikmn}F_{mn} \,. \label{1induc1}$$ Here $H^{ik}$ is the extended excitation tensor. The term $${\cal H}^{ik} \equiv \pi^{ik} DW + {\cal D}^{ikpq}{\cal P}_{pq}
\label{2induc11}$$ does not contain $F_{ik}$ and thus it can be indicated as the spontaneous polarization-magnetization tensor. The term $\phi F^{*ik}$ is typical for the axion electrodynamics (see, e.g., [@ax1]); the four-divergence of this term can be expressed as $F^{*ik}\nabla_k \phi$ due to (\[Emaxstar\]). Finally, the term $${\cal C}^{ikmn} \equiv
C^{ikmn}_{(0)} + \lambda^{ikmn} DW + Q^{ikmnpq} {\cal P}_{pq}
\label{C1}$$ describes the total linear response of the electrodynamic system including pyro-type and striction-type effects induced by the dark energy.
### Pyro- coefficients\
associated with the coupling to dark energy
The skew-symmetric tensor $\pi^{ik}$ describing the pyro- effects can be represented as $$\pi^{ik}= \pi^i U^k - \pi^k U^i - \epsilon^{ik}_{\ \ mn} \mu^{m}
U^n \,,
\label{pi03}$$ thus visualizing the pyro-electric $\pi^i$ and pyro-magnetic $\mu^m$ coefficients, which are orthogonal to the velocity four-vector ($\pi^iU_i {=} 0 {=} \mu^m U_m$). In general case the dark energy can be characterized by three pyro-electric and three pyro-magnetic coefficients. When we consider the dark energy as a spatially isotropic medium, all six pyro coefficients vanish. For the dark energy with an axial symmetry (e.g., in rotationally symmetric Bianchi-I model) there are two non-vanishing piezo-constants: one piezo-electric and one piezo-magnetic (see, e.g., [@Nye; @SSh] for details).
### Piezo-coefficients\
attributed to the coupling to dark energy
The tensor ${\cal D}^{ikpq}$ possesses the following symmetry of indices $${\cal D}^{ikpq} = - {\cal D}^{kipq} = {\cal D}^{ikqp} \,.
\label{d1}$$ Since the symmetric pressure tensor ${\cal P}_{pq}$ is considered to be orthogonal to the velocity four-vector $U^i$, one can conclude that $${\cal D}^{ikpq} U_p = 0 = {\cal D}^{ikpq} U_q \,.
\label{d2}$$ This means that there are $6 \times 6 = 36$ independent coupling constants in the tensor of piezo-coefficients. This tensor can be decomposed with respect to irreducible parts as follows: $${\cal D}^{ikpq} = d^{i(pq)} U^k - d^{k(pq)} U^i - \epsilon^{ik}_{\ \ ls}U^s h^{l(pq)} \,.
\label{d3}$$ Here the piezo-electric coefficients $d^{i(pq)}$ and piezo-magnetic coefficients $h^{l(pq)}$ are defined by $$d^{i(pq)} \equiv {\cal D}^{ikpq} U_k \,, \quad h^{l(pq)} \equiv \frac12 \epsilon^{ls}_{\ \ ik} {\cal D}^{ikpq} U_s \,,
\label{d4}$$ they are symmetric with respect to the indices $p,q$, and are pure space-like, i.e., they satisfy the equalities $$d^{i(pq)} U_i {=} 0 {=} d^{i(pq)} U_p \,, \quad h^{l(pq)} U_l {=} 0 {=} h^{l(pq)} U_p \,.
\label{d5}$$ In other words, in general case, the dark energy influence can be characterized by 18 piezo-electric coefficients $d^{i(pq)}$ and/or by 18 piezo-magnetic coefficients $h^{l(pq)}$. When the dark energy is spatially isotropic, all these coefficients are equal to zero. For the dark energy with axial symmetry there are four non-vanishing piezo-electric and four non-vanishing piezo-magnetic coefficients (see, e.g., [@Nye; @SSh] for details).
### Permittivity tensors\
associated with the coupling to dark energy
Using the medium velocity four-vector $U^i$ one can decompose $ {\cal C}^{ikmn}$ uniquely as $${\cal C}^{ikmn} = \left( \varepsilon^{i[m}U^{n]} U^k {-}
\varepsilon^{k[m} U^{n]} U^i \right) {-}$$ $${-}\frac12 \eta^{ikl}(\mu^{-1})_{ls} \eta^{mns} {+}
\eta^{ikl}U^{[m} \nu_{l}^{\ n]} {+} \eta^{lmn}U^{[i}
\nu_{l}^{\ k]} \,. \label{444}$$ Here $\varepsilon^{im}$ is the tensor of total dielectric permeability, $(\mu^{-1})_{pq}$ is the tensor of total magnetic impermeability, and $\nu_{p \ \cdot}^{\ m}$ is the total tensor of magneto-electric cross-effect induced by dark energy. Keeping in mind the decomposition (\[C1\]) one can divide these quantities into three parts $$\varepsilon^{im} = 2 {\cal C}^{ikmn} U_k U_n =
\varepsilon_{(0)}^{im} {+} \sigma^{im} DW {+} \alpha^{im(pq)}
{\cal P}_{pq} \,,$$ $$(\mu^{-1})^{ab} = - \frac{1}{2} \eta^a_{\ ik} {\cal C}^{ikmn}
\eta_{mn}^{\ \ \ b} =$$ $$= (\mu^{-1})_{(0)}^{ab} + \rho^{ab} DW + \beta^{ab(pq)} {\cal
P}_{pq}\,,$$ $$\nu^{am} {=} \eta^a_{\ ik} {\cal C}^{ikmn} U_n {=} \nu^{am}_{(0)}{+} \omega^{am}DW {+} \gamma^{am(pq)} {\cal P}_{pq}\,. \label{varco}$$ To complete the description of the tensor of cross-effects, we can write the sum of $\nu^{am}$ related to the dark energy contribution (\[varco\]) and $\nu^{am}_{({\rm DM})} {=} \phi \Delta^{am}$ related to the contribution of the axionic dark matter. In the formulas written above we introduced the corresponding spatial tensors as follows: $$\varepsilon_{(0)}^{im} = 2 C_{(0)}^{ikmn} U_k U_n \,, \quad \sigma^{im} = 2\lambda^{ikmn} U_k U_n \,,$$ $$\alpha^{im(pq)} = 2Q^{ikmnpq} U_k U_n \,,$$ $$(\mu^{-1})^{ls}_{(0)} = - \frac12 \eta^l_{\ ik} C_{(0)}^{ikmn}\eta^s_{\ mn}\,,$$ $$\rho^{ls} = - \frac12 \eta^l_{\ ik} \lambda^{ikmn}\eta^s_{\ mn} \,,$$ $$\beta^{ls(pq)} = {-} \frac12 \eta^l_{\ ik} Q^{ikmnpq}\eta^s_{\ mn} \,, \quad \nu^{am}_{(0)} {=} \eta^a_{\ ik} C_{(0)}^{ikmn}U_n \,,$$ $$\omega^{am} {=} \eta^a_{\ ik} \lambda^{ikmn}U_n \,, \quad \gamma^{lm(pq)} {=} \eta^l_{\ ik} Q^{ikmnpq}U_n \,. \label{0nu}$$ As usual, the tensors $\eta_{mnl}$ and $\eta^{ikl}$ are the skew-symmetric and orthogonal to $U^i$; they are defined as $$\eta_{mnl} \equiv \epsilon_{mnls} U^s \,, \quad \eta^{ikl} \equiv
\epsilon^{ikls} U_s \,. \label{47}$$ These tensors are connected by the useful identity $$- \eta^{ikp} \eta_{mnp} = \delta^{ikl}_{mns} U_l U^s = \Delta^i_m
\Delta^k_n - \Delta^i_n \Delta^k_m \,. \label{usefulidentity}$$ Upon contraction, equation (\[usefulidentity\]) yields another useful identity $$\frac{1}{2} \eta^{ikl} \eta_{klm} = - \delta^{il}_{ms} U_l U^s =
- \Delta^i_m \,. \label{50}$$ The quantities $\delta^{ikl}_{mns}$ and $\delta^{il}_{ms}$ are the generalized Kronecker deltas. Clearly, the two-indices tensors $\varepsilon^{im}_{(0)}$, $(\mu^{-1})^{(0)}_{ab}$, $\sigma^{im}$, $\rho^{ab}$ are symmetric and orthogonal to $U_k$; each of them possesses six independent components. The spatial (pseudo)tensors $\nu^{am}_{(0)}$ and $\omega^{am}$ are non-symmetric and thus each of them contains nine independent components.
The spatial tensor $\alpha^{im(pq)}$ possesses the symmetry $$\alpha^{im(pq)} = \alpha^{mi(pq)} = \alpha^{im(qp)} \,,
\label{alpha}$$ and, generally, it has $6 \times 6=36$ independent components. When the medium is spatially isotropic there are only two scalars representing this tensor. The symmetry of the tensor $\beta^{ls(pq)}$ is similar: $$\beta^{ls(pq)} = \beta^{sl(pq)} = \beta^{ls(qp)} \,, \label{beta}$$ and it also has 36 independent components in general case, and only two parameters in the spatially isotropic case. Finally, the tensor $\gamma^{lm(pq)}$ with the symmetry $$\gamma^{lm(pq)} = \gamma^{lm(qp)} \,, \label{nu}$$ is characterized by $9 \times 6 {=} 54$ independent components in general case, and vanishes in the spatially isotropic medium. Using (\[0nu\]), we see that the tensor $Q^{abmnpq}$ is generally characterized by 126 independent components.
Gravity field equations
-----------------------
The extended gravity field equations $$\frac{1}{\kappa} \left[R_{ik}{-}\frac{1}{2}Rg_{ik}\right] =$$ $$= T^{(0)}_{ik} {+}
T^{({\rm DE})}_{ik} {+} T^{({\rm DM})}_{ik} {+} T^{({\rm m})}_{ik}{+}T^{({\rm W})}_{ik}
{+}T^{({\rm P})}_{ik} {+} T^{({\rm S})}_{ik}
\label{EineqMIN2}$$ contain the stress-energy tensor of the electromagnetic field in the material medium $T^{(0)}_{ik}$ defined by (\[TEM\]), the stress-energy tensor of the dark energy $T^{({\rm DE})}_{ik}$ presented by (\[fluid\]), the stress-energy tensor of the standard matter $T^{({\rm m})}_{ik}$ decomposed as (\[Tmatter\]), the stress-energy tensor of the pseudoscalar (axion) field $T^{({\rm DM})}_{ik}$ given by $$T^{({\rm DM})}_{ik} = \nabla_i \phi \nabla_k \phi - \frac12
g_{ik} \nabla^n \phi \nabla_n \phi + \frac12 g_{ik}
{\cal V}(\phi^2) \,,
\label{axTE}$$ and three new interaction terms. The term $T^{({\rm W})}_{ik}$ connected with the pyro-type interactions is of the form $$T^{({\rm W})}_{ik} = DW \left[\frac12 g_{ik} F_{mn}\left(\pi^{mn}{+} \frac12 F_{ls} \lambda^{mnls}\right)
{-} \right.$$ $$\left.
{-}F_{mn}\left(\frac{\delta}{\delta g^{ik}} \pi^{mn} {+} \frac12 F_{ls} \frac{\delta}{\delta g^{ik}} \lambda^{mnls} \right)\right] +$$ $$+ \left[W\left(\frac12 g_{ik}+U_iU_k \right) - 2 {\cal B}_{ikls} U^lU^s \right] \times$$ $$\times \nabla_j \left[U^j F_{mn}\left(\pi^{mn}{+}
\frac12 F_{ls} \lambda^{mnls}\right)\right] -$$ $$- \frac12 F_{mn}\left(\pi^{mn}{+} \frac12 F_{ls}
\lambda^{mnls}\right)U_{(i} \nabla_{k)} W \,. \label{pi195}$$ The term $T^{({\rm P})}_{ik}$ relates to the piezo-type contribution to the total stress-energy tensor of the system; it has the form $$T^{({\rm P})}_{ik} = \frac12 g_{ik} {\cal D}^{mnpq}F_{mn} {\cal P}_{pq}
{+} 2 {\cal D}^{mnpq}F_{mn} {\cal B}_{ikls}\Delta^l_p \Delta^s_q {-}$$ $${-}
F_{mn} {\cal P}_{ls} \frac{\delta}{\delta g^{ik}}\left({\cal D}^{mnpq} \Delta^l_p \Delta^s_q \right)
\,. \label{S99}$$ The last new term describes the contribution of the striction - type interactions; it can be written as $$T^{({\rm S})}_{ik} = \frac14 g_{ik} Q^{abmnpq}F_{ab}F_{mn} {\cal P}_{pq} -$$ $$-\frac12 F_{ab}F_{mn} {\cal P}_{ls} \frac{\delta}{\delta g^{ik}}\left(Q^{abmnpq} \Delta^l_p \Delta^s_q \right)
+$$ $$+ Q^{abmnpq}F_{ab} F_{mn} {\cal B}_{ikls}\Delta^l_p \Delta^s_q
\,. \label{S78}$$ The tensor ${\cal B}_{ikls}$ in the formulas (\[pi195\]), (\[S99\]) and (\[S78\]) is defined as follows $${\cal B}_{ikls} \equiv \frac{1}{\sqrt{-g}} \frac{\delta^2 }{
\delta g^{ik} \delta g^{ls}} \left[\sqrt{-g} \ L_{({\rm
DE})}\right]
\,. \label{S2}$$ This four-indices tensor has the following symmetry: $${\cal B}_{ikls}={\cal B}_{kils}={\cal B}_{iksl}={\cal B}_{lsik}\,.
\label{sym1}$$ It can be decomposed phenomenologically using the similar algebraic procedure as for the decomposition of the stress-energy tensor of the dark energy (\[TAX\]): $${\cal B}_{ikls}= {\cal G} U_i U_k U_l U_s + \left({\cal G}^{(1)}_{ls} U_i U_k + {\cal G}^{(1)}_{ik} U_l U_s
\right) +$$ $$+ \left({\cal G}_i U_k
U_l U_s +{\cal G}_k U_i U_l U_s + {\cal G}_l U_i U_k U_s +{\cal
G}_s U_i U_k U_l \right)+$$ $$+ \left({\cal G}^{(2)}_{ks} U_i U_l + {\cal G}^{(2)}_{kl}
U_i U_s + {\cal G}^{(2)}_{is} U_k U_l + {\cal G}^{(2)}_{il} U_k
U_s \right) +$$ $$+\left({\cal G}_{ikl} U_s + {\cal G}_{iks} U_l + {\cal G}_{kls}
U_i + {\cal G}_{ils} U_k \right) + {\cal G}_{ikls} \,,
\label{sym2}$$ where $${\cal G} \equiv U^aU^b {\cal B}_{abcd} U^cU^d \,, \quad {\cal
G}_s \equiv U^aU^b {\cal B}_{abcd} U^c \Delta^d_s \,,$$ $${\cal G}^{(1)}_{ls} \equiv U^aU^b {\cal B}_{abcd} \Delta^c_l \Delta^d_s
\,,$$ $${\cal G}^{(2)}_{ks} \equiv U^aU^c {\cal B}_{abcd} \Delta^b_k
\Delta^d_s \,, \quad {\cal G}_{kls} \equiv U^a {\cal B}_{abcd}
\Delta^b_k \Delta^c_l \Delta^d_s \,,$$ $${\cal G}_{ikls} \equiv
{\cal B}_{abcd} \Delta^a_i \Delta^b_k \Delta^c_l \Delta^d_s
\,.
\label{sym3}$$ The projected four-indices tensor appeared in (\[S99\]), (\[S78\]) $${\cal B}_{ikls} \Delta_p^l \Delta_q^s = {\cal G}^{(1)}_{pq} U_i
U_k + {\cal G}_{kpq} U_i + {\cal G}_{ipq} U_k + {\cal G}_{ikpq}
\label{S221}$$ contains only four terms. The two-indices tensor contributed to (\[pi195\]) $${\cal B}_{ikls} U^l U^s = {\cal G} U_i U_k + {\cal G}_i U_k
+{\cal G}_k U_i + {\cal G}^{(1)}_{ik} \,,
\label{S229}$$ also includes only four terms. We will calculate directly the tensors ${\cal G}$, ${\cal G}_k$, ${\cal G}^{(1)}_{ls}$, ${G}^{(2)}_{ks}$, ${\cal G}_{ikl}$ and ${\cal G}_{ikls}$ below for the model with spatial isotropy.
Equations for the pseudoscalar (axion) field
--------------------------------------------
Since we represented the dark matter constituent of the dark fluid by a pseudoscalar (axion) field, we can easily derive the evolutionary equation for the dark matter by variation of the action functional (\[actmin\]) with respect to the pseudoscalar $\phi$; this procedure yields $$\left[\nabla^k \nabla_k {+} {\cal V}^{\prime}(\phi^2)\right] \phi
= - \frac{1}{4 \Psi^2_0} F^{*}_{mn}F^{mn} \,.
\label{axion33}$$ In [@BBT1; @BBT2; @BNi] we studied more sophisticated equations describing the interaction between electromagnetic field and axionic dark matter; nevertheless, here we restrict ourselves by this simplest model of the axion-photon coupling.
Short summary
-------------
We derived the set of coupled master equations for the extended model: first, electrodynamic equations (\[1induc1\])-(\[C1\]); second, gravity field equations (\[EineqMIN2\])-(\[S229\]); third, equations for the axion field (\[axion33\]). Of course, the derivation of these equations is only the first step in our program. In the next paper we plan to apply these equations for the description of anisotropic models of early universe (in particular, to the Bianchi-I model with global magnetic field) and to consider an anisotropic dark energy (in analogy with, e.g., [@DEanis]). For these applications pyro- and piezo- effects seem to be important. Below we consider only one application of the established model, namely, the application to the isotropic homogeneous model of the Friedmann type. We hope it will be a good illustration that the established model is worthy of attention.
Master equations for a spatially isotropic medium
=================================================
During the late-time universe evolution the dark fluid is considered as a spatially isotropic substratum. The dark matter is modelled as a cold substratum with vanishing pressure. The pressure tensor of such dark fluid is proportional to the projector, ${\cal P}_{ik} = {-} P \Delta_{ik}$, and the scalar quantity $P$ describes the pressure of the dark energy. Concerning the symmetry of tensor coefficients in the context of isotropic and homogeneous cosmological model, we have to assume that all pyro- and piezo- coefficients are vanishing. In addition, we have to assume that all the non-vanishing tensor coefficients can be constructed using three basic elements: first, pure geometrical quantities (metric $g_{ik}$, Levi-Civita tensor $\epsilon^{ikmn}$, Kronecker deltas $\delta^i_k$, $\delta^{ik}_{mn}$, etc.); second, the dynamic quantity $U^i$ (macroscopic velocity of the dark energy), the projector $\Delta_{ik}$; third, phenomenologically introduced coupling constants (in front of corresponding terms). We can calculate directly the variation of all such quantities with respect to the metric, thus completing the model reconstruction. Let us consider in detail the model with spatial isotropy.
Reduction of the electrodynamic equations
-----------------------------------------
In the spatially isotropic medium we have to use the tensor of linear response in the following form: $$C_{(0)}^{ikmn} {=} \frac{1}{2\mu_{(0)}}\left[g^{ikmn}
{+}(\varepsilon_{(0)}
\mu_{(0)}{-}1)\left(g^{ikmn}{-}\Delta^{ikmn}\right) \right].
\label{C0}$$ Here the scalars $\varepsilon_{(0)}$ and $\mu_{(0)}$ are dielectric and magnetic permittivities of the medium, respectively, in the case when the dark fluid influence on this medium is negligible. Similarly, the tensor $\lambda^{ikmn}$ is of the form $$\lambda^{ikmn} {=} \frac{1}{2}\left[\lambda_2 g^{ikmn}
+(\lambda_1 {-} \lambda_2)\left(g^{ikmn}{-}\Delta^{ikmn}\right) \right],
\label{C011}$$ where two phenomenological constants $\lambda_1$ and $\lambda_2$ introduce contributions of the pyro-type interactions (proportional to $DW$) into the total linear response tensor. Because of spatial isotropy the tensor of pyro-coefficients $\pi^{ik}$ and the tensor of piezo-coefficients ${\cal D}^{ikmn}$ vanish.
The last non-trivial element of the extended theory is the tensor of striction-type activity $Q^{ikmnpq}$; in order to construct it we will write, first of all, the space-like tensors $\alpha^{im(pq)}$ and $\beta^{im(pq)}$ using their symmetry: $$\alpha^{im(pq)} {=} \alpha_{(1)} \Delta^{im} \Delta^{pq} {+}
\alpha_{(2)} (\Delta^{ip}\Delta^{mq}{+} \Delta^{iq}\Delta^{mp}) \,,
\label{abg1}$$ $$\beta^{im(pq)} = \beta_{(1)} \Delta^{im} \Delta^{pq} + \beta_{(2)}
(\Delta^{ip}\Delta^{mq}+ \Delta^{iq}\Delta^{mp}) \,. \label{abg2}$$ As for the (pseudo)tensor $\gamma^{im(pq)}$, keeping in mind the analogy with classical electrodynamics of spatially isotropic continuous media [@Nye; @SSh], we assume that this (pseudo)tensor of cross-effects is equal to zero, i.e., $\gamma^{im(pq)} {=} 0$. In other words, when the medium is spatially isotropic, we deal with four coupling parameters $\alpha_{(1)}$, $\alpha_{(2)}$, $\beta_{(1)}$, $\beta_{(2)}$, which characterize electro- and magneto-striction. The corresponding reconstruction of the tensor $Q^{ikmnpq}$ yields $$Q^{ikmnpq}= \frac12 \left[\alpha_{(1)}\Delta^{pq}
\left(g^{ikmn}{-}\Delta^{ikmn} \right) {+} \right.$$ $$\left.{+}
\alpha_{(2)} U_l U_s
\left(g^{iklp} g^{mnsq}{+} g^{iklq}g^{mnsp}\right) + \right.$$ $$\left. + \beta_{(1)}\Delta^{pq}\Delta^{ikmn} -
\beta_{(2)}(\eta^{ikp}\eta^{mnq}{+}\eta^{ikq}\eta^{mnp})
\right]\,. \label{abg4}$$ Clearly, only the four-indices tensor $Q^{ikmnpq} \Delta_{pq}$ enters the electrodynamic equations, when the pressure tensor of the dark energy is spatially isotropic; it has now the form $$Q^{ikmn} \equiv Q^{ikmnpq}\Delta_{pq} = \frac12 \alpha g^{ikmn} + \frac12 (\beta-\alpha)\Delta^{ikmn}
\,, \label{abg49}$$ i.e., only two effective coupling constants $$\alpha = 3 \alpha_{(1)} + 2
\alpha_{(2)} \,, \quad \beta = 3
\beta_{(1)} + 2 \beta_{(2)} \,, \label{asm22}$$ appeared in this tensor instead of four parameters $\alpha_{(1)}$, $\alpha_{(2)}$, $\beta_{(1)}$ and $\beta_{(2)}$. Total permittivity tensors of the spatially isotropic medium influenced by the dark energy are now the following: $$\varepsilon^{im} = \Delta^{im} \varepsilon \,, \quad \varepsilon =
\varepsilon_{(0)} + \lambda_1 DW - \alpha P\,, \label{e22}$$ $$(\mu^{-1})_{ab} = \frac{1}{\mu} \ \Delta_{ab} \,, \quad
\frac{1}{\mu} = \frac{1}{\mu_{(0)}} + \lambda_2 DW - \beta P \,, \label{m22}$$ $$\nu^{am} = 0 \,. \label{n22}$$ This means that in the spatially isotropic case one can define the scalar refraction index of the medium, $n$, accounting for the influence of the dark energy, yielding $$n^2 \equiv \varepsilon \mu = \frac{n^2_{(0)} + \mu_{(0)}(\lambda_1 DW - \alpha P)}{1 + \mu_{(0)}(\lambda_2 DW - \beta P) } \,, \label{2varco}$$ where $n^2_{(0)} \equiv \varepsilon_{(0)} \mu_{(0)}$ is the square of the refraction index of the medium which does not feel the dark energy influence. The corresponding phase velocity of the electromagnetic waves in the striction-active medium is $$V_{({\rm ph})} \equiv \frac{c}{n}= c \sqrt{\frac{1 + \mu_{(0)}(\lambda_1 DW - \beta P)}{n^2_{(0)} + \mu_{(0)}(\lambda_2 DW - \alpha P)}} \,. \label{v1}$$ The group velocity of the electromagnetic waves is defined as $$V_{({\rm gr})} \equiv c \ \frac{2n}{({n}^2+1)} \,, \label{v2}$$ (see, e.g., [@BBL] for details), and can be easily displayed using (\[2varco\]). Let us mention that the influence of the dark energy provides the spatially isotropic electrodynamic system to possess non-stationary properties, when the universe expands. To be more precise, the refraction index, the phase and group velocities become functions of the cosmological time: $n(t)$, $V_{({\rm ph})}(t)$, $V_{({\rm gr})}(t)$, due to the coupling to the non-stationary dark energy with time dependent energy density $W(t)$ and pressure $P(t)$.
Reduction of the gravity field equations
----------------------------------------
In order to reduce the formulas (\[pi195\])-(\[S2\]) for the spatially isotropic case we have to make the following preliminary steps. First, we put the tensors $\pi^{ik}$, ${\cal D}^{ikpq}$ equal to zero, since now the spontaneous polarization-magnetization is inadmissible because of the model symmetry. The second step is to calculate directly the variation derivatives $ \frac{\delta}{\delta g^{ik}} \lambda^{mnls}$, $\frac{\delta}{\delta g^{ik}}\left(Q^{abmnpq} \Delta^l_p
\Delta^s_q \right)$ using the reduced formulas (\[C011\]), (\[abg4\]) and the auxiliary formulas, which we presented in Appendix. The third step is to derive the formulas for the variation derivatives $ \frac{\delta}{\delta g^{ik}} W$, $\frac{\delta}{\delta g^{ik}} DW$ and $\frac{\delta}{\delta
g^{ik}}P$. Let us consider in detail the third step.
Our ansatz is that the [*spatially isotropic*]{} dark energy can be modelled by a real scalar field $\Psi$ with the Lagrangian $$L_{({\rm DE})} = - \frac12 g^{mn} \partial_m \Psi \partial_n \Psi
+ \frac12 V(\Psi^2)\,, \label{DE1}$$ and the corresponding stress-energy tensor $T^{({\rm DE})}_{ik}$ $$T^{({\rm DE})}_{ik} {=} \partial_i \Psi \partial_k \Psi {-} \frac12
g_{ik} g^{mn} \partial_m \Psi \partial_n \Psi {+} \frac12 g_{ik}
V(\Psi^2)\,. \label{DE2}$$ This idea correlates with the attempt to describe the dark matter in terms of pseudoscalar field $\phi$ (see (\[actmin\]) and (\[axTE\])). The velocity four-vector $U^i$ is defined as an eigen-vector of the tensor $T^{({\rm DE})}_{ik}$, and we obtain readily $$T^{({\rm DE})}_{ik} U^k = W
U_i=$$ $$=\partial_i \Psi D \Psi - \frac12 U_i
g^{mn} \partial_m \Psi \partial_n \Psi + \frac12 U_i V(\Psi^2) \,, \label{DE3}$$ $$W \equiv U^iT^{({\rm DF})}_{ik}U^k {=} (D\Psi)^2 {-} \frac12
g^{mn}
\partial_m \Psi \partial_n \Psi {+} \frac12 V(\Psi^2)\,.
\label{DE4}$$ Thus, $\partial_n \Psi {=} U_n D\Psi$ and we obtain the well-known relationships (see, e.g., [@Odin]): $$W = \frac12 \left[(D\Psi)^2 + V(\Psi^2) \right] \,,$$ $$P \equiv - \frac13 \Delta^{im} T^{({\rm DF})}_{ik}\Delta^{k}_m =
\frac12 \left[(D\Psi)^2 - V(\Psi^2) \right]\,, \label{DE5}$$ which give (see auxiliary formulas in Appendix) $$\frac{\delta}{\delta g^{ik}} W = \frac{\delta}{\delta g^{ik}} P =
\frac12 \partial_i \Psi
\partial_k \Psi =$$ $$= \frac12 U_i U_k (D\Psi)^2 = \frac12 U_i U_k (W+P) \,.
\label{DE6}$$ Similarly, direct calculation of the tensor ${\cal B}_{ikls}$ yields $${\cal B}_{ikls} = \frac14 \left(2 W {-} P \right)U_{i} U_{k} U_{l}
U_{s} +$$ $$+ \frac14 W \left(U_{i} U_{k} \Delta_{ls}+ U_{l} U_{s}
\Delta_{ik}\right) - P U_{(l} \Delta_{s)(k}U_{i)} -$$ $$- \frac14 P \left(
\Delta_{ls}\Delta_{ik} {+} \Delta_{li}\Delta_{ks} {+}
\Delta_{lk}\Delta_{is}\right) \,. \label{DE7}$$ Thus, for this illustrative example we obtain $${\cal G} = \frac14 \left(2 W {-} P \right) \,, \quad {\cal
G}_s = 0 \,,$$ $${\cal G}^{(1)}_{ls} = \frac14 W \Delta_{ls}
\,,
\quad {\cal G}^{(2)}_{ks} = {-} \frac14 P \Delta_{ls} \,, \quad {\cal G}_{kls} = 0 \,,$$ $${\cal G}_{ikls} {=} - \frac14 P \left(
\Delta_{ls}\Delta_{ik} {+} \Delta_{li}\Delta_{ks} {+}
\Delta_{lk}\Delta_{is}\right)
\,.
\label{2sym3}$$ These formulas give the hint to write the working formula $$\frac{\delta}{\delta g^{ik}} {\cal P}_{pq} {=}$$ $${=}\frac12 U_iU_k \left[{\cal P}_{pq} {-}\Delta_{pq} W \right] {-} \frac12 \left[g_{p(i} {\cal P}_{k)q} {+} g_{q(i} {\cal P}_{k)p} \right]
, \label{DE8}$$ and then to summarize the total stress-energy tensor of the electromagnetic field interacting with the dark energy as follows: $$T^{({\rm EM})}_{ik} = T^{(0)}_{ik} + T^{({\rm W})}_{ik} + T^{({\rm
S})}_{ik} =$$ $$= \left[\frac14 g_{ik}F_{mn}-\frac12
\left(g_{im}F_{kn}+g_{km}F_{in} \right) \right]{\cal
C}^{mnab}F_{ab} +$$ $$+\frac18 U_i U_k F_{ab}F_{mn}\left\{(W+P)Q^{abmn} + \right.$$ $$\left. +
\lambda^{abmn}\left[(W+P)\nabla_j U^j -DW \right]\right\} +$$ $$+ \frac18 U_i U_k (W+P)
D\left(\lambda^{abmn}F_{ab}F_{mn} \right)
\,. \label{DE9}$$ Here we used the notations $${\cal C}^{abmn} = C_{(0)}^{abmn} + \lambda^{abmn} \ DW - Q^{abmn}
\ P=$$ $$= \frac{1}{2\mu}\left[g^{abmn} + \left(\varepsilon \mu -1\right)
\left(g^{abmn} - \Delta^{abmn}\right)\right] \,, \label{DE10}$$ where the quantities $\mu$ and $\varepsilon$ are already defined by (\[e22\]) and (\[m22\]).
Model with hidden anisotropy
----------------------------
The spatial isotropy of the space-time happens to be violated, when one considers the model with global magnetic and/or electric fields. For the magnetic field in vacuum one should study the Bianchi-I cosmological model instead of the Friedmann one, since the tensor $\Delta^p_iT_{pq}^{({\rm EM})}\Delta^q_k$ in the right-hand side of the Einstein equations is not spatially isotropic in this case (i.e., $\Delta^{1p}T_{pq}^{({\rm EM})}\Delta^q_1 {=}\Delta^{2p}T_{pq}^{({\rm EM})}\Delta^q_2 \neq \Delta^{3p}T_{pq}^{({\rm EM})}\Delta^q_3$, when $x^3$ is the anisotropy axis). When an electrodynamic system interacts with dark energy, the situation changes essentially: one can find such states of the system, for which the magnetic field is non-vanishing but the spatial isotropy is inherited. For instance, when $$\frac{1}{\mu_{(0)}} + \lambda_2 DW -\beta P =0 \,, \label{TEM121}$$ we obtain immediately that $$\Delta^p_iT_{pq}^{({\rm EM})}\Delta^q_k =$$ $$= (\varepsilon_{(0)}{+}\lambda_1 DW{-}\alpha
P)\left[\frac{1}{2}\Delta_{ik} E_m E^m {-} E_i E_k \right] \,,
\label{TEM112}$$ where $E^i{=}F^{ik}U_k$ is the electric field four-vector; for pure magnetic field $E^i{=}0$, thus $\Delta^p_iT_{pq}^{({\rm
EM})}\Delta^q_k {=}0$. This model with [*hidden*]{} magnetic anisotropy is exotic, since the effective refraction index is now equal to infinity, and the phase and group velocities of the electromagnetic waves are equal to zero for such dark medium. Similarly, when $\varepsilon_{(0)} {+} \lambda_1 DW {-}\alpha P
{=}0$, but $\frac{1}{\mu_{(0)}}\neq \beta P {-}\lambda_2 DW $ we deal with the so-called hidden electric anisotropy. Finally, when $\varepsilon_{(0)}{=}\alpha P{-}\lambda_1 DW$ and $\frac{1}{\mu_{(0)}} {=} \beta P{-}\lambda_2 DW$ simultaneously, the stress-energy tensor $\Delta^p_iT_{pq}^{({\rm EM})}\Delta^q_k
$ is equal to zero identically, and the electromagnetic source of the gravity field disappears from the Einstein equations. Similar results related to a hidden magnetic anisotropy were obtained early in the frameworks of the nonminimal Einstein-Maxwell theory [@BaZim] and extended Einstein-Maxwell theory [@2007A; @2007B].
Cosmological applications: Unlighted epochs
===========================================
We obtained extended master equations for the coupled electromagnetic and gravitational fields, which take into account interactions of pyro-, piezo- and striction- types. In the nearest future we intend to analyze applications of these master equations to the early universe with global magnetic field, and to the problem of late-time universe accelerated expansion. In this work we consider only one illustration of the extended model. To be more precise, in this Section we assume that the space-time is of the Friedmann-Lemaître-Robertson-Walker type with the metric $$ds^2 = c^2dt^2 - a^2(t)(dx^2+dy^2+dz^2) \,, \label{F1}$$ and this space-time is a fixed background for a local (test) electrodynamic system. In other words, here we neglect by the backreaction of the electromagnetic field on the gravity field, but consider the striction-type influence of the spatially isotropic cosmic dark energy on the electrodynamic system. We are interested in the analysis of the so-called unlighted epochs in the universe history, analogs of which were described in [@BBL] in the framework of nonminimal field theory. We use the term “unlighted epochs” for intervals of the universe evolution, for which the square of the effective refraction index $n^2(t)$ (see (\[2varco\])) takes negative values, $n^2(t)<0$. During these periods of time the refraction index is a pure imaginary quantity, and the phase and group velocities of the electromagnetic waves (see (\[v1\])) and (\[v2\]) are not defined. Clearly, the function $n^2(t)$ can change sign at the moments $t_{(s)}$ of the cosmological time when the numerator or the denominator in (\[2varco\]) vanish. When the numerator vanishes, one has $n(t_{(s)}){=}0$, $V_{({\rm
ph})}(t_{(s)}){=}\infty$, and $V_{({\rm gr})}(t_{(s)}){=} 0$. When the denominator vanishes, one has $n(t_{(s)}){=}\infty$, $V_{({\rm
ph})}(t_{(s)}){=}0$, and $V_{({\rm gr})}(t_{(s)}){=} 0$. In both cases the unlighted epochs appear and disappear when the group velocity of the electromagnetic waves vanishes, i.e., at these points the energy transfer stops. We indicate the times $t_{(s)}$ as the unlighted epochs boundary points (see [@BBL] for details). Below we consider three very illustrative examples of the evolution of the cosmic dark energy and of the corresponding behavior of the pressure function $P(t)$ for the case, when the striction coefficients only are non-vanishing.
De Sitter-type models
---------------------
The simplest model of the dark energy is the de Sitter one; for this model the dark energy pressure is constant $P{=}{-} \Lambda$. Clearly, the refraction index for this model is also constant $$n^2 = \frac{n^2_{(0)} + \mu_{(0)} \alpha \Lambda}{1 + \mu_{(0)}
\beta \Lambda } \,, \label{ds1}$$ i.e., unlighted epochs are not available. When $$\beta - \alpha = \frac{n^2_{(0)}-1}{\mu_{(0)} \Lambda} \,,
\label{ds18}$$ we obtain that the universe expansion is characterized by the condition $n^2{=}1$, so that $V_{({\rm gr})} {=} V_{({\rm ph})}
{=} c$.
Anti-Gaussian solution
----------------------
In [@1BB2011; @2BB2011] the exact solution of the Archimedean-type model is obtained, which was indicated as Anti-Gaussian solution, since for this solution the scale factor $a(t)$ is of the form $$a(t) = a(t^*) \exp{\left[\frac{8\pi G}{3\nu}(t-t^*)^2\right]}\,.
\label{pi2}$$ Here we repeated the notations from [@1BB2011] $$\log{\frac{a(t^*)}{a(t_0)}} \equiv - \frac{\nu}{4}[\rho(t_0)+
E_{(0)}] \,,$$ $$t^* = t_0 - \nu \sqrt{\frac{3}{32\pi
G}[\rho(t_0)+ E_{(0)}]} \,. \label{pi4}$$ The parameter $\nu$ is a coupling constant of the Archimedean-type interaction between dark energy and dark matter, the parameters $a(t_0)$, $\rho(t_0)$ and $E_{(0)}$ are the initial data for the scale factor, energy-density of the dark energy and energy-density of the dark matter, respectively. According to that model the dark energy pressure and energy-density (here and below we use the symbols $\Pi(t)$ and $\rho(t)$, respectively, for these quantities in analogy with [@1BB2011; @2BB2011]) are described by the formulas $$\Pi(t)= \Pi(t_0) - \frac{4}{\nu} \log{\left[\frac{a(t)}{a(t_0)}\right]} \,,$$ $$\rho(t)= \rho(t_0) + \frac{4}{\nu} \log{\left[\frac{a(t)}{a(t_0)}\right]} \,.
\label{pi1}$$ The formula for the pressure can be rewritten in the form $$\Pi(t) = \Pi(t^*) - \frac{32\pi G}{3\nu^2}(t-t^*)^2\,,$$ $$\Pi(t^*) = \Pi(t_0) + \rho(t_0) + E_{(0)} \,. \label{pi3}$$ We assume that the late-time universe evolution is characterized by the model with $n^2(t) \to 1$; this assumption provides that at present the light propagates with phase and group velocities equal to the speed of light in vacuum. According to (\[2varco\]) and (\[pi3\]) this requirement at $t \to \infty$ leads to the equality $\beta {=} \alpha$. Also, we put $\mu_{(0)}{=1}$ for simplicity and assume that $n^2_{(0)}{=}\varepsilon_{(0)}>1$. Then we use the replacement $$z^2 \equiv \frac{32\pi G}{3\nu^2}(t-t^*)^2
\label{pi33}$$ and transform (\[2varco\]) into $$n^2 = \frac{z^2-Z_2}{z^2-Z_1} \,, \label{pi5}$$ where $$Z_2 \equiv \Pi(t^*) - \frac{n^2_{(0)}}{\alpha} \,, \quad Z_1
\equiv \Pi(t^*) - \frac{1}{\alpha} \,,$$ $$\alpha \left(Z_1-Z_2
\right) = [n^2_{(0)}-1] >0 \,. \label{pi6}$$ Now we are ready to describe unlighted epochs.
### Models without unlighted epochs
The refraction index $n(t)$ is equal to one identically, when $n^2_{(0)}{=}1$ and thus $Z_1{=}Z_2$. Also, the quantity $n^2(t)$ is positive for [*arbitrary*]{} time, when $Z_1$ and $Z_2$ are negative, i.e., $$\Pi(t^*) < \frac{n^2_{(0)}}{\alpha} \,, \quad \Pi(t^*) <
\frac{1}{\alpha} \,. \label{pi689}$$ In both cases the unlighted epochs can not appear.
### Unlighted epochs of the first type
Let the parameter $Z_1$ be negative, and $Z_2$ be positive. Clearly, it is possible, when $$\frac{n^2_{(0)}}{\alpha} < \Pi(t^*) < \frac{1}{\alpha} \ \ \Rightarrow \ \ \alpha <0 \,, \quad \Pi(t^*) <0 \,.
\label{un1}$$ Thus, the refraction index is imaginary, when $|z|< \sqrt{Z_2}$, or in more details, $$|t-t^*| < \Delta_2 \,, \quad \Delta_2 \equiv \sqrt{\frac{3\nu^2}{32\pi G}\left[\Pi(t^*)- \frac{n^2_{(0)}}{\alpha} \right]}\,.
\label{un2}$$ At the boundary points of this time interval, $t_{(\pm)}{=}t^* \pm \Delta_2$, the refraction index and the group velocity vanish, $n(t_{(\pm)}){=}0$, $V_{({\rm gr})}(t_{(\pm)}){=}0$, and the phase velocity becomes infinite $V_{({\rm ph})}(t_{(\pm)}){=} \infty$. The duration of this unlighted epoch is equal to $2\Delta_2$ (see Panel A of Fig.1).
![[Sketches of basic graphs illustrating unlighted epochs of four types. Unlighted epochs relate to the intervals of cosmological time $t$ for which the function $n^2(t)$, the squared effective refraction index, is negative. Panel A relates to the case, when $Z_1<0$, $Z_2>0$, and thus the denominator of the function $n^2(t)$ is positive (see (\[pi33\])-(\[pi6\])); this panel illustrates the simply connected unlighted epoch of the first type with zeroth values of the function $n^2(t)$ at the boundary points. Panel B illustrates the simply connected unlighted epoch of the second type, which corresponds to the case $Z_1>0$, $Z_2<0$, so that both boundary values of the function $n^2(t)$ are infinite. The graphs of the unlighted epochs of the third type (Panel C) and of the fourth type (Panel D) are doubly-connected; they correspond to the cases $Z_2>Z_1>0$ and $Z_1>Z_2>0$, respectively. When $n^2{=}0$ or $n^2 {=}\infty$, the group velocity of electromagnetic wave (see (\[v2\])) takes zero value, i.e., the energy transfer stops. For all the cases one has that $n^2(t\to \pm \infty) \to 1$ (see (dashed) horizontal asymptotes.)]{}](graph){width="8cm"}
### Unlighted epochs of the second type
Now, let the parameter $Z_2$ be negative, and $Z_1$ be positive. Clearly, it is possible, when $$\frac{1}{\alpha} < \Pi(t^*) < \frac{n^2_{(0)}}{\alpha} \ \ \Rightarrow \ \ \alpha >0 \,, \quad \Pi(t^*) > 0 \,.
\label{un3}$$ Thus, the refraction index is imaginary, when $|z|< \sqrt{Z_1}$, or in more details, $$|t-t^*| < \Delta_1 \,, \quad \Delta_1 \equiv \sqrt{\frac{3\nu^2}{32\pi G}\left[\Pi(t^*)- \frac{1}{\alpha} \right]}\,.
\label{un4}$$ At the boundary points of this time interval, $t_{(\pm)}{=}t^* \pm \Delta_1$, the refraction index is infinite, $n(t_{(\pm)}){=} \infty$, thus the group and phase velocities vanish, $V_{({\rm gr})}(t_{(\pm)}){=}0$, $V_{({\rm ph})}(t_{(\pm)}){=} 0$. The duration of this unlighted epoch is equal to $2\Delta_1$ (see Panel B of Fig.1).
### Unlighted epochs of the third type
Now we consider the case, when both parameters $Z_1$ and $Z_2$ are positive. For positive $\alpha$ this gives the conditions $$\Pi(t^*) > \frac{n^2_{(0)}}{\alpha} \,,
\label{un5}$$ so that $n^2(t)<0$, when $\sqrt{Z_2} <|z|< \sqrt{Z_1}$, or equivalently, $$\Delta_2 < |t-t^*| < \Delta_1 \,.
\label{un6}$$ This unlighted epoch is divided into two separated sub-epochs, the duration of both sub-epochs is $\Delta_1{-}\Delta_2$ (see Panel C of Fig.1). Similarly, when $\alpha<0$ and $\Pi(t^*) > \frac{1}{\alpha}$, we could find two unlighted sub-epochs at $\Delta_1 < |t-t^*| < \Delta_2$ (see Panel D of Fig.1).
Super-exponential solution
--------------------------
In [@1BB2011] the exact solution of the Archimedean-type model is obtained, which was indicated as super-exponential, since for this solution the scale factor is of the form $$\frac{a(t)}{a(t_0)} {=} \exp \left\{ \sqrt{\frac{2 \rho (t_0)}{9 \rho_0}} \sinh{\left[ \sqrt{12 \pi G \rho_0} (t{-}t_0)\right]} \right\},
\label{se1}$$ where the parameter $\rho_0$ is the so-called bag constant. The corresponding dark energy pressure is $$\Pi(t)= \Pi(t_0) +3\left[\rho(t_0) + \Pi(t_0) - \rho_0 \right] \log{\left[\frac{a(t)}{a(t_0)}\right]} -$$ $$-
\frac92 \rho_0 \log^2{\left[\frac{a(t)}{a(t_0)}\right]} \,.
\label{se2}$$ Surprisingly, this model can be reduced to the one considered in the previous section, if we use the following definition for $z$: $$z(t) \equiv \sqrt{\rho(t_0)} \sinh{\left[ \sqrt{12 \pi G \rho_0} \ (t-t_0)\right]} -$$ $$-
\frac{\left[\rho(t_0) +
\Pi(t_0) - \rho_0 \right]}{\sqrt{2\rho_0}} \,.
\label{se4}$$ In order to complete the analogy, we find the parameter $t^*$ from the equation $z(t^*) {=} 0$ (see (\[se4\])) and obtain from (\[se2\]) and (\[se1\]) that $$\Pi(t^*)\equiv \Pi(t_0) + \frac{\left[\rho(t_0) + \Pi(t_0) - \rho_0 \right]^2}{2\rho_0} \,,
\label{se5}$$ with $Z_1$ and $Z_2$ inherited from (\[pi6\]). Thus, we deal again with the analysis of the formula (\[pi5\]) and obtain the similar results, if we make the replacement $$\sqrt{\frac{32\pi G}{3\nu^2}}(t-t^*) \ \ \Rightarrow \ \$$ $$\left\{\sqrt{\rho(t_0)} \sinh{\left[ \sqrt{12 \pi G \rho_0}
(t{-}t_0)\right]} {-} \frac{\left[\rho(t_0) {+} \Pi(t_0) {-}
\rho_0 \right]}{\sqrt{2\rho_0}} \right\}. \label{se6}$$ Again, the model admits the existence of unlighted epochs of four types sketched on Panels 1-4 of Fig.1.
Remarks on stability of the dark energy scalar potential under quantum fluctuations, and constraints on the striction model
---------------------------------------------------------------------------------------------------------------------------
We established pure classical model of striction-type interactions between dark energy and electrodynamic system. The paper context does not allow us to consider quantum aspects of this model, however, we hope to return to this problem in the next work. Here we would like to discuss briefly only three remarks, which could be important for physical understanding of the model consequences.
### How the corrections to the scalar field potential can influence the master equations of the striction-type model?
The crucial point of establishing the total set of master equations of the striction-type model is the finding of the tensor ${\cal B}_{ikls}$ (see (\[S2\]) and (\[sym2\])). Generally, it can not be presented in an explicit form by means of the energy density $W$ and of the dark energy pressure $P$, and its reconstruction requires sophisticated phenomenological decomposition (\[sym2\]). However, when we treat the dark energy as a spatially homogeneous isotropic medium using an analogy with some scalar field $\Psi$, we obtain ${\cal B}_{ikls}$ by direct variation procedure (see (\[DE7\])), thus providing the model to be self-closed. Moreover, we find that, when we restrict our-selves by the striction-type interactions only, i.e., $\lambda^{ikmn}{=}0$, the electrodynamic equations include the scalar $P$ only, and the gravity field equations include $P$ and the combination $W{+}P$ only. It follows from (\[DE5\]) that the sum $W{+}P{=}{(D\Psi)}^2$ does not feel the shape of the potential $V(\Psi^2)$; as for the dark energy pressure $P$, it, clearly, depends on $V(\Psi^2)$. This fact confirms that modeling of striction-type extensions of master equations for the electromagnetic and gravitational fields by means of a scalar dark energy is sensitive to the choice of the scalar field potential.
### One-loop corrections to the scalar field potential\
in the framework of anti-Gaussian model
Based on the method of one-loop corrections to potentials of complex or real scalar fields attributed to the dark energy, dark matter or dark fluid, the authors of the works [@loop1; @loop2] have shown that these scalar field potentials are usually stable under quantum fluctuations, however, a coupling to fermions is very restricted. When we consider an electrodynamic system influenced by a scalar dark energy, we have to take into account these results keeping in mind two aspects. First, the scalar field potential $V(\Psi^2)$ attributed to the dark energy (see (\[DE1\])) can be modified due to one-loop corrections along the line discussed in [@loop1; @loop2]; since this potential enters the formula for the pressure of the dark energy (see (\[DE5\])), these corrections can re-define the coupling constants of the striction-type interaction discussed above. Second, the electrodynamic system inevitably contains electrically charged fermions; when the fermion mass $m_{\rm f}$ depends on the scalar field, $m_{\rm f}(\Psi)$, we can use directly the method and estimations discussed in [@loop1; @loop2]. As for (possible) dependence of the photon mass on the scalar field $\Psi$, this problem is worthy a special attention and will be discussed in a separate paper.
Let us apply the method used in [@loop1; @loop2] to the anti-Gaussian model discussed in Subsection V.B. According to (\[pi1\]) and (\[DE5\])), we readily obtain that $${\dot{\Psi}}^2(t) = \rho(t) {+} \Pi(t)= \rho(t_0) {+} \Pi(t_0) \equiv {\dot{\Psi}}^2(t_0) \,,
\label{loop1}$$ thus providing the scalar field $\Psi$ to be linear function of the cosmological time $t$: $$\Psi(t) = \Psi(t_0) + {\dot{\Psi}}(t_0) \ t \,.
\label{loop2}$$ Then, using (\[DE4\])) and (\[DE5\])) we see that the potential $V(\Psi^2){=} V^{*}(\Psi)$ is a quadratic function of the scalar field: $$V^*(\Psi) = A + 2B \Psi + C \Psi^2 \,,
\label{loop3}$$ where $$A = \rho(t_0)-\Pi(t_0) - 2B \Psi(t_0) - C \Psi^2(t_0) \,,$$ $$B = \frac{16}{\nu {\dot{\Psi}}(t_0)} \sqrt{\frac{\pi G}{6}\left[\rho(t_0){+}E_{(0)} \right]} - C \Psi(t_0) \,,$$ $$C = \frac{64 \pi G}{3 \nu^2 {\dot{\Psi}}^2(t_0)} \,.
\label{loop4}$$ For the function (\[loop3\]) the second derivative of the potential is $V^{*\prime \prime}(\Psi){=} 2C$, thus the formula (2) from [@loop1] gives us the one-loop modification of the scalar potential (\[loop3\]) $$V_{\rm 1-loop}(\Psi) = V^{*}(\Psi) {+} \frac{4 G \Lambda^2_{\rm s}}{3 \pi \nu^2 {\dot{\Psi}}^2(t_0)} {-} \frac{\Lambda^2_{\rm f}}{8\pi^2}[m_{\rm f}(\Psi)]^2
\,,
\label{loop5}$$ where $\Lambda_{\rm s}$ and $\Lambda_{\rm f}$ are the ultra-violet cutoffs of the scalar and fermion fluctuations, respectively; $m_{\rm f}(\Psi)$ is $\Psi$-dependent fermion mass, which appears in the Lagrangian of matter $L_{({\rm m})}$ (see (\[0actmin\])), when the matter is considered to consist of fermions (see (1) in [@loop1]). Clearly, being constant, the second term in the right-hand side of (\[loop5\]) can be absorbed by the constant $A$ appeared in the potential (\[loop3\]); the authors of the works [@loop1; @loop2] indicate such a case as describing the potential stable under scalar fluctuations. As for estimations of the potential stability related to fermion fluctuations, the third term in (\[loop5\]) is exactly the same as the one in (2) of the paper [@loop1]; this means that estimations discussed in Section III of [@loop1] are also valid. Let us repeat, that estimations related to the scalar field coupling to photons and gravitons based on one-loop corrections require a special consideration and, unfortunately, are out of frame of this paper.
### How the corrections to the scalar field potential could be visualized in the cosmic striction-type phenomena?
Let us consider the consequences of the scalar field potential modification, given by the second term in the right-hand side of (\[loop5\]), for the unlighted epochs formation. In fact, this constant one-loop correction to the quadratic potential leads to the following formal re-definition of the parameter $\Pi(t^{*})$ in (\[pi3\]): $$\Pi(t^{*}) \to \tilde{\Pi}(t^{*}) = \Pi(t^{*}) - \frac{2 G \Lambda^2_{\rm s}}{3 \pi \nu^2 {\dot{\Psi}}^2(t_0)} \,.
\label{loop6}$$ Then we have to replace the parameter $\Pi(t^{*})$ with this new parameter $\tilde{\Pi}(t^{*})$ in all inequalities, which predetermine the structure of the unlighted epochs (see Sections V.B1, V.B2, V.B3, V.B4). Let us illustrate the consequences of such pressure re-definition by the examples, for which $\alpha >0$ and $\Pi(t^{*})$ is positive (see, e.g., (\[pi689\]), (\[un3\]), (\[un5\])). Since the second term in (\[loop6\]) is negative, we can imagine that $\tilde{\Pi}(t^{*})$ becomes negative due to such scalar potential correction. This means that the inequalities (\[pi689\]) (with the replacement $\Pi(t^{*}) \to \tilde{\Pi}(t^{*})$) remain valid, so that this veto for the unlighted epoch appearance holds out. The inequalities (\[un3\]) and (\[un5\])) are not valid now, therefore, the corresponding unlighted epochs of the second and third types become forbidden because of scalar fluctuations. In other words, the answer on the question about absence or presence of the unlighted epochs seems to be very sensitive to the choice of the scalar field potential and to its one-loop corrections.
Discussion and conclusions
==========================
1\. A new model of coupling between a cosmic dark fluid and electrodynamic systems is established, i.e., the extended Lagrangian is introduced and the extended master equations are derived for electromagnetic and gravitational fields. What is the novelty of this model from the mathematical point of view? First, we introduced four cross-terms into the Lagrangian, which contain the Maxwell tensor up to the second order, on the one hand, and contain the pressure tensor of the dark energy and the convective derivative of its energy-density scalar, on the other hand. Thus, a modified Lagrangian is not of pure field-type, since these cross-terms are the products of pure field-type elements ($F_{ik}$) and of quantities defined algebraically (see (\[fluid\]), (\[1fluid\]) for the definitions of $U^i$, $W$, ${\cal P}_{pq}$). Second, we described a modified procedure of variation with respect to the metric, which happened to be necessary for these new cross-terms defined algebraically. This modified procedure is based on the rule of variation of the macroscopic velocity four-vector (\[11T1\]) taken from the works [@2007A; @2007B]; on the rule of algebraic decomposition of the second variation of the dark energy Lagrangian (\[S2\])-(\[sym3\]), and on the ansatz about the structure of this decomposition in the case of spatially isotropic medium (see (\[DE6\])-(\[DE8\])).
2\. What is the physical motivation of the Lagrangian extension, which we made? Since we follow the mathematical scheme, which is well-known in the relativistic electrodynamics of continuous media, we indicate the new cross-terms in the extended Lagrangian (\[actmin\]) using the similar terminology: as (dark energy inspired) analogs of terms describing electric and magnetic striction, piezo-electricity and piezo-magnetism, pyro-electricity and pyro-magnetism. This analogy allowed us to interpret new coupling constants as (dark energy induced) pyro-, piezo- and striction coefficients, respectively.
3\. First cosmological application of the model shows that a striction-type interaction between the dark energy and test electrodynamic system provides the phase and group velocities of electromagnetic waves to become sophisticated functions of cosmological time. In the asymptotic regime, at $t
\to \infty$, these functions tend to the speed of light in vacuum, i.e., $V_{({\rm ph})} \to c$ and $V_{({\rm gr})} \to c$. However, during the universe evolution the so-called unlighted epochs can appear, for which the effective refraction index of the cosmic medium is an imaginary quantity. At the boundary points of these unlighted epochs the group velocity of the electromagnetic waves takes zero value, so the electromagnetic energy transfer stops.
4\. The last unlighted epoch (if such epochs have ever existed in the real Universe) should finish before the so-called recombination era, as far as, cosmic microwave background composed of relic photons traveling freely is observable. In our terms this means that the characteristic time $t_{(\rm last UE)}{=}t^{*}{+}\Delta_2$ is less than $t_{({\rm rec})} \simeq 10^{13} {\rm sec}$. Using (\[pi4\]), (\[un2\]), (\[pi3\]) at $t_0{=}t_{({\rm rec})}$ and the condition $t_{(\rm last UE)}< t_{({\rm rec})}$, we obtain a cosmological constraint $\Pi(t_{({\rm rec})})<\frac{n^2(t_{({\rm rec})})}{\alpha}$ linking the phenomenological parameter of a striction-type coupling $\alpha$, the value of the refraction index $n(t_{({\rm rec})})$ at the end of the recombination era, and the value of the dark energy pressure $\Pi(t_{({\rm rec})})$ at that moment. It is only one example of constraints appearing in this model; we intend to discuss other constraints in a special note.
5\. The criteria of existence of the unlighted epochs are very sensitive to the choice of the scalar field potential, which one uses for modeling of the dark energy. We have illustrated this sentence on the example of the anti-Gaussian model. In particular, taking into account one-loop corrections to the dark energy scalar field potential, one can show, that scalar fluctuations are able to avoid the unlighted epoch formation.
6\. We expect that the established model can provide new interesting results in application to the anisotropic Bianchi-I cosmological model, since in this period of the universe evolution pyro-magnetic and piezo-magnetic effects can appear in addition to the magneto- striction effect admissible both on the anisotropic and isotropic stages of the universe expansion.
[**Acknowledgments**]{}
This work is supported by the Russian Foundation for Basic Research (Grant No. 14-02-00598).
Appendix {#appendix .unnumbered}
========
In order to calculate directly the variation derivatives of basic quantities, we have to fix the following auxiliary formulas. We start with the well-known formulas for the variation of the determinant of the metric and metric itself: $$\frac{1}{\sqrt{-g}}\frac{\delta \sqrt{-g}}{\delta g^{ik}} =
-\frac12 g_{ik} \,,$$ $$\frac{\delta g_{pq}}{\delta g^{ik}} = -
\frac12 \left[g_{p(i} g_{k)q} + g_{q(i} g_{k)p}\right] \,,
\label{01T1}$$ and for the four-indices tensor $g^{ikmn} \equiv
g^{im}g^{kn}{-}g^{in}g^{km}$ $$\frac{\delta g^{abmn}}{\delta g^{ik}} = \delta^{[a}_{(i} g_{k)}^{\
b]mn} + g_{\ \ \ \ (k}^{ab[m} \delta^{n]}_{i)} \,. \label{1T6}$$ The formulas for the variation of the velocity four-vector $$\frac{\delta U^l}{\delta g^{ik}} = \frac14 \left(\delta^{l}_i U_k {+} \delta^{l}_k U_i \right) = \frac12 \delta^l_{(i} U_{k)} \,,$$ $$\frac{\delta U_a}{\delta g^{ik}} = - \frac14
\left(g_{ia}U_k {+} g_{ka}U_i \right) = - \frac12 g_{a(i}U_{k)}
\,,
\label{11T1}$$ stand to keep the normalization condition $g^{ik}U_i U_k {=}1$ (see, e.g., [@2007A; @2007B]). Using (\[01T1\]) and (\[11T1\]) it is easy to write the formulas for the variation of the projectors $$\frac{\delta \Delta^{pq}}{\delta g^{ik}} = \delta^{(p}_{(i} \Delta^{q)}_{k)} \,,$$ $$\frac{\delta \Delta_{pq}}{\delta g^{ik}} = - \frac12 \left[g_{p(i} \Delta_{k)q} + g_{q(i} \Delta_{k)p} \right] \,,
\label{1T2}$$ $$\frac{\delta \Delta^{p}_{q}}{\delta g^{ik}} = \frac12 \left[\delta^p_{(i} \Delta_{k)q} - \Delta^p_{(i} g_{k)q}\right] \,.
\label{1T3}$$ For the variation of the four-indices projector $\Delta^{ikmn}
\equiv \Delta^{im}\Delta^{kn}{-} \Delta^{in}\Delta^{km}$ we use the convenient formula $$\frac{\delta \Delta^{abmn}}{\delta g^{ik}} = \delta^{[a}_{(i} \Delta_{k)}^{\ b]mn} + \Delta_{\ \ \ \ (k}^{ab[m} \delta^{n]}_{i)} \,.
\label{1T7}$$ Finally, the variation of the Levi-Civita tensor $\epsilon^{abpq}$ and of the associated tensor $\eta^{abp} \equiv \epsilon^{abpq}U_q
$ yield $$\frac{\delta \epsilon^{abpq}}{\delta g^{ik}} = \frac12 \epsilon^{abpq} g_{ik} \,,$$ $$\frac{\delta \eta^{abp}}{\delta g^{ik}} = \frac12 \left[\eta^{abp} g_{ik} - \epsilon^{abp}_{\ \ \ (i}U_{k)} \right] \,.
\label{1T40}$$
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|
---
abstract: 'In the quantum metrology protocol described by Tacla et al. \[Tacla et al., Phys. Rev. A [**82**]{}, 053636 (2010)\] where a two mode Bose-Einstein condensate (BEC) is used for parameter estimation, the measured quantity is to be obtained by doing a one parameter fit of the observed data to a theoretically expected signal. Here we look at different levels of approximation used to model the two mode BEC to see how the estimate improves when increasing level of detail is added to the theory while at the same time keeping the expected signal computable.'
author:
- Salini Jose
- Anil Shaji
bibliography:
- 'bib.bib'
title: 'Corrections to the expected signal in quantum metrology using highly anisotropic Bose-Einstein Condensates'
---
Introduction \[sec1\]
=====================
Nonlinear interactions between the $N$ elementary units of the quantum probe in particle based quantum limited single parameter estimation schemes can allow the measurement uncertainty to scale as $1/N^{k}$, where $k$ is the degree of nonlinearity [@boixo_generalized_2007]. Without loss of generality the elementary units that make up the quantum probe can be taken to be qubits. The scaling of the minimum possible uncertainty in the value of the classical parameter that is measured with respect to the resources that go into the metrology protocol, as quantified by $N$, is a measure of the efficiency and performance of the measurement scheme. The “Heisenberg limited scaling” of $1/N$ was held to be the absolute limit for the performance of a particle based metrology scheme like Ramsey interferometry [@bollinger; @huelga] until it was clarified that this is so under the assumption that each of the probe qubits undergoes independent parameter dependent evolutions.
With nonlinear parameter dependent couplings between the probe qubits it was shown that even if the initial state of the quantum probe is restricted to being a product state of the $N$ units, the best possible scaling of the uncertainty is $1/N^{k-1/2}$ [@boixo_quantum-limited_2007]. In a sequence of related publications [@boixo_quantum_2008; @boixo_quantum-limited_2009; @Boixo-Tacla] a proposal to use a two mode BEC to perform a proof of principle experiment that demonstrates a scaling of $1/N^{3/2}$ was put forward. Realizations of the experiment in the lab are also being successfully pursued [@egorov_long-lived_2011; @egorov_measurement_2013; @napolitano_interaction-based_2011]. The effective nonlinear interaction modeled as the $g|\psi|^{2}$ potential term in the Gross-Pitaevski equation [@Dalfovo; @legget] describing the condensate under the mean field approximation furnishes a $k=2$ coupling that can be used to design an experiment that estimates the value of a function of the constant $g$ with the measurement uncertainty potentially scaling as $1/N^{3/2}$.
The initial proposal in [@boixo_quantum_2008] was developed further in [@boixo_quantum-limited_2009], [@Boixo-Tacla] as well as in [@taclacaves] to include incrementally the deleterious effects of possible non-ideal conditions in the lab as well as to relax the simplifying assumptions that were made in the theoretical analysis. This Paper is also another step in the same direction where we focus on one of the simplifying assumptions made about the initial state of the BEC and explore how and when corrections due to the relaxation of this assumption that were proposed in [@taclacaves] would come into play.
The motivation for refining the original proposal of the BEC based metrology scheme is to push towards a viable experiment in the lab. In a nutshell the problem can be posed as follows: under non-ideal conditions with none of the simplifying assumptions the effective scaling we get is $1/(N^{3/2}\eta_{N})$ where $\eta_{N}$ can depend on $N$ as well as have higher order dependencies on the measured parameter itself. Enumerating and understanding all the factors that influence $\eta_{N}$ and crucially its $N$ dependence is important for a fool proof interpretation of the proposed experiment as one that surpasses the $1/N$ scaling. The same problem can be viewed from a slightly different perspective also. The signal that is measured in the final step of the experiment is the number of atoms in each one of the two internal states of the two mode BEC. Repeating the experiment a fixed number of times gives us the time dependence of these numbers. In the ideal situation the only unknown quantity on which the time dependence of the population of atoms depends on is the measured parameter. However, in practice it would depend on $\eta_{N}$ as well. Therefore if the strategy of doing a one parameter fit of the measured time evolution of the populations so as to find the value of the parameter is to work, an almost complete knowledge of the dependence of $\eta_{N}$ on the details of the experiment is essential.
Quantum limited measurements using a two mode BEC \[sec2\]
==========================================================
In [@boixo_quantum_2008] and [@boixo_quantum-limited_2009], a two mode BEC at zero temperature is the quantum probe. Limiting the possible electronic (hyperfine) states of the atoms to just two lets us treat each one as a qubit. At zero temperature all the atoms can be assumed to be in the ground state of the condensate with identical overlapping wave functions. The mean field approximation holds under these conditions and if we further assume that all the atoms are initially in one of the two possible hyperfine states then their wave function is given by the ground state solution of the time-inedependent Gross Pitaevski [@Dalfovo; @legget] equation, $$\label{eq:cgp1}
\bigg[ - \frac{\hbar^{2}}{2m} \nabla^{2} + V + g (N-1) |\psi_{N}|^{2} \bigg] \psi_{N} = \mu_{N} \psi_{N},$$ where $N$ is the number of atoms in the BEC, $\mu_{N}$ is the chemical potential, $V$ is the external trapping potential and the coupling constant $g$ is related to the $s$-wave scattering length $a$ and the mass $m$ of the atoms as $$\label{eq:const1}
g=\frac{ 4 \pi \hbar^{2} a}{m}.$$
The proposed experiment proceeds by applying an instantaneous pulse that puts each atom in the condensate in a specific superposition of the two internal states. Assuming that only elastic collisions are allowed between the atoms in the two hyperfine states labeled by $|1\rangle$ and $|2\rangle$ respectively, the Hamiltonian of the system is, $$\begin{aligned}
\label{eq:hamil1}
\hat{H} & = & \sum_{\alpha =1,2} \int d {\mathbf r} \, \bigg[ \frac{\hbar^{2}}{2m} \nabla \hat{\psi}^{\dagger}_{\alpha} \cdot \nabla \hat{\psi}_{\alpha}^{\vphantom{\dagger}} + V(r)\hat{\psi}^{\dagger}_{\alpha} \hat{\psi}_{\alpha} ^{\vphantom{\dagger}} \bigg] \nonumber \\
&& \qquad + \; \frac{1}{2} \sum_{\alpha, \beta} g_{\alpha \beta} \int d{\mathbf r} \, \hat{\psi}^{\dagger}_{\beta} \hat{\psi}^{\dagger}_{\alpha} \hat{\psi}_{\alpha}^{\vphantom{\dagger}}\hat{\psi}_{\beta}^{\vphantom{\dagger}},
\end{aligned}$$ where $\hat{\psi}_{\alpha}$ is the modal annihilation operator. For a zero temperature BEC we can truncate the modal annihilation operator to just one term of the form $$\hat{\psi}_{\alpha} = \psi_{N, \alpha} ({\mathbf r}) \hat{a}_{\alpha}.$$
In [@boixo_quantum_2008] the rather strong assumption that at least for short times after the pulse that puts the atoms in a superposition state, the spatial part of their wave functions are identical, is made so that $$\hat{\psi}_{\alpha} = \psi_{N} ({\mathbf r}) \hat{a}_{\alpha}.$$ With this assumption it was shown that the Hamiltonian in Eq. (\[eq:hamil1\]) can be brought to the form $$\label{eq:hamil2}
\hat{H} = H_{0} + \gamma_{1} \eta_{N} (N-1) \hat{J}_{z} + \gamma_{2} \eta_{N} \hat{J}_{z}^{2},$$ where $$\hat{J}_{z} = \frac{1}{2} (\hat{a}_{1}^{\dagger} \hat{a}_{1}^{\vphantom{\dagger}} - \hat{a}_{2}^{\dagger} \hat{a}_{2}^{\vphantom{\dagger}} ),$$ and $$\label{eq:eta1}
\eta_{N} = \int d{\mathbf r} |\psi_{N}({\mathbf r})|^{4}.$$ $H_{0}$ is a $c$-number energy that depends on $N$ while the constants $\gamma_{1}$ and $\gamma_{2}$ characterize the three elastic scattering processes listed above and are defined as $$\gamma_{1} \equiv \frac{1}{2} ( g_{11} - g_{22}) \quad {\rm and} \quad \gamma_{2} \equiv \frac{1}{2} (g_{11} + g_{22}) - g_{12},$$ with $g_{ij}$ being related to the scattering length $a_{ij}$ of the corresponding process through Eq. (\[eq:const1\]).
The proposed experiment uses $^{87}\,{\rm Rb}$ atoms in the $|F=1; M_{F} =-1\rangle \equiv |1\rangle$ and $|F=2; M_{F} =+1\rangle \equiv |2\rangle$ states. This choice has the added advantage that the ratios $\{a_{22}:a_{12}:a_{11} \} = \{ 0.97:1:1.03\}$ [@williams] mean that $$\gamma_{2} = 0.$$ So for a two mode BEC of $^{87}\,{\rm Rb}$ atoms, $\gamma_{1} \eta_{N} (N-1) \hat{J}_{z} $ is the only term in the Hamiltonian in Eq. (\[eq:hamil2\]) that generates a relative phase between the two states $|1\rangle$ and $|2\rangle$. This phase can be used to estimate the value of the parameter $\gamma_{1}$ from which we can obtain the value of one of the two scattering lengths, say $a_{22}$, given that we know $a_{11}$. Since $\gamma_{1}$ and $a_{22}$ are linearly related through $g_{22}$ with the scaling of the measurement uncertainty in both being identical, in the following, we will consider $\gamma_{1}$ as the measured parameter but we will keep in mind that $a_{11}$ (and $g_{11}$) are known quantities. Comparing with conventional Ramsey interferometry [@Gleyzes] in which the parameter dependent evolution of the probe is generated by a Hamiltonian proportional to $J_{z}$ we can see how the relative phase evolves $N$ times faster (assuming $N\gg 1$). As detailed in [@boixo_quantum-limited_2009] the quantum Cramer-Rao bound on the measurement uncertainty in $\gamma_{1}$ will scale with $N$ as $$\delta \gamma_{1} \sim \frac{1}{\eta_{N} N^{3/2}}.$$ Achieving a $1/N^{3/2}$ scaling requires $\eta_{N}$ not to depend on $N$. From the definition of $\eta_{N}$ in Eq. (\[eq:eta1\]) we see that $\eta_{N}^{-1}$ is proportional to the volume of the ground-state wave function. The primary reason for the dependence of $\eta_{N}$ on $N$ is that as the number of atoms increases the BEC ground state wave function expands. To avoid the expansion of BEC wave function, in [@boixo_quantum_2008] highly anisotropic traps are proposed so that the expansion will be along only a few of the dimensions at least over a range of $N$ that is sufficient to demonstrate a scaling better than $1/N$. It is known that three-dimensional Bose-Einstein condensates confined in highly anisotropic traps exhibit lower dimensional behavior when the number of condensed atoms is well below a critical value [@gorlitz]. Under such conditions, the expansion of the BEC wave function with $N$ along the tightly confined dimensions can be effectively neglected because the characteristic energy scale along those directions far exceeds the scattering energy of the atomic cloud.
Repeating the initial pulse that put the $^{87}\,{\rm Rb}$ atoms in an equal superposition of the two hyperfine levels would convert the relative phase information between the two states into population information. If the populations of atoms in states $|1\rangle$ and $|2 \rangle$ are measured as a function of time through multiple repetitions of the experiment, we expect it to oscillate in time with a frequency $\Omega_{N} = (N-1) \eta_{N} \gamma_{1}/\hbar$ provided all the assumptions detailed above hold. If $\eta_{N}$ is known along with $N$ then a one parameter fit of the observed population versus time data will give us the value of the measured parameter $\gamma_{1}$.
As mentioned before, to model the expected signal from a real experiment, we have to relax the assumptions made previously and obtain a more detailed expression for the time dependence of the two populations. In the next section we discuss the approaches for obtaining an expression for the time dependence.
Time dependence of the atomic populations and estimating eta \[Sec3\]
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Under the mean field approximation the dynamics of the two mode BEC initialized in the state $(|1\rangle + |2\rangle)/\sqrt{2}$ is governed by the time-dependent, coupled GP equations, $$\label{eq:gp2}
i \hbar \frac{\partial \psi_{N, \alpha}}{\partial t} = \bigg(\!\! -\frac{\hbar^{2}}{2m} \nabla^{2} + V + \frac{N-1}{2} \sum_{\beta} g_{\alpha \beta} |\psi_{N, \beta}|^{2} \!\!\bigg) \psi_{N, \alpha},$$ with $\alpha, \beta = 1,2$. In the absence of actual experimental data, we will compare the theoretically expected signal to the atomic populations obtained by numerical integration of the above pair of equations. It must be noted that for this reason, we will not be going beyond the mean field approximation in this Paper and also that we put in the known value of $a_{22}$ into the numerical integration of the coupled GP equation and in this sense, the data obtained is equivalent to the data from a true experiment.
The population of atoms in each of the two hyperfine states is given by [@boixo_quantum-limited_2009] $$\label{eq:pop}
p_{1,2} = \frac{1}{2} [ 1 \pm {\rm Im} (\langle \psi_{N,1} | \psi_{N,2} \rangle)],$$ and is therefore determined by the overlap of the two spatial wave functions, $$\label{eq:Ot}
O(t) = \langle \psi_{N,1} | \psi_{N,2} \rangle = \int d^{3} r \, \psi_{N,1}^{*}({\mathbf r}, t) \psi_{N,2}({\mathbf r}, t) .$$ This overlap will be the main quantity of interest in the rest of the paper. In the simplest and rather ideal case where the spatial part of the two wave functions are assumed to be identical except for a spatially independent relative phase between the two, we have $$p_{1,2} = \frac{1}{2}(1 \pm \sin \Omega_{N}t), \qquad \Omega_{N} = (N-1) \eta_{N} \gamma_{1}/\hbar,$$ as discussed previously.
In order to estimate $\eta_{N}$, in [@boixo_quantum-limited_2009] for an anisotropic trap a product form for the ground state wave function of the condensed atoms is assumed as $$\label{eq:gnd1}
\psi_{0}(\rho, \, z) = \chi_{0}(\rho) \phi_{N}(z),$$ where $z$ labels the $d$ loosely confined “longitudinal” dimensions while $\rho$ labels the remaining $D=3-d$ tightly confined “transverse” dimensions. The main objective of using an anisotropic trap is to find a working range of $N$ for which the spreading of the wave function as $N$ increases is significant only along the longitudinal dimensions thereby limiting the change in $\eta_{N}$ with $N$ and yielding a better than $1/N$ scaling for $\delta \gamma_{1}$. In [@boixo_quantum-limited_2009] two critical atom numbers $N_{L}$ and $N_{T}$ that bound this working range is identified.
For concreteness, from here on, we will restrict our discussion to a quasi-one dimensional BEC in harmonic trapping potentials so that $d=1$. If $N$ is in the effective working range between $N_{L}$ and $N_{T}$ we may assume that the transverse part of the product wave function in Eq. (\[eq:gnd1\]) is just the ground state wave function of the transverse harmonic trapping potential, $$\label{eq:trans1}
\chi_0=\frac{1}{\sqrt{2 \pi\rho_0^2}}\exp \bigg(-\frac{\rho^2}{4 \rho_0^2} \bigg),$$ where $\rho_{0} = \sqrt{\hbar/2 m \omega_{T}}$ with $\omega_{T}$ being the frequency of the transverse trap. The product wave function assumed in Eq. (\[eq:gnd1\]) lets us split $\eta_{N}$ also into a product as $$\label{eq:eta1a}
\eta_{N} = \eta_{T} \eta_{L}.$$ Using $\chi_{0}$ from Eq. (\[eq:trans1\]) above, we get $$\label{eq:etat1}
\eta_{T} =\frac{1}{4 \pi \rho_{0}^{2}}.$$ We can now use $\eta_{T}$ in the one dimensional reduced, time-independent GP equation, $$\label{eq:redGP1}
\bigg[-\frac{\hbar^2}{2m}\frac{d^2}{dz^2}+\frac{1}{2}m \omega_{L}^{2} z^2+g_{11}(N-1)\eta_T|\phi_{N}|^2 \bigg]\phi_{N}=\mu_L\phi_{N},$$ to obtain the longitudinal wave function $\phi_{N}(z)$. Here $\mu_{L} = \mu_{N} - \hbar \omega_{T}$ is the longitudinal part of the chemical potential. We can compute $\eta_{L}$ and its $N$ dependence from $\phi_{N}$.
When $N$ is much larger than $N_L$ but still smaller than $N_{T}$, the kinetic-energy term in the reduced GP equation can be neglected, and the longitudinal wave function is approximated by the Thomas-Fermi solution [@legget] $$\label{eq:lwave}
|\phi_N(z)|^2=\frac{\mu_L-m \omega_{L}^{2} z^2/2}{(N-1)g_{11}\eta_T}$$ Using normalization condition for $\phi_{N}$ we get $\mu_L=m \omega_{L}^{2} z_{N}^2/2$ where $z_{N}$, the Thomas-Fermi size of the longitudinal trapping potential is given by $$\label{eq:halfw}
z_N=\bigg[ \frac{3(N-1)g_{11}\eta_T}{2 m \omega_{L}^{2}} \bigg]^{1/3}$$ In the regime where the Thomas-Fermi approximation holds we obtain, $$\label{eq:etal1}
\eta_L=\int dz |\phi_{N}|^4 = \frac{2}{5} \bigg[ \frac{9\pi m \omega_{L}^{2} \rho_{0}^{2}}{2 (N-1) g_{11}}\bigg]^{\frac{1}{3}}$$ Note that $g_{11}$ on which $\gamma_{1}$ depends on appears in the above equations for $\eta_{T}$ and $\eta_{L}$. However as mentioned before, we assume that $g_{11}$ is known and even though we are treating $\gamma_{1}$ as the parameter that is being measured we are really estimating $g_{22}$ (or equivalently $a_{22}$). Using equations (\[eq:etat1\]) and (\[eq:etal1\]) we can have an estimate of $\eta_{N}$ and use it to compute the expected frequency of oscillation of the atomic populations, $\Omega_{N}$. In Figure \[fig:figure1\] the numerically obtained value of $O_{I}(t) = {\rm Im} \langle \psi_{N,1} | \psi_{N,2} \rangle$ is compared with the expected signal $T_{1}(t) = \sin \Omega_{N}t$ for two different values of $N$. We see that the agreement is reasonable for very short times but breaks down quickly.
The assumption that the transverse part of the wave function is just the ground state wave function of the harmonic trap is valid in the low $N$ limit, while the assumption that the longitudinal wave function is given by the Thomas-Fermi approximation is valid in the large $N$ limit. We therefore expect that the best agreement between the expected signal and the numerical one will be at an intermediate value of $N$ between $N_{L}$ and $N_{T}$ provided the assumption that the wave function has the product form in Eq. (\[eq:gnd1\]) does not break down badly in this regime. To compare the agreement between theory and numerics we use as a measure the root-mean-square deviation of the expected signal from the numerically obtained one averaged over a single “period” of the expected signal, i.e. $$\label{eq:compare1}
D_{1}(N) = \sqrt{\frac{1}{m} \sum_{i=1}^{m;\, t_{m} \leq \tau } \Big[ O_{I}(t_{i}) - T_{1}(t_{i}) \Big]^{2} } ,$$ where $\tau$ is the time between two successive zero crossings in the same direction of the expected signal and $t_{i}$ are the time values at which the overlap $O(t)$ has been numerically computed. The measure $D_{1}(N)$ as a function of $N$ is shown in Figure \[fig:figd1\]. We see that the deviation is slowly increasing with N. There is no optimal intermediate value of $N$ for which the agreement is best in this case.
Position dependent phase
------------------------
Because of the difference in scattering lengths, $a_{11}$, $a_{22}$ and $a_{12}$, atoms in each of the two internal states see slightly different effective potentials due to scattering even if it is arranged so that both sets of atoms see the same external trapping potentials. The assumption that the spatial profile of the wave function of both sets of atoms is identical breaks down very quickly because of the different potentials seen by the atoms and at long enough times, the two sets of atoms end up segregating [@hall_dynamics_1998]. Modeling the segregation of the atoms within the mean field approximation may not be consistent since the differential velocities acquired by atoms of each type may give at least some of them enough kinetic energy to go out of the ground state. So we will not attempt to track down the effect of the segregation of atoms on the metrology protocol. However, prior to the segregation itself, the wave functions of the two modes picks up position dependent phases. Since the differences in the scattering energy are negligible compared to the transverse trap depth, we assume that the position dependent phase develops only in the longitudinal part of the wave function. Keeping the distribution of atoms identical in the longitudinal direction also, $$|\phi_{N}(z, t)|^{2} = q_{0}(z),$$ we obtain the position dependent relative phase between the two modes as [@boixo_quantum-limited_2009] $$\label{eq:rel2}
\delta \theta(z) = \Omega_{N}t \bigg( 1 + \frac{q_{0}(z) - \eta_{L}}{\eta_{L}} \bigg),$$ The overlap integral that gives the expected signal is then given by $$\label{eq:T2}
T_{2}(t) = {\rm Im} \Big[ e^{-i \Omega_{N}t} \int dz \, q_{0} e^{-i \Omega_{N}t (q_{0} - \eta_{L})/\eta_{L}} \Big].$$ The $N$ dependence of the deviation $D_{2}(N)$ defined using Eqs. (\[eq:T2\]) and (\[eq:compare1\]) is plotted in Figure \[fig:figure2\].
The assumptions that led to the expression for $T_{2}(t)$ in Eq. (\[eq:T2\]) again include a product form, $$\label{eq:gnd2}
\psi_{N, \alpha} = \chi_{0}(\rho) \phi_{N, \alpha} (z, t), \qquad \alpha = 1,2.$$ Further $\chi_{0}$ is again the ground state wave function of the trap and $\phi_{N, \alpha}$ are the wave functions for each of the two modes along the longitudinal direction with the position dependent relative phase. To compute $T_{2}(t)$ we further assume that $q_{0}(z)$ is the square of the Thomas-Fermi wave function. Our expectation that the assumptions about both $\chi_{0}$ and $q_{0}$ become approximately valid at an intermediate range of $N$ is borne out by Fig. \[fig:figure2\] where the agreement between the numerically computed signal and the theoretically expected one is best.
Including the motion of the atoms and attended change in the spatial profile of the wave functions of the two modes is beyond the scope of the mean field approximation we consider. So we are restricted to a time regime in which such motion is negligible as far as the proposed experiment goes. Within this regime, in order to get a better theoretically expected signal, we have to improve the expressions for $q_{0}$, $\eta_{T}$ and $\eta_{L}$. In Fig. \[fig:figure3\], the RMS deviation of the absolute value square of the longitudinal part of the numerically computed Gross-Pitaevski ground state wave function for atoms in state $|1\rangle$ from the square of the Thomas-Fermi wave function is plotted as a function of $N$. This again shows that the estimate of $q_{0}$ and $\eta_{L}$ that goes into the computation of $T_{2}(t)$ is off the mark for very low values of $N$ as well as for high values of $N$.
Similary, in Fig. (\[fig:devx\]), the deviation of the transverse part of the GP ground state along the $x$-axis from the harmonic oscillator ground state wave function for the same trapping frequency is shown. Here we see that the deviation monotonically increases as a function of $N$.
\[!htb\]
Perturbative corrections to the initial wave function \[Sec4\]
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The corrections that need to be applied to the initial wave function of the form assumed in Eq. (\[eq:gnd2\]) can be viewed as the emergence of true three dimensional behavior in the quasi-one dimensional wave function we assume that will make the product form invalid. In [@taclacaves] the emergence of three-dimensional behavior in reduced-dimension BECs trapped by highly anisotropic potentials is studied using a perturbative Schmidt decomposition of the condensate wave function between the transverse and longitudinal directions. In this section we see how this level of sophistication to the theory at the mean field level can introduce corrections of the right type that can fix the deviations seen above between the expected and numerically obtained signals. In [@taclacaves] the product form for the ground state wave function is not completely abandoned but rather it is replaced by a sum of products, with each term in the sum being treated as corrections to the previous one as $$\label{eq:schmidt1}
\psi = \sum_{n=0}^{\infty} \epsilon^{n} \chi_{n}(\rho) \phi_{n}(z),$$ where $\psi$ is the solution of the time independent GP equation, $$\begin{aligned}
\label{eq:gp3}
\mu \psi & = & \bigg( -\frac{\hbar^{2}}{2m} \nabla_{\rho}^{2} - \epsilon \frac{\hbar^{2}}{2m} \frac{\partial^{2} \;}{\partial z^{2}} + \frac{1}{2} m \omega_{T}^{2} \rho^{2} \nonumber \\
&& \quad \qquad+\; \epsilon \frac{1}{2} m \omega_{L}^{2} z^{2} + \epsilon (N-1)g |\psi|^{2} \bigg) \psi.\end{aligned}$$ The form for $\psi$ in Eq. (\[eq:schmidt1\]) can equivalently be viewed as a Schmidt decomposition [@schmidt; @schmidt2] with $\{\chi_{n} \}$ and $\{\phi_{n}\}$ forming the Schmidt basis in the transverse and longitudinal directions respectively. In (\[eq:schmidt1\]), the Schmidt decomposition has been re-written as an expansion in powers of $\epsilon$ by absorbing the Schmidt coefficients, $c_{n}$, that appear in the decomposition into the transverse wave functions so that they are normalized as $\langle \chi_{n} | \chi_{m} \rangle = c_{n}^{2} \delta_{nm}$. The longitudinal wave functions are delta function normalized. In the perturbation theory developed in [@taclacaves], the chemical potential as well as the the Schmidt basis functions are expanded in powers of $\epsilon$ as $$\begin{aligned}
\mu & = & \sum_{m=0}^{\infty} \epsilon^{m} \mu_{m}, \nonumber \\
\chi_{n} & = & \sum_{m=0}^{\infty} \epsilon^{m} \chi_{nm}, \nonumber \\
\phi_{n} & = & \sum_{m=0}^{\infty} \epsilon^{m} \phi_{nm}. \end{aligned}$$
If we include corrections to first order to the product wave function in Eq. (\[eq:gnd2\]), we have $$\begin{aligned}
\psi_{1}(\rho, z) & = & [\chi_{00} (\rho) + \epsilon \chi_{01} (\rho)][\phi_{00}(z) + \epsilon \phi_{01}(z)] \\
&& \qquad \qquad +\; \epsilon \chi_{10}(\rho) \phi_{10}(z).
\end{aligned}$$ The last term makes the wave function corrected to first order entanglement. We are interested in comparing the expected signal with the position dependent phase in Eq. (\[eq:T2\]) with the numerically obtained one when the perturbative corrections are added. When we compute the longitudinal distribution $q_{0}$ by integrating out the transverse part, the contribution from the term containing $\phi_{10}$ is of higher order and so we will not consider this term in the following. Without this term, $\psi_{1}$ retains the product form.
Consistent with our development so far, we take $\chi_{00}$ to the Gaussian ground state wave function, $\xi_{0}$, of the transverse harmonic trap while $\phi_{00}$ is the longitudinal Thomas-Fermi wave function from Eq. (\[eq:lwave\]). We can improve the computation of the expected signal by using a numerical solution to the reduced GP equation in (\[eq:redGP1\]). However we restrict to the Thomas-Fermi approximation since it allows the theoretical computation to be done without using numerical integration while at the same time allowing us meet the objective of this Paper of seeing how each layer of approximation improves the estimate of the measured parameter, $a_{22}$. The correction to the transverse wave function we consider is given by $$\label{eq:trans01}
\chi_{01} = -\eta_{L}g_{11}(N-1) \sum_{n=1}^{\infty} \xi_{n} \frac{\langle \xi_{n} | \chi_{00}^{3} \rangle}{E_{n} - \mu_{0}},$$ where $\xi_{n}$ are the eigenfunctions of the two dimensional, transverse harmonic trap with corresponding energies $E_{n}$ and $\mu_{0} = E_{0}$ where $E_{0}$ is the ground state energy of the transverse trap. Defining $$\phi_{0}(z) = \phi_{00}(z) + \phi_{01}(z) \qquad {\rm and} \qquad \tilde{\mu}_{L} = \mu_{L} + \mu_{1},$$ the first order correction $\phi_{01}$ can be obtained from the reduced Gross Pitaevski like equation for $\phi_{0}$, $$\label{eq:long01}
\bigg[ \frac{1}{2} m \omega_{L}^{2} z^{2} + (N-1)g_{11} \eta_{T} \phi_{0}^{2} - 3 g_{11}^{2} (N-1)^{2} \Gamma_{T} \phi_{0}^{4}\bigg] \phi_{0} = \tilde{\mu}_{L} \phi_{0},$$ where $$\label{eq:GammaT}
\Gamma_{T} = \sum_{n=1}^{\infty} \frac{\langle \xi_{n} | \xi_{0}^{3} \rangle^{2}}{E_{n} - \mu_{0}}.$$ Note that in Eq. (\[eq:long01\]) we have ignored the kinetic energy term to be consistent with the Thomas-Fermi wave function we are using for $\phi_{00}$.
For a cigar shaped BEC, we have from [@taclacaves], $$\chi_{01} (\rho) = -a_{11} \eta_{L} (N-1) \sum_{n_{r} = 1}^{\infty} \frac{\xi_{n_{r} 0} (\rho)}{2^{n_{r}} n_{r}},$$ where $n_{r}$ is the radial quantum number that appears when the eigenfunctions of the two dimensional, transverse, harmonic potential is written in plane polar coordinates. The eigenfunctions with azimuthal quantum number $m$ equal to zero which have finite overlap with ground state wave function and and its powers are given by $$\xi_{n_{r}0} = e^{-\rho^{2}/2 \rho_{0}^{2}} L_{n_{r}}(\rho^{2}/\rho_{0}^{2}) \sqrt{\pi} \rho_{0},$$ where $L_{n_{r}}(x)$ are the Laguerre polynomials. For a cigar shaped trap, we also have, $$\Gamma_{T} = \frac{\eta_{T}^{2}}{2 \hbar \omega_{T}} \ln \frac{4}{3}.$$ The algebraic, fourth order equation (\[eq:long01\]) for $\phi_{0}$ has solutions $$(\phi_{0})^{2} = \frac{\eta_{T} \pm \eta_{T} \big[ 1 - 12 \Gamma_{T} \eta_{T}^{-2} (\tilde{\mu}_{L} - m\omega_{L}^{2} z^{2}/2 )\big]^{1/2}}{6 g_{11} (N-1) \Gamma_{T}}.$$ The solution with the minus sign is consistent with the requirement that when the term containing $\Gamma_{T}$ in Eq. (\[eq:trans01\]) is absent, the solution reduces to the Thomas-Fermi wave function in Eq. (\[eq:lwave\]). The normalization of $\phi_{0}$ is used to find the unknown quantity $\mu_{1}$ that determines $\tilde{\mu}_{L}$. From $\phi_{0}$ we get $\phi_{01} (z) = \phi_{0}(z) - \phi_{00} (z)$. In Fig. (\[fig:figure5\]), plots of $\phi_{0}$ and the correction $\phi_{01}$ are shown and we see that for atom numbers between $N_{L}$ and $N_{T}$, the correction narrows down the Thomas-Fermi wave function as expected and consequently increases $\eta_{L}$. We also see that because we are keeping only the first order correction to the longitudinal wave function, there is a tendency to over-correct the wave function as $N$ increases as noted in [@taclacaves]. This can be mitigated by going to higher orders but then the wave function will not remain separable between the transverse and longitudinal dimensions.
With $\eta_{T}$ and $\eta_{L}$ computed as $$\begin{aligned}
\eta_{T} & = & \int d \rho \, |\chi_{00} (\rho) + \chi_{01}(\rho) |^{4} , \\
\eta_{L} & = & \int dz\, |\phi_{00}(z) + \phi_{01} (z) |^{4},\end{aligned}$$ we can compute the expected signal with position dependent phase as $$\label{eq:T3}
T_{3}(t) = {\rm Im} \Big[ e^{-i \Omega_{N}t} \int dz \, |\phi_{0}|^{2} e^{-i \Omega_{N}t (|\phi_{0}|^{2} - \eta_{L})/\eta_{L}} \Big].$$ The $N$ dependence of the deviation $D_{3}(N)$ defined using Eqs. (\[eq:T3\]) and (\[eq:compare1\]) is compared with $D_{1}(N)$ and $D_{2}(N)$ in Figure \[fig:figure6\].
We see that adding the perturbative correction further reduces the cumulative RMS deviation between the theoretically expected signal and the actual one. The smaller deviation translates to a better estimate of $a_{22}$. The deviation can be further reduced by dropping the Thomas-Fermi approximation and solving the reduced GP equation for the longitudinal wave function numerically with and without the perturbative corrections.
Conclusion \[concl\]
====================
The main question we have addressed in this paper is the choice of function with one free parameter to fit the data from a proof-of-principle quantum metrology experiment using a two mode BEC in a highly anisotropic, cigar shaped trap. We have looked at three analytical models based on different simplifying assumptions that give theoretically expected signals which when fitted to the observations will yield the value of the measured parameter. We computed the deviation of the theoretically expected signal from simulated data produced by numerically integrating the coupled GP equation describing the system. In the numerical integration of the GP equation we assign a value to the measured parameter. We computed the theoretical signal for the same value of the parameter and quantified the deviation between the theoretical and simulated curves by taking root-mean-squared difference between the two. We showed that the perturbative approach to finding the initial state of the BEC in the anisotropic trap proposed in [@taclacaves] led to a better theoretical fit for certain ranges of atom numbers in the BEC. It is possible to have a hybrid approach as in [@Boixo-Tacla] and use the values of $\eta_{T}$ and $\eta_{L}$ obtained from the numerically computed initial state of the BEC, which does not depend on $a_{22}$, in Eq. (\[eq:rel2\]). However our focus is on how well the theoretical models predict the behavior of the metrology setup and hence we do not include this approach in the present discussion. In [@taclacaves2] extending the perturbative approach to obtain corrections to the time evolution of the two mode BEC is discussed. The measured parameter appears in the perturbative equations themselves and not just in the solutions which are theoretically expected signals. Going beyond the results presented in this Paper, the theoretically expected signal can be further improved by solving the perturbative equations in [@taclacaves2] treating $a_{22}$ as a fitting parameter.
The authors thank Alexandre B. Tacla for a critical reading of the manuscript and valuable comments. This work is supported in part by a grant from the Fast-Track Scheme for Young Scientists (SERC Sl. No. 2786), and the Ramanujan Fellowship programme (No. SR/S2/RJN-01/2009), both of the Department of Science and Technology, Government of India.
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abstract: 'The goal of Author Profiling (AP) is to identify demographic aspects (e.g., age, gender) from a given set of authors by analyzing their written texts. Recently, the AP task has gained interest in many problems related to computer forensics, psychology, marketing, but specially in those related with social media exploitation. As known, social media data is shared through a wide range of modalities (e.g., text, images and audio), representing valuable information to be exploited for extracting valuable insights from users. Nevertheless, most of the current work in AP using social media data has been devoted to analyze textual information only, and there are very few works that have started exploring the gender identification using visual information. Contrastingly, this paper focuses in exploiting the visual modality to perform both age and gender identification in social media, specifically in Twitter. Our goal is to evaluate the pertinence of using visual information in solving the AP task. Accordingly, we have extended the Twitter corpus from PAN 2014, incorporating posted images from all the users, making a distinction between tweeted and retweeted images. Performed experiments provide interesting evidence on the usefulness of visual information in comparison with traditional textual representations for the AP task.'
author:
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Miguel A. [A]{}lvarez-Carmona$^1$, Luis Pellegrin$^1$, Manuel Montes-y-Gómez$^1$,\
Fernando Sánchez-Vega$^1$, Hugo Jair Escalante$^1$, A. Pastor López-Monroy$^2$,\
Luis Villaseñor-Pineda$^1$, Esaú Villatoro-Tello$^3$\
$^1$*Computer Science Department*\
*Instituto Nacional de Astrofísica, Óptica y Electrónica*\
*Luis Enrique Erro 1, Puebla 72840, México*\
$^2$ *Research in Text Understanding and Analysis of Language Lab*\
*University of Houston*\
4800 Calhoun Road, Houston, TX 77004, USA\
$^3$ *Language and Reasoning Research Group*\
*Information Technologies Department, Universidad Autónoma Metropolitana*\
*Unidad Cuajimalpa (UAM-C), Ciudad de México, 05348, México*
title: 'A visual approach for age and gender identification on Twitter[^1]'
---
**Keywords:** Visual author profiling, age identification, gender identification, social media, twitter, CNN representation.
Introduction
============
Nowadays there is a tremendous amount of information available on the Internet. Specifically, social media domains are constantly growing thanks to the information generated by a huge community of active users. Such information is available in several modalities, including text, image, audio and video. The availability of all this information plays an important role in designing appropriate tools for diverse tasks and applications. Particularly, during recent years, the Author Profiling (AP) task has gained interest among the scientific community. AP aims at revealing demographic information (e.g., age, gender, native language, personality traits, cultural background) of authors through analyzing their written texts [@koppel2002automatically]. The AP task has a wide range of applications and it could have a broad impact in a number of problems. For instance, in forensics, profiling authors could be used as valuable additional evidence; in marketing, this information could be exploited to improve targeted advertising.
As known, social media data has a multimodal nature (e.g., text, images, audio, social interactions), however, most of the previous research on AP has been devoted to the analysis of the textual modality [@burger2011discriminating; @koppel2002automatically; @nguyen2013old; @peersman2011predicting; @schler2006effects], disregarding information from the other modalities that could be potentially useful for improving the performance of AP methods. Accordingly, some works have begun to exploit distinct modalities for approaching the AP problem [@Can13; @Merler15; @taniguchi; @You13]. The visual modality has resulted particularly interesting, mostly because it is, to some extent, language independent nature. In fact, previous work has found a relationship between images and users’ interests, opinions and thoughts [@Cristani13; @Eftekhar14; @Hum11; @Wu15; @YangHE15; @You16; @You14].
Although visual information is particularly appealing for AP, it is just recently that some authors began to pay attention to the content of images shared by users. For example, for gender identification, some authors have exploited the information provided by the colors adopted by users in their profiles [@Lovato14]. In [@Azam16] authors used state of the art face-gender recognizers over user profile pictures. Nonetheless, the most common strategy so far consist in exploiting the *posting behavior*, which implies the manual classification of posted images in order to analyze histograms of classes/objects posted by users [@Hum11].
Despite previous efforts for including visual information in the AP task, only the gender recognition problem has been studied, leaving the age identification problem unexplored. In addition, many of the previous research considers an scenario where manually tagged images are provided for training, resulting in an impractical and unrealistic scenario for AP systems.
In order to overcome these limitations, we present a thorough analysis on the pertinence of visual information for approaching the AP problem, targeting both, age and gender identification. Our study comprises an analysis on the discriminative capabilities of tweeted and retweeted images by users. As part of the study, a method for AP using images is proposed in this paper. The proposed method relies on a representation derived from a pre-trained convolutional neural network. Through our study, we aim to bring some light into: *i)* the importance of just the visual information for solving the AP tasks (age and gender), and *ii)* how complementary are both textual and visual information for the age and gender identification problems. The main contributions of this paper are as follows:
- We built an extended *multimodal* version of the PAN 2014 AP corpus, a reference data set for AP in social media. For this, we incorporated all of the images from the users’ profiles contained in the original corpus.
- We propose a method for AP from images based on state-of-the-art (CNNs) representation learning techniques, which have not been previously used for this task.
- We propose a methodology for addressing the age identification problem using posted images in Twitter. To the best of our knowledge, this is the first effort in approaching this task by using purely visual information.
- We provide a comparative analysis on the importance of using textual and visual information for age and gender identification.
- We evaluate the usefulness of images in the AP task, whether they are tweeted or retweeted by the users.
The remainder of this paper is organized as follows. Section \[relwork\] reviews related work on AP using textual, visual and multimodal approaches. Section \[visadop\] describes the proposed methods for AP using images. Section \[dataset\] describes the adopted methodology for building a multimodal corpus for AP in Twitter. Section \[expres\] presents our experimental results, which comprise quantitative and qualitative evaluation. Finally, Section \[conclus\] outlines our conclusions and future work directions.
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------- --------------------- ----------------------------
*Gender* *Age* *Personality* *Interests* *Others*
[@argamon2003gender; @argamon2007mining; @argamon2009automatically; @burger2011discriminating; @cheng2011author; @goswami2009stylometric; @herring2006gender; @koppel2002automatically; @Lopez-Monroy15; @mukherjee2010improving; @Ortega16; @otterbacher2010inferring; @peersman2011predicting; @rao2010classifying; @sarawgi2011gender; @schler2006effects; @yan2006gender] [@argamon2003gender; @argamon2007mining; @argamon2009automatically; @goswami2009stylometric; @Lopez-Monroy15; @nguyen2013old; @nguyen2011author; @Ortega16; @peersman2011predicting; @rao2010classifying; @schler2006effects] [@Litvinova16; @Litvinova15] [@Li2015; @penas13] sentiment anal. [@Rosso16]
[@Azam16; @Hum11; @Ma14; @Shigenaka16; @You14] — [@Cristani13; @Eftekhar14; @Lovato14; @Wu15] [@YangHE15; @You16] retweet predict. [@Can13]
[@Merler15; @taniguchi] — — — sentiment anal. [@You13]
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------- --------------------- ----------------------------
Related Work {#relwork}
============
According to the literature, AP in social media has two main subtasks: age and gender detection (see [@argamon2003gender; @argamon2007mining; @argamon2009automatically; @burger2011discriminating; @cheng2011author; @goswami2009stylometric; @herring2006gender; @koppel2002automatically; @Lopez-Monroy15; @mukherjee2010improving; @Ortega16; @otterbacher2010inferring; @peersman2011predicting; @rao2010classifying; @sarawgi2011gender; @schler2006effects; @yan2006gender] for gender, and [@argamon2003gender; @argamon2007mining; @argamon2009automatically; @goswami2009stylometric; @Lopez-Monroy15; @nguyen2013old; @nguyen2011author; @Ortega16; @peersman2011predicting; @rao2010classifying; @schler2006effects] for age). Related tasks include personality prediction [@Litvinova16; @Litvinova15], interests identification [@Li2015; @penas13] (a.k.a. genres), sentiment and emotion recognition [@Rosso16], among others. Generally speaking, AP has been approached as a single-label classification problem, where the target profiles (e.g., males *vs.* females) stand for the target classes.
To accurately model target profiles it is necessary to extract general demographic-features that apply to heterogeneous groups of authors, and indicate, to some extent, how they use them given their native language, genre, age, etc. [@argamon2003gender]. Thus, the AP task in social media is particularly challenging given the nature of Internet interactions and the informality of written language.
Table \[tab:relwor\] provides a summary of related work on AP. We can distinguish three broad approaches for addressing the AP task: textual, visual and multimodal. Notice that both visual and multimodal approaches have been less studied (see last two rows in Table \[tab:relwor\]). Regarding the textual approach, authors have proposed combinations of textual attributes ranging from lexical features (e.g., content words [@argamon2003gender] and function words [@koppel2002automatically]) to syntactical features (e.g., POS-based features [@bergsma2012stylometric], personal phrases [@Ortega16], and probabilistic context-free grammars [@sarawgi2011gender]). Concerning the visual approach, most of the research had focused on gender recognition [@Azam16; @Ma14; @Shigenaka16; @You14], and have only considered some general statistics about the shared images as features [@Hum11]. Similarly, some works have considered visual information for the task of personality prediction [@Cristani13; @Eftekhar14; @Lovato14; @Wu15]. For instance, in [@Lovato14], authors try to determine users’ behavioral biometric traits analyzing their favorite images in Flickr. As features, authors considered the average size of the regions, colorfulness, and wavelet textures among others. They conclude that images are very good elements for determining people’s behavior. Similarly, in [@Cristani13] authors proposed a method for identifying personality traits from Flickr users. Their obtained results showed a strong correlation between the personality and the favorite image of users.
Regarding multimodal approaches, it is worth mentioning that this type of strategies has just recently starting to gain attention. For example, in [@taniguchi] authors proposed a weighted text-image scheme for gender classification of twitter users. The basic idea consist on identifying a set of image categories associated for male and female classes. This process is done through the use of a CNN, which is used for determining a score for every user’s image. At the end, computed scores are averaged and combined with textual information for approaching the gender identification problem.
Contrary to previous research, this paper focuses in exploiting the visual modality to perform both age and gender identification in social media. Accordingly, we have extended a well-known benchmark corpus by means of incorporating images into users’ profiles. Thus, we are able to perform a comparative analysis on the importance of using textual and visual information for the age and gender identification problem. In addition, we also evaluate the utility of images when they are original images (i.e., tweeted by the user) or reused images (i.e., retweeted from other user’s post). Last, but not least, our visual approach to AP is based on state of the art visual descriptors (CNN-based).
Author profiling from images {#visadop}
============================
In this section we describe the proposed visual approach for AP. Firstly, Section \[representation\] describes the adopted image representation. Then, Section \[methods\] introduces two different methods for performing AP in Twitter using information from posted images.
CNN-based image representation {#representation}
------------------------------
Defining robust and discriminative features for images is not a trivial task. A direct way for defining these features is manually, through handcrafted features provided by experts or using a mechanical turk approach [@Sorokin08]. As it is possible to imagine, this approach is highly inviable due to the number of images that would have to be labeled in a social media domain. Hence, we adopted an alternative solution based on feature learning by using a pre-trained deep learning model[^2] [@Oquab14; @Yosinski14]. Deep models are composed of multiple processing layers that allow to learn representations of data with multiple levels of abstraction [@Lecun15]. For instance, when an image is propagated in a pre-trained deep model, it is processed layer-by-layer transforming an array of pixel values (an image) into a representation that amplifies important aspects of the input and suppress irrelevant variations for discrimination [@Lecun15]. This methodology has reported outstanding results in a number of computer vision tasks. Our intuition is that this type of representation can be beneficial in solving the posed task.
As mentioned before, our dataset is composed of images from Twitter. Thus, we choose a pre-trained model general enough to cover the visual diversity of target images. As known, a pre-trained model will perform better, under a transfer learning scenario, when the target dataset shares a similar distribution with the source dataset [@transferDL]. Accordingly, we used the 16-layer CNN-model called VGG [@Simonyan14]. This model was trained on the ImageNet dataset [@Russakovsky15], a large visual database designed for visual object recognition, including classification and detection of objects and scenes.
Every image’s representation is obtained by passing its raw input (pixels) through the ConvNet model using the Caffe Library [@Jia14]. Then the activation of an intermediate layer is used as the representation of the feed image. For our experiments, we choose the 4096 activations produced in the last hidden layer of the net.
The activation of this layer produces similar values when similar images are introduced [@Krizhevsky12]. Note that our CNN representation did not rely on the last layer of the net, which produces detection scores over 1000 different classes. The reason is that transferability is negatively affected by the specialization of higher layer neurons to their original task at the expense of performance on the target task [@Yosinski14]. In our case, an abstract representation over 4096 neurons would be reduced to 1000 different classes. Therefore, for representing images we use the last hidden layer, and the final layer is employed only for the qualitative analysis reported in Section \[analysis\].
AP from visual and multimodal information {#methods}
-----------------------------------------
This section describes two different visual-based methods for AP in social media. Also, it presents an effective multimodal approach that jointly uses textual and visual information, as well as a baseline method exclusively based on the use of textual information.
1. **Visual methods**. The two proposed methods for AP from images are illustrated in Figure \[approach\]. Both of them use the same input information and apply the same process for building the images’ representation; this is, given the set of images from a user (profile), each image is passed through the pre-trained deep model obtaining its vector representation (see upper box in the figure). Next, the obtained information can be exploited in two different ways, deriving in our two proposed strategies:
1. *Individual classification*: first, each image from a user is classified individually, then, the AP class of the user is determined by means of a majority-vote strategy.
2. *Prototype classification*: first, each user is represented by a prototype vector built by averaging the CNN representations from all his/her images. Then, this prototype is feed to a standard classier that outputs the AP class of the user.
![Two visual approaches for AP in Twitter: (a) individual-based classification and (b) prototype-based classification.[]{data-label="approach"}](figures/approach){width="8cm"}
2. **Multimodal method**. This method follows the same pipeline that 1.(b) for building the images’ representation, however, it is called a multimodal representation since it combines visual and textual information. Specifically, we build the multimodal prototype of each user by concatenating the visual prototype with a traditional BOW representation from all the tweets that the user has posted.
3. **Textual method**. As before, a prototype representation is built for each user, however, for this method we consider only textual information. Two different BOW representations were used in the experiments, containing the 2k and 10k most frequent terms from the training data respectively.
More details regarding the implementation of these methods are given in Section \[expres\]. Next, we introduce the multimodal corpus specifically assembled for evaluating our proposed approach.
A multimodal corpus for twitter AP {#dataset}
==================================
Images shared by social media users tend to be strongly correlated to their thematic interests as well as to their style preferences. Motivated by these facts we tackled the task of assembling a corpus considering text and images from twitter users. Mainly, we extended the PAN-2014 [@rangel2014overview] dataset by harvesting images from the already existing twitter users.
The PAN-2014 twitter dataset considers gender profiles (female vs. male), and five non-overlapping age profiles. It includes tweets (only textual information) from English and Spanish users. Based on this dataset we harvested more than 150,000 images, corresponding to a subset of 450 user profiles, 279 profiles in English and 171 in Spanish[^3]. The images associated to all of the users were downloaded and lined to existing user profiles, resulting in a new multimodal twitter corpus for the AP task. Next subsections present detail information from the new corpus.
Statistics of the multimodal twitter corpus
-------------------------------------------
Table \[tab:stats\] presents general statistics of the new multimodal AP corpus, which includes around 85,000 images for the English users and approximately 73,000 for the Spanish profiles. Given our interest in studying the discriminative capabilities of tweeted and retweeted images, we have separated both kinds of images; approximately 50% of the collected images correspond to each kind. It is worth noting that although there is a considerable amount of images per profile, there is a high standard deviation in both corpora, indicating the presence of some users with very few images in their profiles. Table \[tab:stats\] also shows that users from the Spanish corpus posted more images (40% more) than the users from the English corpus.
**EN** **SP**
-- ------------------ ---------------- ----------------
by *profile* 304 ($\pm$340) 428 ($\pm$409)
in *tweet* set 159 ($\pm$239) 208 ($\pm$304)
in *retweet* set 144 ($\pm$188) 220 ($\pm$223)
: General statistics of the images from the English (EN) and Spanish (SP) corpora.[]{data-label="tab:stats"}
Tables \[tab:stages\] and \[tab:stgenre\] present additional statistics on the values that both variables, gender and age can take, respectively. On the one hand, Table \[tab:stages\] divides profiles by age ranges, i.e., 18-24, 25-34, 35-49, 50-64 and 65-N. Both languages show an important level of imbalance, being the 35-49 class the one having the greatest number of users, while extreme ages are the ones with the lowest. Nonetheless, the users from the 65-N range are the ones with the greatest number of posted images as well as the lower standard deviation values. It is also important to notice that, in both corpora, the users belonging to the 50-64 range share in average a lot of images, but show a large standard deviation, indicating the presence of some users with too many and very few images.
-- -- -------- ---------------- ---------------- ----------------
**\#** *by profile* *in tweets* *in retweets*
17 246 ($\pm$80) 141 ($\pm$50) 105 ($\pm$35)
78 286 ($\pm$202) 148 ($\pm$118) 137 ($\pm$109)
123 301 ($\pm$253) 154 ($\pm$155) 147 ($\pm$138)
54 334 ($\pm$238) 174 ($\pm$168) 160 ($\pm$120)
7 441 ($\pm$102) 291 ($\pm$76) 150 ($\pm$53)
12 254 ($\pm$99) 123 ($\pm$58) 131 ($\pm$45)
36 331 ($\pm$198) 154 ($\pm$101) 177 ($\pm$109)
85 414 ($\pm$341) 207 ($\pm$234) 207 ($\pm$170)
32 565 ($\pm$308) 258 ($\pm$197) 307 ($\pm$179)
6 808 ($\pm$173) 440 ($\pm$116) 368 ($\pm$87)
-- -- -------- ---------------- ---------------- ----------------
: Statistics of images shared by each age category, in both English (EN) and Spanish (SP) corpora.[]{data-label="tab:stages"}
On the other hand, Table \[tab:stgenre\] reports some statistics for each gender profile. It is observed a balanced number of male and females users in both corpora as well as a similar number of shared images.
-- -- -------- ---------------- ---------------- ----------------
**\#** *by profile* *in tweets* *in retweets*
140 162 ($\pm$294) 83 ($\pm$182) 79 ($\pm$158)
139 141 ($\pm$274) 75 ($\pm$192) 65 ($\pm$144)
86 228 ($\pm$372) 104 ($\pm$236) 124 ($\pm$205)
85 200 ($\pm$347) 104 ($\pm$242) 96 ($\pm$178)
-- -- -------- ---------------- ---------------- ----------------
: Statistics of images shared by each gender category, in both English (EN) and Spanish (SP) corpora.[]{data-label="tab:stgenre"}
Experimental results {#expres}
====================
This section presents experimental results on the multimodal twitter corpus introduced previously. This section is divided in two parts: (1) a comparison among different methods for performing AP, followed by (2) a discussion based on a purely visual evaluation. Overall, our aim is to show how useful the images are to approach the AP task.
-- --------------------------- ----------- ------------- ----------- -----------
*methods* age (EN) gender (EN) age (SP)
T1: BoW (2k) 0.394 0.741 0.481 0.601
T2: BoW (10k) 0.409 0.755 **0.505** **0.703**
V3: LL-CNN (all-imgs) 0.349 0.526 0.481 0.524
V4: LL-CNN AVG (all-imgs) 0.390 0.700 0.380 0.650
M3: T1+V4 0.414 0.775 0.451 0.685
M6: T2+V4 **0.423** **0.778** 0.433 0.642
-- --------------------------- ----------- ------------- ----------- -----------
Comparison of textual, visual and multimodal methods for AP
-----------------------------------------------------------
This subsection compares the performance (classification accuracy) of the methods introduced in Section \[visadop\] when approaching the AP task. Evaluation is carried out by profile, allowing us to make a fair comparison among the different approaches. In order to provide comparable results, using the profile ID in the PAN-2014 twitter corpus, we construct 10 subject-independent partitions (including at least one subject from each class in each partition) and we adopt 10-fold cross-validation strategy for evaluation. As expected, partitions are unbalanced with respect to number of images. We used a SVM using LibLinear [@Fan08] for classification, making a direct comparison among evaluated approaches. Thereby, although different representations are used by the methods, the information came from the same profiles. Besides, the evaluated representations used by the methods do not include any preprocessing, this is to perform a fair comparison among different modalities.
Considering the proposed visual methods, four variants were evaluated: three of them using individual classification, considering all-images (i.e. V3 in Table \[tab:compare\]); and one using prototype classification in the all-images subset (V4 in Table \[tab:compare\]). In this last case, we decided to include only the variant with the biggest performance.
Table \[tab:compare\] presents the obtained results, highlighting in bold the best obtained result by column. On the one hand, surprisingly, the evaluation in English corpus reveals that using a multimodal approach is better for detecting age and gender, when all images are collapsed into a prototype. Regarding the visual methods, they perform poorly when compared to the textual one (except V4).
On the other hand, the best results on the evaluation in the Spanish corpus are achieved by textual approaches, especially when more terms in BoW are considered, i.e. 10k. However, it is worth noticing that for gender identification, the V4 method is very competitive (as in the case of the English corpus). The multimodal approaches obtained quite competitive performance as well, but they were not able to outperform the text-only results.
With the goal of studying the discriminative properties of tweeted and retweeted images, we performed an additional experiment in which both types of images were separated and then we evaluated our proposed visual and multimodal approaches. The obtained results reveals an accuracy performance of 0.432 for the V3 method using retweeted images, surpassing the results obtained in age identification for English corpus. However, none other improvements were observed.
How much a single image says about the user
-------------------------------------------
This scenario aims to evaluate globally the visual information in images. In this setting, every image from a profile is an individual instance where the label is the same that was assigned to the profile. We have evaluated gender and age for the two corpora in PAN-2014. For the sake of the evaluation, we have included the probability for each class evaluated.
From Table \[tab:eval0\], we can confirm the usefulness of visual information for performing gender identification, obtaining in most of the cases better results than the class probability in both corpora. Besides, a better performance is observed in the female gender. Instead, the age recognition task seems to be more challenging under this scenario, it seems that images posted by users are more diverse in terms of age.
Interestingly, although, the highest performances are reached for the majority classes, these are only higher than the class probability for the Spanish corpus. Whereas for the English language, results for three age intervals surpassed the class probability.
-- -- -- --
-- -- -- --
: Accuracy performance on age and gender identification.[]{data-label="tab:eval0"}
\
### The importance of tweeted and retweeted images
In view of the results obtained earlier, we repeated the experiment (same evaluation protocol), but this time separating the source of the images, i.e. tweeted and retweeted. This distinction aimed at answering the following questions:
1. Do the posted (tweet) and re-shared (retweet) images express author’s interests in the same way?
2. Do any of these ways of sharing images give more information about the user’s profile?
Thus, three specific scenarios were defined according to the source of the images: **(a)** testing all-images (from both sources) and training with one source, i.e. tweeted images, retweeted images, and all-images; **(b)** training with all-images and testing in one of the two sources; and finally **(c)** testing and training using one of the two sources. The obtained results are presented in Table \[tab:eval1\].
From Table \[tab:eval1\] we can stress the following:
- Results for scenario **(a)** indicate that only using an image source for training is comparable and even better to use both sources together, in age and gender identification (see results from scenario **(a)** when compared with results from Table \[tab:eval0\]). It is interesting that English and Spanish take advantage of different sources, English presents better performances using retweeted images for training, whilst Spanish from tweeted images.
- Scenario **(b)** allows us to compare the results obtained in each source by including all-images as training. In the case of age, only the retweeted images in Spanish have presented a considerable increment in performance, i.e. from 0.29 to 0.34.
- Scenario **(c)** allows us to compare the results obtained with those obtained in scenario **(b)**. Here, the training is reduced to individual sources in comparison to scenario **(b)** that uses all-images. In general, we can see decrements in age and gender identification over English, with an exception in retweeted images where it has obtained better results. Instead, the tweeted images for gender identification in Spanish has been the only with an increment in performance.
Summarizing the obtained results from the experiments in this section, we have showed that it is feasible, using only the visual information from posted images, to approach the gender identification task and in some cases the age identification as well. Moreover, we have found that, in fact, the image source matters (i.e. tweeted or retweeted), and it is possible to exploit it for achieving better results on the age and gender identification.
A picture is worth a thousand words
-----------------------------------
Inspired by the saying *’a picture is worth a thousand words’*, we decided to classify individual profiles taken randomly. Some works have followed the same idea, for instance [@Hum11] has performed a statistical analysis by genders, and [@You16] has constructed a dataset from Pinterest with the aim to classify images from a user into categories. Instead, here explicitly each profile is defined by a tweet composed of only 1000 words, and after classified. In return, images from the same profile are classified. Both, classification schemes are compared trying to answer whether it is possible to say more with a picture than with thousand words.
In order to approximate the same conditions for both classification scenarios, the training instances are provided from the same profiles but of course using its respective representation. For the case of textual approach, samples of 1000 words are used for textual representation, repeating as many subsets of this size can be extracted from the profile. Instead, all images in profiles are classified for the visual approach. Obtained results are presented in Tables \[tab:1img\_age\] and \[tab:1img\_gen\] for age and gender identification, respectively.
In general, results from Table \[tab:1img\_age\] indicate that it is possible to identify with reasonable accuracy some age ranges by using images only, this holds for both languages. Especially, minority classes where using only 1000 words it is not enough. Even more interesting is to observe that the image source is important, presenting better results than using all-images without making a distinction.
-- ------------ ---------- ----------- ----------- ----------- -----------
***ages***
0% 0% 4.6% **12.6%** 4%
0% 14.2% **29.7%** 13.8% 17%
**100%** **100%** 30% 41.8% 33.1%
14.2% 14.2% 38.5% **41.8%** 33.1%
0% 0% 0% **8.3%** 0%
18-24 0% 0% 3.1% 3.1% **14.2%**
25-34 0% 11.1% 28.1% **39.6%** 31.4%
35-49 33% **50%** 38.3% 27.6% 8.5%
50-64 18.1% 18.1% 20.1% 29.1% **29.4%**
65-N 0% **14.2%** 0.3% 1.4% 9.1%
-- ------------ ---------- ----------- ----------- ----------- -----------
: Accuracy performance for age ranges.[]{data-label="tab:1img_age"}
On the other hand, results obtained by gender identification indicate that it is kind of easier to determine whether a profile belongs a female/male person by their images than by their words. Interestingly, for female gender it is better using the tweeted images for performing the identification, while for the male gender using retweeted images works better.
-- -------------- ----------- ----------- ----------- -----------
*BoW (2k)* 40% 0% 20% 25%
*BoW (10k)* 60% 20% 20% 50%
*all-images* 54.6% 55.5% 52.5% 55.7%
*tweets* **75.7%** 37% **95.7%** 5.3%
*retweets* 55.2% **59.2%** 38.9% **62.8%**
-- -------------- ----------- ----------- ----------- -----------
: Accuracy performance for gender.[]{data-label="tab:1img_gen"}
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Qualitative analysis of the posted images {#analysis}
-----------------------------------------
This section presents qualitative experiments in order to show how useful are images for exploring information in Twitter. Our aim to perform this study is to show the visual evidence left by users in a lapse of time.
For this experiment we represented images with the final layer of the considered CNN. The dimensions correspond to the 1000 categories in ImageNet [@Russakovsky15], thus the higher the value is, then the more likely the corresponding category appears. Of course it is unlikely that images posted in a ’wild’ scenario could be represented by only 1000 classes. However, through this kind of experiments we can define user preferences using denotative descriptions, i.e. labels assigned.
Hence, in order to analyze the content in images posted by the users, we labeled all-images using these 1,000 ImageNet categories [@Simonyan14]. After classification, images in a specific gender or age range are concentrated in normalized histograms. A similar study has been carried out in [@YangHE15] using Pinterest images under the *Travel* category, nevertheless, the authors intended to answer whether user-generated visual contents had predictive capabilities for users’ preferences beyond labels.
Figure \[words\_manvsfemale\] shows a list of words with frequency order (top to down), comparing how often they are used by gender in both corpora, i.e. male versus female. Besides, a sample of images posted accompany both genders. For producing such word list, the difference is calculated over normalized histograms of genders. We have taken 20 words with the biggest difference considering equal number of words in favor of each gender.
On the one hand, in the English corpus (see \[fig:manen\], \[fig:genen\] and \[fig:femen\] in Figure \[words\_manvsfemale\]), the male gender users seem to post images associated to topics as sports (i.e. ’mountain bike’, ’scoreboard’, etc.), machines and vehicles (i.e. ’airship’, ’trailer truck’, ’streetcar’, etc.). Whereas, for female gender users, there is a topic related to beauty products (’hair spray’, ’perfume’, ’hand blower’), and another associated to fashion (i.e. ’velvet’, ’wig’, ’pajama’, etc.).
On the other hand, in the Spanish corpus (see \[fig:manes\], \[fig:genes\] and \[fig:femes\] in Figure \[words\_manvsfemale\]), the male gender users are prone to post images that include sports (i.e. ’scoreboard’ and ’digital clock’) as well as technology (i.e. ’monitor’, ’hand-held computer’, ’iPod’, etc.). While the female gender users posted images related to fashion (i.e. ’wig’, ’miniskirt’, ’bonnet’, etc.).
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A second qualitative evaluation is presented in Figure \[words\_most\_labeled\], this time showing the most posted words by age ranges in both corpora. For this purpose, each row presents an age range by a sample of images and its respectively cloud words. Here the size of the word indicates its frequency. The idea is to allow the reader to judge the visual information that we can extract from images posted in Twitter.
Conclusions and Future Work {#conclus}
===========================
This paper explored the use of visual information to perform both age and gender identification in social media, specifically in Twitter. Novel methods for AP using visual information were proposed, as well as techniques based on multimodal techniques (text+images) for approaching the task. The models incorporating visual information rely on a CNN for feature extraction. The usefulness of images for AP was also explored by contrasting performance when using tweeted and retweeted images for the predictive models. Furthermore, we extended a benchmark data set for AP (PAN-2014 Twitter dataset) to include visual information by incorporating images from the users’ profiles. The release of the later dataset is an important contribution of this work to the state-of-the-art on AP, as it will motivate further research on visual and multimodal approaches to AP.
Using the extended benchmark, we conducted an extensive evaluation including textual, visual and multimodal methods for AP. The obtained results represent relevant evidence on the usefulness of visual information for AP. On the one hand, experimental results suggested that approaches based on multimodal information result in better performance, when compared to single modality approaches in both, age and gender prediction. On the other hand, results indicated that images tend to be more relevant than text for determining the gender of Twitter users. We also found that the usefulness of visual information is somewhat dependent on the language of tweets.
Regarding the analysis on the discriminative capabilities of tweeted and retweeted images, the obtained results did not allow us to formulate a convincing conclusion. However, results seemed to indicate that the image source matters. For example, results showed that females gender identification was more accurate when using tweeted images, whereas for male gender identification using retweeted images worked better. For future work, we plan to extend this study by incorporating more advanced techniques for modeling both visual and textual information. We also consider evaluating the usefulness of other information modalities for AP, such as the posting behavior and video sharing.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by CONACyT under scholarships 401887, 214764 and 243957; project grants 247870, 241306 and 258588; and the Thematic Networks Program (Language Technologies Thematic Network, project 281795).
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[^1]: Preprint of [@thispaper]. The final publication is available at IOS Press through <http://dx.doi.org/10.3233/JIFS-169497>
[^2]: Features learned by a deep model on a generic and very large dataset are transferred to another task.
[^3]: Note that the PAN-2014 corpus includes more profiles in both languages, however, for some twitter users it was impossible to download their associated images.
|
---
abstract: 'We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.'
address:
- 'CIMAT, Callejón Jalisco S/N, Mineral de Valenciana, Guanajuato 36240, Mexico\'
- 'Statistics Program, University of Delaware, 213 Townsend Hall Newark, DE 19716, USA\'
author:
-
-
title: A note on a maximal Bernstein inequality
---
*Dedicated to the memory of Sándor Csörgő*
Introduction and statement of main result
=========================================
Let $X_{1},X_{2},\dots,$ be a sequence of independent random variables such that for all $i\geq1$, $EX_{i}=0$ and for some $\kappa>0$ and $v>0$ for integers $m\geq2$, $E\vert X_{i}\vert ^{m}\leq
vm!\kappa^{m-2}/2$. The classic Bernstein inequality (cf. [@SW], page 855) says that, in this situation, for all $n\geq1$ and $%
t\geq0,$$$\mathbf{P}\Biggl\{ \Biggl\vert \sum_{i=1}^{n}X_{i}\Biggr\vert
>t\Biggr\} \leq2\exp\biggl\{ -\frac{t^{2}}{2vn+2\kappa t}\biggr\} .$$ Moreover (cf. [@R], Theorem B.2), its maximal form also holds; that is, we have $$\mathbf{P}\Biggl\{ \max_{1\leq j\leq n}\Biggl\vert
\sum_{i=1}^{j}X_{i}\Biggr\vert >t\Biggr\} \leq2\exp\biggl\{ -\frac{t^{2}}{%
2vn+2\kappa t}\biggr\} .$$ It turns out that, under a variety of assumptions, a sequence of not necessarily independent random variables $X_{1},X_{2},\dots,$ will satisfy a generalized Bernstein-type inequality of the following form: For suitable constants $A>0$, $a>0$, $b\geq0$ and $0<\gamma<2$ for all $m\geq0$, $n\geq1$ and $t\geq0$, $$\label{assumpgamma}
\mathbf{P}\{|S(m+1,m+n)|>t\}\leq A\exp\biggl\{ -\frac{at^{2}}{n+bt^{\gamma}}%
\biggr\} ,$$ where, for any choice of $1\leq k\leq l<\infty$, we denote the partial sum $%
S(k,l)=\sum_{i=k}^{l}X_{i}.$ Here are some examples.
\[ex1\] Let $X_{1},X_{2},\dots,$ be a stationary sequence satisfying $EX_{1}=0$ and . For each integer $n\geq1$ set $S_{n}=X_{1}+\cdots+X_{n}$ and $B_{n}^{2}=\operatorname{Var}( S_{n}) $. Assume that for some $\sigma_{0}^{2}>0$ we have $B_{n}^{2}\geq\sigma_{0}^{2}n $ for all $n\geq1$. Statulevičius and Jakimavičius [@SJ] and Saulis and Statulevičius [@SS] prove that the partial sums satisfy (\[assumpgamma\]) with constants depending on a Bernstein condition on the moments of $X_{1}$ and the particular mixing condition that the sequence may fulfill. In fact, all values of $1\leq\gamma<2$ are attainable. Their Bernstein-type inequalities are derived via a result of [@BR] relating cumulant behavior to tail behavior, which says that for an arbitrary random variable $\xi$ with expectation $0$, whenever there exist $\gamma\geq0$, $H>0$ and $\Delta>0$ such that its cumulants $%
\Gamma_{k}( \xi) $ satisfy $\vert \Gamma_{k}( \xi) \vert \leq( k!/2)
^{1+\gamma}H/\Delta^{k-2}$ for $k=2,3,\dots,$ then for all $x\geq0$ $$\label{BR}
\mathbf{P}\{ \pm\xi>x\} \leq\exp\biggl\{ -\frac{x^{2}}{2( H+( x/\Delta^{1/(
1+2\gamma) }) ^{( 1+2\gamma) /(1+\gamma)}) }\biggr\} .$$
In Example \[ex1\], $\xi=S_{n}/B_{n}$ and $\Delta=d\sqrt{n}$ for some $d>0$.
Doukhan and Neumann [@DN] have shown, using the result in (\[BR\]), that if a sequence of mean zero random variables $%
X_{1},X_{2},\dots,$ satisfies a general covariance condition, then the partial sums satisfy (\[assumpgamma\]). Refer to their Theorem 1 and Remark 2, and also see [@KN].
Assume that $X_{1},X_{2},\dots,$ is a strong mixing sequence with mixing coefficients $\alpha( n) $, $n\geq1$, satisfying for some $c>0$, $\alpha( n) \leq\exp( -2cn)
$. Also assume that $EX_{i}=0$ and for some $M>0$ for all $i\geq1$, $%
\vert X_{i}\vert \leq M$. Theorem 2 of Merlevéde, Peligrad and Rio [@MPR] implies that for some constant $C>0$ for all $t\geq0$ and $n\geq1$, $$\label{exin2}
\mathbf{P}\{ |S_{n}|>t\} \leq\exp\biggl( -\frac{Ct^{2}}{%
nv^{2}+M^{2}+tM( \log n) ^{2}}\biggr) ,$$ with $S_{n}=\sum_{i=1}^{n}X_{i}$ and where $v^{2}=\sup_{i>0}( \operatorname{Var}(
X_{i}) +2\sum_{j>i}\vert \operatorname{cov}( X_{i},X_{j}) \vert )
>0.$
To see how the last example satisfies (\[assumpgamma\]), notice that for any $0<\eta<1$ there exists a $D_{1}>0$ such that for all $t\geq0$ and $n\geq1$, $$\label{ex2}
nv^{2}+M^{2}+tM( \log n) ^{2}\leq n( v^{2}+M^{2}) +D_{1}t^{1+\eta}.$$ Thus (\[assumpgamma\]) holds with $\gamma=1+\eta$ for suitable $A>0$, $a>0$ and $b\geq0$.
For any choice of $1\leq i\leq j<\infty$ define$$M(i,j)=\max\{|S(i,i)|,\ldots,|S(i,j)|\}.$$ We shall show, somewhat unexpectedly, that if a sequence of random variables $X_{1},X_{2},\ldots,$ satisfies a Bernstein-type inequality of the form ([assumpgamma]{}), then, without any additional assumptions, a modified version of it also holds for $M(1+m,n+m)$ for all $m\geq0$ and $n\geq1$.
\[T1\] Assume that, for constants $A>0$, $a>0$, $b\geq0$ and $\gamma\in(0,2)$, inequality (\[assumpgamma\]) holds for all $m\geq0,n\geq1$ and $%
t\geq0$. Then for every $0<c<a$ there exists a $C>0$ depending only on $A,a$, $b$ and $\gamma$ such that for all $n\geq1$, $m\geq0$ and $t\geq0$, $$\label{Cineq}
\mathbf{P}\{M(m+1,m+n)>t\}\leq C\exp\biggl\{ -\frac{ct^{2}}{n+bt^{\gamma}}\biggr\} .$$
\[RA\] Notice that though $c<a$, $c$ can be chosen arbitrarily close to $a$.
\[R1\] Theorem \[T1\] was motivated by Theorem 2.2 of Móricz, Serfling and Stout [@MSS], who showed that whenever for a suitable positive function $g( i,j) $ of $( i,j) \in \{
1,2,\dots \} \times \{ 1,2,\dots \} $, positive function $%
\phi ( t) $ defined on $( 0,\infty ) $ and constant $%
K>0$, for all $1\leq i\leq\break j<\infty $ and $t>0$, $$\mathbf{P}\{|S(i,j)|>t\}\leq K\exp \{ -\phi ( t) /g( i,j) \} ,$$then there exist constants $0<c<1$ and $C>0$ such that for all $n\geq
1$ and $t>0$, $$\mathbf{P}\{M(1,n)>t\}\leq C\exp \{ -c\phi ( t) /g( 1,n) \} .$$Earlier, Móricz [@M79] proved that in the special case when $\phi ( t) =t^{2}$ one can choose arbitrarily close to $1$ by making $C>0$ large enough. This inequality is clearly not applicable to obtain a maximal form of the generalized Bernstein inequality.
\[R\] We do not know whether there exist examples for which ([assumpgamma]{}) holds for some $0<\gamma<1$ and $b>0$. However, since the proof of our theorem remains valid in this case, we shall keep it in the statement.
\[R2\] The version of Theorem \[T1\] obtained by replacing everywhere $%
|S(m+1,m+n)|$ by $S(m+1,m+n)$ and $M(m+1,m+n)$ by $M^{+}( m+1,m+n)
=\max_{m+1\leq j\leq n+m}( S(m+1,j)\vee0) $ remains true with little change in the proof.
\[R3\] Theorem \[T1\] also remains valid for sums of Banach space valued random variables with absolute value $\vert \cdot\vert $ replaced by norm $\Vert \cdot\Vert $.
\[Statistics\] In statistics, maximal exponential inequalities are crucial tools to determine the exact rate of almost sure pointwise and uniform consistency of kernel estimators of the density function and the regression function. The literature in this area is huge. See, for instance, [@DM92; @DM94; @EM00; @EM05; @GG; @St] and the references therein. These results only treat the case of i.i.d. observations. Dependent versions of our maximal Bernstein inequalities should be useful to determine exact rates of almost sure consistency of kernel estimators based on data that possess a certain dependence structure. In fact, some work in this direction has already been accomplished in Section 4.2 of [@DN]. To carry out such an application in the present paper is well beyond its scope.
Theorem \[T1\] leads to the following bounded law of the iterated logarithm.
\[C1\] Under the assumptions of Theorem \[T1\], with probability $1$, $$\label{BLIL}
\limsup_{n\rightarrow\infty}\frac{|S(1,n)|}{\sqrt{n\log\log n}}\leq\frac {1}{\sqrt{a}}.$$
\[R5\] In general, one cannot replace “$\leq$” by “$=$” in (\[BLIL\]). To see this, let $Y$, $Z_{1},Z_{2},\dots$ be a sequence of independent random variables such that $Y$ takes on the value $0$ or $1$ with probability $1/2$ and $Z_{1},Z_{2},\dots$ are independent standard normals. Now define $X_{i}=YZ_{i}$, $i=1,2,\ldots.$ It is easily checked that assumption (\[assumpgamma\]) is satisfied with $A=2,$ $a=1/2$, $b=0$ and $%
\gamma=1.$ When $Y=1$ the usual law of the iterated logarithm gives with probability $1$, $$\label{ac}
\limsup_{n\rightarrow\infty}|S(1,n)|/\sqrt{n\log\log n}=\sqrt{2}=1/\sqrt {a},$$ whereas, when $Y=0$ the above limsup is obviously $0.$ This agrees with Corollary \[C1\], which says that with probability $1$ the limsup is ${\leq}\sqrt{2}$. However, we see that with probability $1/2$ it equals $\sqrt{2} $ and with probability $1/2$ it equals $0$.
Theorem \[T1\] is proved in Section \[sec2\] and the proof of Corollary \[C1\] is given in Section \[sec3\].
Proof of theorem {#sec2}
================
The case $b=0$ is a special case of Theorem 1 of [@M79]. Therefore we shall always assume that $b>0$. Choose any $0<c<a.$ We prove our theorem by induction on $n$. Notice that by the assumption, for any integer $n_{0}\geq1$ we may choose $C>An_{0}$ to make the statement true for all $1\leq n\leq n_{0}$. This remark will be important, because at some steps of the proof we assume that $n$ is large enough. Also, since the constants $A$, $a$, $b$ and $%
\gamma$ in (\[assumpgamma\]) are independent of $m$, we can assume $m=0$ without loss of generality in our proof.
Assume the statement holds up to some $n\geq2$. (The constant $C$ will be determined in the course of the proof.)
*Case* 1: Fix a $t > 0$ for which $$\label{alpha}
t^{\gamma} \leq\alpha n$$ for some $0<\alpha<1$ to be specified later. (In any case, we assume that $%
\alpha n\geq1$.) Using an idea of [@MSS], we may write for arbitrary $%
1\leq k\leq n$, $0\leq q\leq1$ and $p+q=1$ the inequality $$\begin{aligned}
&&\mathbf{P}\{M(1,n+1)>t\}\\
&&\quad\leq\mathbf{P}\{M(1,k)>t\}+\mathbf{P}\{|S(1,k)|>pt\}+\mathbf{P}%
\{M(k+1,n+1)>qt\}.\end{aligned}$$
Let $$u=\frac{ n + t^{\gamma} b (q^{\gamma}-q^{2})}{1+q^{2}}.$$ Notice that $$\label{eq}
\frac{t^{2}}{u+ b t^{\gamma}}=\frac{q^{2}t^{2}}{n-u + b q^{\gamma}t^{\gamma}}.$$ Set $$\label{k}
k=\lceil u\rceil .$$ Using the induction hypothesis and (\[assumpgamma\]) we obtain $$\begin{aligned}
\label{mainineqgamma}
&&\mathbf{P}\{M(1,n+1)>t\}\nonumber\\ [-8pt]\\ [-8pt]
&&\quad\leq C\exp\biggl\{ -\frac{ct^{2}}{k+ b t^{\gamma}}\biggr\} + A \exp\biggl\{ -%
\frac{a p^{2}t^{2}}{k + b p^{\gamma}t^{\gamma}}\biggr\} +C\exp\biggl\{ -\frac{%
c q^{2} t^{2}}{n - k + b q^{\gamma}t^{\gamma}}\biggr\} .\nonumber\end{aligned}$$ Notice that we chose $k$ to make the first and third terms in the right-hand side of ([mainineqgamma]{}) almost equal, and since by (\[k\]) $$\frac{t^{2}}{k+ b t^{\gamma}}\leq\frac{q^{2}t^{2}}{n - k + b q^{\gamma
}t^{\gamma}},$$ the first term is greater than or equal to the third.
First we handle the second term in (\[mainineqgamma\]), showing that for $%
0\leq t\leq(\alpha n)^{1/\gamma}$, $$\exp\biggl\{ -\frac{ap^{2}t^{2}}{k+bp^{\gamma}t^{\gamma}}\biggr\} \leq \exp\biggl\{
-\frac{ct^{2}}{n+1+bt^{\gamma}}\biggr\}.$$ For this we need to verify that for $0\leq t\leq(\alpha n)^{1/\gamma}$, $$\label{eq1}
\frac{ap^{2}}{k+bp^{\gamma}t^{\gamma}}>\frac{c}{n+1+bt^{\gamma}},$$ which is equivalent to $$ap^{2}(n+1+bt^{\gamma})>c(k+bp^{\gamma}t^{\gamma}).$$ Using that $$k=\lceil u\rceil\leq u+1=1+\frac{1}{1+q^{2}}[ n+b(q^{\gamma}-q^{2})t^{%
\gamma}] ,$$ it is enough to show $$n\biggl( ap^{2}-\frac{c}{1+q^{2}}\biggr) +t^{\gamma}\biggl(
ap^{2}b-cbp^{\gamma}-\frac{cb}{1+q^{2}}(q^{\gamma}-q^{2})\biggr) +ap^{2}-c>0.$$ Note that if the coefficient of $n$ is positive, then we can choose $\alpha$ in (\[alpha\]) small enough to make the above inequality hold, even if the coefficient of $t^{\gamma}$ is negative. So in order to guarantee (\[eq1\]) (at least for large $n$) we only have to choose the parameter $p$ so that $ap^{2}-c>0$ – which implies that $$\label{assump-c-1}
ap^{2}-\frac{c}{1+q^{2}}>0$$ holds – and then select $\alpha$ small enough.
Next we treat the first and third terms in (\[mainineqgamma\]). By the remark above, it is enough to handle the first term. Let us examine the ratio of $C\exp\{-ct^{2}/(k+bt^{\gamma})\}$ and $C\exp\{-ct^{2}/(n+1+bt^{%
\gamma})\}$. Notice again that since $u+1\geq k$, $$\begin{aligned}
n+1-k & \geq& n-u=n-\frac{n+b(q^{\gamma}-q^{2})t^{\gamma}}{1+q^{2}} \\
& =& \frac{q^{2}n-b(q^{\gamma}-q^{2})t^{\gamma}}{1+q^{2}} \\
&\geq& n\frac{q^{2}-\alpha b(q^{\gamma}-q^{2})}{1+q^{2}} \\
& =&\!: c_{1}n.\end{aligned}$$ At this point we need that $0<c_{1}<1$. Thus we choose $\alpha$ small enough so that $$q^{2}-\alpha b(q^{\gamma}-q^{2})>0. \label{assump-c-2}$$ Also, using $t\leq(\alpha n)^{1/\gamma}$, we get the bound $$(n + 1 + b t^{\gamma}) ( k + b t^{\gamma}) \leq n^{2} ( 1 + \alpha
b)^{2} =: c_{2} n^{2},$$ which holds if $n$ is large enough. Therefore, we obtain for the ratio $$\exp\biggl\{ -ct^{2}\biggl( \frac{1}{k+b t^{\gamma}}-\frac{1}{n+1+b t^{\gamma }%
}\biggr) \biggr\} \leq\exp\biggl\{ -\frac{c c_{1} t^{2}}{c_{2} n}\biggr\} \leq%
\mathrm{e}^{-1},$$ whenever $c c_{1} t^{2}/(c_{2} n) \geq1$, that is, $t\geq\sqrt{c_{2} n/ (c c_{1})}$. Substituting back into (\[mainineqgamma\]), for $t\geq\sqrt{%
c_{2} n / (c c_{1})}$ and $t\leq(\alpha n)^{1/\gamma}$ we obtain $$\begin{aligned}
&&\mathbf{P}\{M(1,n+1)>t\}\\
&&\quad\leq\biggl( \frac{2}{\mathrm{e}}C + A \biggr) \exp\{-ct^{2}/(n+1+ b
t^{\gamma})\}\leq C\exp\{-ct^{2}/(n+1+ b t^{\gamma})\},\end{aligned}$$ where the last inequality holds for $C> A \mathrm{e}/(\mathrm{e}-2)$.
Next assume that $t<\sqrt{c_{2}n/(cc_{1})}$. In this case, choosing $C$ large enough, we can make the bound $>1$, namely $$C\exp\biggl\{ -\frac{ct^{2}}{n+1+bt^{\gamma}}\biggr\} \geq C\exp\biggl\{ -%
\frac{cc_{2}n}{cc_{1}n}\biggr\} =C\mathrm{e}^{-c_{2}/c_{1}}\geq1,$$ if $C>\mathrm{e}^{c_{2}/c_{1}}$.
*Case* 2: Now we must handle the case $t>(\alpha
n)^{1/\gamma }$. Here we apply the inequality $$\mathbf{P}\{M(1,n+1)>t\}\leq\mathbf{P}\{M(1,n)>t\}+\mathbf{P}%
\{|S(1,n+1)|>t\}.$$ Using assumption (\[assumpgamma\]) and the induction hypothesis, we have $$\mathbf{P}\{M(1,n+1)>t\}\leq C\exp\biggl\{ -\frac{ct^{2}}{n+bt^{\gamma}}%
\biggr\} +A\exp\biggl\{ -\frac{at^{2}}{n+1+bt^{\gamma}}\biggr\} .$$ We will show that the right-hand side $\leq
C\exp\{-ct^{2}/(n+1+bt^{\gamma})\}$. For this it is enough to prove $$\label{less}
\exp\biggl\{ -ct^{2}\biggl( \frac{1}{n+bt^{\gamma}}-\frac{1}{n+1+bt^{\gamma}}\biggr) \biggr\}
+\frac{A}{C}\exp\biggl\{ -\frac{t^{2}(a-c)}{n+1+bt^{\gamma }}\biggr\} \leq1.$$
First assume that $\gamma\leq1$. Using the bound following from $t>(\alpha n)^{1/\gamma}$, we get $$\frac{t^{2}}{(n+bt^{\gamma})(n+bt^{\gamma}+1)}\geq\frac{t^{2}}{(\alpha
^{-1}+b)(2\alpha^{-1}+b)t^{2\gamma}}=:t^{2-2\gamma}c_{3}\geq c_{3}.$$ We have that the right-hand side of (\[less\]) for $a\geq c$ is less than $$\mathrm{e}^{-cc_{3}}+\frac{A}{C}\leq1$$ for $C$ large enough.
For $1<\gamma<2$ we have to use a different argument. For $t$ large enough (i.e., for $n$ large enough, since $t>(\alpha n)^{1/\gamma}$) we have $$\exp\biggl\{ -\frac{ct^{2}}{(n+bt^{\gamma})(n+bt^{\gamma}+1)}\biggr\}
\leq\exp\{ -cc_{3}t^{2-2\gamma}\} \leq1-\frac{cc_{3}t^{2-2\gamma }}{2}.$$ We also have for $C>A$, $$\frac{A}{C}\exp\biggl\{ -\frac{t^{2}(a-c)}{n+1+bt^{\gamma}}\biggr\} \leq\exp\biggl\{
-t^{2-\gamma}\frac{a-c}{2\alpha^{-1}+b}\biggr\} .$$ It is clear that since $a>c$, for $t$ large enough, that is, for $n$ large enough, $$\frac{cc_{3}t^{2-2\gamma}}{2}>\exp\biggl\{ -t^{2-\gamma}\frac{a-c}{%
2\alpha^{-1}+b}\biggr\} .$$ The proof is complete.
Proof of corollary {#sec3}
==================
Choose any $\lambda>1$ and set $m_{r}=\lceil \lambda^{r}\rceil $ for $r=1,2,\ldots.$ Now, using inequality (\[Cineq\]), we get $$\begin{aligned}
&&\mathbf{P}\bigl\{ M(1,m_{r}) >\sqrt{c^{-1}m_{r+1}\log\log m_{r}}\bigr\}\\
&&\quad\leq C \exp\biggl\{ -\frac{m_{r+1}\log\log m_{r}}{m_{r}+b(
c^{-1}m_{r+1}\log\log m_{r}) ^{\gamma/2}}\biggr\} .\end{aligned}$$ Since as $r\rightarrow\infty$ $$\frac{m_{r+1}\log\log m_{r}}{m_{r}+b( c^{-1}m_{r+1}\log\log m_{r})
^{\gamma/2}}=\bigl( 1+\mathrm{o}( 1) \bigr) \lambda\log r,$$ it is readily checked that for $r_{0}$ large enough so that $\log\log
m_{r_{0}}>0,$ $$\sum_{r=r_{0}}^{\infty}\mathbf{P}\bigl\{ M(1,m_{r}) >\sqrt{%
c^{-1}m_{r+1}\log\log m_{r}}\bigr\} <\infty$$ and thus, since $m_{r+1}/m_{r}=\lambda+\mathrm{o}( 1) $, we get by the Borel–Cantelli lemma that with probability $1$ $$\label{aa}
\limsup_{r\rightarrow\infty}\frac{ M(1,m_{r})}{\sqrt{m_{r}\log\log m_{r}}}%
\leq\sqrt{\lambda c^{-1}}.$$ Next we see that for all $r\geq r_{0}$ $$\max_{m_{r}\leq n<m_{r+1}}\frac{|S(1,n)|}{\sqrt{n\log\log n}}\leq \frac{%
M(1,m_{r+1})}{\sqrt{m_{r}\log\log m_{r}}}.$$ Thus by (\[aa\]), with probability $1,$ $$\begin{aligned}
&&\limsup_{r\rightarrow\infty}\max_{m_{r}\leq n<m_{r+1}}\frac{|S(1,n)|}{\sqrt{%
n\log\log n}}\\
&&\quad\leq\limsup_{r\rightarrow\infty}\frac{M(1,m_{r+1})}{\sqrt{m_{r}\log\log
m_{r}}}\\
&&\quad=\limsup_{r\rightarrow\infty}\frac{M(1,m_{r+1})}{\sqrt{m_{r+1}\log\log
m_{r+1}}}\frac{\sqrt{m_{r+1}\log\log m_{r+1}}}{\sqrt{m_{r}\log\log m_{r}}}%
\leq\lambda\sqrt{c^{-1}}.\end{aligned}$$ Hence, since $\lambda>1$ can be chosen arbitrarily close to $1$ and $c<a$ arbitrarily close to $a$, we have proved (\[BLIL\]).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to the referee for a number of penetrating comments and suggestions that greatly improved the paper. Kevei’s research was partially supported by the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences and Mason’s by an NSF grant.
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|
---
abstract: 'We describe the CO Luminosity Density at High-z (COLDz) survey, the first spectral line deep field targeting CO(1–0) emission from galaxies at $z=1.95-2.85$ and CO(2–1) at $z=4.91-6.70$. The main goal of COLDz is to constrain the cosmic density of molecular gas at the peak epoch of cosmic star formation. By targeting both a wide ($\sim$51 arcmin$^2$) and a deep area ($\sim$9 arcmin$^2$), the survey is designed to robustly constrain the bright end and the characteristic luminosity of the CO(1–0) luminosity function. An extensive analysis of the reliability of our line candidates, and new techniques provide detailed completeness and statistical corrections as necessary to determine the best constraints to date on the CO luminosity function. Our blind search for CO(1–0) uniformly selects starbursts and massive Main Sequence galaxies based on their cold molecular gas masses. Our search also detects CO(2–1) line emission from optically dark, dusty star-forming galaxies at $z>5$. We find a range of spatial sizes for the CO-traced gas reservoirs up to $\sim40$ kpc, suggesting that spatially extended cold molecular gas reservoirs may be common in massive, gas-rich galaxies at $z\sim2$. Through CO line stacking, we constrain the gas mass fraction in previously known typical star-forming galaxies at $z=2$–3. The stacked CO detection suggests lower molecular gas mass fractions than expected for massive Main Sequence galaxies by a factor of $\sim3-6$. We find total CO line brightness at $\sim34\,$GHz of $0.45\pm0.2\,\mu$K, which constrains future line intensity mapping and CMB experiments.'
author:
- |
Riccardo Pavesi$^\dagger$, Chelsea E. Sharon, Dominik A. Riechers, Jacqueline A. Hodge, Roberto Decarli, Fabian Walter, Chris L. Carilli, Emanuele Daddi, Ian Smail, Mark Dickinson, Rob J. Ivison,\
Mark Sargent, Elisabete da Cunha, Manuel Aravena, Jeremy Darling, Vernesa Smolčić,\
Nicholas Z. Scoville, Peter L. Capak, Jeff Wagg
bibliography:
- 'biblio\_CODF.bib'
title: 'The CO Luminosity Density at High-z (COLDz) Survey: A Sensitive, Large Area Blind Search for Low-[*J*]{} CO Emission from Cold Gas in the Early Universe with the Karl G. Jansky Very Large Array'
---
Introduction
============
Although the process of galaxy assembly through star formation is believed to have reached a peak rate at redshifts of $z=2$–3 (i.e., $\sim$10–11 billion years ago), the fundamental driver of this evolution is still uncertain [@MadauDickinson]. In order to understand the physical origin of the cosmic star formation history (i.e., the rate of star formation taking place per unit comoving volume), we need to quantify the mass of cold, dense gas in galaxies as a function of cosmic time, because this gas phase controls star formation [@KennicuttEvans]. In particular, the evolution of the cold gas mass distribution can provide strong constraints on models of galaxy formation by simultaneously measuring the gas availability and, through a comparison to the star formation distribution function, the global efficiency of the star formation process (see @CarilliWalter for a review). In this work, we carry out the first fully “blind" deep-field spectral line search for CO(1–0) line emission, arguably the best tracer of the total molecular gas mass at the peak epoch of cosmic star formation, by taking advantage of the greatly improved capabilities of NSF’s Karl G. Jansky Very Large Array (VLA).
To date, observations of the immediate fuel for star formation, i.e., the cold molecular gas, have mostly been limited to follow-up studies of galaxies that were pre-selected from optical/near-infrared (NIR) deep surveys (and hence based on stellar light) or selected in the sub-millimeter based on dust-obscured star formation as sub-millimeter galaxies (SMGs; for reviews see, e.g., @Blain2002 [@Casey14]). In particular, optical/NIR color-selection techniques (e.g., “BzK", “BM/BX"; @Daddi04 [@Steidel04]) have explored significant samples of massive, star forming galaxies at $z\sim$1.5 to 2.5 [@Daddi08; @Daddi10a; @Tacconi10; @Tacconi13] and the sub-mm selection has been particularly effective in identifying the most highly star-forming galaxies at this epoch for CO follow-up (e.g., @Bothwell13). Although such targeted CO studies are fundamental to explore the properties of known galaxy populations, they need to be complemented by blind CO surveys that do not pre-select their targets, which may potentially reveal gas-dominated and/or systems with uncharacteristically low star formation rate missed by other selection techniques.
Targeted CO studies have found more massive gas reservoirs at $z\sim2$ compared to local galaxies. Cold molecular gas is therefore believed to be the main driver for the high star formation rates of normal galaxies at these redshifts (e.g., @Greve2005 [@Daddi08; @Daddi10a; @Daddi10b; @Tacconi10; @Genzel10; @Bothwell13]). Recent studies have claimed tentative evidence for an elevated star formation efficiency, i.e., star formation rate generated per unit mass of molecular gas, at $z\sim2$ compared to local galaxies [e.g., @Genzel15; @Scoville16; @Schinnerer16; @Scoville17; @Tacconi17]. Such an elevated star formation efficiency could be related to massive, gravitationally unstable gas reservoirs. The interstellar gas content of galaxies therefore appears to be the main driver of the star formation history of the Universe, during the epoch when galaxies formed at least half of their stellar mass content (e.g., @MadauDickinson). Although targeted molecular gas studies currently allow to observe larger galaxy samples more efficiently than blind searches, their pre-selection could potentially introduce an unknown systematic bias. Critically, such studies may not uniformly sample the galaxy cold molecular gas mass function. The best way to address such potential biases, and thus, to complement targeted studies, is through deep field blind surveys, in which galaxies are directly selected based on their cold gas content. Although some targeted CO(1–0) deep studies have previously been attempted (most notably @Aravena12 and @Rudnick17), these studies have typically targeted overdense (proto-)cluster environments. Hence, a blind search approach, to sample a representative cosmic volume is needed, in order to assess the statistical significance of such previous studies.
CO(1–0) line emission is one of the most direct tracers of the cold, molecular inter-stellar medium (ISM) in galaxies[^1]. Its line luminosity can be used to estimate the cold molecular gas mass by means of a conversion factor ($\alpha_{\rm CO}$; see @Bolatto_review for a review). Although other tracers of the cold ISM have been utilized to date, including mid-[*J*]{} CO lines and the dust continuum emission, these are less direct tracers because they require additional, uncertain conversion factors (e.g., CO excitation corrections and dust-to-gas ratios). Specifically, while the ground state CO(1–0) transition traces the bulk gas reservoir, mid-[*J*]{} CO lines such as CO(3–2) and higher-[*J*]{} lines are likely to preferentially trace the fraction of actively star-forming gas. Hence, their brightness requires additional assumptions about line excitation, in order to provide a measurement of the total gas mass. Furthermore, different populations of galaxies may be characterized by significantly different CO excitation conditions (e.g., BzK, SMGs and quasar hosts; @Daddi10b [@Riechers06; @Riechers11a; @Riechers11b; @Ivison_GN19; @Bothwell13; @CarilliWalter; @Narayanan14]), which also show considerable individual scatter (e.g., @Sharon16).
Long-wavelength dust continuum emission has been suggested to be a measure of the total gas mass, and is utilized to great extent in recent surveys with ALMA to investigate large samples of far-infrared (FIR)-selected galaxies [@Eales12; @Bourne13; @Groves15; @Scoville16; @Scoville17; @ASPECS2]. Nonetheless, there remain substantial uncertainties in the accuracy of the calibration for this method at high redshift especially below the most luminous, most massive sources.[^2] Another caveat to using FIR continuum emission instead of CO comes from the finding that the dust emission measured by ALMA may not always trace the bulk of the gas distribution. This is made clear by the small sizes of the dust-emitting regions compared to the star forming regions and the gas as traced by CO emission (e.g., @Riechers_GN19 [@Riechers14; @Ivison_GN19; @Simpson15; @Hodge16; @Miettinen17; @Chen2017]).
Disentangling the causes for the observed increased star formation activity at $z\sim2$ is not straightforward, since an increased availability of cold gas may be difficult to distinguish from increased star formation efficiency due to the uncertainty in deriving gas masses, for representative samples of galaxies. Now, thanks to the unprecedented sensitivity and bandwidth of the VLA and the Atacama Large (sub-)Millimeter Array (ALMA), CO deep field studies can be carried out efficiently, and these are ideal to address such potential selection effects. Previous deep field studies, with the Plateau de Bure Interferometer (PdBI; now the NOrthern Extended Millimeter Array, NOEMA) in the HDF-N [@Decarli14; @Walter14] and ALMA in the HUDF (ALMA SPECtroscopic Survey in the Hubble Ultra-Deep Field Pilot or ASPECS-Pilot, @ASPECS1 [@ASPECS2]), have provided the first CO blind searches covering mid-[*J*]{} transitions such as CO(3–2)[^3], which are accessible at millimeter wavelengths. These studies have yielded crucial constraints on the molecular gas mass function at $z\sim1$–3, subject to assumptions on the excitation of the CO line ladder to infer the corresponding molecular gas content.[^4] They have found broad agreement with models of the CO luminosity evolution with redshift by finding an elevated molecular gas cosmic density at $z>1$ in comparison to $z\sim0$, but they may suggest a tension with luminosity function models at $z\gtrsim$1 by finding a larger number of CO line candidates than expected [@ASPECS2].
In order to more statistically characterize the molecular gas mass function in galaxies at $z=2$–3 and 5–7 than previously possible, while avoiding some of the previous selection biases, we have carried out the COLDz survey[^5] a blind search for CO(1–0) and CO(2–1) line emission using the fully upgraded VLA[^6]. The main objective of this survey is to constrain the CO(1–0) luminosity function at $z=2$–3, which provides the most direct census of the cold molecular gas at the peak epoch of cosmic star formation free from excitation bias, and based on a direct selection of the cold gas mass in galaxies. As such, the COLDz survey is highly complementary to millimeter-wave surveys like ASPECS and targeted studies. The CO(1–0) intensity mapping technique explored by [@Keating15; @Keating16] is complementary to our approach. Intensity mapping offers sensitivity to the aggregate line emission signal from galaxies, but only measures the second raw moment of the luminosity function (therefore not distinguishing between the characteristic luminosity and volume density). While the intensity mapping technique allows to cover significantly larger areas of the sky, it does not directly measure gas properties of individual galaxies, and is therefore complementary to direct searches such as COLDz.
In a previous paper (@Lentati15; Paper 0) we have described a first, interesting example of the galaxies identified in this survey. In this work (Paper [slowromancap1@]{}), we describe the survey, present the blind search line catalog, analyze the results of line stacking, and outline the statistical methods employed to characterize our sample. In Paper [slowromancap2@]{}, we present the analysis of the CO luminosity functions and our constraints on the cold gas density of the Universe at $z=2$–7, (Riechers et al., submitted). In Section 2 of this work, we describe the VLA COLDz observations, the calibration procedure and the methods to mosaic and produce the signal-to-noise cubes. In Section 3, we describe our blind line search through Matched Filtering in 3D. In Section 4, we present our “secure" and “candidate" CO line detections in both the deeper (in COSMOS) and wider (in GOODS-N) fields. In Section 5, we utilize stacking of galaxies with previously known spectroscopic redshifts, to provide strong constraints on their CO luminosity. In Section 6, we derive constraints to the total CO line brightness at $\sim$34 GHz. In Section 7, we discuss the implications of our results in the context of previous surveys. We conclude with the implications for future surveys with current and planned instrumentation. A more detailed analysis of the line search methods, the statistical characterization of the candidate sample properties, and upper limits found for additional galaxy samples are presented in the Appendix.
In this work we adopt a flat, $\Lambda$CDM cosmology with $H_0=70\,$km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{\rm M}=0.3$ and a Chabrier IMF.
-------------------- ---------- --------------- ------------------- --------------- -------------------
Field Pointing D D$\rightarrow$DnC DnC DnC$\rightarrow$C
configuration configuration configuration configuration
Baseline range (m) 40–1000 40-2100 40–2100 40–3400
COSMOS 1–7 82 hr 11 hr
GOODS-N 1–7 13 hr
GOODS-N 6 3 hr
GOODS-N 8–14 15 hr
GOODS-N 15–21 15 hr
GOODS-N 22–28 14 hr 1.4 hr
GOODS-N 29–35 3 hr 12 hr
GOODS-N 36–42 11 hr 3 hr
GOODS-N 43–49 14 hr 1.3 hr
GOODS-N 50–56 10.5 hr 3.5 hr
GOODS-N 57 2 hr
-------------------- ---------- --------------- ------------------- --------------- -------------------
\[obs\_table\]
Observations
============
In order to constrain both the characteristic luminosity, $L^*_{\rm CO}$, or “knee" of the CO(1–0) luminosity function and the bright end, we have optimized our observing strategy following the “wedding cake" design, to cover a smaller deep area and a shallower, wide area. We have used the wide-band capabilities of the upgraded VLA to obtain a continuous coverage of 8 GHz in the Ka band (PI: Riechers, IDs 13A-398; 14A-214) in a region of the COSMOS field (centered on the dusty starburst AzTEC-3 at $z=5.3$ as a line reference source, @Capak11 [@Riechers14]) and in the GOODS-N/CANDELS-Deep field, in order to take advantage of the availability of excellent multi-wavelength data [@Grogin; @Koekemoer; @Giavalisco2004].
The COSMOS data form a 7-pointing mosaic (center: R.A.=10h 0m 20.7s, Dec.=2$^{\circ}$35’17”) with continuous frequency coverage between 30.969 GHz and 39.033 GHz. The GOODS-N data form a 57-pointing mosaic (center: R.A.=12h 36m 59s, Dec.=62$^{\circ}$13’43.5”) with continuous coverage between 29.981 GHz and 38.001 GHz (Figs. \[fig:coverage\],\[fig:pointings\]). The total on-source time was approximately 93 hrs in the COSMOS field and 122 hrs in the GOODS-N field. The frequency range targeted in this project covers CO(1–0) at $z=1.95$–2.85 and CO(2–1) at $z=4.91$–6.70, such that the space density of CO(2–1) line emitters is expected to be smaller than for CO(1–0) (Fig. \[fig:coverage\]; e.g., @Popping14 [@Popping16]). Both the large redshift spacing and the expected redshift evolution of the space density of CO emitters lessens the severity of the redshift ambiguity in our survey compared to previous studies.
At 34 GHz the VLA primary beam can be described as a circular Gaussian with FWHM$\sim$80$^{\prime\prime}$, so our pointing centers were optimized to achieve a sensitivity that is approximately uniform in the central regions of the mosaics by choosing a spacing of 55$^{\prime\prime}$ ($<80^{\prime\prime}/\sqrt{2}$) in a standard hexagonally packed mosaic [@Condon98]. During each observation, we targeted a set of 7 pointings in succession, alternating through phase calibration. We performed pointing scans at the beginning of each observation, with additional pointing observations throughout for observations longer than 2 hours. Most of the COSMOS and GOODS-N data were taken in the D configuration of the VLA. Some of the observations, especially for the GOODS-N pointings, were fully or partially carried out in DnC configuration, in re-configuration from D to DnC (D$\rightarrow$DnC) and in re-configuration from DnC to C (DnC$\rightarrow$C). Pointings are named sequentially from GN1 to GN57 (groups of GN1–7, GN8–14 etc. were observed together; Table \[obs\_table\]).
The total area imaged, down to a sensitivity of $\sim30\%$ of the peak, in COSMOS is 8.9 arcmin$^2$ at 31 GHz and 7.0 arcmin$^2$ at 39 GHz. In GOODS-N the total area is 50.9 arcmin$^2$ at 30 GHz and 46.4 arcmin$^2$ at 38 GHz. The correlator was set-up in 3-bit mode, at 2 MHz spectral resolution (corresponding to $\sim18\,$km s$^{-1}$ at 34 GHz), to simultaneously cover the full 8 GHz bandwidth for each polarization (Fig. \[fig:coverage\]). Tuning frequency shifts between tracks, and sometimes in the same track, were used to mitigate the edge channels noise increase in order to achieve a uniform depth across the frequency range (Table 2).
COSMOS observations
-------------------
The dataset in the COSMOS field consists of 46 dynamically scheduled observations between 2013 January 26 and 2013 May 14, each about 3 hours in duration. Flux calibration was performed with reference to 3C286, and J1041+0610 was observed for phase and amplitude calibration. Three frequency tunings offset in steps of 12 MHz were adopted to cover the gaps between spectral windows and to obtain uninterrupted bandwidth.
GOODS-N observations
--------------------
The GOODS-N dataset consists of 90 observations between 2013 January 27 and 2014 September 27, each about 2 hours in duration. Pointing 6, which covers the 3 mm PdBI pointing of the CO deep field in [@Decarli14] in the HDF-N, was observed both as part of the 1–7 pointing set, and in two additional, targeted observations to achieve better sensitivity. Pointing GN57 was observed for 3 hours (127 min on source) in D-array configuration on 18 December 2015, in order to follow up the most significant negative line feature in GN1–56 (see Appendix D for details). J1302+5748 was used for phase calibration, and the flux was calibrated by observing either 3C286 (in 7 observations) or in reference to the phase calibrator (in the remaining observations). An average phase calibrator flux at 34 GHz of [*S=*]{}0.343 Jy and spectral index of $-0.2$ was assumed in the observations in 2013 and [*S=*]{}0.21 Jy and spectral index of $-0.6$ was assumed in 2014, as regularly measured in the tracks where a primary flux calibrator was observed. Based on track-to-track variations of the calibrator flux, we estimate a $\sim$20% total flux calibration uncertainty. The spectral setup employed uses two dithered sets of spectral windows, with a relative shift of 16 MHz, in order to fully cover the 8 GHz bandwidth available without gaps.
![Frequency coverage of the VLA COLDz survey, in the Ka band. The frequency range covers CO(1–0) at $z=1.95$–2.85, and CO(2–1) at z=4.91–6.70.[]{data-label="fig:coverage"}](Fig1.pdf){width=".5\textwidth"}
[ c c c c c c ]{} Transition & $\nu_0$ & $z_{min}$ & $z_{max}$ & $\langle z \rangle$ & Volume\
& \[GHz\] & & & & \[$\rm{Mpc}^3$\]\
\
CO(1–0) & 115.271 & 1.953&2.723& 2.354&20,189\
CO(2–1) & 230.538 & 4.906& 6.445 &5.684 &30,398\
\
CO(1–0) & 115.271 & 2.032&2.847&2.443&131,042\
CO(2–1) &230.538 & 5.064&6.695 &5.861&193,286\
Data Processing
---------------
Data calibration was performed in version 4.1, using the VLA data reduction pipeline (v.1.2.0). [casa]{} version 4.5 was used to re-calculate visibility weights using the improved version of `statwt` that excludes flagged channels when calculating weights, and for imaging and mosaicking [@CASA]. The pipeline radio-frequency interference (RFI) flagging, which uses [casa]{} `rflag` to identify transient lines, was switched off, as recommended by the developers, since it can potentially remove narrow spectral lines and because there is little RFI in the Ka-band (with the exception of the 31.487–31.489 GHz range, which we flag prior to running the [casa]{} pipeline). The pipeline was further modified to only flag the first and last channel of each spectral window (instead of 3 channels), regardless of proximity to baseband edges, to minimize the gap between sub-bands. We find that the bandpass is sufficiently flat that this choice gives the best trade-off between sensitivity in the end channels and additional noise, although some noise increase at the band edges is visible in Figure \[fig:Noise\]. After executing the pipeline, we visually inspected the visibilities in the calibrator fields to identify any necessary additional flagging. We then re-executed the pipeline to obtain a final calibration. In addition, for most GOODS-N observations we modify the pipeline to flux-calibrate in reference to the gain calibrator (whenever a primary flux calibrator was not observed).
We identified a small number of noisy spectral channels in our observations that are not removed by the calibration pipeline. The noisy channels were initially discovered as narrow spikes of a small number of channels in amplitude vs. frequency plots of visibilities from the science target fields, and they are mostly associated with single antennas. Being very narrow in frequency (one or two 2-MHz channels), the noise spikes are not significantly reduced by the statistical weights obtained from `statwt`, which minimizes the effects of all other noise features, since the weights are computed per spectral window. Including one of these noisy channels for the affected antenna during the imaging of a single pointing of the mosaic from a single observation track increases the rms noise by $\sim20\%$ in that frequency channel. Selecting channels whose standard deviation exceeds the mean standard deviation in that spectral window for that antenna by $3\sigma$ is a sufficient criterion to exclude most of the problematic noise spikes (these are only of order $\sim0.2\%$ of all channel-antenna combinations). This method is partially redundant to the algorithms in `rflag` (which we did not execute as part of the pipeline), but it reduces the risk of removing real spectral lines since the noisy channels are selected for individual antennas. We also found that many noisy channels in the same antenna repeat over time during an observation, and would therefore be more problematic if left in the data cube. We find a concentration of noise spikes in roughly four peaks over the frequency range, which correlate with peaks in the weighted calibrated amplitudes as a function of frequency. We consider this to be indicative of random electronic problems that manifest as increased noise and thus are more prevalent in certain hardware components of the correlator than others. The presence of four peaks is likely associated with the underlying basebands, since there appears to be one peak in each baseband, but no precise correlation of the noise peak frequencies to the baseband edges could be identified. The feature is stochastic and does not appear to preferentially affect any particular subset of antennae. These noise spikes are at least twice as narrow as the narrowest blindly selected line candidates (which are rare among all candidates) and therefore residual anomalous noise spikes are believed not to measurably affect our line search.
Calibrated data from each pointing were imaged separately without any `CLEAN` cycles, because the fields do not contain strong continuum or line sources (see Section 4). We imaged the total intensity (sum of the two polarizations) using natural weighting and choosing a pixel size of 0.5$^{\prime\prime}$ consistently in the two fields. The smallest adopted channel width is 4 MHz, equivalent to 35 km s$^{-1}$ mid-band, which is less than the typical line width from galaxies. With this choice, our data cubes have $\sim$2000 channels after averaging polarizations. A crucial aspect of the imaging procedure, necessary for blind line searches, is to avoid any frequency regridding by interpolation, therefore we image using the nearest channel, rather than interpolating. Interpolation would introduce correlations between the noise of adjacent channels, undermining the statistical basis of the search for spectral lines, and producing a significant number of spurious noise lines. Disabling frequency interpolation when imaging visibilities may introduce a very small, less than half a channel, frequency error that we consider negligible because 4 MHz channels (35 km s$^{-1}$) are smaller than the typical linewidth.
The geometric average synthesized beam size for COSMOS ranges between 2.2$^{\prime\prime}$ and 2.8$^{\prime\prime}$ as a function of frequency and the beam axes ratio is in the range of 0.8–1, while for GOODS-N the synthesized beam size differs more significantly from pointing to pointing. In particular, the main difference is between the subset of “high resolution pointings" (GN20–GN42) and the rest (GN1–GN19 and GN43–GN57). The geometric average beam size ranges between approximately 1.3$^{\prime\prime}$ to 2.0$^{\prime\prime}$ for the high resolution pointings, and between 2.1$^{\prime\prime}$ and 3.1$^{\prime\prime}$ for the rest. The beam axes ratio is in the range of 0.6–0.9. The individual pointing cubes are subsequently smoothed to a common beam size of 3.38$^{\prime\prime}\times$ 2.91$^{\prime\prime}$ for COSMOS and 4.1$^{\prime\prime}\times$ 3.2$^{\prime\prime}$ for GOODS-N, using the [casa]{} task `imsmooth`. This compromise in resolution and signal-to-noise is necessary in order to mosaic all pointings together, and to search for line emission in a uniform manner. This is called the Smoothed-mosaic. Separately, we have also mosaicked the pointings with their native resolution (after removing the beam information from the headers), and this Natural-mosaic (where the resolution is set by natural weighting) was used to exclusively search for spatially un-resolved sources, for which the spatial size information is not important. Fig. \[fig:pointings\] shows the spatial coverage provided by the individual pointings in our two mosaicked fields.
The [casa]{} function `linearmosaic` was used to mosaic the images of our COSMOS data together. We wrote a custom script to optimally mosaic the images of the GOODS-N data, using Equation. \[mosaic\_formula\], which takes into account the different noise levels in different pointings, per channel, in order to compute optimal weights for mosaicking: $$I=\frac{\sum_p I_p A({\bf x}-{\bf x_p})/\sigma_p^2}{\sum_p A({\bf x}-{\bf x_p})^2/\sigma_p^2},
\label{mosaic_formula}$$ where $A$ is the primary beam function, ${\bf x_p}$ are the pointing center positions, $I_p$ represents the specific intensity data from pointing p and $\sigma_p$ is the noise level in pointing p (computed on a per channel basis).
All COSMOS pointings were always observed in every execution, and for comparable amounts of time. The GOODS-N pointings were observed in blocks of 7, over the course of several months due to scheduling constraints. Therefore, they have slightly different noise levels, partly due to the upgraded 3-bit samplers in the later (2014) observations. Furthermore, some GOODS-N data were not taken in the D configuration but rather in a combination of the DnC configuration, in transition between the D and the DnC and in transition between the DnC and the C configuration. Therefore, when smoothed to a common beam, these pointings have higher noise, because the information on the the longest baselines is effectively discarded. For these reasons, the noise is significantly spatially varying in the smoothed version of the GOODS-N mosaic, which we take into account when analyzing the data (Fig. \[fig:Noise\] shows the noise before and after smoothing). All pointings suffer a noise increase due to smoothing, because the targeted beam for the smoothing process has to be larger than every beam in any pointing, at any frequency, and also includes those beams that have different position angles. In the case of the COSMOS mosaic, the [casa]{} function `linearmosaic` produces the mosaic edge at the 30% of peak level sensitivity (per-channel). For consistency, we therefore apply the same criterion in our GOODS-N mosaics. In order to do this, we define a mask which produces the mosaic edge at 30% of peak sensitivity in the Natural-mosaic, and utilize the same mask for the Smoothed-mosaic for consistency.
Constructing the Signal-to-Noise cubes
--------------------------------------
In order to search for emission lines in our data, we produce a signal-to-noise ratio (SNR) cube by calculating a noise value for each pixel and in each frequency channel of the mosaics. Spatial variations in the noise are introduced by mosaicking pointings with different noise levels and primary beam corrections. The noise in the resulting mosaic can be calculated assuming statistical independence of the noise in different pointings, and can therefore be calculated by summing their standard deviations in quadrature, with weights given by Eqn. \[mosaic\_formula\]: $$\sigma(x)=\frac{1}{\sqrt{\sum_p A({\bf x}-{\bf x_p})^2/\sigma_p^2}},
\label{mosaic_noise}$$
where $\sigma_p$ is the measured noise in the individual pointing images, $A$ is the primary beam function and ${\bf x_p}$ are the pointing center positions. In the special case of pointings with approximately equal noise (as in our COSMOS data) we can use a simplified expression, where the denominator is simply the square root of the sensitivity map, output from [casa]{}’s `linearmosaic` function: $$\sigma(x)=\frac{\sigma}{\sqrt{\sum_p A({\bf x}-{\bf x_p})^2}}.$$
The frequency variation of the noise is accounted for by measuring the noise in each frequency channel, in the individual pointings. In COSMOS, where the noise variations from pointing to pointing can be neglected, we calculate the signal-to-noise ratio by multiplying the signal cube by the square root of the sensitivity map (which gives spatially uniform noise), and then dividing each channel map by its standard deviation to normalize the pixel value distribution. In GOODS-N, we measure the noise in each pointing and apply Eqn. \[mosaic\_noise\] to compute noise and signal-to-noise ratio cubes.
Line search methods
===================
The main objective of this survey is to carry out a blind search for CO(1–0) and CO(2–1) emission lines in the COLDz dataset. No other bright lines are expected to contaminate the 30–39 GHz frequency range (Fig. \[fig:coverage\]). The line brightness sensitivity is approximately equal for the low and high redshift bins (corresponding to CO(1–0) and CO(2–1), respectively; Figure \[fig:sensitivity\]). We therefore expect a higher source density in the low redshift bin due to the expected evolution of the cosmic gas density [@Popping14; @Popping16]. Hence, we will assume that all detected features correspond to CO(1–0) unless data at other wavelengths suggest that they belong to the higher redshift bin. In order to detect emission lines in our data cubes, we have implemented a previously published method ([spread]{}, @Decarli14), and developed three new methods to explore the differences between different detection algorithms (see Appendix A for details).
The objective of a line search algorithm is to systematically assess the significance (expressed as Signal-to-Noise ratio, SNR) of candidate emission lines in the data. The relevant information available to us is the strength of the signal, the number of independent samples that make up the line, and the spatial and frequency structure (for which we have priors based on previous samples of CO detections at high redshift). In particular, we expect most CO sources to be either unresolved or resolved over a few beams at most at the $\sim3^{\prime\prime}$ resolution of our mosaics (which corresponds to $\sim$25 kpc at $z\sim2.5$ and $\sim17$ kpc at $z\sim6$), and we expect the line FWHM to be in the range of 50 to 1000 km s$^{-1}$ [@CarilliWalter].
Our line search method of choice, Matched Filtering in 3D (MF3D), expands on the commonly used spectral Matched Filtering (MF1D; e.g, [aips]{} `serch`). Matched Filtering corresponds to convolving the data with a filter, or template, which is “matched" to the sources of interest in order to attenuate the noise and concentrate the full signal-to-noise of the source in the peak pixel. A detailed description of the MF3D method is presented in Appendix A.
We also implement and test some of the previously used methods on our data, in particular [spread]{} and Matched Filtering in the spectral domain, i.e., in 1D. The main limitation of [spread]{} is that it does not employ the full spatial information available, but only utilizes signal strength. While Matched Filtering in 1D is arguably the optimal search method for completely unresolved sources (for which the spectrum at the peak spatial pixel contains the full information), it still requires a prescription for identifying pixels belonging to the same source, and it needs to be generalized to account for the possibility that some sources may be slightly extended. Besides accounting for extended sources, the 3D Matched Filtering also captures the spreading of the signal-to-noise over different spatial positions in different frequency channels, which is at least in part a consequence of moderate SNR. For this reason, it is natural to use the spatial information by using templates that include a spatial profile. Therefore we extended the method to Matched Filtering with 3D templates. A description of the detailed implementation of all line search methods and a more detailed comparison is presented in Appendix A.
Results of The Line Search
==========================
The 3D Matched-Filtering procedure provides an output including the maximal SNR for each line candidate, the position in the cube where that maximal SNR is achieved, the number of templates for which the candidate has $>4 \sigma$ significance, and the template size (spatial and frequency width) where the highest SNR is achieved. We run the line search down to a low SNR threshold of $4\sigma$. The number of identified features is very large, due to the large number of statistical elements in our data cubes. Specifically, we estimate approximately $2.8\times 10^6$ and $1.7\times 10^7$ independent elements for the COSMOS and GOODS-N fields, respectively, by dividing the mosaic area by the beam area and dividing by a line FWHM of 200 km s$^{-1}$. However, we caution that naively estimating the extent of the noise tails from these numbers does not provide a good estimate, as previously described by [@Vio16; @Vio17] (also see Appendix F.2 for more details).
We mask radio continuum sources in our fields, which contaminate the line candidates: one in the COSMOS field at 10:00:20.67 +02:36:01.5 with a flux of 0.024 mJy beam$^{-1}$, and three in the GOODS-N field at 12:36:44.42 +62:11:33.5 with a flux of 0.3 mJy beam$^{-1}$, 12:36:52.92 +62:14:44.5 with a flux of 0.17 mJy beam$^{-1}$ and 12:36:46.34 +62:14:04.46 with a flux of 0.07 mJy beam$^{-1}$ (Hodge et al., in prep.). Even though the continuum fluxes of these sources only have low significance in the individual channels ($<0.3\sigma$ and $<2\sigma$ per 4 MHz channel for the brightest source in COSMOS and GOODS-N, respectively), we remove any candidate within 2.5$^{\prime\prime}$ of the spatial positions of these sources, because they are likely spurious and caused by noise superposed to the continuum signal. Specifically, once we remove the continuum flux from their spectra, the significance of those line candidates becomes lower than $\sim4.5\sigma$, indicating that they likely correspond to noise peaks.
In Table \[tab\_lines\], we present the list of the secure line emitters in COSMOS and in GOODS-N which were independently, spectroscopically confirmed. While we are confident that our highest SNR ($>6.4\sigma$) candidates correspond to real CO emission lines because they all have identified multi-wavelength counterparts, we also define a longer lists of line candidates which have significantly lower purity ($\sim$5%-40%) as a statistical sample in Table \[tab\_lines\_appendix\] in Appendix E, as described below. Although only a fraction of those tabulated sources are real emission lines, they provide statistical information once we account for their fractional purity, and therefore they may be used to constrain the CO luminosity function. While a fraction of these lower significance candidates may be expected to correspond to real CO emission, we advise caution in interpreting these lower significance candidates on a per-source basis until they are independently confirmed.
In order to determine the reliability of the line candidates presented in Table \[tab\_lines\_appendix\], we compare the SNR distribution to that for “negative" line candidates, following the standard practice (e.g., @Decarli14 [@ASPECS1]) which relies on the symmetry of interferometric noise. We provide a detailed description of our candidate purity estimation in Appendix F, but we point out that an excess of positive candidates over the negatives, for signal-to-noise ratios above a threshold is an indication that at least a fraction of those positive candidates may correspond to real sources, rather than due to noise. By adopting this criterion, we determined the SNR thresholds for our candidate lists consistently for both fields by cutting at the SNR level that includes as many negative line candidates as unconfirmed positive candidates. Thus, we exclude from the count the high signal-to-noise, confirmed sources (4 in COSMOS and 2 in GOODS-N, Table \[tab\_lines\]), and we require that the number of unconfirmed sources is greater than the number of negative lines down to the same SNR threshold, thereby constituting an excess. This procedure determines SNR thresholds on the candidates catalog of 5.25$\sigma$ for the smaller COSMOS field and 5.5$\sigma$ for the wider GOODS-N field (Table \[tab\_lines\_appendix\]). The threshold is chosen to be higher in the wide, GOODS-N mosaic because the larger number of statistical elements produces more pronounced noise tails.
Measuring line candidate properties
-----------------------------------
After selecting the blind search line candidates, we separately measure their line properties using a standard method described in the following. The statistical corrections were computed adopting identical methods in the artificial source analysis (Appendix F.3).
In order to extract the spectrum of the line candidates, we fit a 2D-Gaussian to the velocity-integrated line maps and extract the flux in elliptical apertures with sizes equal to the FWHM of the fitted Gaussians. For the integrated line maps, we use a velocity range equal to the FWHM of the template that maximizes the SNR. This procedure is expected to provide the highest SNR of the extracted flux. In the infinite SNR case, this aperture choice includes half of the total flux, and we therefore correct the extracted flux scale of the spectrum by a factor of two. We then fit a Gaussian line profile to the aperture spectrum and measure its peak flux and velocity width, from which we derive the integrated fluxes reported in Tables \[tab\_lines\] and \[tab\_lines\_appendix\]. We also measure peak fluxes for the candidates, which are expected to best represent the correct flux for unresolved sources. For the peak fluxes, we extract the spectrum at the highest pixel in the integrated line map. We find that the peak fluxes are compatible with aperture fluxes for point-like sources, and so we choose to adopt the aperture fluxes because they measure the full flux of extended sources at the expense of slightly larger uncertainties. We calculate the positional and size uncertainty of the 2D Gaussian fitting using the [casa]{} task `imfit`, applied to the same integrated line maps described above. The positional uncertainty is relevant when establishing counterpart associations (as detailed in Appendix E for the full candidate list). It is dominated by the detection SNR and the spatial size of the synthesized beam or extended emission. In the COSMOS field, we can measure aperture fluxes in the Natural-mosaic, to make full use of the highest SNR (the fluxes are typically within 20% of the values measured in the Smoothed-mosaic). Specifically, the 7 pointings of the mosaic have an approximately equal beam size. This allows us to calculate an average beam size for each channel and hence, to correctly measure aperture fluxes. These are the fluxes we report in Tables \[tab\_lines\] and \[tab\_lines\_appendix\] for the most significant candidates, which we also use for the luminosity function.
In the GOODS-N field, on the other hand, we are limited to measuring aperture fluxes for resolved objects in the Smoothed-mosaic, because of the strong beam size variations across the mosaic which make it impossible to precisely define a beam in the Natural-mosaic. Nonetheless, since most of the candidates are unresolved in the original data (show highest SNR in the Natural-mosaic), in those cases we report the peak fluxes, measured in the Natural-mosaic, without concern for missing any flux, and without being affected by the beam size variations.
In the GOODS-N field, there is another beam size effect that needs to be taken into account even in the Smoothed-mosaic. The measured beam size is actually larger than the formal $4.1^{\prime\prime} \times 3.2^{\prime\prime}$ size which was targeted with the [casa]{} task `imsmooth`, and is slightly pointing-dependent, as explained in Appendix C. The measured beam area is $\sim1.4$ times larger in the D-array only pointings, and $\sim1.7$ times larger in the higher resolution pointings than the target size for the smoothing procedure, because of the precise $uv$-plane coverage and the effect of tapering. Therefore, we measure the correct beam size after smoothing, by Gaussian-fitting to the smoothed dirty beam, in each pointing, for each channel. We correct the aperture flux for each candidate line detection in the Smoothed-mosaic by calculating an effective beam area given by a weighted average of the beams of the overlapping pointings, weighted by the square of the primary beams (the same weighted average that determines the flux in the mosaic). We calculate aperture fluxes in this way in the Smoothed-mosaic, and confirmed that the peak pixel flux in unresolved sources matches this corrected aperture flux, within the uncertainties.
The measured CO line fluxes are affected by the effect of a warmer cosmic microwave background (CMB) at the redshift of our sources, which is a uniform background (hence invisible to an interferometer) at the small scales of galaxy sizes [@daCunha13]. While we do not expect corrections for our $z=2$–3 sources to be significant ($\sim20-25$%) a larger correction (up to a factor $\sim2$) may be required if the gas kinetic temperature were lower than expected. On the other hand, the CO(2–1) line luminosity from the $z>5$ sources may be underestimated by up to a factor of $\sim2-5$ [@daCunha13]. We do not apply any of these corrections to the measured line flux values reported here. These effects will be further discussed in Paper [slowromancap2@]{}, in the context of the CO luminosity function.
Individual candidates
---------------------
We have identified 26 line candidates in the COSMOS field down to a SNR threshold of 5.25, and 31 candidates in the GOODS-N field down to a SNR threshold of 5.5 (Tables \[tab\_lines\] and \[tab\_lines\_appendix\]). The top four sources in COSMOS and two among the highest SNR sources in GOODS-N have been independently confirmed through additional CO transitions (@Daddi09 [@Riechers10a; @Riechers_GN19]; and in prep.; @Ivison_GN19; Pavesi et al., in prep.). Furthermore, we include COLDz.GN.31 in this set of independently confirmed sources (Table \[tab\_lines\]), although it is slightly below the formal 5.5$\sigma$ cutoff, because it corresponds to CO(2–1) line emission from HDF850.1 [@Walter12]. This line source does not contribute to our evaluation of the CO(2–1) luminosity function because it does not satisfy the significance threshold to be included in the statistical sample (Paper [slowromancap2@]{}). For reference, we here briefly describe these individual secure candidates and we show their CO line maps and spectra in Figs. \[fig:spectra\_COS1\] and \[fig:spectra\_GN\_main\]. Line maps and spectra of the complete statistical sample are presented in Appendix E for reference.
We detect four previously known dust-obscured massive starbursting galaxies, and three secure sources in the COSMOS field that lie within the scatter of the high-mass end of the Main Sequence at $z\sim2$ (@Lentati15; Pavesi et al., in prep.) These galaxies may be representative of a galaxy population that has not been well studied to date, due to our novel selection technique.
**COLDz.COS.0**: We identify the brightest candidate in the COSMOS field with CO(2–1) from the z=5.3 sub-millimeter galaxy AzTEC-3, detected at a SNR of 15, and which was chosen to be near the center of our survey region. This galaxy is known to reside in a massive proto-cluster [@Riechers10a; @Riechers14; @Capak11]. The line flux is compatible with the previously measured value of $0.23\pm0.03$ Jy km s$^{-1}$ [@Riechers10a] within the relative flux calibration uncertainty. This source is also detected at 3 GHz, with a flux of $20\pm3\,\mu$Jy [@Smolcic17], and by SCUBA-2 at $850\mu$m as part of the S2COSMOS survey with a significance of $9.3\sigma$ and a flux of $8.1^{+1.1}_{-1.3}$ mJy (J.M. Simpson, et al. in prep).
**COLDz.COS.1**: This high signal-to-noise detection is matched in position (offset $0.3^{\prime\prime}\pm0.3^{\prime\prime}$) and CO(1–0) redshift to a source with photometric redshift ($z_{phot}$=2.6–2.9) in the COSMOS2015 catalog [@Laigle16]. We have confirmed its redshift with ALMA through a detection of the CO(3–2) line (Pavesi et al., in prep.). This source is also detected at 3 GHz, with a flux of $15\pm2\,\mu$Jy [@Smolcic17], and at $850\mu$m with a significance of $6.0\sigma$ and a flux of $4.9_{-1.2}^{+1.1}$ mJy (J.M. Simpson, et al. in prep).
**COLDz.COS.2**: This high signal-to-noise detection is matched in position (offset $0.3^{\prime\prime}\pm0.3^{\prime\prime}$) to a source in the COSMOS2015 catalog [@Laigle16]. We have confirmed its redshift with ALMA through a detection of the CO(3–2) line (Pavesi et al., in prep.), and some of its properties were previously presented in [@Lentati15]. The photometric redshift in the COSMOS2015 catalog is highly uncertain, and not compatible with the CO redshift of 2.477 within $1\sigma$ ($z_{phot}$=2.9–4.4). This source is also detected at 3 GHz, with a flux of $19\pm3\,\mu$Jy [@Smolcic17], and at $850\mu$m with a significance of $5.9\sigma$ and a flux of $4.0_{-1.0}^{+0.9}$ mJy (J.M. Simpson, et al. in prep).
**COLDz.COS.3**: This high signal-to-noise detection, is a significantly spatially extended CO source with a deconvolved size of $(4.0^{\prime\prime}\pm1.1^{\prime\prime})\times(1.8^{\prime\prime}\pm1.2^{\prime\prime})$. It is matched in position to two galaxies in the COSMOS2015 catalog (@Laigle16; offsets of $0.14^{\prime\prime}\pm0.3^{\prime\prime}$ and $1.8^{\prime\prime}\pm0.3^{\prime\prime}$). We have confirmed its CO(1–0) redshift with ALMA through a detection of the CO(4–3) line (Pavesi et al., in prep.). The cataloged photo-$z$ for both galaxies ($z_{phot}$=1.8–1.9) is not compatible with the CO redshift of 1.97 within $1\sigma$. This source is also detected at 3 GHz, with a flux of $27\pm3\,\mu$Jy [@Smolcic17]. The S2COSMOS survey shows a weak signal at $850\mu$m with a significance of $3.7\sigma$. The formal $4\sigma$ limit on the deboosted flux is $<4.0$ mJy, and the tentative detection suggests a potential source at a flux level of $\sim2-3$ mJy (J.M. Simpson, et al. in prep).
**COLDz.GN.0**: We identify the brightest candidate in the GOODS-N field with CO(2–1) line emission from GN10, a massive, bright dust-obscured starbursting galaxy [@Pope06; @Dannerbauer08; @Daddi09]. We find a CO redshift of $z=$5.3, showing that the previous redshift determination ($z$=4.04) was incorrect. Its properties are described in Riechers et al., in prep. This source is also detected at 1.4 GHz with a flux of $36\pm4\,\mu$Jy [@Morrison10], and by SCUBA-2 at $850\mu$m in the SCUBA-2 Cosmology Legacy Survey (S2CLS) with a significance of $9.2\sigma$ and a flux of $7.5\pm1.5$ mJy [@Geach17].
**COLDz.GN.3**: We identify this source with CO(1–0) line emission from GN19, a merger of two massive, bright dust-obscured starbursting galaxies at $z$=2.49 found by [@Pope06] and characterized in detail by [@Tacconi06; @Tacconi08], [@Riechers_GN19], and [@Ivison_GN19]. It is detected by the 5.5 GHz eMERGE survey, with a flux of $9.6\pm1.7\,\mu$Jy [@eMerge]. Its line flux is compatible with the previously measured total flux of $0.33\pm0.04$ Jy km s$^{-1}$ from [@Riechers_GN19]. This source is also detected at 1.4 GHz, with a flux of $28\pm4\,\mu$Jy and $33\pm4\,\mu$Jy for the W and E components, respectively [@Morrison10], and at $850\mu$m with a significance of $7.9\sigma$ and a flux of $6.5\pm1.1$ mJy [@Geach17].
**COLDz.GN.31**: We also detect CO(2–1) line emission from the bright, dust-obscured starbursting galaxy HDF850.1 ($z$=5.183), with a moderate significance of SNR=5.3. We include this line detection here given the known match, but we do not include it in the statistical analysis because it does not reach the significance threshold for detection by the blind line search. The measured flux is compatible with the previously reported flux of $0.17\pm0.04$ Jy km s$^{-1}$ [@Walter12]. It is detected by the 5.5 GHz eMERGE survey, with a flux of $14\pm3\,\mu$Jy [@eMerge], but is not detected at 1.4 GHz [@Morrison10]. This source is also detected at $850\mu$m with a significance of $7.1\sigma$ and a flux of $5.9\pm1.3$ mJy [@Geach17].
The other line candidates identified by our blind line search with moderate significance are to date not independently confirmed (Table \[tab\_lines\_appendix\]). Thus, we only use their properties in a statistical sense in the following, to place more detailed constraints on the CO luminosity function. We point out that three out of the seven secure, confirmed sources in our blind search belong to the high redshift bin, and therefore suggest caution in interpreting the indicated CO(1–0) redshift, especially for those line candidates without strong counterparts. We describe the complete candidate sample in Appendix E, where we also discuss potential counterpart associations. In Appendix F, we develop novel statistical techniques to evaluate the purity and completeness of this statistical sample, which yield the best constraints to the CO(1–0) luminosity function at $z\sim$2–3 to date (Paper [slowromancap2@]{}).
Statistical counterpart matching
--------------------------------
All SNR$>$6.4 candidates in COSMOS and GOODS-N have optical, NIR and/or radio/(sub)-mm counterparts (in addition to GN19 and HDF850.1). At lower SNR, it becomes more difficult to establish definitive counterparts due to the modest precision of photometric redshifts and potential (apparent or real) spatial offsets of the emission. Our purity analysis (Appendix F) suggests that the contamination from noise is considerable. As an example, for the candidates shown below SNR=6, we may expect only 1 or 2 out of 10 to be real CO line emitters due to the large sizes of the data cubes. Therefore, we consider the lack of counterparts as a possible indication that a line candidate may be due to noise. On the other hand, the very objective of a blind search for CO emitting galaxies is to address a potential bias against optical/NIR-faint galaxies. Possible explanations for the lack of counterparts are: 1) the stellar light could be too dust-obscured to be visible in the rest-frame optical/NIR; 2) the CO line may correspond to the [*J*]{}=2–1 transition; placing the galaxy at $z>5$, such that counterparts may only exist below the detection limit; 3) a CO-bright emitter may be gas-rich but have low star formation rate and/or stellar mass, which would make it optically “dark".
### Optical-NIR counterparts
We here consider the uncertain line candidates near and below the SNR threshold only. If we match all $5<SNR<6$ candidates in COSMOS (60 in total) to the COSMOS2015 photometric catalog [@Laigle16], by requiring a spatial separation of $<2^{\prime \prime}$ and a $z_{\rm CO10}$ or $z_{\rm CO21}$ within the 68th percentile range of the photometric redshifts, we find 10 matches. This is $\sim2.7\sigma$ higher than the number of matches found for random displacements of the positions of our candidates (randomly expecting $\sim4.7\pm2.0$ associations). We therefore conclude that some ($\sim3-7$) of the 10 associations (out of these top 60 candidates) are likely to be real physical counterparts to real CO line emitters, in agreement with our typical purity estimate of order $\sim10\%$ for the statistical sample in this SNR interval (Appendix F). Consistently, we also find a $1.8\sigma$ excess of positional matches within $<2^{\prime \prime}$ for this extended candidates sample, 20 matches with a $13.8\pm 3.4$ false positive rate, by spatially associating to the [*Spitzer*]{}/IRAC-based catalog by the deep SEDS survey [@Ashby2013]. This confirms that at least a fraction of our line candidates in the COSMOS field at these lower SNR levels may have real counterpart associations, to be confirmed by future spectroscopic observations.
We repeat the same procedure in GOODS-N, for the candidates with SNR$>$5.4, excluding the independently confirmed ones (51 in total). We employ the best redshifts available from [@Skelton14] and [@Momcheva16], using the same selection criteria with a separation requirement of $<2^{\prime \prime}$. The grism spectroscopy does not significantly impact our matched counts, as almost all of the potential counterparts are too faint and only have photometric redshifts. We only find a slight excess relative to chance associations ($\sim1.1\sigma$), by finding 9 associations at an expected chance rate of $6.3\pm2.5$. The latest “super-deblended" GOODS-N catalog from [@superdeblended] does not yield any additional associations besides the secure sources corresponding to GN10 and GN19. In addition, we search for positional matches within $<2^{\prime \prime}$ for this extended candidate sample by searching for spatial associations in the [@Ashby2013] [*Spitzer*]{}/IRAC-based catalog from the deep SEDS survey. We do not find any excess of matches over the expected false positive rate .
The counterpart association signal in GOODS-N does not constitute a significant excess, perhaps due to contamination by chance associations with low redshift galaxies. However, at least approximately $\sim6-10$ line candidates out of the top $\sim200$ have a very close [*Spitzer*]{}/IRAC counterpart ($<1^{\prime\prime}$) and a photometric redshift estimate which is compatible with the CO(1–0) line candidate, as would be expected for real counterpart matches.
In the following, we evaluate the implications of a lack of $3.6\,\mu$m counterparts for some of our lower SNR CO line candidates. The deep [*Spitzer*]{}/IRAC images in Fig. \[fig:spectra\_COS1\] and Appendix E are derived from the [splash]{} observations [@Steinhardt14], while the [*Spitzer*]{}/IRAC images in GOODS-N were obtained as part of the legacy GOODS program [@Giavalisco2004]. Due to the moderate resolution of [*Spitzer*]{} observations, these images are sometimes contaminated by lower redshift galaxies or stars, reducing our ability to detect counterparts at higher redshift and hence, in those cases the following limits may not apply. In order to asses the implications of a counterpart non-detection in the IRAC $3.6\mu$m images, we use template spectral energy distributions (SEDs) for star forming galaxies from [@BruzualCharlot], redshift them to $z\sim2.3$ and convolve them with the IRAC $3.6\mu$m filter curve, using [magphys]{} to estimate the stellar mass limits placed by a lack of detection in COSMOS or GOODS-N [@magphys08; @magphys15]. The expected mass-to-light ratio at this wavelength depends on the stellar population ages and star formation histories, as well as on the degree of dust extinction. The following estimates are thus only indicative. We estimate that the lack of IRAC $3.6\,\mu$m counterparts at the $\sim 0.2\,\mu$Jy and $\sim 0.06\,\mu$Jy limits ($\sim 3\sigma$; @Ashby2013) of the COSMOS and GOODS-N data correspond to approximate stellar mass upper limits of $\sim6\times 10^{9}$ and $\sim2\times 10^{9}\,{\rm M}_\odot$ respectively at $z\sim2.3$ for a representative $A_V\sim2.5$[^7]. These limits suggest that a lack of infrared counterparts implies either a very low stellar mass, or a high degree of dust obscuration. The stellar mass limits would be significantly higher for a line candidate associated with CO(2–1) emission at $z>5$. Indeed, repeating the same calculations for $z\sim5.8$ we obtain significantly less constraining stellar mass limits of $\sim1.3\times 10^{11}$ and $\sim4\times 10^{10}\,{\rm M}_\odot$ for a representative $A_V\sim2.5$, in COSMOS and GOODS-N, respectively.
### Radio counterparts
We also searched for counterpart matches in the deep COSMOS 3 GHz continuum catalog [@Smolcic17], only finding associations for COLDz.COS0, COS1, COS2 and COS3 by using a 3$^{\prime\prime}$ search radius. Of the 18 sources from the [@Smolcic17] catalog located within the boundaries of our mosaic, our secure sources represent the only ones with a redshift estimate (photometric or spectroscopic when available) falling within our survey volume. All the remaining sources from the [@Smolcic17] catalog within our survey area lie in the range $z=$0.1–1.6. We performed an equivalent search in the catalog from the eMERGE 5.5 GHz survey of the GOODS-N field [@eMerge], finding a single association for GN19 by using a 3$^{\prime\prime}$ search radius. We also searched the VLA 1.4 GHz catalog of the GOODS-N field from [@Morrison10] with the same criteria, finding two matches for GN19 and a radio counterpart for GN10. We also used these radio catalogs to search for counterpart associations with CO candidates to a lower significance of $5\sigma$. We found one candidate at a SNR=5.13 which satisfies the requirement of close association with a radio source (within $3^{\prime\prime}$), and with a CO redshift which is compatible with the $1\sigma$ interval for the photometric redshift listed by the optical-NIR photometric catalogs. This candidate, named COLDz.GN.R1 in the following, is at J2000 12:37:02.53 +62:13:02.1, and has an offset of $1^{\prime\prime}.7\pm0^{\prime\prime}.6$ from the radio source. We show this candidate in Fig. \[fig:radio\_cand\_figs\]. The photometric redshift estimate is 4.73–5.30, which is compatible with the CO(2–1) redshift of $z_{\rm 21}=5.277\pm0.001$ implied by the COLDz data. We measure a CO(2–1) line luminosity of $(8\pm3)\times10^{10}\,$K km s$^{-1}$pc$^2$, which implies a gas mass of $(2.9\pm1.1)\times10^{11}\,{\rm M}_\odot$ for a standard $\alpha_{\rm CO}$=3.6 M$_\odot$ (K km s$^{-1}$ pc$^2$)$^{-1}$ and $r_{\rm 21}=1$. The [@Skelton14] catalog reports a stellar mass of $2.8\times10^{11}\,$M$_\odot$, which suggests a molecular gas mass fraction of $\sim1$. The radio continuum fluxes are S$_{\rm 1.4\,GHz}=(26\pm4)\,\mu$Jy and S$_{\rm 5.5\,GHz}=(20 \pm4)\,\mu$Jy (@Morrison10 [@eMerge], respectively). This suggests a star formation rate of $\sim200-400\,$M$_\odot$ yr$^{-1}$ when applying the radio-FIR correlation [@Delhaize17]. Although this line candidate has a higher probability of corresponding to real emission than implied by its SNR, we do not include it in the statistical analysis to preserve the un-biased (i.e., CO SNR-limited) nature of our selection.
The deep radio catalogs by [@Smolcic17] in COSMOS and by [@Morrison10] in GOODS-N have a $5\sigma$ sensitivity limit of $\sim11\,\mu$Jy at 3 GHz and 20 $\mu$Jy at 1.4 GHz, respectively which can be converted to upper limits on the $L_{\rm FIR}$ for radio counterparts to our line candidates through the radio-IR correlation. By adopting the relationship from [@Delhaize17], we deduce a detection limit of $L_{\rm FIR}<4-7\times10^{11}\,L_{\odot}$ in the $z=1.953$–2.847 redshift range. On the other hand, recent results have suggested that the radio-FIR correlation in disk-dominated star-forming galaxies may not show a redshift evolution as used by [@Delhaize17] [@Molnar]. If true, this would suggest less constraining limits of $L_{\rm FIR}<1-3\times10^{12}\,L_{\odot}$. The $5\sigma$ sensitivity limit of the eMERGE catalog at 5.5 GHz is approximately 15 $\mu$Jy, and corresponds to limits of $L_{\rm FIR}<8-14\times10^{11}\,L_{\odot}$ according to [@Delhaize17], and to $L_{\rm FIR}<2.5-6\times10^{12}\,L_{\odot}$ according to [@Molnar]. These limits may be constraining, because our measured $L'_{\rm CO}$ would imply median $L_{\rm FIR}\sim10^{12}\,L_{\odot}$ and $\sim4\times10^{12}\,L_{\odot}$ based on the star-formation law [@Daddi10b; @Genzel10] for our unconfirmed line candidates in the COSMOS and GOODS-N fields, respectively. Possible reasons for the lack of radio counterparts may be due to fainter radio fluxes in our sample than expected from the radio-IR correlation, lower star formation rates than expected based on the gas masses, or that candidates may correspond to CO(2–1) emission at $z>5$. Alternatively, line candidates may not be real and be due to noise. The possibility of gas-rich, low star-formation rate galaxies would be particularly interesting, because surveys like the one reported here may be the only way to uncover such a hidden population.
Identification and stacking of galaxies with previous spectroscopic redshifts
=============================================================================
------------------ -------------------- ------------------ ------------------ ------------------ -------------------- --------------------------
ID Preferred Redshift PdBI preferred PdBI covered This survey Low-[*J*]{} Flux $L'_{\rm CO}$ constraint
Mid-[*J*]{} line Mid-[*J*]{} line Low-[*J*]{} line or 3$\sigma$ limit
(Jy km s$^{-1}$)
ID.01 1.88 2–1 5–4 2–1 $<$0.05 r$_{52}>$1.6
ID.02 1.81 2–1 5–4 2–1 $<$0.02 r$_{52}>$2.6
ID.03 1.78 (secure) 2–1 5–4 2–1 $<$0.05 r$_{52}>$1.8
ID.04 1.71 2–1 5–4 2–1 $<$0.03 r$_{52}>$0.9
ID.05 2.85 3–2 5–4 2–1 $<$0.06 r$_{52}>$1.2
ID.08 (HDF850.1) 5.19 (secure) 5–4 5–4\* 2–1 $0.17\pm0.06$ r$_{52}=0.40\pm0.16$
ID.10 2.33 3–2 3–2\*/6–5 1–0/2–1 $<$0.03 r$_{31}\&r_{62}>$0.7
ID.11 2.19 3–2 3–2\*/6–5 1–0/2–1 $<$0.04 r$_{31}\&r_{62}>$0.9
3–2 7–6 2–1 $<$0.03 r$_{72}>$1.0
ID.12 2.19 3–2 3–2\*/6–5 1–0/2–1 $<$0.05 r$_{31}\&r_{62}>$0.6
3–2 7–6 2–1 $<$0.03 r$_{72}>$0.7
ID.13 2.18 3–2 3–2\*/6–5 1–0/2–1 $<$0.04 r$_{31}\&r_{62}>$0.6
3–2 7–6 2–1 $<$0.03 r$_{72}>$0.7
ID.14 2.15 3–2 3–2\*/6–5 1–0/2–1 $<$0.04 r$_{31}$or r$_{62}>$0.8
3–2 7–6 2–1 $<$0.03 r$_{72}>$0.7
ID.15 2.15 3–2 3–2\*/6–5 1–0/2–1 $<$0.04 r$_{31}\&r_{62}>$1.2
3–2 7–6 2–1 $<$0.03 r$_{72}>$1.2
ID.17 (HDF850.1) 5.19 (secure) 6–5 6–5\* 2–1 $0.17\pm0.06$ r$_{62}=0.24\pm0.10$
ID.18 2.07 3–2 3–2\*/6–5 1–0/2–1 $<$0.05 r$_{31}\& r_{62}>$1.3
3–2 7–6 2–1 $<$0.04 r$_{72}>$1.3
ID.19 2.05 (secure) 3–2 3–2\*/6–5 1–0/2–1 $<$0.07 r$_{31}$ & r$_{62}>$0.7
3–2 7–6 2–1 $<$0.04 r$_{72}>$0.9
ID.20 2.05 3–2 3–2\*/6–5 1–0/2–1 $<$0.05 r$_{31}\& r_{62}>$0.8
3–2 7–6 2–1 $<$0.03 r$_{72}>$1.1
ID.21 3.04 4–3 7–6 2–1 $<$0.04 r$_{72}>$0.8
------------------ -------------------- ------------------ ------------------ ------------------ -------------------- --------------------------
\[Decarli\_counterpart\]
**Note** Preferred redshifts are quoted from [@Decarli14]. Although we systematically constrain every possible J line assignment that would place a CO line in our data, the asterisk marks those cases where the assignment is preferred by [@Decarli14]. ID.03 has a secure redshift identification which places it outside our redshift coverage. ID.19 has a secure redshift which places it within our coverage.
Identification and stacking of previous mid-[*J*]{} blind CO surveys
--------------------------------------------------------------------
We searched the GOODS-N dataset for low-[*J*]{} CO counterparts to the candidate mid-[*J*]{} CO detections from our previous CO blind survey in the HDF-N with the PdBI [@Decarli14]. We find a single match in our candidate list, which corresponds to CO(2–1) line emission from HDF850.1. We systematically searched for every possible mid-[*J*]{}/low-[*J*]{} CO line combination that would place a low-[*J*]{} CO line in our surveyed volume (Table \[Decarli\_counterpart\]). Several of these possible mid-[*J*]{}/low-[*J*]{} CO line combinations are not the preferred line identifications by [@Decarli14]. Therefore, our non-detections are consistent with their preferred redshift in those cases constraining or ruling out several alternative redshift solutions allowed by the PdBI data alone. In order to search for lower significance candidate lines, we extract spectra at the mid-[*J*]{} candidate positions, and evaluate the significance of any features or place $3\sigma$ upper limits to the line fluxes. By assuming the same line FWHM as the candidate mid-[*J*]{} CO lines, we then derive limits on the line brightness temperature ratios (Table \[Decarli\_counterpart\]). We evaluate the signal-to-noise by spectral (1D) match-filtering of individual spectra, extracted in the central pixel and within 5 frequency channels around the expected position of the lines. We do not find any significant detections above a $3\sigma$ threshold. Some of our upper limits imply super-unity line brightness temperature ratios for the mid-[*J*]{} CO candidates. While a lower-[*J*]{} level can be less populated than a higher-[*J*]{} level, or low optical depths can cause such high line ratios, the physical conditions that give rise to such ratios are rare. In cases where super-unity line ratios are found for redshifts (i.e., mid-[*J*]{}/low-[*J*]{} CO line combinations) disfavored by [@Decarli14], our data provide supporting evidence for the preferred redshifts identified by [@Decarli14], under the assumption that those line candidates are real. As an example, in the case of ID.03, multiple line detections have determined a secure redshift. Since this redshift does not lie within our surveyed volume, our non-detection is consistent with the redshift identification by [@Decarli14]. On the other hand, candidate ID.19 was confirmed to lie at $z=$2.0474 based on optical grism spectroscopy [@Decarli14]. Therefore, the candidates lies in our survey volume and our line ratio limit ($r_{31}>0.7$) is significant. This suggests moderately elevated CO excitation compared to the average ratio found for a sample of Main Sequence galaxies at $z=$1.5 ($r_{31}=0.42$, @Daddi15), and even compared to the average ratio ($r_{31}=0.52\pm0.09$) found for a sample of sub-millimeter galaxies [@Bothwell13].
For the mid-[*J*]{} line candidates ID.15 and ID.18, our constraints on the line ratios are higher than unity. This suggests that an alternative lower redshift mid-[*J*]{} line assignment of CO(2–1) in the PdBI data may be more likely (since it would imply a redshift outside our survey volume), if the line candidate were confirmed to correspond to real emission.
We also stack the extracted spectra to obtain more sensitive limits. In particular, we select random subsets of candidates for stacking, to take into account a possible mis-identification of the correct [*J*]{} value for some CO lines. To search for lines in the stacked spectra, we match-filter using the same set of spectral templates as the main line search (Table \[template\_sizes\]). We find no line signal in the stacks above $3\sigma$ significance. Assuming an average line FWHM of 300 km s$^{-1}$, we therefore obtain a sensitive $3\sigma$ upper limit of $0.014\pm0.002$ Jy km s$^{-1}$ to the line fluxes, where the quoted uncertainties on the limit depend on the number of stacked spectra. From the same sample, we also separately stack the nine candidates for which the lines were identified as likely CO(3–2) emission by [@Decarli14], and whose redshift would place their CO(1–0) line in our data cube. We obtain a $3\sigma$ limit of $\sim0.019$ Jy km s$^{-1}$, which implies a constraining limit of $r_{31}>2.0$ when using the mean CO(3–2) flux in the limit (0.34 Jy km s$^{-1}$, using the same weights as in our stack). We estimate how many of these stacked spectra need to be removed in order for the line luminosity ratio to become smaller than unity. We find that at least six of them may not correspond to real emission, subject to the stated assumptions. In summary, we find some tentative evidence suggesting that mid-[*J*]{} blind CO searches may preferentially select galaxies with relatively high CO excitation. An alternative interpretation may be that some of the candidate mid-[*J*]{} CO emitters considered here, may be spurious and do not correspond to real CO emission. In order to more strongly differentiate between these possibilities, more sensitive 3 mm observations need to be carried out.
Identification and stacking of galaxies with optical redshifts
--------------------------------------------------------------
In order to obtain additional constraints on the CO luminosity of galaxies that remain individually undetected in the volume covered by our survey, we utilize the available optical/NIR spectroscopic redshift information for galaxies in our well studied target fields for stacking. We extract single pixel spectra of the sources described in the following, and we stack them with a weighted average. As weights, we used the inverse of the variance of the local noise following [@ASPECS2]. We present additional, less constraining, stacks of galaxies in Appendix G, where we consider galaxies with grism redshifts and galaxies at higher redshifts, for which CO(2–1) may lie within our data.
### Spectroscopic redshifts in the COSMOS field
Only seven galaxies have known ground-based optical spectroscopic redshifts that place them within our COSMOS data cube, all of which were obtained as part of the zCOSMOS-deep survey (Lilly et al., in prep.). These galaxies have relatively low stellar masses ($\lesssim10^{10}\,M_\odot$). Therefore, we do not expect to detect their CO emission individually. We also do not detect their averaged CO line emission down to a deep $3\sigma$ limit of $<$0.008 Jy km s$^{-1}$ (assuming a line FWHM of 300 km s$^{-1}$), after stacking spectra extracted at their positions (stacked spectrum shown in Fig. \[fig:stacked\_spectra\]). This limit implies $L^\prime_{\rm CO}<1.7-2.7\times 10^{9}\,$K km s$^{-1}$ pc$^2$ for different redshifts within our surveyed range. In order to determine the implications of this limit, we perform SED fitting of the same galaxies with [magphys]{} [@magphys08; @magphys15] to estimate their stellar masses, finding that they are compatible with the tabulated values in COSMOS2015 whenever the photo-$z$ is similar to the spectroscopic redshift (only 3/7 cases). These stellar masses are in the range $10^9\,{\rm M}_\odot-10^{10}\,{\rm M}_\odot$. Assuming that these galaxies lie on the Main Sequence, we use the fitting functions from [@Speagle] to determine star formation rates (SFR) $\sim2-25\,{\rm M}_\odot \, {\rm yr}^{-1}$. These values are consistent with the lack of detections by the 3 GHz survey by [@Smolcic17], which implies SFR$<40-70\,{\rm M}_\odot \, {\rm yr}^{-1}$ based on the radio-FIR correlation[^8] estimated by [@Delhaize17]. The SFRs estimated by [magphys]{} for these galaxies (with great uncertainty due to the lack of FIR detections) span the 6–150$\,{\rm M}_\odot \, {\rm yr}^{-1}$ range, with a mean of 55$\,{\rm M}_\odot \, {\rm yr}^{-1}$. Assuming the star formation law found for Main Sequence galaxies at high redshift[^9] [@Daddi10b; @Genzel10], we can use the star-formation rate estimates to infer an expected $L^\prime_{\rm CO}$ $\sim8\times 10^9 \,$K km s$^{-1}$ pc$^2$, which is higher than our measured limit, and therefore not fully consistent. The adopted chain of scaling relations, including SED fitting and the star formation law have large scatter, and therefore introduce large uncertainties in the $L^\prime_{\rm CO}$ estimate. The apparent tension with our upper limit would disappear if the average SFR were $\sim10-15\,{\rm M}_\odot \, {\rm yr}^{-1}$. The non-detection of the stacked CO(1–0) emission therefore provides a valuable constraint on the CO luminosity of faint, modestly massive $z=2$–3 galaxies, as our best estimates for their SFR appear to be in tension with the expectated $L^\prime_{\rm CO}$ based on the star formation law by [@Daddi10b; @Genzel10].
### Spectroscopic redshifts in the GOODS-N field
The 3D-[*HST*]{} catalog [@Skelton14; @Momcheva16] provides 67 galaxies in the region in GOODS-N covered in our survey area with ground-based optical spectroscopic redshift, whose CO(1–0) line is covered by our data. We also include 13 more galaxies with more recent spectroscopic redshifts from the catalog by [@superdeblended] in our analysis. One of the galaxies in this combined sample corresponds to GN19 and is individually detected. Therefore, we exclude it from further investigation here. One more galaxy with a spectroscopic redshift of $z_{\rm spec}=2.320$ is perfectly matched, in position and redshift with a SNR=4.6 CO line candidate at coordinates J2000 12:36:49.10 +62:18:14.0 and $z_{\rm 10}=2.3192\pm0.0003$, which we name COLDz.GN.S1 (Figure \[fig:radio\_cand\_figs\]). This CO line candidate is significantly extended (FWHM of $\sim9.0^{\prime\prime}\pm0.5^{\prime\prime}\sim74\pm4$ kpc) and appears to be associated with two, potentially interacting, galaxies with a projected separation of $\sim20$ kpc. The galaxy to the south is associated with the spectroscopic redshift measurement, while the galaxy to the North has a compatible photometric redshift estimate. The total aperture CO flux of GN.S1 is $(0.22\pm0.11)$ Jy km s$^{-1}$, corresponding to $L'_{\rm CO}=(5\pm3)\times10^{10}\,$ K km s$^{-1}$ pc$^2$. We find a molecular gas mass of $(2.0\pm1.1)\times10^{11}\,$M$_\odot$ when assuming $\alpha_{\rm CO}=3.6\,{\rm M}_\odot$ (K km s$^{-1}$ pc$^2$)$^{-1}$. The stellar mass of the southern (brightest) component is $\sim1.2\times10^{11}\,$M$_\odot$, and that of the northern component $\sim3\times10^{9}\,$M$_\odot$, suggesting a high gas mass fraction $\sim1.7$. This gas fraction may be elevated due to the galaxy interaction, although the star formation rate reported by [@superdeblended] of $\sim160$ M$_\odot$ yr$^{-1}$ is approximately $\sim2-3\times$ lower than what may be expected from the total CO luminosity based on the star formation law [@Daddi10b; @Genzel10]. Therefore, we may be witnessing a gas-rich early phase of the merger, which may precede a starburst. This source is also tentatively detected in the S2CLS map, with a significance of $\sim3.3\sigma$ and a $850\,\mu$m flux of $3.2\pm1.0$ mJy [@Geach17], which is compatible with the moderate star formation rate estimate. Allowing for an offset of $<2^{\prime\prime}$ and $<500\,$km s$^{-1}$ results in four more potential candidate association, but they are not likely to be real due to apparent offsets in the emission and because they are not associated with the most massive, or most star-forming galaxies in the sample. In the set of galaxies with spectroscopic redshifts covered by our data, nine galaxies have stellar mass estimates of $>5\times10^{10}\,{\rm M}_\odot$, corresponding to a gas fraction ${\rm M}_{\rm gas}/{\rm M}_*\sim1$ at our approximate $3\sigma$ sensitivity limit of $L'_{\rm CO}\sim1.5\times10^{10}\,$ K km s$^{-1}$ pc$^2$. These galaxies may therefore be expected to be individually detectable. Excluding GN19 and GN.S1, the remaining seven galaxies remain undetected, implying M$_{\rm gas}/{\rm M}_*<1$. Previous samples of Main Sequence galaxies at $z=2-3$ have shown typical molecular gas mass fractions of order M$_{\rm gas}/{\rm M}_*\sim1-1.5$ in this stellar mass range [@Genzel15; @Scoville17], i.e., higher than those limits. We note that adopting the same CO conversion factor as utilized by [@Genzel15], including the correction for a stellar-mass dependent metallicity, would result in approximately 50% higher molecular gas mass estimates. Overall, the observed limits may be consistent with previous observations of the molecular gas fractions in Main Sequence galaxies, although they appear to be at the low end of the expected scatter of the relation.
Stacking all 78 spectra, i.e, excluding GN19 and COLDz.GN.S1, yields a tentative ($\sim3.4\sigma$) CO line detection in the deep stack ($6\pm3\times 10^{-3}$ Jy km s$^{-1}$; Fig. \[fig:stacked\_spectra\]). The noise in this stacked spectrum is $\sim23\,\mu$Jy beam$^{-1}$ in 35 km s$^{-1}$-wide channels. The stacked galaxies in this sample have a wide range of stellar masses, and are therefore expected to show a range of CO luminosities. While constraining the average CO luminosity for this sample, we note that such an average does not represent common properties of the stacked galaxies. The measured flux in the stacked spectrum corresponds to an average CO luminosity for this galaxy sample of $L'_{\rm CO}=(1.5\pm0.8)\times10^9$ K km s$^{-1}$ pc$^2$ at an average $z\sim2.4$. These 78 galaxies have a mean stellar mass of $2.1\times10^{10}\,M_\odot$ and an average star formation rate of $\sim45\,{\rm M}_\odot$ yr$^{-1}$ (with quartiles of 3.6, 22 and 66 M$_\odot$ yr$^{-1}$, respectively), according to [@superdeblended] when available, and to [@Skelton14] otherwise. We also stack the subset of 34 spectra corresponding to the most massive galaxies with M$_*>10^{10}\,$M$_\odot$, expecting them to contribute the strongest CO signal. We detect emission in this sub-stack with a significance of 3.5$\sigma$, corresponding to a line flux of $1.2\pm0.7\times 10^{-2}$ Jy km s$^{-1}$ and a line FWHM of $200\pm80$ km s$^{-1}$ (Fig. \[fig:stacked\_spectra\]). The line flux in the stacked spectrum corresponds to $L'_{\rm CO}=(3.2\pm1.8)\times10^9$ K km s$^{-1}$ pc$^2$ at an average $z\sim2.4$, which corresponds to a molecular gas mass of M$_{\rm gas}=(1.2\pm0.6)\times 10^{10}\,$M$_\odot$ according to our earlier choice of $\alpha_{\rm CO}$. This gas mass should be compared to the mean stellar mass $\sim3.6\times10^{10}\,$M$_\odot$ of this sub-sample (with a median of $\sim2.8\times10^{10}\,$M$_\odot$) and the average star formation rate of $\sim66\,{\rm M}_\odot$ yr$^{-1}$ (median of $\sim52\,{\rm M}_\odot$ yr$^{-1}$) [@superdeblended]. To further investigate these star formation rate estimates, we also stack SCUBA-2 S2CLS 850 $\mu$m images and derive a 3$\sigma$ upper limit of $<0.7$ mJy at the positions of the sample galaxies [@Geach17]. Utilizing the average FIR SED from the ALESS sample, at approximately the same redshift as these galaxies, the sub-mm flux upper limit implies a star formation rate constraint of $\lesssim60-100$ M$_\odot$ yr$^{-1}$ [@ALESS_daCunha]. A modified black body model with a dust temperature of $T_d=35$K also implies comparable limits. We find that the star formation rate of this set of galaxies is compatible with random scatter around the star-forming Main Sequence reported by [@Speagle]. The average expected CO luminosity based on the star formation law [@Daddi10b; @Genzel10] may be higher than our measurement by about a factor of 3 ($L'_{\rm CO}\sim9.6\times10^9$ K km s$^{-1}$ pc$^2$).
### Implications
Figure \[fig:gas\_frac\] summarizes our constraints on the molecular gas mass fraction in the analyzed samples of galaxies. In order to convert CO luminosity to molecular gas mass, we consider two different assumptions for $\alpha_{\rm CO}$. First, we assume a constant value of $\alpha_{\rm CO}=3.6\,{\rm M}_\odot$ (K km s$^{-1}$ pc$^2$)$^{-1}$ adopted by some previous studies (e.g., @Daddi10b [@Decarli14; @ASPECS1]). We then also consider a metallicity-dependent conversion factor, evaluated by assuming a redshift and stellar mass-metallicity relation [@Genzel15]. Previous studies investigating optically and FIR-selected galaxy samples have estimated the relationship between gas mass fraction, stellar mass, redshift and SFR-offset from the Main Sequence (e.g., @Genzel15 [@Scoville17]). While the PHIBSS project estimated molecular gas masses by measuring the CO(3–2) line emission [@Tacconi13; @Genzel15], [@Scoville16; @Scoville17] have used the flux on the Rayleigh-Jeans tail of the dust continuum emission to estimate the total gas masses. We here assume that the samples of galaxies plotted, although not complete to any degree due to their pre-selection for having a spectroscopic redshift, may be somewhat representative of the star-forming Main Sequence (Figure \[fig:gas\_frac\]). Their star formation rates are consistent with scatter around the Main Sequence and appear to include as many galaxies above and below the Main Sequence estimated by [@Speagle]. Although our CO detections (both blind and with previous spectroscopic redshifts) are indicative of gas fractions compatible with or above (GN19) expectations for Main Sequence galaxies, the individual CO non-detections and the stacked signal appear to be systematically lower than the predicted averages, suggesting lower gas mass fractions than might be expected (Figure \[fig:gas\_frac\]). We can quantify the apparent deficit in stacked signal relative to expectations for the ${\rm M}_*\gtrsim10^{10}\,{\rm M}_\odot$ sample, by calculating expected gas masses for the individual stacked galaxies, predicted as a function of their redshift, stellar masses and star formation rates. The expected sample average molecular gas mass is $5.8\times10^{10}\,{\rm M}_\odot$ adopting the best fit relation by [@Genzel15] and $7.5\times10^{10}\,{\rm M}_\odot$ according to the relation by [@Scoville17]. The constant CO luminosity conversion factor above would imply a ratio between expected and observed stacked CO luminosity of $4.8\pm2.4$ and $6.3\pm3.1$ according to the relations by [@Genzel15] and by [@Scoville17], respectively. Applying instead the metallicity dependent CO conversion factor suggested by [@Genzel15] to individual galaxies would somewhat reduce the tension, implying ratios of $3.0\pm1.7$ and $3.8\pm2.1$ according to the relations by [@Genzel15] and by [@Scoville17], respectively. While the constraints for low stellar mass galaxies may be compatible with an evolving CO conversion factor due to low metallicity, this is unlikely to resolve the apparent conflict at the high mass end, and may point to lower than expected gas masses.
Total CO line brightness at 34 GHz
==================================
One additional key measurement that becomes possible with the COLDz survey is to determine the total CO line brightness at 30–39 GHz in the survey volume. We follow the simple procedure outlined by [@Carilli16], and as a first conservative estimate, include only the independently confirmed line candidates, for each field in order to derive secure lower bounds. We derive lower brightness temperature limits (since we only include the securely detected sources, without any completeness correction) at the average frequency of 34 GHz of $T_B\gtrsim0.4\pm0.2\,\mu$K for the COSMOS field and $T_B\gtrsim0.05\pm0.04\,\mu$K for the GOODS-N field, respectively. The uncertainties are dominated by Poisson relative uncertainties, due to the limited number of sources considered. Sources near the knee of the CO luminosity function (Paper [slowromancap2@]{}) dominate the total surface brightness, as expected. Since the two measurements are sensitive to different parts of the CO luminosity function, we add the two values to obtain our best estimate for a lower limit on the average surface brightness of $T_B\sim0.45\pm0.2\,\mu$K. Next, we attempt to include a longer list of candidates, down-weighted by their purities (evaluated in Appendix F), to estimate a plausible uncertainty range. In the COSMOS field, also including all moderate SNR candidates presented in Table \[tab\_lines\_appendix\], we obtain $T_B\sim0.48\,\mu$K and $T_B\sim0.57\,\mu$K without and with the completeness corrections evaluated in Appendix F, respectively. In the GOODS-N field, also including all candidates in Table \[tab\_lines\_appendix\], we obtain $T_B\sim0.18\,\mu$K and $T_B\sim0.3\,\mu$K without and with the completeness corrections, respectively. Because the complete candidate list in GOODS-N overlaps in flux ranges with the candidates in COSMOS, it is not clear that the best estimate for this case may simply be derived by adding the two contributions; the plausible range of values from our data should therefore be considered to be the full range $T_B\sim0.2-0.6\,\mu$K, with a likely lower limit of $T_B\sim0.45\pm0.2\,\mu$K. These measurements are consistent with that of $T_B\sim0.94\pm0.09\,\mu$K at 99 GHz by [@Carilli16] within the expectation that the total (all CO lines) average surface brightness may slightly increase between 34 GHz and 99 GHz due to adding more CO transitions together (e.g., @Righi08). Our measurement of the average surface brightness is in agreement with theoretical predictions (e.g, @Righi08 [@Pullen13]) which suggest a range of $T_B=0.3-1\,\mu$K. Our constraints on the total CO brightness at 34 GHz suggest that the CO signal will be an important contribution to CMB spectral distortion at these frequencies, which is relevant for upcoming experimental efforts. In particular, as shown in Figure 2 by [@Carilli16], our constraints at 34 GHz are already higher than the PIXIE sensitivity limit [@Kogut11; @Kogut14] and, while lower than the low-redshift Compton distortion component, it is higher than the relativistic correction to the low-redshift signal, the primordial Silk damping distortion, and the imprint of primordial hydrogen and helium recombination radiation contributions. A measurement of these important cosmological probes will therefore necessarily require a subtraction scheme that will remove the CO line signal (also see @Carilli16).
Discussion and Conclusions
==========================
In this work, we have carried out the first blind search deep field targeting CO(1–0) line emission at the peak epoch of cosmic star formation at $z=2$–3. This allowed us to provide the least biased measurement of the molecular gas content in a representative sample of galaxies at this epoch. One of our main findings is the absence of a population of massive, gas-rich galaxies with suppressed star formation in our high signal-to-noise sample, which would have been missed by previous selection techniques. The lower signal-to-noise, and hence lower purity, CO line candidate sample includes several candidates without clear multi-wavelength counterparts, which are therefore possible candidates for such a population of “dark", gas-rich galaxies. Nonetheless, the low purity of such candidate lines requires further, independent confirmation as the absence of a counterpart is more likely to indicate that the line feature may be spurious.
Interestingly, the CO line sources detected with confidence in this study include a mix of different galaxy populations. In particular, our sub-sample of independently confirmed CO emitters contains previously known starbursts like AzTEC-3 (by design), GN10, GN19, and HDF850.1 but also COLDz.COS.1, 2 and 3 and COLDz.GN.S1 which belong to the massive end of the Main Sequence at $z\sim2$ (Pavesi et al., in prep.). This highlights the CO(1–0)-based selection, which does not preferentially select outliers in star formation such as starbursts as preferentially selected by sub-millimeter continuum selected samples. Also, the total gas mass is accurately traced by these measurements, without the extinction biases that affect optical/NIR selected samples.
Most studies of molecular gas in galaxies at high redshift to-date have targeted mid-[*J*]{} CO lines. Although these lines have higher fluxes than the ground state, [*J*]{}=1–0 transition, and therefore are typically easier to detect, their higher critical densities and level energies imply that they do not always faithfully trace the bulk of the gas mass, but that they can be biased towards the dense and warm fraction of the gas reservoir. Therefore, in order to derive gas masses from those mid-[*J*]{} CO lines, an excitation correction needs to be assumed (i.e. a ratio of those lines to the CO(1–0) line brightness), which introduces a source of uncertainty. Previous blind CO searches have targeted mid-[*J*]{} CO lines [@Decarli14; @ASPECS1], and therefore relied on similar excitation correction assumptions in order to derive constraints to the total molecular gas mass. In this study, we have shown that blind CO(1–0) searches, selecting galaxies uniquely through their total gas masses, find a varied sample of galaxies belonging to a mix of different populations, which may be characterized by significant differences in CO excitation (e.g., starbursting and Main Sequence galaxies; @Daddi10b [@CarilliWalter; @Riechers06; @Riechers11a; @Riechers11b; @Ivison_GN19; @Bothwell13]). We also find significant excitation differences among the individual sources (to be described in detail by Pavesi et al., in prep.). Furthermore, our limits on the CO(1–0) line luminosities in the candidates previously selected by [@Decarli14] indicate that the corresponding galaxies either have substantially elevated CO excitation, or that a large fraction of them may not correspond to real line emission.
The so called “wedding cake" design of the COLDz survey, targeting a shallow wide field and a deep narrower field, allows us to provide valuable, independent constraints to different parts of the CO luminosity function (Paper [slowromancap2@]{}) which would not have been possible with a single field due to the limited accessible volume and depth. While the sensitivity of our deeper field (in COSMOS) is within a factor of two of the sensitivity that was previously achieved by ASPECS through ALMA in a comparable redshift bin (after correcting for CO excitation), the volume that we could sample in that field is six times larger. Furthermore, the volume covered in both fields combined is $>$50 times as large as that covered by ASPECS-Pilot and $>60$ times as large as that carried out in the HDF-N with the PdBI, given the $>60$ times larger survey area ($\sim$60 vs $\sim$1 arcmin$^2$).
In this study, we have also significantly further developed the methods utilized to carry out blind searches for emission lines in interferometric datasets. In particular, we have generalized the Matched Filtering technique that is commonly used in the spectral dimension to identify spectral lines, to the regime of interferometric data cubes where sources may be spatially extended. By taking advantage of this new source selection method, we have blindly detected significantly extended CO(1–0) line sources like COLDz.COS.3, which hosts a very large cold gas reservoir ($\sim30-40$ kpc). Furthermore, one of our highest SNR line emitters in the GOODS-N field (GN19) and a galaxy with optical spectroscopic redshift (COLDz.GN.S1), also appear extended ($\sim40-70$ kpc) in CO observations due to a major gas-rich merger in this galaxy (see also @Riechers_GN19 [@Ivison_GN19]). The high incidence, two out of the eight most significant CO(1–0) sources, suggests that extended CO(1–0) sources may in fact be prevalent, in agreement with previous findings [@Riechers_GN19; @Ivison_GN19]. Indeed, through the blind search we have selected other CO candidates which may be significantly extended, some of which might have been missed by previous blind line search techniques searching only for unresolved sources [@Decarli14; @ASPECS1; @ASPECS2]. The candidate CO lines span a large range in line FWHM, from $\sim60$ km s$^{-1}$ to $\sim800$ km s$^{-1}$, although the narrowest ones have not yet been independently confirmed. This demonstrates the need for the inclusion of a broad range in line width templates in order not to miss a significant fraction of the signal. The spread and distribution in the line FWHM we find is comparable to those previously measured (e.g., @Tacconi13), although the occurrence of a particularly broad line (in COLDz.COS.2) in our limited, highest quality sample, suggest that there may be a larger incidence of broad lines in blindly selected CO sources, compared to optical/NIR selections. Nonetheless, due to the limited number of sources, this finding requires further, independent study.
Although the molecular gas fraction estimates for the CO detections are comparable to expectations, the lack of CO detections for a number of massive galaxies with good quality spectroscopic redshifts and the detection of their stacked CO signal suggests that molecular gas mass fractions for typical Main Sequence galaxies may be somewhat lower than expected (Figure \[fig:gas\_frac\]; @Genzel15 [@Scoville17]). A possible caveat to this interpretations may come from higher systematic uncertainty than expected of the optical spectroscopic redshifts used in the stacking, which may lead to missing a fraction of the CO flux in the stack. While this analysis could only be carried out on samples of galaxies with previous spectroscopic redshift measurements, its conclusion is in agreement with the finding from the blind search. In particular, if the gas mass fraction and hence the CO luminosity of the known galaxies (in addition to all other galaxies in the observed cosmic volume without spectroscopic redshifts) were closer to expectations, the number of blind CO detections would have been higher. Empirical predictions based on SED fitting and scaling relations suggest an expected number of CO emitters in the range 10–20 for our field in COSMOS and 5–15 for our field in GOODS-N [@daCunha_predict]. In addition, the high mass end of the galaxy distribution in our cosmic volume provides the strongest result for lower than expected gas masses, but also the full sample over the shown stellar mass range provides important constraints, compatible with a metallicity evolution in the CO conversion factor (Figure \[fig:gas\_frac\]). The significant detection of CO emission from galaxies in the stack suggests that our dataset is rich in additional signal, that is too faint to be reliably blindly identified at the current depth, but which can be mined through spectroscopic observations at other wavelengths. We have therefore demonstrated the power of stacking the CO signal from galaxies with spectroscopic redshifts, in order to fully take advantage of the information in CO deep field data. We have also developed statistical methods (presented in Appendix F) to evaluate the purity, completeness and recovered candidate properties with higher accuracy than previous techniques. This enables us to infer the best constraints to date on the CO(1–0) luminosity function at $z\sim$2–3 (Paper [slowromancap2@]{}).
With this CO deep field study, we also further demonstrate that blind CO searches are sensitive to “optically dark", dust-obscured galaxies at very high redshift, such as GN10 and HDF850.1. In particular, the massive molecular gas reservoirs of these galaxies are among the largest in our field (Riechers et al., in prep.). Our sample of new, high-SNR CO(1–0) spectra for COLDz.COS1, 2 and 3 and for GN.S1 provides a significant contribution to the state of current CO(1–0) measurements of Main Sequence galaxies at $z>2$[^10] (see Pavesi et al., in prep., for details).
Finally, the Next Generation VLA (ngVLA) is necessary to significantly improve the constraints presented here (e.g., @Casey_ngVLA [@Carilli_ngVLA; @ngVLA_McKinnon; @ngVLA_memo]). In particular, an equivalent survey in the 30–38 GHz range with five to ten-fold sensitivity improvement for point-source detection as provided by the ngVLA will allow reaching the depth of these observations in a small fraction of the time ($\sim1/50$), therefore routinely reaching depths of $\log(L'_{\rm CO}/L_\odot)\sim9.5$ in one to two hours of observation. The high survey speed of the ngVLA will uniquely enable the deep, wide area surveys which are necessary to build large statistical samples, currently inaccessible to the VLA. These future surveys will constrain the luminosity function to well below the knee, with percent precision, for a comparable observing effort as the present survey. A significant benefit of the ngVLA will also come from the planned smaller antennas, which increase the field of view for a fixed total collecting area, therefore enhancing the survey speed. In addition, the vast bandwidth of the ngVLA will allow us to simultaneously cover CO(1–0) emission over a large fraction of the age of the Universe, and will therefore allow us to probe CO(1–0) over the almost complete redshift range up to $z\sim$10.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. DR and RP acknowledge support from the National Science Foundation under grant number AST-1614213 to Cornell University. RP acknowledges support through award SOSPA3-008 from the NRAO. We thank Tom Loredo for helpful discussion. IRS acknowledges support from the ERC Advanced Grant DUSTYGAL (321334), STFC (ST/P000541/1) and a Royal Society/Wolfson Merit Award. VS acknowledges support from the European Union’s Seventh Frame-work program under grant agreement 337595 (ERC Starting Grant, CoSMass) RJI acknowledges ERC funding through Advanced Grant 321302 COSMICISM. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work is based on observations taken by the CANDELS Multi-Cycle and 3D-[*HST*]{} Treasury Program (GO 12177 and 12328) with the NASA/ESA [*HST*]{}, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. We acknowledge funding towards the 3-bit samplers used in this work from ERC Advanced Grant 321302, COSMICISM. JAH acknowledges support of the VIDI research program with project number 639.042.611, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO). MTS acknowledges support from the Science and Technology Facilities Council (grant number ST/P000252/1).
Additional details on the line search methods
=============================================
In this section we provide additional details of our line search methods in interferometric data cubes, which were used to carry out the blind line search presented in Section 4 of the main text. We first provide a more complete description of our method of choice, Matched Filtering in 3D[^11] (MF3D, which extends Matched Filtering in the spectral dimension; i.e., MF1D), and then we compare its performance to three alternative methods that we have also investigated.
Matched filtering in 3-D interferometric data cubes
---------------------------------------------------
Since we do not expect the CO line emission in $z=2$–3 galaxies to be resolved over more than a few beams at most, we expect our sources to be spatially well described by a family of 2-D Gaussian templates. Therefore, under the prior of source shape, Matched Filtering is theoretically an optimal detection method.
The Matched Filtering method can be thought of as concentrating all of the extended (spatially and in frequency) signal to a peak pixel that captures both the overall strength of the original signal and how closely this matches the template shape. At the same time, the smoothing of the noise in regions without signal allows us to reliably measure the noise level on the scale probed by the template size. In this way, the problem of finding emission lines with structure is effectively reduced to the problem of just examining peak heights to assess their significance.
We compute templates that are Gaussians in frequency and circular 2D Gaussians spatially (sizes given in Table \[template\_sizes\]). We then convolve the signal-to-noise cube with these templates by multiplication in Fourier space to produce multiple Matched-Filtered cubes, one for each template. The main difference between the traditional application of Matched Filtering in astronomical images and our application to interferometric data comes from the spatially correlated nature of the noise in interferometric images. In the case of un-correlated (i.e., white) noise, the matched filter simply corresponds to the expected source shape and size, but correlated noise introduces deviations from this matching, as described in the following.
The frequency width of a template approximately matches the line width that it selects, because the noise in different channels is uncorrelated. On the other hand, spatially the noise has a non-zero correlation length, as determined by the synthesized beam. Therefore, the “matching" to a template is not the intuitive relation for which spatial template size matches the size that it selects. As an example, for un-resolved sources, i.e. sources whose image is beam-sized, the maximal SNR is realized at the peak pixel rather than over an extended area. To calculate the relationship between template size and selected size, we therefore considered the idealized problem of circular Gaussian beams and Gaussian sources, which can be treated analytically. We calculate the correspondence between template size which maximizes the SNR and source size (see Appendix B). To carry out the calculation, we have to make the approximation of source positions being known a priori, evaluating the signal-to-noise at this position. This is not what is done in practice, since positions are unknown. The pixel with the locally highest SNR is utilized instead. We briefly discuss the effects of this approximation on the recovered SNR in Appendix B.
The results of this calculation show that template “matching" (i.e. providing the maximum SNR) takes place approximately when: $$\sigma_A^2=\sigma_h^2+2\sigma_b^2$$ where $\sigma_A$ is the size (radial standard deviation) of the source in the image (which is given by the sum in quadrature of the real source size and the beam), $\sigma_h$ is the size of the template Gaussian, and $\sigma_b$ is the beam size. For source sizes smaller than $\sqrt{2}\sigma_b$, the template size that maximizes the SNR is an infinitely narrow source template. Therefore, we include a single-pixel template (we call this “0-size", being the limit where the radius of the spatial Gaussian tends to 0), which implements a single-pixel search (i.e., MF1D as a subset of MF3D) and therefore selects un-resolved and very slightly resolved sources. The analytical expression above only provides an indication of the matching dependence, but we do not make use of it in the following. The main simplification comes from measuring the SNR at the (in practice unknown) real position of the source rather than at the local maximum. In Appendix F, we explore the analysis of simulated artificial sources through our MF3D algorithm, and we use those simulations to numerically estimate a probabilistic connection between template sizes and injected source sizes.
The detailed steps of the blind search are summarized by the flowchart in Fig. \[fig:flow\_MF3D\], and in the following. As described in Section 2, in order to correctly mosaic different fields together we smooth every pointing to a common, larger, beam. This procedure reduces the SNR for point sources (see Fig. \[fig:Noise\]). We therefore also run a single-pixel Matched Filtering search on the Natural-mosaic, which was obtained without any smoothing. While the lack of a common beam in the Natural-mosaic would, strictly speaking, imply that the spatial structure may not be accurately calculated, this effect is negligible in the COSMOS data, where the different pointings have roughly equal resolution. Furthermore, the Natural-mosaic is sufficient for a search for un-resolved sources, where the flux at the peak pixel represents the total flux, and is therefore correctly recorded in the Natural-mosaic. We treat the result of this Natural-mosaic Matched Filtering step as an additional “spatial template", one for which less smoothing was done than even the single-pixel search in the Smoothed-mosaic. We therefore refer to it as the “-1 pixel" template. In the end, we combine the results from this search with those of the other templates, as detailed in the following.
The signal-to-noise ratio of a detection corresponds to the ratio of the height of the peak in the Matched-Filtered cube to the standard deviation of empty regions. We initially normalize our templates such that the sum of the squares of the template values equals one. For independent pixel noise (which applies to the frequency channels, but not to the spatial pixels), this normalization choice would imply that the noise after convolving would be the same as the noise before. This implies that the peak height corresponds to the total SNR of the candidate. For the 3D case, in particular for spatially extended templates, we need to account for the fact that the synthesized beam size results in small scale spatial noise correlations. While the calculation for the noise in the smoothed cube is close to the measured values (see Appendix B), we decide to measure the noise in the convolved cubes directly from the standard deviation of pixel values. Since our dataset is mostly free of signal, we estimate the noise in each Matched-Filtered cube in the COSMOS field simply by taking the standard deviation of the whole cube, and normalize by dividing each pixel by this value. In the case of the GOODS-N mosaic though, the beam size is not uniform across the mosaic, even after smoothing. In particular, the beam size in the pointings that had higher native resolution, ends up being larger after smoothing than in the other pointings due to the particular nature of the $uv$ coverage (see Appendix C for details). The main consequence of this slightly spatially varying beam, affecting the Smoothed-mosaic for the GOODS-N field, is that during Matched Filtering, the noise in the Matched-Filtered cubes is not uniform. This is expected, as the noise change during convolution is a function of the ratio of the pre- and post-convolution beam size. We therefore measure the noise in the Matched-Filtered cubes, separately for two sets of pointings (GN29–GN42; and all the others), and use these sets to construct an approximate noise map for the Matched-Filtered mosaics in GOODS-N, for each template (see Appendix C for details).
--------- --------------- ------------------
Spatial FWHM Frequency FWHM
(arcsec) (4 MHz-channels)
COSMOS 0, 1, 2, 3, 4 4, 8, 12, 16, 20
GOODS-N 0, 1, 2, 3, 4 4, 8, 12, 16, 20
--------- --------------- ------------------
: Template Sizes for MF3D Line Search Technique
\[template\_sizes\]
**Note** Gaussian template sizes utilized in our Matched Filtering in 3D. A spatial size of 0 stands for a single-pixel spatial extent, and implements the single-pixel search that is optimal for un-resolved sources. These sizes represent a uniform sampling of the parameter space that we conservatively expect to represent CO sources. 4 MHz correspond to $\sim$35 km s$^{-1}$, mid-band.
In order to blindly identify line features in our data and evaluate their significance using the Matched-Filtered cubes (one for each template), we need to locate the peaks and determine the template which provides the highest SNR to each candidate, thereby identifying the template that best matches the feature shape. The first stage is to identify, in each Matched-Filtered cube, the peaks, i.e. the local maxima above some significance threshold. In order to find the significance and position of a peak, we select all voxels (i.e., volume pixels) in the data cube above a fixed threshold, and then retain those voxels that are local maxima by comparing to the values in a small surrounding box of 12 channels by 8 pixels by 8 pixels.
Next, we cross-match objects identified in the different Match-Filtered cubes (obtained from the different templates) in order to remove repeat identifications of the same object. We form a master list of all objects selected from all templates, sorted by SNR. We then parse through each entry, from highest to lowest SNR, and form clusters characterized by their SNR-weighted average positions and the template with the highest SNR. We add a candidate object to a cluster if it resides within a 5.3 voxel radius of the cluster center (this threshold was found to be appropriate for our pixel size and channel width; more generally the frequency and spatial separation thresholds may need to be different) and only if the template under consideration differs from the other templates in the cluster (to avoid clustering features identified in the same template since they are most likely independent objects). By moving down the SNR-sorted list, we guarantee that clusters are built from their highest significance members to their lowest. This method ensures that spatially extended/broad objects that are also identified at lower significance in smaller templates are included in the appropriate cluster as members. For neighboring point-like sources (with high significance in the smallest templates), this method maintains both objects as separate clusters, and allows their corresponding low-significance/extended template candidate to be associated with both clusters. In this way, we avoid grouping separate objects into the same cluster, and we avoid splitting single objects into multiples. Each cluster then corresponds to a single galaxy candidate in our final catalog.
In order to choose the clustering thresholds, and to asses the independence of the result from their precise values, we test how well the algorithms performs in not clumping too much or too little by computing distances to closest neighbors. We test this both for the first stage of clump-finding in the Matched-Filtered cubes to check the method to identify clump peaks works, and for the second stage of matching features across different templates. We inspect the distribution of the neighbor distances, and check that they behave as expected, without splitting clumps into different components (which would show up as many objects having a very close neighbor that would look like part of the same clump to visual inspection), and without including different clumps (by changing the clustering thresholds and looking for any significant changes). We do not find significant issues in either phase of clustering. This technique was therefore used to refine our choice of clustering thresholds. A few objects are objectively difficult to distinguish as one or more parts and so the algorithm performance is at a comparable level to what could be achieved through manual inspection. Overall, the method does very well in finding local peaks, appropriately splitting separate objects even when they are close together. The second stage of associating entries across different templates, while more challenging to evaluate, appears to be largely insensitive to the precise value of the thresholds within a few voxels range.
Comparison to Matched Filtering in 1D
-------------------------------------
A simpler version of our line search algorithm, which we call Matched Filtering in 1D (MF1D), corresponds to extracting a spectrum at each spatial pixel and running a spectral line search on each spectrum with 1D Gaussian templates. As emission lines at high-$z$ are typically approximated as Gaussians, we note that assumptions of square profile templates are less optimal matches in the frequency dimension and therefore do not maximize the SNR for candidate emission lines, although we find the difference to be small.
We have investigated the MF1D approach, which is frequently used in the case of single-dish data, in order to provide a check on the results of our line search and to evaluate its performance. [@ASPECS1] utilized a version of this method, which is effectively matched filtering in 1D with square line templates. The main difference between the method utilized there and our implementation consists in our estimating the noise through the standard deviation of the full SNR cube, rather than individual binned channel maps. This is not expected to cause a significant difference. This method, like MF3D, also requires some prescription for recognizing clusters of significant voxels as belonging to the same candidate when they are close together. We achieved this by building lists of clumps with running average positions, and clustering up to a radius of 9 voxels. In the case of unresolved sources, where the peak pixel contains the maximum signal-to-noise, this method performs just as well as our more general method, since the set of templates used in this technique is a subset of those in MF3D (“0-size" templates). However, it will miss a large fraction of the extended sources by underestimating their true SNR. Although we do not expect a large fraction of resolved sources, a blind search should be as agnostic as possible with regards to the properties of the galaxies that may be selected. Indeed, since CO(1–0) traces the total cold, dense gas mass, it is precisely the tracer that may reveal extended gas reservoirs. One of our top candidates, COLDz.COS.3, harbors a very extended gas reservoir, with SNR peaking in the 2$^{\prime\prime}$ template. A single-pixel search assigns this line a SNR=8.2 rather than 9.2, which would imply a discrepancy of -10%. While this error would not significantly affect the significance of this candidate, such an error would be enough to move a moderate significance candidate with SNR=5.5 , to 4.9, and therefore would effectively be missed by our search. Another advantage of the MF3D method over the 1D is that it allows to capture a larger fraction of the signal, for broader lines, because in that case the peak signal may be substantially spread over several spatial pixels, in different frequency channels, due to noise. While this spreading of the signal over different pixels, in different channels, causes an ambiguity between spectral-SNR and moment-map-SNR for single-pixel methods, this ambiguity is resolved when the full 3D information is taken into account through MF3D. Therefore, we conclude that this method can be absorbed into our more general, improved MF3D framework and that it can be considered a subset of that technique.
Comparison to the `Source Extractor` method
-------------------------------------------
We also considered modifications of existing source finding software, such as `Source Extractor`, which can effectively capture the spatial information of a line candidate while avoiding merging adjacent independent peaks [@SExtractor]. We used the spatial source detection part of SExtractor on individual channel maps with varying frequency binnings. We combined the detections across different binnings and at different frequencies and then established prescriptions to identify lines and their aperture-integrated SNR. These prescriptions made the results very dependent on the precise criteria used to evaluate the significance of a line. The principle is somewhat similar to Matched Filtering. It requires binning data cubes to multiple different velocity widths, and these binnings correspond to templates of different frequency width. Then the method relies on SExtractor for the spatial source extraction (recognizing clusters of high pixels as one unique object). It also requires finding the correct binning that maximizes the signal-to-noise ratio. A challenge for this method is the choice of an aperture size for the flux extraction in the channel maps, to be used in separately evaluating signal and noise. Combining different aperture sizes, which imitates the range of spatial templates in MF3D, introduces additional difficulties with precisely evaluating aperture flux noise. This hybrid technique is sub-optimal in the frequency dimension, because binning is equivalent to filtering with a rectangular function, which is a worse match to the expected spectral line profile than a Gaussian shape, although the difference is small. Our tests show that this line search method, can have similar outcomes in selecting lines to Matched Filtering in 3D for our data. In particular, $>85\%$ of the top $\sim100$ candidates are matched between both methods. Comparing lists of candidates, we find that objects that were assigned a high SNR by the SExtractor method but were less significant in MF3D appeared to be less plausible candidates to visual inspection because of improbable line shapes. The SExtractor method provides a valuable check for our use of extended templates in MF3D. In particular, the extended templates in MF3D allow finding those sources that would be missed by MF1D. The SExtractor method, which is sensitive to extended structure, confirms our extended candidate selection. Therefore, we conclude that our MF3D method coherently combines the results from single-pixel methods and other methods which are biased towards extended sources, like the use of SExtractor with fixed aperture sizes.
Comparison to the [spread]{} technique
--------------------------------------
We have also explored [spread]{}, an algorithm developed by [@Decarli14] for the PdBI blind field line search, to find emission lines in our VLA observations. This method corresponds to binning the data set in frequency, and identifying channel maps with an excess of signal compared to the Gaussian noise pixel intensity distribution. This method does not take advantage of the spatial information (neither spatial extent nor position), but only of the total flux. The excess signal in a channel map does not need to come from a single source, because the [spread]{} statistic is a global value that characterizes the whole channel map. This method did not perform reliably on our data set, since it relies on the small number pixel statistics on the tails of the noise distribution, which are necessarily subject to large fluctuations. The [spread]{} statistic was able to isolate the same top candidate sources as our other methods, but it loses discriminating power below a SNR of $\sim8$, since the [spread]{} statistic does not track SNR and loses the ability to locate moderate significance features. We conclude that MF3D captures any useful information obtained from [spread]{}.
Comparison to Duchamp
---------------------
We also compare our method to the sophisticated line-searching tool [*Duchamp*]{}, which was developed for SKA-precursor data cubes [@Duchamp1]. [*Duchamp*]{} was extensively tested by [@Duchamp_test1; @Duchamp_test2], and found to provide a good blind search algorithm for both unresolved and extended emission. Because our survey is only expected to detect unresolved or slightly resolved CO emission, much of the power of [*Duchamp*]{} (e.g., “a trous" wavelet reconstruction) is not optimized for our targets of interest. The smoothing (convolution) pre-processing offered by [*Duchamp*]{} is equivalent to Matched Filtering with Gaussian templates in the spatial dimension and Hanning templates in the frequency dimension, although [*Duchamp*]{} only allows specifying one template size at a time, and not combining results from different templates. We find that smoothing along the frequency axis is necessary in order to recover even the most significant line emitters in our cubes, as expected due to the wide line-widths relative to channel widths. On the other hand, [*Duchamp*]{} does not allow to smooth in the frequency and spatial dimensions simultaneously, thereby preventing optimal recovery of the full signal-to-noise for slightly extended sources. While the “a trous" wavelet reconstruction is designed to perform well on extended structure of a general shape, it is not optimal for recovering only slightly extended spatial structure, and hence does not yield the same signal-to-noise recovery as Matched Filtering in this specific case of interest. [*Duchamp*]{} offers two choices for peak identification algorithms leading to candidate identification, with pixel clustering being predominantly carried out spatially rather than spectrally. While both of these algorithms perform equally well in recovering all of our top line candidates, the simple 3D peak identification algorithm implemented in MF3D simultaneously utilizes the full 3D information.
Based on all these considerations, we find that the best use of [*Duchamp*]{} in our data is achieved by manually adopting different frequency-width smoothing templates and combining the resulting signal-to-noise ratios, to select unresolved line candidates of different velocity widths. This procedure directly mimics our MF3D method, and therefore we do not adopt [*Duchamp*]{} for the COLDz survey data.
Matched-Filtering interferometric images
========================================
We here discuss the analytical results of our investigation of Matched Filtering in 3D, in the specific case when it is adapted to interferometric images. We study idealized noise and source conditions, in order to derive an approximate relationship between the template spatial size and the “matched" size of a feature that would display the highest SNR for that template. The purpose is to demonstrate the effect of correlated interferometric noise on the sizes that are selected through this technique.
If for simplicity we assume the synthesized beam to be a circular Gaussian, with standard deviation $\sigma_b$, then the noise correlation function can be shown to be $\langle n({\bf x}) n({\bf x'})\rangle=\sigma_0^2 e^{-|{\bf x}-{\bf x'}|^2/4\sigma_b^2}$, where $\sigma_0$ is the noise of the image. Let us define our idealized data as containing a Gaussian source of peak intensity [*s*]{} and convolved size $\sigma_A$, in addition to additive, zero-mean Gaussian noise in the image. We also assume the spatial template to be a circular Gaussian of size $\sigma_h$, i.e. $h({\bf x})=\frac{1}{2\pi\sigma_h^2} e^{-x^2/2\sigma_h^2}$. The expectation value of the template-convolved image at the (assumed known) position of the source is then given by: $\frac{s\sigma_A^2}{\sigma_A^2+\sigma_h^2}$. Furthermore, the standard deviation of the convolved image is given by $\frac{\sigma_0 \sigma_b}{\sqrt{\sigma_b^2+\sigma_h^2}}$. Therefore, the signal-to-noise ratio measured in the Matched-Filtered image is given by $SNR=\frac{s}{\sigma_0} \frac{\sigma_A^2 \sqrt{\sigma_b^2+\sigma_h^2}}{\sigma_b (\sigma_A^2+\sigma_h^2)}$. For a fixed source size $\sigma_A$, this SNR has a maximum at $\sigma_h^2=\sigma_A^2-2\sigma_b^2$ or $\sigma_h=0$ (i.e. the delta function limit of a Gaussian, corresponding to a single-pixel template) in case $\sigma_A^2<2\sigma_b^2$, i.e, if the intrinsic deconvolved source size is smaller than the beam size.
We have run simulations to compare these analytical results to the discretized case of pixels, and did not find significant differences. We also explored the effect of the realistic implementation of Matched Filtering, i.e. where the source position is not known a-priori but the peak of the convolved image is taken instead. The main result appears to be that for $\sigma_A^2$ up to $2\sigma_b^2$ the signal-to-noise is almost flat as a function of template size. Therefore, a single-pixel template and slightly extended templates maximize the signal-to-noise with a smooth and slow transition, as the source size becomes more important relative to the beam size. For the purpose of our measurement, a precise formula for the match between template size and source size is not needed, and a probabilistic assignment based on artificial source recovery suffices. In particular, the results from our artificial sources show that in the full 3D case the matching may be complex and it depends on signal-to-noise as well as on line velocity width. The extra dependence on the line width can be understood as due to the use of the peak value, rather than the value at the known source position, because a wider-velocity line allows for a larger area over which the peak may be found (due to the combination to positive noise and real signal). To conclude, the matching of sources and templates at the basis of Matched Filtering can be approximately estimated from the previous calculation. For unresolved or slightly resolved sources, the SNR is a weak function of templates size. As the $\sigma_A^2\sim2\sigma_b^2$ threshold is approached and crossed, the SNR becomes a rapidly increasing function of template size, with a clear peak for extended templates. Therefore, in order to avoid missing extended sources in blind line searches in interferometric data, we recommend the inclusion of extended (hence 3D) templates, as described in this work.
Accounting for the beam inhomogeneity in our GOODS-N mosaic
===========================================================
In order to mosaic pointings together, it is preferable to smooth all pointings to a common beam. This is not straightforward for the wide-area part of this survey (in the GOODS-N field), due to the large number of pointings observed over the course of several months, which caused a range of array configurations to be utilized.
For the pointings in the COSMOS field, each pointing was observed in every track. Therefore, the mosaic has uniform beam size properties. In the GOODS-N field, on the other hand, the beam differences potentially cause non-uniformity in the mosaic. This is most significant for pointings GN29 to GN42, which were mostly observed in the DnC configuration. In preparation for mosaicking the individual pointings are all smoothed in the image plane to a common beam size with [casa]{} `imsmooth`, but pointings for which a larger amount of smoothing was required end up with slightly larger beams than the target beam size.
The reason why smoothing DnC data seems to necessarily produce slightly different beams than D-configuration data can be appreciated from a look at the Fourier Transform of a typical image from these pointings, effectively their $uv$ coverage, in Fig. \[fig: Smoothing\_GN\]. Smoothing multiplies the $uv$-plane by a tapering Gaussian of the appropriate size, calculated from the size of the starting beam size and the target beam size. However, the D-configuration $uv$ coverage, does not look the same as a Gaussian-tapered DnC-configuration $uv$ coverage, hence the final beam is always going to be an imperfect match (unless both datasets are smoothed to a very large beam, at which point the initial shape of either $uv$-coverage does not matter).
While the slight spatial inhomogeneity of the beam size is inconsequential in producing the Signal-to-Noise ratio cube (as the noise in the mosaic can be calculated analytically and accounted for by Eqn. \[mosaic\_noise\]), the beam size difference causes spatially varying noise in the Matched-Filtered cubes (see Appendix B). The main difference is between the set of 14 pointings (GN29–GN42) and the rest, so we also mosaic and Match-Filter them separately, in addition to working with mosaics with and without this set. Exploiting the improved uniformity within these sub-mosaics, we can measure the noise post-Matched-Filtering. The objective is using the noise in the Matched-Filtered sub-mosaics to calculate the noise in the Matched-Filtered full GOODS-N mosaic. The Signal-to-Noise ratio in the full mosaic is related to the sub-mosaics by:
$$\begin{split}
SNR_{TOT}=\\
&\hspace*{-60pt} \frac{\sqrt{\sum_A A_i^2/\sigma_i^2}}{\sqrt{\sum_{TOT} A_i^2/\sigma_i^2}} SNR_A+\frac{\sqrt{\sum_B A_i^2/\sigma_i^2}}{\sqrt{\sum_{TOT} A_i^2/\sigma_i^2}} SNR_B \doteq \\
& \hspace*{-60pt} f_A SNR_A+f_B SNR_B,
\end{split}
\label{appendix_eqn}$$
where $TOT=A \bigcup B$ represent the set of GN29–GN42 pointings and the set of the remaining pointings. Match-Filtering then corresponds to convolving with template [*h*]{}. This can be expressed as:
$$\begin{split}
(f \cdot SNR) \ast h=\int f(x) SNR(x) h(y-x) dx \simeq \\
& \hspace*{-60pt} f(y) \int SNR(x) h(y-x),
\end{split}$$
with $f(x)\simeq f(y)+ O(f'\cdot FWHM_h) $. To zeroth order, we can take [*f*]{} as constant over the scale of template [*h*]{}, which allows to pull it out of the integral. This approximation is appropriate, because the fraction functions [*f*]{} change slowly over the size of a template. Therefore, the noise after convolving with the template is: $$std[ (f_A(x) SNR_A) \ast h ] \simeq f_A \cdot std[ SNR_A \ast h ],$$ and we can calculate the noise in the Matched-Filtered mosaic by summing the standard deviations from the two terms in Eqn. \[appendix\_eqn\] in quadrature. This method only requires measuring the noise in the Matched-Filtered sub-mosaics, in which the noise is uniform, and the fraction functions: $f_A$ and $f_B$, which can be calculated. Therefore, this process allows us to calculate noise maps of the Matched-Filtered cubes in GOODS-N, thereby accounting for the noise inhomogeneity due to spatially varying beam sizes.
![Top: Aperture spectra of the most significant negative feature in the GOODS-N data in the original data (yellow histogram; where the feature was selected) and in the newer observations (red; pointing GN57). Bottom: The feature, a putative formaldehyde absorption line against the CMB (contours), appeared to be compatible with the 72.4 GHz ($5_{14}$–$5_{15}$) line of formaldehyde at the photo-$z\sim1.13$ of the galaxies shown in the [*HST*]{} H-band image. The left image shows the line map in the original data, the right image shows the same frequency range in the newer data, where the line is not present, which suggests that it was simply due to noise. The contours are shown in steps of $1\sigma$ starting from $\pm2\sigma$ with negative signal as solid contours to show absorption.[]{data-label="fig:formald_spectra_before"}](formald_before_mod.pdf "fig:"){width=".28\textwidth"}![Top: Aperture spectra of the most significant negative feature in the GOODS-N data in the original data (yellow histogram; where the feature was selected) and in the newer observations (red; pointing GN57). Bottom: The feature, a putative formaldehyde absorption line against the CMB (contours), appeared to be compatible with the 72.4 GHz ($5_{14}$–$5_{15}$) line of formaldehyde at the photo-$z\sim1.13$ of the galaxies shown in the [*HST*]{} H-band image. The left image shows the line map in the original data, the right image shows the same frequency range in the newer data, where the line is not present, which suggests that it was simply due to noise. The contours are shown in steps of $1\sigma$ starting from $\pm2\sigma$ with negative signal as solid contours to show absorption.[]{data-label="fig:formald_spectra_before"}](formald_after_mod.pdf "fig:"){width=".23\textwidth"}
Search for Negative features as potential Formaldehyde absorption
=================================================================
Putative feature
----------------
To better understand the characteristics of the noise in our survey data, we have also used our MF3D line searching algorithm to detect negative line features. In order to constrain spurious line features due to noise we take advantage of the symmetry around zero of interferometric noise, in the absence of strong sources in the field. Although negative line features can usually be assumed to be due to noise, line absorption against the uniform CMB has been suggested to be a potential source of such negative lines. In particular, formaldehyde in dense molecular gas in galaxies has been confirmed, at low-$z$, to have the potential to produce such absorption against the CMB [@Zeiger10; @Darling12].
The most significant negative feature in the initial GOODS-N data cube (pointings GN1–GN56) had a high significance of $\sim6.6\sigma$, and it appeared to be coincident with a pair of local interacting galaxies, GOODS J123702.92+620959.0 with a photo-$z$ of $z$=$1.13\pm0.05$ (Fig. \[fig:formald\_spectra\_before\]). Intriguingly, the strong absorption feature would be consistent with the 72.4 GHz ($5_{14}$–$5_{15}$) line of formaldehyde (${\rm H_2CO}$) at $z\sim$1.13. The energy-level structure of formaldehyde allows collisional population anti-inversion in dense molecular clouds, making the line excitation temperature lower than the CMB temperature and producing absorption against the CMB itself. Formaldehyde silhouettes of galaxies in absorption offer both a novel probe of cosmological size and distance, and a measurement of dense gas masses, density and excitation properties [@Darling12]. Therefore, in order to investigate the possibility that our most significant negative feature may be a real absorption line against the CMB, we obtained additional VLA data, both at the same frequency (as an additional pointing, GN57 as part of our main survey) and at 22.6 GHz, in order to target the $4_{13}$–$4_{14}$ line of formaldehyde. We observed this additional tuning with the VLA K-band (project ID: 15B-370; PI: Pavesi) on 6 November 2015. The observations lasted approximately 3 hours (130 min on source) in D-array configuration, with a spectral setup consisting of a single tuning of the two 1-GHz 8-bit samplers (2 GHz total, dual polarization), with central frequencies of 21.58 GHz and 22.5815 GHz for the two intermediate frequencies (IFs), respectively. The same calibrators were observed as for the main survey observations. We calibrated the data using [casa]{} v.4.5 using the VLA pipeline and minor manual flagging, and we imaged the visibilities using natural weighting. The data cube was produced with 1 MHz channels, corresponding to $\sim$13 km s$^{-1}$, which is small compared to the expected linewidth. The data cube has a beam size of $4.4^{\prime\prime}\times 3.4^{\prime\prime}$, and an [*rms*]{} noise of $\sim$0.2 mJy beam$^{-1}$ in 1 MHz-wide channels.
We did not detect the lower frequency line (Fig. \[fig:formald\_spectra\_after\]), and the new observations in pointing GN57 at the same frequency do not show any evidence for absorption at the same position and frequency (Fig. \[fig:formald\_spectra\_before\]). We therefore rule out the presence of an absorption line, and we conclude that the original feature was simply due to noise. By excluding the possibility that the most significant negative feature may correspond to real absorption we strengthen our confidence in the assumption that all (or at least most) negative line features are due to noise, which is crucial for the purity assessment of our positive line candidates.
${\rm H_2CO}$ deep field limits
-------------------------------
Our lack of detections of significant formaldehyde absorption lines allows us to place some of the first constraints on the cosmic abundance of such absorption lines. By assuming a line FWHM of 200 km s$^{-1}$, we derive median $6\sigma$ limits (with no negative line candidates found above this threshold) of 0.18 mJy beam$^{-1}$ and 0.55 mJy beam$^{-1}$ for the COSMOS and GOODS-N fields, respectively, corresponding to $\Delta T_{\rm Obs}$ of $-0.03$ K and $-0.11$ K, at the average frequency and beam size of our survey. We note that the beam size of our observations ($\sim3^{\prime\prime}$) is likely to be larger than the absorbing molecular regions ($\sim0^{\prime\prime}.25-1^{\prime\prime}.25$ at $z\sim1$; @Darling12), implying a dilution of the expected signal strength due to the beam filling factor of $\sim0.025-0.1$. We use the absence of significant negative detections to infer a probability distribution for their space abundance, by assuming a uniform, uncorrelated distribution of sources over the cosmic volume covered by our survey, and therefore, a Poisson number count. The probability distribution for the space abundance is then an exponential distribution with a mode at zero, and a mean equal to the inverse of the volume sampled (the 68th percentile upper limit to the space density is listed in Table \[formald\_table\]). We use the model results from [@Darling12] to derive, from our $\Delta T_{\rm Obs}$ limit, a constraint on the line optical depth. These models imply that, at $z\sim1$, the maximal expected temperature decrement with respect to the CMB is $\Delta T_{\rm Obs}/(1-\exp^{-\tau})\sim-1.2\,$K, for the $5_{14}$–$5_{15}$ and $4_{13}$–$4_{14}$ lines covered by our survey. This implies limits on the line optical depth of $\tau\lesssim$0.025 and 0.1 for the COSMOS and GOODS-N fields, respectively. Although these values are comparable to the optical depths previously measured for the lower frequency formaldehyde transitions, our results may be weaker by an order of magnitude or more, due to the beam filling factor [@Mangum08; @Darling12; @Mangum13].
[ c c c c c c c c ]{} Transition & $\nu_0$ & $z_{min}$ & $z_{max}$ & $\langle z \rangle$ & Volume&$\Delta T_{\rm Obs}$ limit&Volume density\
& \[GHz\] & & & & \[$\rm{Mpc}^3$\]&\[K\]& \[$\rm{Mpc}^{-3}$\]\
\
$4_{13}$–$4_{14}$&48.285& 0.24&0.56&0.44&1,850&$-0.03$& $<6.2\times10^{-4}$\
$5_{14}$–$5_{15}$&72.409 &0.86&1.34 &1.12&9,253&$-0.03$& $<1.2\times10^{-4}$\
\
$4_{13}$–$4_{14}$&48.285&0.27&0.61& 0.47&13,690&$-0.11$& $<8.3\times10^{-5}$\
$5_{14}$–$5_{15}$&72.409 &0.90&1.42&1.18&62,329&$-0.11$&$<1.8\times10^{-5}$\
\[formald\_table\]
![VLA K-band spectrum at the expected redshifted frequency of the 48.3 GHz $4_{13}$–$4_{14}$ line of formaldehyde, at $z\sim1.13$. If the absorption feature we detected in our original data had been a real formaldehyde absorption line we would expect to detect strong absorption, which is not seen. []{data-label="fig:formald_spectra_after"}](formald_lowf.pdf){width=".5\textwidth"}
Description of the individual line candidates
=============================================
In this section, we briefly describe the remaining CO line candidates and potential counterpart associations. These candidates are currently not independently confirmed, and thus, are only used in our statistical analysis. Because we only expect a small fraction of these candidates to correspond to real CO line emission, we advise caution in interpreting these lower significance candidates on a per-source basis until they are independently confirmed. Quoted photometric redshift ranges are the $1\sigma$ uncertainties reported in the COSMOS2015 [@Laigle16] and the CANDELS catalogs [@Brammer; @Skelton14; @Momcheva16]. The positional search radius considered is $3^{\prime\prime}$, which is dictated by the positional uncertainties (which are larger for extended sources) and the possibility of real physical offsets in the stellar emission, e.g., due to differential dust obscuration. The visual counterpart inspection was carried out utilizing [*HST*]{} (H-band in GOODS-N and I-band in COSMOS, where H-band was not available) and IRAC 3.6 $\mu$m images (band 1), which are shown in Figs. \[fig:spectra\_COS2\] and \[fig:spectra\_GN1\]. We have also inspected images from the other IRAC bands and find no evidence for additional counterpart matches relative to IRAC band 1.
COSMOS
------
**COLDz.COS.4**: This is the highest signal-to-noise candidate in COSMOS without a secure counterpart. There is a potential match at $0.5^{\prime\prime}\pm0.6^{\prime\prime}$ to the NW in the COSMOS2015 catalog, with an uncertain photo-$z$=1.5–2.3, matching the CO(1–0) redshift of $z$=2.30. Two I-band and 3.6$\mu$m sources are aligned with the elongated CO candidate emission (Fig. \[fig:spectra\_COS2\]).
**COLDz.COS.5**: No counterpart is found in the COSMOS2015 catalog or the images at the position of this candidate.
**COLDz.COS.6**: The images show an IRAC 3.6$\mu$m source $3^{\prime\prime}\pm0.2^{\prime\prime}$ to the SE of the candidate which has a photo-$z$=2.9–3 and might therefore be associated with our candidate although the offset appears significant (the CO(1–0) redshift is $z$=2.60). At the CO position there is an I-band source with photo-$z$=0.44–0.48, although its 3.6$\mu$m image is contaminated by the brighter, higher-z galaxy. It is unclear if the CO candidate may be related.
**COLDz.COS.7**: The CO(1–0) redshift of this candidate is $z$=2.22. It is at the position of an [*HST*]{} I-band source, which is not in the COSMOS2015 catalog, and is not visible in the 3.6$\mu$m image.
**COLDz.COS.8**: Candidate is at the position of a faint [*HST*]{} I-band and IRAC 3.6$\mu$m source, which is listed in the COSMOS2015 catalog $2.3^{\prime\prime}\pm0.4^{\prime\prime}$ to the N ($z_{phot}$=0.2–0.7).
**COLDz.COS.9**: Candidate shows spatially extended CO emission, centered on an [*HST*]{} I-band source with a photo-$z$=0.1–0.8, which is therefore unlikely to be associated with the candidate.
**COLDz.COS.10**: Candidate is spatially extended and is co-spatial with multiple faint [*HST*]{} I-band galaxies. The COSMOS2015 catalog only reports a faint galaxy $1.3^{\prime\prime}\pm0.7^{\prime\prime}$ to the SW, with a very uncertain photo-$z$ of 0.8–4.3, which may be associated with our candidate.
**COLDz.COS.11**: Candidate is near the position of a low-$z$ galaxy ($z_{phot}$=0.32–0.35). There is a brighter IRAC 3.6$\mu$m source $1.6^{\prime\prime}\pm0.4^{\prime\prime}$ to the NW, ($z_{phot}$=1.5–1.6), which may be related to the CO candidate with a CO redshift of 2.0.
**COLDz.COS.12**: No counterpart is found in the COSMOS2015 catalog or the images at the position of this candidate.
**COLDz.COS.13**: No counterpart is found in the COSMOS2015 catalog or the images at the position of this candidate.
**COLDz.COS.14**: Candidate is affected by foreground contamination which prevents any counterpart assessment. In particular, there is a bright photo-$z$=0.9 galaxy at $1.6^{\prime\prime}\pm0.3^{\prime\prime}$ to the NW.
**COLDz.COS.15**: Candidate has a potential counterpart match. The COSMOS2015 catalog lists a galaxy $1.8^{\prime\prime}\pm0.4^{\prime\prime}$ to the NW with a photo-$z$=1.8–2.8, which is very faint in the I band and IRAC 3.6$\mu$m images. Assuming CO(1–0) would place this candidate at $z$=2.32.
**COLDz.COS.16**: Candidate has a potential counterpart match, but it appears confused with a bright galaxy $1.6^{\prime\prime}$ to the SE, with a photo-$z$=1.0–1.2. The potential counterpart has photo-$z$=1.3–2.6 and is located about $2^{\prime\prime}\pm0.5^{\prime\prime}$ to the NE.
**COLDz.COS.17**: No counterpart is found in the COSMOS2015 catalog or the images.
**COLDz.COS.18**: Candidate has a potential counterpart match, $1.6^{\prime\prime}\pm0.4^{\prime\prime}$ to the N, with a photo-$z$=2.4–2.5; assuming CO(1–0) would place it at $z$=2.68. This candidate is contaminated by a local bright galaxy to the NE.
**COLDz.COS.19**: Candidate does not appear to have a counterpart. An M star is located $0.8^{\prime\prime}\pm0.7^{\prime\prime}$ to the NE, and partly prevents counterpart identification.
**COLDz.COS.20**: Candidate does not have a counterpart. The COSMOS2015 catalog lists two galaxies at separations of $2.2^{\prime\prime}\pm0.3^{\prime\prime}$ and $2.3^{\prime\prime}\pm0.3^{\prime\prime}$, respectively. The first galaxy has a photo-$z$=0.6–0.9 and the second one is at photo-$z=$1.9–2.5. This latter galaxy may be associated with our candidate, which has a CO(1–0) redshift of $z_{10}$=2.67.
**COLDz.COS.21**: No counterpart is found in the COSMOS2015 catalog or the images, at the position of this spatially extended candidate, but the IRAC 3.6$\mu$m images are contaminated by bright nearby stars and galaxies.
**COLDz.COS.22**: A faint galaxy is visible in the [*HST*]{} I-band image, $1.5^{\prime\prime}\pm0.7^{\prime\prime}$ to the NW, which may be associated with our line candidate. The catalog lists a very uncertain photo-$z$=1.5–5.5, which is compatible with the CO(1–0) redshift of $z_{10}$=2.27.
**COLDz.COS.23**: Candidate has a potential counterpart association. This is a galaxy $1.5^{\prime\prime}\pm0.5^{\prime\prime}$ to the SW, which is compatible with the position of at least part of the slightly spatially extended line emission. The photometric redshift for this galaxy is photo-$z$=1.7–2.8, which is compatible with the CO(1–0) redshift of $z_{10}$=1.99.
**COLDz.COS.24**: Candidate has a potential counterpart association. The potential counterpart is only $0.8^{\prime\prime}\pm0.9^{\prime\prime}$ to the N and has a photometric redshift of $z_{\rm phot}$=1.9–2.0 which is close to the CO(1–0) redshift of $z_{10}$=2.26. A second galaxy is seen, $1.6^{\prime\prime}\pm0.9$ to the E, which has a photometric redshift of photo-$z$=0.89–0.92 and which contaminates the emission in the IRAC 3.6$\mu$m images.
**COLDz.COS.25**: This spatially extended candidate has a potential counterpart. This potential counterpart is $1.4^{\prime\prime}\pm0.7^{\prime\prime}$ to the SE and has a photometric redshift of photo-$z$=2.6–3.0 which is close to the CO(1–0) redshift of $z_{10}$=2.43. The IRAC 3.6$\mu$m images are contaminated by a nearby star, which makes it difficult to identify faint sources reliably.
GOODS-N
-------
**COLDz.GN.1**: This spatially extended line candidate has potential counterpart matches. There are multiple galaxies which are compatible with the line emission position, blended in the IRAC 3.6$\mu$m image but visible in [*HST*]{} H-band, with photometric redshifts in the CANDELS catalog [@Skelton14]. The closest catalog match has a separation of only $0.4^{\prime\prime}\pm0.6^{\prime\prime}$ to the NE and has an uncertain photo-$z$=0.8–2.2. The catalog lists three more galaxies within $3^{\prime\prime}$ (separations of $1.3^{\prime\prime}\pm0.6^{\prime\prime}$, $2^{\prime\prime}\pm0.6^{\prime\prime}$ and $2.6^{\prime\prime}\pm0.6^{\prime\prime}$), with photo-$z$s of 1.5–1.7, 0.9–2.5 and 1.1–1.8 respectively. The CO(1–0) redshift of our candidate is $z_{10}$=2.08, which makes it compatible with at least two of these potential counterparts.
**COLDz.GN.2**: This is the highest SNR candidate in GOODS-N without a clear counterpart. The CANDELS catalog lists a faint source $2.6^{\prime\prime}\pm0.3^{\prime\prime}$ to the SE, with uncertain photo-$z$=1.0–2.0 [@Skelton14]. The CO(1–0) redshift of our candidate ($z$=2.54) makes it a possible, although unlikely counterpart.
**COLDz.GN.4**: Candidate is unlikely to have a counterpart. There are no galaxies in the CANDELS catalog within $3^{\prime\prime}$, and no galaxies are visible in the [*HST*]{} H-band or IRAC 3.6$\mu$m images.
**COLDz.GN.5**: There are two galaxies in the images within $2^{\prime\prime}$ of the line candidate, with separations of 1.5$^{\prime\prime}\pm0.3^{\prime\prime}$ and 1.7$^{\prime\prime}\pm0.3^{\prime\prime}$, respectively. They are unlikely to be counterparts because they have photometric redshifts of $z_{phot}=$1.1–1.3 and 0.4–0.5 respectively, while the CO(1–0) redshift of our candidate is $z_{10}$=2.1.
**COLDz.GN.6**: Candidate has a potential match $2.8^{\prime\prime}\pm0.4^{\prime\prime}$ to the SE, in the direction where the CO emission is slightly spatially extended. The catalog lists a photo-$z$ of 2.4–2.5 which is compatible with the CO(1–0) redshift of $z_{10}$=2.51, suggesting a possible counterpart match.
**COLDz.GN.7**: Candidate has an unlikely, but possible match $2.9^{\prime\prime}\pm0.3^{\prime\prime}$ to the SE, which appears to be at a significant offset. The galaxy has a photo-$z$=1.8–2.0, which is not compatible with the CO(1–0) redshift of $z_{10}$=2.76, therefore we do not consider this to be a match.
**COLDz.GN.8**: This spatially extended CO candidate has a possible match $3.0^{\prime\prime}\pm0.8^{\prime\prime}$ to the SE, with an uncertain photo-$z$ of 1.4–2.4 which is compatible with the CO(1–0) redshift of $z_{10}$=2.05. This is a potential match, because the line emission appears to be very spatially extended, and may be compatible with coming from a dust-obscured part of the optical galaxy.
**COLDz.GN.9**: Candidate is unlikely to have a counterpart. It appears near a spec-$z$=0.516 galaxy, which is $2.8^{\prime\prime}\pm0.3^{\prime\prime}$ to the NW. The catalog also lists a faint, photo-$z$=1.9–2.1 galaxy $2.4^{\prime\prime}\pm0.3^{\prime\prime}$ to the SW (which appears to be significantly offset from the CO line emission), which could be consistent with the CO(1–0) redshift of $z_{10}$=2.17.
**COLDz.GN.10**: This slightly spatially extended candidate is unlikely to have a counterpart. The closest catalog association is $1.5^{\prime\prime}\pm0.5^{\prime\prime}$ to the NE and has a photo-$z$ of 4.4–5. The CO(2–1) redshift for our candidate would be $z_{21}$=6.0, and is therefore an unlikely match. The CANDELS catalog lists two more galaxies, just below $3^{\prime\prime}$ to the NE with photo-$z$s of 1.7–2.0 and 1.9–2.4, respectively, which may be compatible with the CO(1–0) redshift of $z_{10}$=2.51.
**COLDz.GN.11**: Candidate appears spatially extended and elongated. No objects are seen in the [*HST*]{} H-band and IRAC 3.6 $\mu$m images. The CANDELS catalog lists a galaxy 1.8$^{\prime\prime}\pm0.7^{\prime\prime}$ to the NW which has a photo-$z$ of 0.6–1.6. This is inconsistent with the CO(1–0) redshift of $z_{10}$=2.29, so a match is unlikely.
**COLDz.GN.12**: Candidate has no likely match. The catalog lists a faint galaxy, $1.7^{\prime\prime}\pm0.4^{\prime\prime}$ to the NE, with photo-$z$=4.1–4.4, which may potentially be associated. The counterpart status is difficult to evaluate due to blending with the bright local (spec-$z$=0.784) galaxy at a separation of just $2.4^{\prime\prime}\pm0.4^{\prime\prime}$.
**COLDz.GN.13**: Candidate is spatially extended and elongated, and is unlikely to have a counterpart association. The CANDELS catalog lists two potential matches within $3^{\prime\prime}$, with separation 1.7$^{\prime\prime}\pm0.5^{\prime\prime}$ and 3$^{\prime\prime}\pm0.5^{\prime\prime}$, respectively. The photometric redshifts listed by the catalog are $z_{phot}=$0.2–0.4 and 0.6–1.4, respectively, which make them unlikely counterparts given the CO(1–0) redshift of our candidate ($z_{10}$=2.14).
**COLDz.GN.14**: This is a spatially extended CO candidate, and it has a possible counterpart, which is faint but visible in the IRAC 3.6 $\mu$m image. We identify this counterpart with the catalog listing of a photo-$z$=2.4–2.7 galaxy which is displaced by $2.8^{\prime\prime}\pm0.7^{\prime\prime}$ to the SW. This counterpart is compatible with the CO(1–0) redshift of $z_{10}$=2.38. The offset may not be significant, because the IR-detected galaxy appears to be compatible with the position of this spatially extended candidate.
**COLDz.GN.15**: Candidate may have a counterpart. The 3.6 $\mu$m image is partly blended with a spec-z=0.453 galaxy $2.3^{\prime\prime}\pm0.6^{\prime\prime}$ to the W which makes the identification difficult. The catalog lists two possible counterparts, with photo-$z$=1.6–1.8 and 3.1–3.9, offset respectively $0.9^{\prime\prime}\pm0.6^{\prime\prime}$ and $1.7^{\prime\prime}\pm0.6^{\prime\prime}$ to the NW and NE, which are not compatible with the CO(1–0) redshift of $z_{10}$=2.46.
**COLDz.GN.16**: Candidate may have a counterpart. The image is partly blended with a spec-$z$=0.961 galaxy $2.3^{\prime\prime}\pm0.6^{\prime\prime}$ to the NE, which makes identification difficult. The CANDELS catalog lists three more galaxies within $2^{\prime\prime}$ and $3^{\prime\prime}$ from our candidate, with photo-$z$=0.9–2.0, 2.2–2.4 and 1.4–2.3, all of which may be compatible with the CO(1–0) redshift of $z_{10}$=2.51.
**COLDz.GN.17**: Candidate appears spatially extended and elongated. The catalog lists a galaxy $1.5^{\prime\prime}\pm0.5^{\prime\prime}$ to the SE, which is visible in the IRAC 3.6 $\mu$m images. This galaxy has a photo-$z$ of 1.2–1.9, which is only somewhat inconsistent with the CO(1–0) redshift of $z_{10}$=2.47. Therefore a counterpart association cannot be ruled out.
**COLDz.GN.18**: Candidate appears to be closely associated with other, lower significance, candidates which are visible in the line maps. It has a potential match, a faint galaxy with photo-$z$=1.3–2.2 only $0.8^{\prime\prime}\pm0.5^{\prime\prime}$ to the SE, which is compatible with the CO(1–0) redshift of $z_{10}$=2.12. The catalog also lists three galaxies $0.4^{\prime\prime}\pm0.5^{\prime\prime}$, $1.8^{\prime\prime}\pm0.5^{\prime\prime}$ and $2.7^{\prime\prime}\pm0.5^{\prime\prime}$ to the SE, with photo-$z$s of 0.3–0.9, 2.5–2.8 and 4.3–5.2, respectively, which may be associated in case of incorrect photometric redshifts.
**COLDz.GN.19**: This spatially extended candidate is blended with a local foreground galaxy at $z$=0.564. No continuum emission is detected in our data at this position and therefore we exclude the possibility that the line candidate may be spurious and due to noise superposed to continuum emission. The presence of the bright foreground contaminates the [*HST*]{} H-band and the IRAC 3.6 $\mu$m images making it difficult to evaluate the counterpart status. Lensing of a faint $z$=2.20 galaxy by the foreground galaxy is a possibility.
**COLDz.GN.20**: Candidate may have a counterpart. It appears close to a foreground galaxy, which partly contaminates the IRAC 3.6 $\mu$m image. The [*HST*]{} H-band image shows a potential match which the catalog identifies as a galaxy $1.16^{\prime\prime}\pm0.3$ to the NW with a photo-$z$ of 1.6–2.5. The association is not ruled out by our CO(1–0) redshift of $z_{10}$=2.75.
**COLDz.GN.21**: Candidate is unlikely to have a counterpart. The images show a galaxy $1.0^{\prime\prime}\pm0.6^{\prime\prime}$ to the NE, which has a grism-$z$ of 0.86–0.94 from [@Momcheva16]. This makes it incompatible with the CO(1–0) redshift of $z_{10}$=2.27.
**COLDz.GN.22**: No counterpart is found in the images or the CANDELS catalog at the position of this candidate.
**COLDz.GN.23**: No counterpart is found in the images or the CANDELS catalog at the position of this candidate.
**COLDz.GN.24**: This spatially extended candidate may have a counterpart, which is visible in the IRAC 3.6 $\mu$m image but not in the [*HST*]{} H-band image. The separation is $2.9^{\prime\prime}\pm0.5^{\prime\prime}$ to the S, but this may not be significant due to the extent of the emission. The photo-$z$ is uncertain and ranges from 0.9 to 2.3, therefore an association is not strongly ruled out by our CO(1–0) redshift of $z_{10}$=2.7.
**COLDz.GN.25**: Candidate has a potential counterpart, $1.7^{\prime\prime}\pm0.7^{\prime\prime}$ to the NE. The galaxy is well visible in H-band and IRAC 3.6 $\mu$m images and has a photo-$z$ of 1.9–2.2, which is compatible with the CO(1–0) redshift of $z_{10}$=2.24.
**COLDz.GN.26**: This spatially extended CO candidate has potential matches, which appear very faint in the IRAC 3.6 $\mu$m image. The catalog lists two galaxies at $1.8^{\prime\prime}\pm0.5^{\prime\prime}$ and $1.9^{\prime\prime}\pm0.5^{\prime\prime}$ to the NE, with uncertain photo-$z$s of 0.6–4.2 and 2.1–3.7 respectively, which makes them compatible with the CO(1–0) redshift of $z_{10}$=2.14. The spatial offset may not be significant because of the spatial extent of the emission, and these are therefore potential counterpart matches.
**COLDz.GN.27**: Candidate has three potential counterparts in the catalog with close photo-$z$s. The first is $2.3^{\prime\prime}\pm0.4^{\prime\prime}$ to the SE, with a photo-$z$ of 1.5–2.3. The second is $2.7^{\prime\prime}\pm0.4^{\prime\prime}$ to the SE, with a photo-$z$ of 1.5–2. Both of these are compatible with the CO(1–0) redshift of $z_{10}$=2.08. The third potential counterpart is located $2.9^{\prime\prime}\pm0.4^{\prime\prime}$ to the NE and has a photo-$z$ in the range 5.2–5.7 which is consistent with the CO(2–1) redshift of $z_{21}$=5.16.
**COLDz.GN.28**: The [*HST*]{} H-band and IRAC 3.6 $\mu$m images show a potential match $1.6^{\prime\prime}\pm0.3^{\prime\prime}$ to the S, but this galaxy was reported to have a spec-$z$=2.932 [@Skelton14]. Assuming CO(1–0) would imply $z_{10}$=2.57, which implies either a lack of counterpart or an incorrect spectroscopic redshift.
**COLDz.GN.29**: Candidate has a possible counterpart. Both the [*HST*]{} H-band and the IRAC 3.6 $\mu$m images show multiple sources within 2$^{\prime\prime}$. The catalog lists two faint galaxies, with photo-$z$=0.8–3.5 and 1.1–1.7 just $1.4^{\prime\prime}\pm0.4^{\prime\prime}$ and $1.5^{\prime\prime}\pm0.4^{\prime\prime}$ to the SE and NE respectively. The first of these is compatible with the CO(1–0) redshift of $z_{10}$=2.51.
**COLDz.GN.30**: This spatially extended candidate does not appear to have counterparts. The IRAC 3.6 $\mu$m image is blended with a bright foreground galaxy, and the only catalog association (offset by $2.7^{\prime\prime}\pm0.9^{\prime\prime}$ to the SW) has a photo-$z$=0.7–0.9, which is incompatible with the CO(1–0) redshift of $z_{10}$=2.68.
Statistical properties of the candidate CO emitter sample
=========================================================
In order to extract as much statistical information as possible from our CO candidate list, we have to evaluate: 1) the probability of each line candidate to be real, 2) the line luminosity probability function and, 3) for each luminosity bin, the completeness of our line search, i.e. the probability that a galaxy would in fact be detected by our line search, as a function of the line emission luminosity, spatial size and velocity width.
In the following sub-sections, we will describe the methods we have developed to evaluate each of these separate components, which enter the luminosity function calculation (Paper [slowromancap2@]{}).
Reliability analysis
--------------------
The purpose of a reliability (also called purity or fidelity) analysis is to consistently assign probability estimates to each line candidate to represent a real line source. In this section, we attempt to provide a general solution to the problem of evaluating purities in the case of blind interferometric line searches, which builds the foundation for our analysis.
The most accurate way to tackle this problem is to utilize the symmetry around zero of the noise distribution provided by interferometric data. This is subject to the caveat of imperfect calibration and of sidelobes of bright sources which, however, should be negligible in our case, because the continuum sources in our field are not very bright ($<0.3\sigma$ and $<2\sigma$ per 4 MHz channel for the brightest source in COSMOS and GOODS-N, respectively). An alternative approach would be to try and reproduce many instances of the noise distribution by a well defined, simplified noise model, and to evaluate the rate of false positive detections as a function of SNR. However, this procedure may be strongly dependent on precisely capturing the statistical correlation properties of the noise [e.g., @frontiers]. We therefore run an equivalent blind line search for negative line features in our data, in order to estimate the contamination due to noise. We show the comparison of the distributions of SNR for positive and negative lines in Fig. \[fig:SNR\_histo\], which were used in the following to estimate the reliability for each positive line candidate. In the following, we apply Bayesian techniques to obtain estimates of the purities that are subject to well controlled assumptions.
The basic idea is to estimate the significance of the excess of positive over negative features at a given SNR. Any excess can be considered indication that a fraction of the positive features may correspond to real line signal. Some previous studies have taken a “cumulative" approach to this problem, and used the ratio of the number of positive and negative features with SNR greater than the SNR of the line under consideration, utilizing this ratio to estimate purities (e.g., @ASPECS1). This may cause a substantial bias for purities that refer to individual candidates. In particular, the presence of high SNR real candidates would raise the purity of moderate SNR positive features. We therefore choose a “differential" approach, but we also choose not to use bins in SNR. This choice is motivated by the small number of candidates in the bins of interest, which would make the results highly dependent on the precise binning of the SNR axis. We therefore model the occurrence rate of lines as an inhomogeneous Poisson process along the SNR axis, with a parametrized mean occurrence rate per unit SNR interval (see Section 14.5 of @Gregory_book, for an introduction). We can then use the machinery of Bayesian inference to study the posterior probability distribution for the rate of real sources and noise spikes, and therefore infer purities for each line candidate.
In order to derive our final likelihood function, we first consider a case where we group line candidates in bins of SNR. While the result of this calculation already has wide applicability and offers certain benefits (e.g., by avoiding any parametric assumptions for the source and noise distributions), binning introduces an unnecessary dependence on bin choice, and does not allow to capture the intrinsic continuity of the source and noise rates as a function of SNR. Therefore, we will follow the standard procedure and take the limit in which the bins are small, such that each bin contains at most one detection, thereby eliminating the bias introduced by binning (e.g., @GregoryLoredo). In each SNR bin, the task at hand is to determine the probability distribution for the fraction of line detections that are real sources rather than noise.
As a starting point, we infer a model for the noise distribution by fitting a Poisson process to the distribution of negative line features. Complex modeling for the noise feature occurrence rate is not necessary for estimating purities because in the moderate SNR regime of interest, the uncertainty will be dominated by shot noise due to the small number of candidate features. We therefore assume the Poisson rate (i.e., the expected number of negative lines per bin) to be well described by the tail of a Gaussian as a function of SNR, centered at zero. We fit for the normalization and width, of this Gaussian and thereby obtain a probabilistic description of the noise. The adopted two-parameter Gaussian tail model provides an excellent fit to the distribution of negative features. We stress that this method does not rely on the assumption of a Gaussian noise distribution, but it rather represents a convenient fitting function which takes advantage of the smoothness of the underlying noise distribution as a function of SNR. This method avoids using discontinuous bins or discontinuous cumulative functions, and allows us to exploit the symmetry between positive and negative noise features to generalize the noise realization provided by the negative features, and to estimate the probability of any positive line candidate to also be due to noise.
In the following, we derive purities using SNR bins. We then consider the continuum limit, as explained above. The quantity of interest is the probability of having $N_{s,i}$ real sources in the i-th SNR bin, given that we observed $N_{o,i}$ lines, $p(N_{s,i}| N_{o,i}, \mu_{b,i})$. Here, $\mu_{b,i}$ is the mean number of noise lines expected in the i-th bin. By explicitly introducing the dependence on the real source rate (for the Poisson process), $\mu_{s,i}$, we can calculate this probability as follows: $$\begin{split}
p(N_{s,i}| N_{o,i}, \mu_{b,i})= \\
& \hspace*{-60pt} \int \textrm{d}\mu_{s,i} \, p(N_{s,i} | N_{o,i}, \mu_{b,i}, \mu_{s,i})\, p(\mu_{s,i} | N_{o,i}, \mu_{b,i}).
\end{split}
\label{eqn_purity1}$$
The first term, i.e. the probability of $N_{s,i}$ real sources once we assume a source rate, is the same as the product probability for $N_{s,i}$ sources given a source rate $\mu_{s,i}$ times $N_{o,i}-N_{s,i}$ noise features, given a noise rate of $\mu_{b,i}$: $$\begin{split}
p(N_{s,i} | N_{o,i}, \mu_{b,i}, \mu_{s,i})= \\
& \hspace*{-60pt} \frac{Pois(N_{s,i},\mu_{s,i})\cdot Pois(N_{o,i} - N_{s,i},\mu_{b,i})}{\sum_{k=0}^{N_{o,i}} [Pois(k,\mu_{s,i})\,Pois(N_{o,i}-k,\mu_{b,i})]}.
\end{split}
\label{eqn_pur3}$$ Here, $Pois(N, \mu)$ stands for the Poisson probability for N events, given a mean $\mu$, and the denominator in the previous expression is a normalization factor. The second term in Eqn. \[eqn\_purity1\] is the probability for the source rate, given the observed number and noise rate, and it is therefore given by $$\begin{split}
p(\mu_{s,i} | N_{o,i}, \mu_{b,i})\propto \\
& \hspace*{-60pt} p(N_{o,i} | \mu_{s,i}, \mu_{b,i})\,p(\mu_{s,i})=Pois(N_{o,i}, \mu_{s,i}+\mu_{b,i}) \,p(\mu_{s,i})
\end{split}
\label{eqn_pur4}$$ by a straightforward application of Bayes theorem.
We then follow the standard prescription for inhomogeneous Poisson processes, considering it as the case where the equally-distributed bins are so small that each bin either contains a single line or not. In this section, we use the term rate of the Poisson process to indicate the number of line feature occurrences per unit SNR interval. In the limit of small bins, containing at most one line detection, the probability for a Poisson rate $\mu$, (which can be assumed to take the form of a parametric function of SNR) given the list of detection SNR, is calculated by the standard formula for the likelihood of an inhomogeneous Poisson process: $$\begin{split}
\log p(\{SNR_i\} | \mu)=\\
& \hspace*{-60pt} \sum_i \log \mu(SNR_i)-\int_a^b \textrm{d}(SNR') \, \mu(SNR').
\end{split}
\label{eqn_purity2}$$ Here, $\{SNR_i\}$ refers to the list of line detection signal-to-noise ratios, the [*a*]{} and [*b*]{} integration limits reflect the range of SNR that is considered for fitting, and $\mu$ is our parametric model function for the rate of lines as a function of SNR.
In the next steps, we use the occurrence rate of background, noise lines, measured from the negatives by maximizing the likelihood for the noise model. A more complex approach would include the full probability distributions for the noise model parameters in the purity evaluation. We have tested this approach and confirmed that it does not affect our purity results. In particular, using MCMC samples from the probability distribution for the noise model parameters, we have evaluated the purity of one of our moderate SNR candidates. We found that the median purity coincides with the purity evaluated with our simpler method, and that the relative scatter in the purity introduced by this uncertainty on the noise model is $\lesssim10\%$. This is much smaller than our conservative estimate of the systematic uncertainty, which we adopt in the following. Therefore, we maximize the probability for the complete set of negative lines (range of the integral $SNR \in [4,\infty)$), while assuming a Gaussian tail model for the rate function $\mu_b=N \, \exp(-\frac{SNR^2}{2\sigma_b^2})$, in order to determine the parameters $N$ and $\sigma_b$. To determine the purity/reliability of each object, we calculate the probability that its “small bin" contains one real source and zero noise lines. Eqn. \[eqn\_pur3\] therefore gives $$p(N_{s}=1 | N_{o}=1, \mu_b, \mu_s)=\frac{\mu_s}{\mu_s+\mu_b},$$ and hence Eqn. \[eqn\_purity1\] becomes $$\begin{split}
\mathrm{purity_k}=p(SNR_k\, \mathrm{is\, real} | \{SNR_i \}, \mu_b)=\\
& \hspace*{-180pt} \int \textrm{d}\mu_s(SNR_k) \, \frac{\mu_s(SNR_k)}{\mu_s(SNR_k)+\mu_b(SNR_k)}\, p(\mu_s | \{SNR_i\}, \mu_b).\\
\end{split}
\label{eqn_pur10}$$ The last term is important, and represents the probability distribution for the source rate parameters (replacing Eqn. \[eqn\_pur4\]). It can be written as the product of the probability in Eqn. \[eqn\_purity2\] (for a rate equal to $\mu_b+\mu_s$) multiplied by priors on the source rate parameters (i.e., the last term above).
In order to compute these purities, we therefore implement a posterior probability function for the source rate $\mu_s$, computed by Eqn. \[eqn\_purity2\], as a function of the model parameters. We sample it using an MCMC technique, making use of the python package `emcee` [@emcee]. The integral in Eqn. \[eqn\_pur10\], which corresponds to the purity of the [*k*]{}-th detection, is equivalent to averaging the ratio $$\frac{\mu_s(SNR_k)}{\mu_s(SNR_k)+\mu_b(SNR_k)}$$ over these MCMC samples of source rate parameters. It may be seen as a weighted average of this ratio, weighted by the posterior probability for $\mu_s$.
The simple parametrization adopted for $\mu_s (SNR)$ is $\mu_{s0}(\frac{SNR}{6})^{-\alpha}$. Thus, we normalize the occurrence rate of real sources at a SNR=6 and allow for a shallow power-law increase of the rate toward lower SNR values, as we expect that there may be more real faint sources than bright sources. We impose uniform, unconstraining priors on $\mu_{s0}$ and $\alpha$. This parametrization is intended to only accurately describe the source rate over a small range of SNR, because the line candidates of dominant interest for the purity estimation are those with $5<$SNR$<6.5$.
By applying this procedure, we face a choice of the SNR range to be fitted. In the COSMOS field, we start by including all the line candidates with $SNR>5$. This results in purities of 100% for the top candidates (with secure counterparts) and $<7\%$ for the next objects down the list. This is caused by the large gap between SNR=5.7 and 9, where no candidates were found, and which favors a low source rate, for our assumed model for the source distribution. Our simplified Poisson model, with slowly varying source rates as a function of SNR, may only be assumed to be an accurate description of the data over a limited range in SNR. We therefore also attempt to exclude the brightest sources, and the large SNR gap without detections in the model fitting. Therefore, to obtain an upper limit on the purities, we exclude the brightest candidates and only fit the range $5<SNR<5.8$. This yields an upper limit on the purities of up to $\sim10\%-20\%$ of the top few remaining objects to be real (Fig. \[fig:purities\]). In the GOODS-N field, there is no gap in the SNR distribution of the line candidates (the highest SNR source is GN10 at z$>$5). Therefore we include all candidates in the range $5<SNR<6.4$. The procedure we have described would attribute a purity of 70% for the candidate COLDz.GN.1, of 50% to GN19 (which we manually correct to be 100% because we know it to be a real line) and in the 30%–50% range for the other SNR$\sim$6 candidates, subsequently decreasing to about 7% at SNR=5.5 (Fig. \[fig:purities\]).
When utilizing these purities to assemble the CO luminosity function, we consider two possible alternative strategies which allow to estimate the effects of the systematic uncertainties introduced by our purity computation. In the first approach, we treat these purities as having 100% uncertainty, i.e. we will draw purities (for the Monte Carlo sampling used to estimate the allowed range of the luminosity function) as independent random numbers, normally distributed around the estimated values with standard deviation equal to the purity estimate themselves and truncating at zero. The alternative approach is to implement these purities as upper limits, and to draw purities from a uniform distribution between zero and the calculated values. The latter provides a more conservative purity estimation. Therefore, the luminosity function constraints are somewhat lower in this method, although compatible between the two methods. This conservative approach attempts to implement the additional information coming from the lack of clear multi-wavelength counterparts to our moderate SNR candidates. We will present the detailed results of both approaches in Paper [slowromancap2@]{}.
The SNR thresholds adopted in Section 4 and Table \[tab\_lines\] correspond to approximate purities of $\sim$4% and $\sim$7% for COSMOS and GOODS-N, respectively. We emphasize that previously employed definitions of purity have differed significantly. In particular, we attempt to assess the fidelity defined by [@ASPECS1] for our candidate selection. The comparison is not straightforward, because the definition of fidelity used in that work relies on the details of their line search algorithm, but an approximate implementation of their method indicates an equivalent fidelity of approximately 80%–90% for COSMOS and 50%–60% for GOODS-N in their method.
Estimating noise tail extent from data cube sizes
-------------------------------------------------
Due to the short-scale noise correlation intrinsic to interferometric noise (over the synthesized beam length-scale), the calculation of the highest expected SNR due to noise (both positive and negative) is not straightforward as the counting of “independent elements" is non-trivial. [@Vio16] and [@Vio17] have independently discussed a similar analysis of this case. We have reached the same conclusions, although we take a slightly different approach as we describe below. A detailed analysis of extreme value statistics in the case of smooth Gaussian random fields (which is a good approximation for interferometric noise) was developed by [@Bardeen86] and [@Bond87], among others, and was expanded upon by [@Colombi11]. Here we only summarize the main results as relevant to our data, and discuss the implications. The objective is a description of the probability distribution function for the highest SNR in a data cube which is uniquely due to noise, and how this varies as a function of cube “size". If we consider the original data cube, then noise is not correlated across different channels and a noise realization is equivalent to a 2D case, with spatial correlation only, and an area equivalent to the total area across the full cube (i.e., the sum of the areas over the independent channels). In this case, the approximate cumulative distribution function for the highest SNR ($\nu$) to be expected from such a noise realization is given by: $$P(\nu_{max}<\nu)\simeq\exp(-\frac{1}{4\sqrt{2\pi}}N_{n} \nu e^{-\nu^2/2}),$$ where $N_n$ is the “naive" counting given by the total area divided by the “beam area" (defined by a radius equal to the beam standard deviation). The second case regards the case where correlation of the noise across channels has been introduced (for example by convolution with a spectral template in order to Matched-Filter) and is also relevant to line searches in the form of the noise properties of Matched Filtered cubes. In this case the correlation takes place in 3D and a slightly different approximate formula describes the cumulative distribution function for the highest SNR ($\nu$) to be expected: $$P(\nu_{max}<\nu)\simeq\exp(-\frac{1}{6\pi\sqrt{2}}N_{n} \nu^2 e^{-\nu^2/2}),$$ where $N_n$ is the “naive" counting given by the total cube volume divided by the “effective beam volume" (an ellipsoid with a radius equal to the beam standard deviation in the spatial dimension and the standard deviation of the template used in the spectral dimension). We have verified that the highest significance noise peaks measured as negative features in our data are compatible with these probabilistic predictions. We note that these distribution functions are quite broad, and only predict the highest SNR expected due to noise to be approximately in the SNR=5.5–6 range for our deeper mosaic and SNR=5.7–6.4 for our wider mosaic. This is a manifestation of the strong intrinsic stochasticity of the noise tails. We also note that the “effective number of independent elements" implied by these estimates is $\sim10$ and $\sim20$ times higher than the naive counting in the 2D and 3D cases, respectively, and that these ratios are themselves increasing functions of data cube size. The naive counting of independent elements would therefore lead to a significant underestimation of the extent of the noise tails. This conclusion is compatible with the results of [@Vio16] and [@Vio17]. However, we here report equations that explicitly describe the distribution of the maximum SNR to be expected from noise rather than implicitly, through the probability distribution function of local maxima. The analysis above is only approximately equivalent to the analysis presented in [@Vio17], because they express the distribution function of interest as a function of $N_p$, i.e., the number of local maxima in the noise realization, which is itself a random variable with its own probability distribution.
Artificial source analysis
--------------------------
COSMOS GOODS-N
------------------- --------------------------------- ---------------------------------
Intrinsic spatial $\sim$0.5, $\sim$3.0, $\sim$4.7 1.0, 2.5, 4.5
size (arcsec)
Convolved spatial 2.6, 4.0, 5.3 2–3, 3–4,4.8–5.4
size (arcsec)
Frequency width 23.2, 46.8, 70 23.2, 46.8, 70
(MHz)
Velocity width $\sim$200, $\sim$400, $\sim$600 $\sim$200, $\sim$400, $\sim$600
(km s$^{-1}$)
: Artificial sources injected sizes
\[injected\_sizes\]
[**Note:**]{} Gaussian sizes utilized for the injected artificial sources. All sizes refer to the Gaussian FWHM. The convolved sizes are the injected sizes in the Natural-mosaic. These are fixed in COSMOS while in GOODS-N, because of the larger beam differences across the mosaic, we injected sources of fixed intrinsic sizes and convolved them to the local beam size, appropriate for each mosaic position.
In order to estimate the completeness and biases introduced by our line search and flux extraction methods, we perform an extensive probabilistic analysis of artificially injected sources into our maps. The main goals of this analysis is to establish a probabilistic connection between recovered candidate properties and intrinsic properties such as spatial size, velocity width and line flux. This will provide some control over the uncertainties that affect the analysis of the CO luminosity function (Paper [slowromancap2@]{}). We also develop a method to correct the luminosity function by the completeness of our line search, which avoids a purely “per-source" completeness estimation as far as possible (due to the bias of “per-source" corrections), while avoiding assumptions that would significantly affect the result.
Since the large majority of the data cube contains very little signal, we use the data themselves as our model for the noise, and inject artificial sources of varying size, velocity width and fluxes at random positions in the data cube (Table \[injected\_sizes\]). We inject sources in each cube (500 in COSMOS and 2500 in GOODS-N), estimating that this will not cause crowding of the field, therefore not causing overlaps between different sources, during the line search and effectively simulating the recovery of each injected source individually. We then analyze each injected cube following the same steps of Matched Filtering in 3D that we applied to the real data, and in the end, we search for the injected sources to determine the recovered SNR, and the line parameters that would have been measured. We define the “flux-factor" as the ratio of the measured line flux to the injected flux. Therefore, the distribution of flux-factors captures both flux corrections, and uncertainties on our flux estimations. The purpose of the flux-factor analysis is not just to correct for potential biases in our flux extraction procedure, but also to estimate the uncertainty of the flux recovery. We subsequently utilize these flux probability distributions to inform our luminosity function estimates (Paper [slowromancap2@]{}). We ignore dependencies on frequency or position of the injected sources flux-factors (determined as function of local SNR) and completeness, thereby obtaining average values that correctly sample the data for an approximately uniform distribution of real sources in our cube.
Both the completeness and the flux-factors are dependent on the source size and line FWHM. Since we only inject sources of three spatial sizes and three frequency widths, we develop a probabilistic framework to relate each line candidate to the different injected sizes (Table \[injected\_sizes\]). For each detected candidate, based on the template size and velocity width where their signal-to-noise peaks, we determine a probability distribution of belonging to each category of “injected" spatial size and line width, therefore matching in a continuous and probabilistic way the measured sizes to a discrete grid of intrinsic properties, as explained in detail in the following.
### Flux-factors based on artificial sources
The artificial source analysis allows us to estimate how well our measured fluxes correspond to the injected flux, for candidates of different SNR, spatial size and velocity width. The objective of the flux-factor analysis is to characterize the uncertainty and bias of our flux estimates, in order to correctly estimate the uncertainty of our luminosity function measurement.
We confirm that aperture fluxes have a slight bias toward higher fluxes, because positive noise adjacent to a candidate source tend to enlarge the fitted sizes, and therefore to contribute spurious flux to the candidate [@Condon97]. We thus need to estimate the magnitude of this bias, at the SNR range of interest ($\sim5$–6), to correct the measured fluxes accordingly. A correction factor relies on an estimate of how likely a measured extended source is to be due to noise rather than real extended structure. In order to determine this bias, we need to assume an expected approximate size distribution for our sources, to be combined with information from the artificial sources, regarding how the flux is affected by the interplay of real and measured sizes.
We use the artificial sources to determine the probabilities of spatial extension (probability of being like spatial bin 0, 1 or 2 of the injected sources, Table \[injected\_sizes\]) given the measured size, as traced by the size of the spatial template which gives the highest SNR. This probability is used in the following to relate the measured properties of each line candidate to the flux-factor and completeness, which are computed for the bins of injected properties. We use Bayes theorem to relate the probability of a given real size, conditional to a measured size: [*p*]{}(real-injected size $|$ measured size) to the probability distribution that we can measure from the artificial sources, which is the probability of measuring a given size, conditional to a certain injected size [*p*]{}(measured size $|$ injected size), by employing a prior on the expected real size distribution[^12]. We must employ a prior for the probabilities of the real sizes that captures our expectation that most sources would be unresolved, while allowing for a fraction of resolved sources coming from extended gas reservoirs, merging and/or blended objects. We adopt 88%, 10% and 2% for the size bins in Table \[injected\_sizes\], respectively. We stress that although these relative fractions are uncertain, their precise choice does not significantly affect any of the results, because their effect is simply to modulate our assignment between line candidates and injected sources. The main effect of this assignment is in the estimation of completeness corrections, where the uncertainty introduced by these priors is small compared to the systematic uncertainty introduced by choosing a weighting based on the detected size distribution.
We estimate [*p*]{}(measured size $|$ injected size), by measuring the fraction of injected sources of a given size, recovered at different match-filter spatial template sizes. In this way, we compute the final posterior probability for a given measured size, to originate from an unknown “injected" size, by combining this with the prior (Fig. \[fig:spatial\_prob\]): $$\begin{split}
p(\textrm{real/injected size} | \textrm{measured size, SNR}) \propto \\
&\hspace*{-200pt} p(\textrm{real size}) \times p(\textrm{measured size} | \textrm{injected size, SNR}).\\
\end{split}$$
We follow the same procedure for relating the measured velocity width (from the peak template) to the injected line widths, by assuming a flat prior for the line width over the three injected bins (Table \[injected\_sizes\]). These are required to compute the completeness of line candidates by relating their measured properties to the injected sources. The derived probability distribution functions are shown in Fig. \[fig:FWHM\_prob\].
In order to calculate flux-factors from the artificial sources, we employ an analogous technique. We calculate probability distributions of the flux-factors, given source spatial template size and SNR, by weighing the distribution of flux-factors found for given measured sizes, by the probability that the given measured size originates from the different possible injected sizes. Specifically, the correction ratio depends both on the injected and the measured size, as a larger ratio is needed to correct for a compact source that appears extended: [^13] $$\begin{split}
p(\textrm{flux ratio}|\textrm{measured size, SNR})=\\
&\hspace*{-150pt} \sum_{injected} p(\textrm{flux ratio}|\textrm{measured, injected, SNR}) \times\\
&\hspace*{-150pt}\times p(\textrm{injected}|\textrm{measured, SNR}).\\
\end{split}$$
These flux-factor distributions (Fig. \[fig:flux\_prob\]) can be approximated by log-normal distributions, peaking near a factor of one, but with a tail to larger ratios to correct for the bias towards larger spatial size, which is introduced by including positive noise as part of the candidate source. We stress that this is just a convenient parametrization of the measured distributions. We calculate the probability distribution of flux ratios for each measured spatial size in bins of SNR. Since the distributions are noisy due to the limited number of artificial sources, and since they do not strongly depend on SNR in the narrow 5–6 range of interest for our candidates, we consider the mean distribution over the $5<$SNR$<6$ range (Fig. \[fig:flux\_prob\]). The fitted log-normal curves to these mean distributions, which provide a good interpolation to the noisy distribution estimates, will be utilized in the construction of the luminosity function. We can understand the general trend seen in these shapes as follows: the smallest template selects point sources. Therefore, the distribution peaks near one. Slightly extended sources have a larger mean flux correction, reflecting the finding that they are most likely noise-smeared point sources and so their flux needs to be reduced. Slightly larger (intermediate-size) sources then require less correction, because it becomes more likely that they are somewhat extended in reality. Even larger sources show a long tail of larger flux-factors, because it is extremely unlikely that the real source is very extended. Therefore their fluxes need to be significantly corrected (or rather, there is significant uncertainty as to their real flux, and we need to account for this in constructing the luminosity function).
### Completeness
In order to estimate the completeness of our detection process, we utilize the artificial sources to measure the fraction of the injected sources that are detected. The objective of the completeness correction is to account for the fraction of the mosaic volume where a given line candidate would be detectable, and to account for the fraction of objects of given intrinsic line luminosity that would be missed by a fixed SNR threshold.
We assume that the fraction of detected lines (a proxy for the probability of detection) only depends on the integrated line flux, on the spatial size, and on the velocity width. By injecting artificial sources that uniformly sample random positions within the edges of the mosaic, we derive completeness corrections that account for the effects of the spatial and frequency variation of the sensitivity, as previously adopted by [@ASPECS1]. While completeness is a property of the overall number counts in a luminosity bin, we partially adopt the spirit of the $1/V_{max}$ method of calculating a “per source" completeness only in so far as this depends on the line velocity width and especially spatial size of the detection. While this may potentially introduce a bias[^14] (see e.g., @Hogg10), it saves us from additional assumptions about the size distribution. We therefore point out the important caveat that our luminosity function estimate does not correct for missed objects due to poor sampling of the size distribution. The effect can be seen by noticing, for example, that the completeness for extended objects at low flux values drops very quickly. This is reflected in the fact that all of our low-flux objects are point sources (therefore, the completeness correction in the lowest line luminosity bins misses the potential contribution from undetected extended sources).
The completeness is measured from the artificial sources as a function of “injected" properties, i.e. injected integrated flux, and spatial size and frequency width (in three bins each, see Table \[injected\_sizes\]), as the ratio of injected sources recovered with SNR above a threshold value of 5$\sigma$ to the total number of injected sources (the precise choice of a threshold does not change the result appreciably). These measured completeness values are shown in Figure \[fig:completeness\], together with the interpolating functions that we use in deriving the luminosity function: the two-parameter ($I_0$ and $I_1$) family of functions $1-e^{-I/I_0}/(I+I_1)$, where I is the integrated line flux. While this chosen family of interpolating functions has no specific significance, we found it to provide an appropriate description of the measured completeness.
The optimal way to correct for the completeness of a luminosity function bin would require calculating the mean completeness within the bin, over the full “internal" parameter space (in our case these are spatial and velocity line sizes, frequency, and precise luminosity within the bin), weighted by the model expectation for the distribution of sources within this space. We adopt an intermediate approach between this “mean completeness" approach and a purely “per-candidate" approach. We assume that the distribution of line luminosity within a bin and the frequency distribution are uniform and average over this subset of the internal space by randomly sampling it. On the other hand, we do not assume an intrinsic spatial size and velocity width distribution, in order to avoid biasing our result. We therefore adopt the sizes and line widths of the candidates to calculate the appropriate completeness, thereby letting the data determine the size and velocity width distributions.
In summary, for each detected line candidate, the size properties (i.e. the spatial size and frequency width) of the “real" underlying source are estimated probabilistically based on the measured size and width (from the peak template). Then, probability-weighted completenesses are the factors that enter the evaluation of the luminosity function. The dependence of these completenesses on the line flux are mean values evaluated for the full luminosity function bin, rather than depending on the candidate line flux measurement. This hybrid approach helps us mitigate the bias which would derive from a purely “per-source" correction (e.g., @Hogg10).
Implementation of the statistical corrections
---------------------------------------------
In order to assemble the luminosity function, we use a variation on the method used by [@Decarli14; @ASPECS2]. We weigh the contribution of each line candidate to the luminosity bin by its purity and inversely by its completeness, and use the total cosmic comoving volume covered within the edges of the mosaic. The completeness correction converts this volume to an effective $V_{max}$, for each galaxy, also accounting for the spatial variation in sensitivity. In order to estimate the systematic uncertainty introduced by our assumptions, we evaluate the luminosity function with many random realizations of flux-factor, purity assignment and luminosity bin widths and boundaries. One of the differences between our approach and the approach employed by [@Decarli14; @ASPECS2] consists in including a larger number of candidates. We have also calculated the luminosity function using the same method employed by [@ASPECS2], and the result is consistent with our more extensive method. The advantage of our approach consists in relying less heavily on the properties of the few moderate SNR, individual candidates that happen to be located near the top of the SNR list, but that still have a limited probability of being real. Although a fraction of the moderate SNR candidates are expected to be real, it is not clear that those near the top (of the set of uncertain candidates) of the SNR list have a significantly greater likelihood of being real given the limited range in SNR considered ($\sim5-6$). By utilizing a larger sample of candidates in deriving constraints to the luminosity function, down-weighted by appropriate purities, we do not introduce additional bias, but rather better explore the implications of the systematic uncertainties. In particular, the statistical justification for a “per-source" purity and completeness correction (which we cannot fully avoid) only holds for large enough samples. By better sampling the “internal" space of possible candidate sizes and line widths, and by adopting average per-bin completenesses (i.e., not using the uncertain, measured fluxes), we aim to achieve a more accurate completeness correction and, evaluation of the systematic uncertainties introduced by these factors.
In detail, for each $L'_{\rm CO}$ bin we calculate a completeness factor appropriate for each of the nine bins in the spatial size-frequency width grid, by averaging over 1000 random realizations of values of $L'_{\rm CO}$ in each bin, using random redshifts to calculate the corresponding integrated flux, and hence the appropriate completeness correction for each. We average over this frequency distribution and precise $L'_{\rm CO}$ within each bin. We therefore enforce a uniform prior, and maintain the dependence on the spatial-frequency size information separate. We subsequently apply them, for each line candidate, as weighted by the candidate probability distribution for its spatial-frequency size assignment. In this way, we use the measured relative occurrence of different spatial and velocity sizes as weights for the appropriate completeness, for each luminosity function bin.
In order to explore the range of luminosity function values allowed by our systematic uncertainty, we use 10,000 Monte Carlo realizations of the luminosity function calculation, for each bin width and shift, where we vary the purity assignment independently for each candidate, and the flux-factor to be applied. We therefore “move candidates around" among adjacent luminosity bins, simulating the effect of the uncertainty in their intrinsic fluxes. We separately implement the purity in one of the two ways that we previously described in Section F.1, either as normally or uniformly distributed. We also implement the flux correction as taking a random value drawn from the appropriate log-normal distribution, which was derived for each spatial size in the previous section. We also add a normal uncertainty of $20\%$ to the measured flux, to reproduce the uncertainty in our flux calibration.
Finally, in order to describe the range of values for the luminosity function (in log comoving volume density space) spanned by our 10,000 Monte Carlo realizations, we calculate the median value for each bin and a measure of the scatter around the median. We evaluate the scatter conservatively by quoting luminosity function ranges that include 90% of the probability. We also evaluate the statistical Poisson uncertainty as appropriate for each bin as a relative uncertainty of $1/\sqrt N$, where [*N*]{} corresponds to the number of candidates in a $L'_{\rm CO}$ bin.
Additional stacking results of galaxies with optical redshifts
===============================================================
In addition to the stacking of individual sets of galaxies described in the main text, we have attempted to stack three more sets of galaxies, based on optical redshift information. We do not detect significant signal in these stacked spectra.
COSMOS proto-cluster at $z=5.3$
-------------------------------
We search for CO(2–1) from nine member galaxies of the AzTEC-3 proto-cluster, as identified through the Lyman Break (LBG) technique and color selection by [@Capak11]. While two of them show a hint of a CO emission line signal ($\sim2.5\sigma$), all others (including LBG-1, the one with the strongest \[C[ii]{}\] emission; @Riechers14 and in prep., @Capak15), are consistent with noise. The positions of the LBGs with tentative CO detections are J2000 10:00:21.96 +02:36:08.5; and 10:00:20.13 +02:35:53.9. Assuming a line FWHM of 250 km s$^{-1}$ (i.e., the width of the \[C[ii]{}\] line in LBG-1) we derive CO line fluxes of approximately $0.03\pm0.012$ Jy km s$^{-1}$, corresponding to $L'_{\rm CO21}=7\pm3\times 10^9$ K km s$^{-1}$ pc$^2$. We do not claim any detections due to the low significance and only quote these as approximate limits for reference. We also stack all spectra at the positions of LBGs (limited to the seven galaxies that are not contaminated by emission from the bright CO(2–1) line in AzTEC-3 at the resolution of our survey), and do not detect a significant signal (Fig. \[fig:stacked\_spectra\_appendix\]). We therefore place a $3\sigma$ limit of $<$0.012 Jy km s$^{-1}$ on their average CO(2–1) emission, corresponding to $L'_{\rm CO21}<3\times 10^9$ K km s$^{-1}$ pc$^2$. The average stellar mass for the stacked LBGs, as reported by the COSMOS2015 catalog, is $4\times 10^9\,{\rm M}_\odot$. If we assume an $\alpha_{\rm CO}=3.6\, {\rm M}_\odot$ (K km s$^{-1}$ pc$^2$)$^{-1}$, we obtain a gas mass upper limit of $<1.1\times10^{10}\,{\rm M}_\odot$, and therefore a gas fraction of ${\rm M}_{\rm gas}/{\rm M}_*<3$ which is not strongly constraining. None of the LBGs are detected in 3 GHz radio continuum emission [@Smolcic17]. This would place a limit of $<165 \,{\rm M}_\odot \, {\rm yr}^{-1}$ on their star formation rate if we adopted the redshift evolution of the radio-FIR correlation measured by [@Delhaize17]. This may be converted to an expected limit of $L'_{\rm CO}<2.2\times10^{10}$ K km s$^{-1}$ pc$^2$ by assuming the star formation law [@Daddi10b; @Genzel10]. This limit is higher than what we derive from our CO non-detection, implying that our deep observations provides strong constraints on the CO luminosity of $z>5$ LBGs.
Grism redshifts in the GOODS-N field
------------------------------------
The 3D-[*HST*]{} catalog [@Momcheva16] contains 694 galaxies in GOODS-N with grism redshifts, for which the CO(1–0) line is covered by our data. Nevertheless, the majority of these grism spectra do not significantly improve the redshift determination over the photometric redshift, and are therefore not usable for stacking. We search for matches to our line search candidates (down to $4\sigma$) within a radius of $2^{\prime\prime}$ and $\sim500$ km s$^{-1}$. We find 16 potential matches and assess the contamination by chance association by also matching our blind detection catalog to 694 random positions within the signal data cube. We find that the distribution due to random associations is well described by a Gaussian with mean 8 associations and a standard deviation of 3.5 associations. Therefore, the majority of our line associations are likely to be random, but some may be expected to be real. The measured line fluxes of these candidate counterparts would imply gas masses that are sometimes larger than the stellar masses. This is possible, but we consider it more likely that those cases may be random noise associations.
We also stack the spectra extracted at the positions of the galaxies with high quality grism redshift (Fig. \[fig:stacked\_spectra\_appendix\]). 37 of them have redshift uncertainties less than 500 km s$^{-1}$ (based on the 95th percentile of the redshift probability distribution reported by @Momcheva16), and 35 more have redshift uncertainties less than 700 km s$^{-1}$. At this level of uncertainty, it would be likely that a fraction of the line signal contributes to the stacked spectrum. Both of these stacks show no detection. Assuming a line FWHM of 300 km s$^{-1}$, this implies $3\sigma$ upper limits of $<$0.011 and $<$0.008 Jy km s$^{-1}$ ; corresponding to $\lesssim$3 and 2 $\times 10^9$ K km s$^{-1}$ pc$^2$, respectively.
HDF850.1 $z=5.2$ galaxy overdensity in GOODS-N
----------------------------------------------
We also search for potential CO(2–1) emission from galaxies in the $z\sim5.2$ overdensity around the sub-millimeter galaxy HDF850.1 [@Walter12], taking advantage of the abundance of available spectroscopic redshifts in this region. 24 of the 105 galaxies with spectroscopic redshifts presented in that work fall within our data, but none of them are individually detected. We stack the spectra to obtain an average spectrum (Fig. \[fig:stacked\_spectra\_appendix\]). No significant emission is found after stacking. Assuming a line FWHM of 300 km s$^{-1}$ implies a $3\sigma$ upper limit of $<$0.015 Jy km s$^{-1}$, and $L'_{\rm CO21}<3.5\times 10^9$ K km s$^{-1}$ pc$^2$ at $z\sim5.2$. We match these 105 galaxies to galaxies within $1^{\prime\prime}$ in the photometric catalog by [@Skelton14], finding 83 matches, but we do not adopt their stellar mass estimates because the redshifts of these galaxies were often greatly under-estimated.
[^1]: In this work, CO always refers to the most abundant isotopologue $^{12}$CO
[^2]: The dust continuum method to determine gas masses may be affected by the metallicity dependence of the dust-to-gas ratio [@Sandstrom13; @Berta16], by trends in dust temperature with redshift (e.g., @Magdis12), or with galaxy population (e.g., @Faisst17).
[^3]: The ASPECS-Pilot survey simultaneously covered the CO(2–1) line in the redshift range $z\sim$1.0–1.7, the CO(3–2) line at $z\sim$2.0–3.1, and higher-[*J*]{} CO transitions at higher redshift.
[^4]: A key challenge in these studies is the uncertainty in assigning candidate emission lines to the correct CO transition, in cases where the redshift of the observed line candidates is not independently known.
[^5]: The COLDz survey data, together with complete candidate lists, and analysis routines may be found online at coldz.astro.cornell.edu.
[^6]: The recently expanded VLA, with its new Ka-band detectors, the new 3-bit samplers, the simultaneous 8 GHz bandwidth, and its improved sensitivity, for the first time, enables carrying out this survey study.
[^7]: For reference, the limits would be $\sim1.6\times 10^9$ and $5\times 10^8\,{\rm M}_\odot$ for $A_V<0.5$, and $\sim 2\times10^{10}$ and $7\times10^{9}\,{\rm M}_\odot$ at $A_V\sim5$
[^8]: Recent work by [@Molnar] suggests that radio-FIR correlation for disk-dominated galaxies may not show redshift evolution, and would imply a less constraing limit of SFR$<100-300\,{\rm M}_\odot \, {\rm yr}^{-1}$
[^9]: These star formation law estimates were mostly based on CO(3–2) observations, therefore variations in the r$_{31}$ line ratio may contribute additional uncertainty.
[^10]: The number of high SNR CO(1–0) detections in non-quasar hosts or sub-millimeter selected galaxies is very limited to-date [@Riechers10b; @Aravena10; @Aravena12; @Aravena14; @Bolatto15; @Spilker16].
[^11]: We provide a Python implementation for this algorithm at https://github.com/pavesiriccardo/MF3D
[^12]: Note that the artificial sources can only be used to estimate distributions conditional to a given injected size, because the relative frequency of injected sizes that was utilized (uniform) is not representative of the expected distribution of real sizes.
[^13]: The only distribution which can be directly estimated through counting artificial sources is conditional to injected size. We cannot marginalize over those sizes without specifying a distribution of expected source sizes first.
[^14]: Regions of parameter space that have very low completeness tend to be poorly sampled and hence cannot properly be accounted for in a completeness calculation that is weighted by the detected candidates
|
---
abstract: 'In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural Hilbert space –a weighted sequence space– on which this operator is symmetric. Then the corresponding eigenfunctions are polynomials, which are the little $q$-Jacobi polynomials. These polynomials form a family in the $q$-Askey scheme, and so many important properties are well known. In particular, we show how the orthogonality relations and the three-term recurrence for the little $q$-Jacobi polynomials can be obtained using only the factorisation of the corresponding operator. As a next step we consider the $q$-hypergeometric operator in general, which leads to the little $q$-Jacobi functions. We sketch the derivation of the corresponding orthogonality using the connection between various eigenfunctions. The link between the $q$-hypergeometric operators with different parameters is studied in general using $q$-analogues of fractional derivatives, and this gives transmutation properties for this operator. In the final parts of these notes we consider partial extensions of this approach to orthogonal polynomials and special functions. The first extension is a brief introduction to the Askey-Wilson functions and the corresponding integral transform. The second extension is concerned with a matrix-valued extension of the $q$-hypergeometric difference equation and its solutions.'
address: 'Radboud Universiteit, IMAPP, FNWI, PO Box 9010, 6500 GL Nijmegen, the Netherlands'
author:
- Erik Koelink
date: |
Final version, August 10, 2018. Lecture notes for June 2018 summer school OPSFA-S8, Sousse, Tunisia.\
`http://www.essths.rnu.tn/OPSF-S8/acceuil.html`
title: |
$q$-special functions, basic hypergeometric series\
and operators
---
Introduction {#sec:Introduction}
============
Basic hypergeometric series have been introduced a long time ago, and important contributions go back to Euler, Heine, Rogers, Ramanujan, etc. The importance and the history of the basic hypergeometric series is clearly indicated in Askey’s foreword to the book on basic hypergeometric series by Gasper and Rahman [@GaspR]. Since the work of Askey, Andrews, Ismail, and coworkers many new results on classes of special functions and orthogonal polynomials in terms of basic hypergeometric series have been obtained. The relation to representation theory of quantum groups and related structures in e.g. mathematical physics and combinatorics has given the topic a new boost in the recent decades.
In these lecture notes we focus on the basic hypergeometric series of type ${}_2\varphi_1$ by studying the corresponding $q$-difference operator to which these series are eigenfunctions. The study of general $q$-difference operators go back to Birkhoff and Trjitzinsky in the 1930s. In §\[sec:BHS\] we first introduce the basic hypergeometric series, and we derive some elementary summation and transformation formulas needed in the sequel. However, we will not prove all the necessary transformation formulas, but refer to Gasper and Rahman [@GaspR] when necessary. Section \[sec:BHS\] is based on the book [@GaspR] by Gasper and Rahman, which is the basic reference for basic hypergeometric series. In §\[sec:BHS-qdiff\] we then discuss the corresponding $q$-difference operator in more detail, by studying the solutions obtained by Frobenius’s method. We also look at the decomposition of the operator using the standard $q$-difference operator. Next, in §\[sec:BHS-qdiff-pol\] we consider a special case of the $q$-difference operator, namely the one which can be related to polynomial eigenfunctions for functions supported on $q^\N$. This essentially leads to the little $q$-Jacobi polynomials, for which we derive the natural orthogonality measure, the corresponding orthogonality relations, and the three-term recurrence relation by using the shift operators. These shift operators are the operators in factorisations of the difference operator.
In §\[sec:BHS-qdiff-nonpol\] we study a more general case. This leads to general orthogonality for ${}_2\varphi_1$-series, which we derive by calculating the spectral measure of the corresponding measure. In §\[sec:transmutation\] we study the transmutation properties of the basic hypergeometric $q$-difference operator. In §\[sec:AW-level\] we then lift this to the level of Askey-Wilson polynomials and the Askey-Wilson functions.
In §\[sec:MVextensions\] we make a first start in order to lift the results on little $q$-Jacobi polynomials of §\[sec:BHS-qdiff-pol\] and little $q$-Jacobi functions of §\[sec:BHS-qdiff-nonpol\] to the matrix-valued extensions.
There are many related results available in the literature, and we indicate several developments in the notes to each section. In particular, it is not clear if the results of §\[sec:BHS-qdiff-pol\] and §\[sec:BHS-qdiff-nonpol\] can be extended to the level of the Askey-Wilson functions as in §\[sec:AW-level\] or the matrix-valued analogues of §\[sec:MVextensions\].
By $\N$ we denote the natural numbers starting at $0$. The standing assumption on $q$ is $0<q<1$.
**Acknowledgement.** I am much indebted to the organisers, Hamza Chaggara, Frej Chouchene, Imed Lamiri, Neila Ben Romdhane, Mohamed Gaied, of the summer school for their work on the summer school and their kind hospitality. I also thank the participants for their interest and discussions. The main sections of the lecture notes are based on previous papers [@KoelR-RMJM02], [@KoelS-IMRN2001], joint with Hjalmar Rosengren and Jasper Stokman, respectively. The lecture notes do not contain any new results, except that the description of the solutions of the matrix-valued $q$-hypergeometric series in §\[sec:MVextensions\] have not appeared before. These solutions have been determined by Nikki Jaspers in her BSc-thesis [@Jasp] under supervision of Pablo Román and the author.
Basic hypergeometric series {#sec:BHS}
===========================
The basic hypergeometric series are analogues of the much better known hypergeometric series and hypergeometric functions. The hypergeometric series ${}_2F_1(a,b;c;z)$ as well as the analogous Thomae series ${}_{r+1}F_r$ and the more general hypergeometric ${}_rF_s$-series are discussed in detail in e.g. [@AndrAR], [@Bail], [@Isma], [@KoekLS], [@KoekS], [@Rain], [@Slat], [@Temm] and many other standard textbooks. Recall the notation for that standard hypergeometric function $$\label{eq:def2F1Pochhammer}
{}_2F_1(a,b;c;z)\, = \,
{\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;z \right)} \, =\, \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n\, n!} z^n,
$$ for this series (and for its sum when it converges) assuming $c\not= 0,-1,-2,\cdots$. This is the *(ordinary) hypergeometric series* or the *Gauss hypergeometric series*. The series converges absolutely for $|z|<1$, and for $|z| = 1$ when $\Re(c-a-b) >0$, see Exercise \[ex:Raabe2F1\]. Many important functions, such as the logarithm, arcsin, exponential, classical orthogonal polynomials can be expressed in terms of Gauss hypergeometric series. $(a)_n$ denotes the *shifted factorial* or *Pochhammer symbol* or *raising factorial* defined by $$\label{eq:1.2.8}
(a)_0 =1,\qquad (a)_n = a(a+1) \cdots (a+n-1) = \frac{\Gamma
(a+n)}{\Gamma (a)},\quad n=1,2,\ldots \ .$$ More generally, one can define hypergeometric series with more parameters.
Around the mid 19th-century Heine introduced the series $$\label{eq:1.2.12}
1 + \frac{(1-q^a)(1-q^b)}{(1-q)(1-q^c)}z
+\frac{(1-q^a)(1-q^{a+1})(1-q^b)(1-q^{b+1})}
{(1-q)(1-q^2)(1-q^c)(1-q^{c+1})}\;z^2 + \cdots ,$$ where it is assumed that $q \ne 1$, $c\not= 0, -1, -2, \ldots$ and the principal value of each power of $q$ is taken. This series converges absolutely for $|z|<1$ when $|q| <1$ and it tends termwise to Gauss’ series as $q\rightarrow 1$, because $$\label{eq:1.2.13}
\lim_{q\to 1} \frac{1-q^a}{1-q} = a.$$ The ratio $(1-q^a)/(1-q)$ considered in is called a $q$-*number* (or *basic number*) and it is denoted by $[a]_q$. One should realise that other notations for $q$-numbers, such as $\frac{q^a-q^{-a}}{q-q^{-1}}$, are also in use. It is also called a $q$-analogue, $q$-deformation, $q$-extension, or a $q$-generalization of the complex number $a$. In terms of $q$-numbers the $q$-*number factorial* $[n]_q!$ is defined for a nonnegative integer $n$ by $[n]_q! = \prod^n_{k=1} [k]_q$, and the corresponding $q$-*number shifted factorial* is defined by $[a]_{q;n} = \prod^{n-1}_{k=0} [a+k]_q$. Clearly, $[a]_{q;n} = (1-q)^{-n} (q^a;q)_n$, with the notation and $\lim_{q\to 1} [a]_{q;n} =
(a)_n$. The series in is usually called *Heine’s series* or, in view of the base $q$, the *basic hypergeometric series* or $q$-*hypergeometric series*, or simply a $q$-*series*.
Notation for basic hypergeometric series {#ssec:BHS-notation}
----------------------------------------
Analogous to Gauss’s notation for the hypergeometric function, Heine used the notation $\vp(a,b,c,q,z)$ for his series. However, since one would like to also be able to consider the case when $q$ to the power $a,b,$ or $c$ is replaced by zero, it is now customary to define the *basic hypergeometric series* $$\label{eq:1.2.14}
\begin{split}
\vp (a,b;c;q,z) \equiv \; {}_2\vp_1 (a,b;c;q,z)
\equiv
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;q,z \right)}
=\; \sum_{n=0}^{\infty} \frac{(a;q
)_n(b;q)_n}{(q;q)_n(c;q)_n}\;z^n,
\end{split}$$ where $$\label{eq:1.2.15}
(a;q)_n = \begin{cases} 1,& n=0,\\
(1-a)(1-aq)\cdots(1-aq^{n-1}),& n=1,2,\ldots ,
\end{cases}$$ is the $q$-*shifted factorial* and for general $a$ and $b$ it is assumed that $c\not= q^{-m}$ for $m = 0, 1, \ldots\ $. Some other notations that have been used in the literature for the product $(a;q)_n$ are $(a)_{q,n}, [a]_n$ (not to be confused with $[a]_q$).
Unless stated otherwise, when dealing with nonterminating basic hypergeometric series we shall assume that $|q| <1$ and that the parameters and variables are such that the series converges absolutely. Note that if $|q|>1$, then we can perform an inversion with respect to the base by setting $p= q^{-1}$ and using the identity $$\label{eq:1.2.24}
(a;q)_n = (a^{-1};p)_n (-a)^np^{-{{\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)}}}$$ to convert the series to a similar series in base $p$ with $|p| <1$, see . The inverted series will have a finite radius of convergence if the original series does. More generally, we call the series $\sum_{n=0}^\infty u_n$ a (unilateral) *hypergeometric series* if the quotient $u_{n+1}/u_n$ is a rational function of $n$. Similarly, a series $\sum_{n=0}^\infty v_n$ a *basic hypergeometric series* (with base $q$) if the quotient $v_{n+1}/v_n$ is a rational function of $q^n$ for a fixed base $q$. The most general form of the quotient is $$\label{eq:1.2.26}
\frac{v_{n+1}}{v_n} = \frac{(1-a_1q^n)(1-a_2q^n)\cdots
(1-a_rq^n)}{(1-q^{n+1}) (1-b_1q^n)\cdots
(1-b_sq^n)}\;\left(-q^n\right)^{1+s-r}z.$$ normalising $v_0 =1$. Generalising Heine’s series, we define an $_r\vp_s$ *basic hypergeometric series* by $$\label{eq:1.2.22}
\begin{split}
{}_r\vp_s (a_1, a_2, \ldots, a_r; b_1,\ldots,b_s;q,z)
&\equiv\;
{\,_{r}\vp_{s} \left( \genfrac{.}{.}{0pt}{}{a_1, a_2,\ldots , a_r}{b_1,\ldots ,b_s}
\ ;q,z \right)} \\
&= \sum_{n=0}^{\infty} \frac{(a_1;q)_n (a_2;q)_n\cdots
(a_r;q)_n}{(q;q)_n (b_1 ;q)_n \cdots (b_s ;q)_n} \left[(-1)^n
q^{{\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)}}\right]^{1+s-r}z^n
\end{split}$$ with ${\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)} = n(n-1)/2$, where $q\ne 0$ when $r> s+1$.
\[rmk:convergencerphis\] If $0 < |q| <1$, the $_r\vp_s$ series converges absolutely for all $z$ if $r\leq s$ and for $|z| <1$ if $r = s+1$. This series also converges absolutely if $|q| >1$ and $|z| < |b_1b_2\cdots b_sq|/|a_1a_2\cdots a_r|$. It diverges for $z\not= 0$ if $0 < |q|
<1$ and $r>s+1$, and if $|q| >1$ and $|z| > |b_1b_2\cdots b_sq|/|
a_1 a_2 \cdots a_r|$, unless it terminates.
Since products of $q$-shifted factorials occur so often, to simplify them we shall frequently use the more compact notations $$\label{eq:1.2.41}
(a_1,a_2,\ldots, a_m;q)_n = (a_1;q)_n (a_2;q)_n\cdots (a_m;q)_n,
\qquad n\in\mathbf{N}. $$
As is customary, the $_r\vp_s$ notation is also used for the sums of these series inside the circle of convergence and for their analytic continuations (called *basic hypergeometric functions*) outside the circle of convergence. To switch from base $q$ to base $q^{-1}$ we note $$\label{eq:ex1.4i}
{\,_{r}\vp_{s} \left( \genfrac{.}{.}{0pt}{}{a_1, \ldots, a_r}{b_1, \ldots, b_s}
\ ;q,z \right)}
= \sum^\infty_{n=0} \frac{(a^{-1}_1, \ldots, a^{-1}_r; q^{-1})_n}{(q^{-
1}, b^{-1}_1, \ldots, b_s^{-1};q^{-1})_n} \left(\frac{a_1 \cdots
a_rz}{b_1 \cdots b_s q}\right)^n$$ assuming the upper and lower parameters are non-zero.
It is important to note that in case one of the upper parameters is of the form $q^{-n}$ for $n\in\textbf{N}$ the series in terminates. From now on, unless stated otherwise, whenever $q^{-j},
q^{-k}, q^{-m}, q^{-n}$ appear as numerator parameters in basic series it will be assumed that $j$, $k$, $m$, $n$, respectively, are nonnegative integers. For terminating series it is sometimes useful to switch the order of summation, which is given by $$\label{eq:ex1.4ii}
\begin{split}
{\,_{r+1}\vp_{s} \left( \genfrac{.}{.}{0pt}{}{a_1, \ldots, a_r, q^{-n}}{b_1, \ldots,
b_s}
\ ;q,z \right)}
= &\frac{(a_1, \ldots, a_r ;q)_n}{(b_1\ldots, b_s;q)_n} \left(\frac{z}{q}\right)^n \left((-1)^n q^{{\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)}} \right)^{s-r-1}\\[4pt]
&\times
\sum^n_{k=0} \frac{(q^{1-n}/b_1, \ldots, q^{1-n}/b_s, q^{-
n};q)_k}{(q, q^{1-n}/a_1, \ldots, q^{1-n}/a_r;q)_k}
\left(\frac{b_1\cdots b_s}{a_1\cdots a_r} \frac{q^{n+1}}{z}\right)^k
\end{split}$$ for non-zero parameters.
Observe that the series has the property that if we replace $z$ by $z/a_r$ and let $a_r\rightarrow \infty$, then the resulting series is again of the form with $r$ replaced by $r-1$. Because this is not the case for the $_r\vp_s$ series defined without the factors $\left[(-1)^nq^{{\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)}}\right]^{1+s-r}$ in the books of Bailey [@Bail] and Slater [@Slat] and we wish to be able to handle such limit cases, we have chosen to define the series ${}_r\vp_s$ as in . There is no loss in generality since the Bailey and Slater series can be obtained from the $r=s+1$ case of by choosing $s$ sufficiently large and setting some of the parameters equal to zero.
For negative subscripts, the $q$-*shifted factorials* as defined in are defined by $$\label{eq:1.2.28}
(a;q)_{-n} = \frac{1}{(1-aq^{-1})(1-aq^{-2})\cdots (1-aq^{-n})} =
\frac{1}{(aq^{-n};q)_n} = \frac{(-q/a)^nq^{{\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)}}}{(q/a;q)_n},$$ where $n = 0,1, \ldots\ $. We also define $$\label{eq:1.2.29}
(a;q)_\infty = \prod^\infty_{k=0} (1-aq^k)$$ for $|q| <1$. Since the infinite product in diverges when $a\not= 0$ and $|q| > 1$, whenever $(a;q)_\infty$ appears in a formula, we shall assume that $|q| < 1$. In particular, for $|q|<1$ and $z$ an integer $$\label{eq:AI.6}
(a;q)_z = \frac{(a;q)_\infty}{(aq^z;q)_\infty},$$ which is a notation that we also employ for complex $z$, where we take standard branch cut for the complex power.
The basic hypergeometric series $${\,_{r+1}\vp_{r} \left( \genfrac{.}{.}{0pt}{}{a_1, a_2,\ldots ,a_{r+1}}{b_1,\ldots\
,b_r}
\ ;q,z \right)}$$ is called $k$-*balanced* if $b_1b_2\cdots b_r
=q^k a_1a_2\cdots a_{r+1}$ and $z = q$, and a 1-balanced basic hypergeometric series is called *balanced* (or *Saalschützian*). The basic hypergeometric series ${}_{r+1}\vp_r$ is *well-poised* if the parameters satisfy the relations $$qa_1 = a_2b_1 = a_3b_2 = \cdots = a_{r+1}b_r;$$ *very-well-poised* if, in addition, $a_2 = qa_1^\hf, a_3 = -qa_1^\hf$. For very-well-poised series the following notation is in use: $$\label{eq:defWseries}
\begin{split}
{}_{r+1}W_r \left(a_1; a_4, a_5, \ldots,
a_{r+1};q,z\right) =&
{\,_{r+1}\vp_{r} \left( \genfrac{.}{.}{0pt}{}{a_1, qa_1^\hf, -qa_1^\hf,
a_4,\ldots,a_{r+1}}{a_1^\hf, -a_1^\hf,
qa_1/a_4,\ldots,qa_1/a_{r+1}}
\ ;q,z \right)} \\
=& \sum_{k=0}^\infty \frac{1-a_1q^{2k}}{1-a_1}
\frac{(a_1, a_4,\cdots, a_{r+1};q)_k}{(q, qa_1/a_4, \cdots, qa_{r+1}/a_1} z^k.
\end{split}$$ The $q$-binomial coefficient is defined as $$\label{eq:AI.39}
{\left[\genfrac{.}{.}{0pt}{}{n}{k}\right]_{q}}\, = \, {\left[\genfrac{.}{.}{0pt}{}{n}{n-k}\right]_{q}} \, = \, \frac{(q;q)_n}{(q;q)_k\, (q;q)_{n-k}}$$ and satisfies the following recurrences $$\label{eq:AI.45}
{\left[\genfrac{.}{.}{0pt}{}{n+1}{k}\right]_{q}} \,=\, q^k\, {\left[\genfrac{.}{.}{0pt}{}{n}{k}\right]_{q}} + {\left[\genfrac{.}{.}{0pt}{}{n}{k-1}\right]_{q}} \, =\,
{\left[\genfrac{.}{.}{0pt}{}{n}{k}\right]_{q}} + q^{n+1-k}\, {\left[\genfrac{.}{.}{0pt}{}{n}{k-1}\right]_{q}}.$$ The generalized $q$-binomial coefficient is defined for complex $\alpha$, $\beta$ by $$\label{eq:AI.40}
{\left[\genfrac{.}{.}{0pt}{}{\alpha}{\beta}\right]_{q}}\, = \, \frac{(q^{\beta+1}, q^{\alpha-\beta+1};q)_\infty}
{(q,q^{\alpha+1};q)_\infty}$$ and then remains valid for complex $\alpha$.
We end Section \[ssec:BHS-notation\] with some useful identities for $q$-shifted factorials; $$\begin{gathered}
(a;q)_n = \frac{(a;q)_\infty}{(aq^n;q)_\infty} \\
(a^{-1}q^{1-n};q)_n = (a;q)_n (-a^{-1})^n q^{-{{\left(\genfrac{.}{.}{0pt}{}{n}{2}\right)}}},
(a;q)_{n-k} = \frac{(a;q)_n}{(a^{-1}q^{1-n};q)_k}\;
(-qa^{-1})^kq^{{\left(\genfrac{.}{.}{0pt}{}{k}{2}\right)}-nk}, \\
(a;q)_{n+k} = (a;q)_n (aq^n;q)_k, \\
(aq^n;q)_k = \frac{(a;q)_k (aq^k;q)_n}{(a;q)_n}, \\
(aq^k;q)_{n-k} = \frac{(a;q)_n}{(a;q)_k},\\
(aq^{2k};q)_{n-k} = \frac{(a;q)_n(aq^n;q)_k}{(a;q)_{2k}}, \\
(q^{-n};q)_k = \frac{(q;q)_n}{(q;q)
_{n-k}}(-1)^kq^{{\left(\genfrac{.}{.}{0pt}{}{k}{2}\right)}-nk}, \\
(aq^{-n};q)_k = \frac{(a;q)_k (qa^{-1};q)_n}{(a^{-1}q^{1-k};q)_n} q^{-nk}, \\
(a;q)_{2n} = (a;q^2)_n (aq;q^2)_n, \\
(a;q)_{3n} = (a;q^3)_n (aq;q^3)_n (aq^2;q^3)_n, \\
(a^2;q^2)_n = (a;q)_n (-a;q)_n, \\
(a^3;q^3)_n = (a;q)_n (\omega a;q)_n (\omega^2 a;q)_n, \qquad \omega = e^{2\pi i /3} \\\end{gathered}$$ and similar expressions for $(a;q)_{kn}$ and $(a^k;q^k)_n$, $k=4,5,\cdots$.
Some summation and transformation formulae {#ssec:BHS-summationtransformation}
------------------------------------------
There are many summation and transformation results for basic hypergeometric series available, and we only give a few basic results. We give precise references in case we need more advanced summation or transformation formulae.
The most fundamental result is the $q$-binomial theorem, stating $$\label{eq:qbinomialthm}
{\,_{1}\vp_{0} \left( \genfrac{.}{.}{0pt}{}{a}{-}
\ ;q,z \right)} =\frac{(az;q)_\infty}{(z;q)_\infty}, \qquad |z|<1.$$ Its terminating version reads $$\label{eq:qbinomial-term}
{\,_{1}\vp_{0} \left( \genfrac{.}{.}{0pt}{}{q^{-n}}{-}
\ ;q,z \right)} = (q^{-n}z;q)_n, \qquad n\in \N.$$ The proof is sketched in Exercise \[ex:qbinomthm\]. We discuss a few consequences of the $q$-binomial theorem.
First, we write $$\begin{gathered}
\frac{(az;q)_\infty}{(z;q)_\infty} \frac{(z;q)_\infty}{(bz;q)_\infty} =
\frac{(az;q)_\infty}{(bz;q)_\infty} \quad \Longrightarrow\quad
{\,_{1}\vp_{0} \left( \genfrac{.}{.}{0pt}{}{a}{-}
\ ;q,z \right)} {\,_{1}\vp_{0} \left( \genfrac{.}{.}{0pt}{}{1/b}{-}
\ ;q,bz \right)} = {\,_{1}\vp_{0} \left( \genfrac{.}{.}{0pt}{}{a/b}{-}
\ ;q,bz \right)}\end{gathered}$$ and this is a product of analytic functions, so that the coefficients of the power series have to be equal. This gives $$\label{eq:qChuVDM}
\sum_{k+p=n} \frac{(a;q)_k}{(q;q)_k} \frac{(1/b;q)_p}{(q;q)_p} b^p =
\frac{(a/b;q)_n}{(q;q)_n} b^n \quad \Longrightarrow \quad
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{q^{-n},a}{c}
\ ;q,q \right)} = \frac{(c/a;q)_n}{(c;q)_n}a^n$$ after relabeling. This is the $q$-Chu-Vandermonde formula.
Another application of the $q$-binomial formula is Heine’s transformation formula. Heine showed $$\label{eq:1.4.1}
_2\varphi_1 (a,b;c;q,z) = \frac{(b,az;q)_\infty}{(c,z;q)_\infty}\;_2\varphi_1 (c/b, z; az;q,b),$$ where $|z| <1$ and $|b| <1$. By iterating the result $$\label{eq:1.4.4}
\begin{split}
_2\varphi_1 (a,b;c;q,z) &= \frac{(c/b, bz;q)_\infty}{
(c,z;q)_\infty}\;_2\varphi_1(abz/c,b;bz;q,c/b) \\[3pt]
&= \frac{(abz/c;q)_\infty}{(z;q)_\infty}\;_2\varphi_1
(c/a,c/b;c;q,abz/c).
\end{split}$$ with appropriate conditions on the parameters for the last two series to be convergent. Heine’s formula can directly be obtained from the $q$-binomial theorem ; $$\begin{split}
_2\varphi_1 (a,b;c;q,z) \, =&\, \frac{(b;q)_\infty}{(c;q)_\infty} \sum_{n=0}^{\infty}
\frac{(a;q)_n(cq^n;q)_\infty}{(q;q)_n (bq^n;q)_\infty}z^n\\[3pt]
=&\, \frac{(b;q)_\infty}{(c;q)_\infty} \sum_{n=0}^{\infty}
\frac{(a;q)_n}{(q;q)_n} z^n \sum_{m=0}^{\infty} \frac{(c/b;q)_m}{(q;q)_m}
(bq^n)^m\\[3pt]
=&\, \frac{(b;q)_\infty}{(c;q)_\infty} \sum_{m=0}^{\infty}
\frac{(c/b;q)_m}{(q;q)_m} b^m \sum_{n=0}^{\infty}
\frac{(a;q)_n}{(q;q)_n} (zq^m)^n\\[3pt]
=&\, \frac{(b;q)_\infty}{(c;q)_\infty} \sum_{m=0}^{\infty} \frac{(c/b;q)_m}{(q;q)_m} b^m
\frac{(azq^m;q)_\infty}{(zq^m;q)_\infty}\\[3pt]
=&\, \frac{(b,az;q)_\infty}{(c,z;q)_\infty} {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{c/b, z}{az}
\ ;q, b \right)},
\end{split}$$ which gives . The implied convergence of the series above is assumed to hold. Limit cases of Heine’s transformation formulas , are $$\label{eq:1.4.1limita}
\begin{split}
(c;q)_\infty {\,_{1}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a}{c}
\ ;q,z \right)} &\, =\, (a,z;q)_\infty {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{c/a, 0}{z}
\ ;q, a \right)} \\
&\, = \, (c/a;q)_\infty {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{az/c, a}{0}
\ ;q, \frac{c}{a} \right)} \\
&\, = \, (az/c, c;q)_\infty {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{c/a, 0}{c}
\ ;q, \frac{az}{c} \right)} \\
&\, = \, (z;q)_\infty {\,_{1}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{az/c}{z}
\ ;q, c \right)},
\end{split}$$ so that in particular $(c;q)_\infty \, {}_1\varphi_1(0;c;q,z) = (z;q)_\infty \, {}_1\varphi_1(0;z;q,c)$ is symmetric in $c$ and $z$. This symmetry is observed by Koornwinder and Swarttouw [@KoorS-TAMS] in their study of the $q$-Hankel transform for the ${}_1\varphi_1$-$q$-Bessel functions. Taking a limit in we obtain $$\label{eq:1.4.1limitb}
(c;q)_\infty {\,_{0}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{-}{c}
\ ;q,z \right)}
\, = \, (z/c, c;q)_\infty {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{0, 0}{c}
\ ;q, \frac{z}{c} \right)}
\, = \, {\,_{1}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{z/c}{0}
\ ;q, c \right)} .$$
Consider the $q$-integral on an interval $[0,a]$, defined by $$\label{eq:qintegral0a}
\int_0^a f(t) \, d_q(t) = (1-q)a \sum_{k=0}^\infty q^k f(aq^k)$$ whenever the function $f$ is such that the series in converges. Note that we can view as a Riemann sum for $\int_0^a f(t)\, dt$ on a non-equidistant partition of the interval $[0,a]$. Using the notation we can rewrite as $$\begin{gathered}
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;q,z \right)} = \frac{(b,c/b, z;q)_\infty}{(q,c,z;q)_\infty}
\sum_{k=0}^\infty \frac{(q^{k+1},azq^k;q)_\infty}{ (q^kc/b, zq^k;q)_\infty} b^k \\
= \frac{(b,c/b, z;q)_\infty}{(q,c,z;q)_\infty} \frac{1}{1-q} \int_0^1
\frac{(qt,azt;q)_\infty}{ (tc/b, tz;q)_\infty} t^{-1+\log_q b}\, d_qt\end{gathered}$$ which can be considered as a $q$-analogue of Euler’s integral representation $${\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;z \right)} =\frac{\Ga(c)}{\Ga(b)\Ga(c-b)}\int_0^1 t^{b-1} (1-t)^{c-b-1} (1-tz)^{-a}\, dt,
\qquad \Re c> \Re b>0.$$ for the hypergeometric series, see e.g. [@AndrA], [@Temm].
Another integral representation is the Watson integral representation. In Watson’s formula we assume $0<q<1$, and then $$\label{eq:Watsonint2phi1}
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;q,z \right)} =
\frac{-1}{2\pi} \frac{(a,b;q)_\infty}{(q,c;q)_\infty}
\int_{-i\infty}^{i\infty} \frac{(q^{1+s}, cq^s;q)_\infty}{(aq^s, bq^s;q)_\infty}
\frac{\pi (-z)^s}{\sin(\pi s)} \, ds$$ for $|z|<1$ and $|\arg(-z)|<\pi$. The contour runs from $-i\infty$ to $i\infty$ via the imaginary axis with indentations such that the poles of $1/\sin(\pi s)$ lie to the right of the contour and the poles of $1/(aq^s, bq^s;q)_\infty$ lie to the left of the contour. Then Watson’s formula follows by a residue calculation and estimates on the behaviour of the integrand, see [@GaspR §4.2]. By then flipping the contour, and evaluating the integral using the residues at the poles of $1/(aq^s, bq^s;q)_\infty$ and performing the right estimates shows the connection formula, see [@GaspR §4.3]; $$\label{eq:4.3.2}
\begin{split}
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;q,z \right)}\,
= &\, \frac{(b,c/a;q)_\infty (az,q/az;q)_\infty}
{(c,b/a;q)_\infty(z,q/z;q)_\infty}
\; {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a,aq/c}{aq/b}
\ ;q,cq/abz \right)}
\\ &\, + \, \frac{(a,c/b;q)_\infty}{(c,a/b;q)_\infty} \frac{(bz,q/bz;q)_\infty}
{(z,q/z;q)_\infty}\;
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{b,bq/c}{bq/a}
\ ;q,cq/abz \right)}.
\end{split}$$ which gives the analytic continuation to the region $|\text{arg} (-z)| < \pi$, with $c$ and $a/b$ not integer powers of $q$, and $a,b, z \not= 0$. Note that the coefficients in are related to theta functions,. Indeed, because of Jacobi’s triple product identity, see [@GaspR §1.6], $$\label{eq:thetafunction}
\theta(z) = (z,q/z;q)_\infty \qquad \Longrightarrow
\qquad \theta(q^kz) = (-z)^{_k} q^{-\frac12 k(k-1)} \theta(z)$$ $\theta$ is a renormalized Jacobi theta function.
Exercises
---------
1. \[ex:Raabe2F1\] Use Raabe’s test to show that ${}_2F_1(a,b;c,z)$, $|z|=1$, converges absolutely for $\Re(c-a-b)>0$.
2. Prove the statements on convergence of the basic hypergeometric series as in Remark \[rmk:convergencerphis\].
3. Prove .
4. Prove .
5. Prove the useful identities for $q$-shifted factorials.
6. \[ex:qbinomthm\] Askey’s proof of the $q$-binomial theorem goes as follows. Denote the ${}_1\vp_0$-series by $h_a(z)$ and show that $$\begin{gathered}
h_a(z) \, - \, h_{aq}(z)\, = \, -az\, h_{aq}(z), \qquad
h_a(z)\, - \, h_a(qz)\, = \, (1-a)z\, h_{aq}(z)\quad \Longrightarrow \\
h_a(z) \, = \, \frac{1-az}{1-z}\, h_a(qz).\end{gathered}$$ Iterate and use the analyticity and the value at $z=0$ to finish the proof.
Notes {#notes .unnumbered}
-----
The basic reference for basic hypergeometric series is the standard book [@GaspR] by Gasper and Rahman, or the first edition of [@GaspR]. The book by Gasper and Rahman contains a wealth of information on basic hypergeometric series. There are older books containing chapters on basic hypergeometric series, e.g. Bailey [@Bail], Slater [@Slat], as well as the Heine’s book –the second edition of *Handbuch der Kugelfunktionen* of 1878, see references in [@GaspR]. More modern books on special functions having chapters on basic hypergeometric series are e.g. [@AndrAR], [@Isma]. Another useful reference is the lecture notes by Ismail [@Isma-LN].
Basic hypergeometric $q$-difference equation {#sec:BHS-qdiff}
============================================
An important aspect of the hypergeometric series ${}_2F_1$ is that it can be used to describe the solutions to the *hypergeometric differential equation* $$\label{eq:hypergeometricdiffeqtn}
z(1-z) \frac{d^2f}{dz^2}(z) + (c-(a+b+1)z) \frac{df}{dz}(z) - ab f(z) =0,$$ see e.g. [@AndrAR], [@Isma], [@Rain], [@Temm]. In particular, $$u_1(z) = {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;z \right)}, \ \ c\not= 0,-1,-2,\cdots$$ solves the hypergeometric differential equation as can be checked directly by plugging the power series expansion. Other solutions expressible in terms of hypergeometric series are e.g. $$\begin{gathered}
u_2(z) = z^{1-c} {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a-c+1,b-c+1}{2-c}
\ ;z \right)},\ \ c\not=2,3, \cdots \\
u_3(z) = z^{-a} {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a,a-c+1}{a-b+1}
\ ;\frac{1}{z} \right)}, \ \ a-b\not= -1, -2, \cdots \\
u_4(z) = z^{-b} {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{b,b-c+1}{b-a+1}
\ ;\frac{1}{z} \right)}, \ \ b-a\not= -1, -2, \cdots.\end{gathered}$$ The differential equation is a Fuchsian differential equation with three regular singular points at $0$, $1$ and $\infty$. So one usually also considers the similar solutions in terms of power series around $z=1$, but these solutions do not have appropriate $q$-analogues.
So in general we have two linearly independent solutions in terms of power series around $0$ and two linearly independent solutions in terms of power series around $\infty$. Since the solution space of is $2$-dimensional, there are all kinds of relations between these solutions. One of the classical relations between hypergeometric series is given by $$\label{eq:rel2F1at0andinfty}
\begin{split}
{\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a,b}{c}
\ ;z \right)} = \frac{\Ga(c)\Ga(b-a)}{\Ga(b)\Ga(c-a)}
&(-z)^{-a} {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{a,a-c+1}{a-b+1}
\ ;\frac{1}{z} \right)} \\
+ &\frac{\Ga(c)\Ga(a-b)}{\Ga(a)\Ga(c-b)}
(-z)^{-b} {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{b,b-c+1}{b-a+1}
\ ;\frac{1}{z} \right)}
\end{split}$$ for $|\arg(-z)|<\pi$.
The Jacobi polynomials are special cases of the hypergeometric series ${}_2F_1$; explicitly $$\label{eq:defJacobipols}
P_n^{(\al,\be)}(x) = \frac{(\al+1)_n}{n!} {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{-n, n+\al+\be+1}{\al+1}
\ ;\frac12(1-x) \right)}.$$ So in particular, the Jacobi polynomials are eigenfunctions to a second-order differential operator. This differential operator can then be studied on the weighted $L^2$ spaces with respect to the beta-weight $(1-x)^\al(1-x)^\be$ on $[-1,1]$. The differential operator is a self-adjoint operator on a suitable domain with compact resolvent. The orthogonality of the Jacobi polynomials is related to the orthogonality of the eigenvectors of the corresponding differential operator.
The Jacobi functions are $$\label{eq:defJacobifunctions}
\phi^{(\al,\be)}_\la (t) = {\,_{2}F_{1} \left( \genfrac{.}{.}{0pt}{}{\frac12(\al+\be+1+i\la),\frac12(\al+\be+1-i\la)}{\al+1}
\ ;-\sinh^2t \right)}$$ and these are eigenfunctions of a related second order differential operator, after a change of variables. The corresponding Jacobi function transform arises from the spectral decomposition of the differential operator, see [@Koor-Jacobi] for more information as well as the link to representation theory of non-compact symmetric spaces of rank one.
Basic hypergeometric $q$-difference equation {#ssec:BHS-qdiff}
--------------------------------------------
For fixed $q\not=1$, the $q$-derivative operator $D_q$ is defined by $$\label{eq:1.3.21}
D_qf \, (x) \, = \, \frac{f(x) - f(qx)}{(1-q)x}, \quad x\not=0,$$ and $D_qf \, (0)=f'(0)$ assuming the derivative exists. Then $D_qf(x)$ tends to $f'(x)$ as $q\to 1$ for differentiable $f$. We can iterate; $D_q^nf \, = \, D_q (D_q^{n-1}f)$, $n=1,2,\cdots$. The $q$-difference operator $D_q$ applied to the ${}_2\vp_1$-series: $$\label{eq:ex1.12ii}
D^n_q \, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a, b}{c}
\ ;q, z \right)}
\, = \, \frac{(a,b;q)_n}{(c;q)_n(1-q)^n}\,
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{aq^n, bq^n}{cq^n}
\ ;q, z \right)}$$ which can be checked directly. Moreover, $u(z) = \; _2\vp_1 (a,b;c;q,z)$ satisfies (for $|z| < 1$ and in the formal power series sense) the second order $q$-difference equation $$\label{eq:ex1.13}
z(c-abqz) D^2_q u + \left[ \frac{1-c}{1-q} + \frac{(1-a)(1-b) -
(1-abq)}{1-
q}z\right] D_q u\\[3pt]
- \frac{(1-a)(1-b)}{(1-q)^2} u = 0,$$ which is a $q$-analogue of the hypergeometric differential equation . Indeed, replacing $a$, $b$, $c$ with $q^a$, $q^b$, $q^c$ and taking formal limits, shows that tends to as $q\to 1$. Explicitly, is $$\label{eq:ex1.13a}
(c-abz)\, u(qz) \,+\, \left(-(c+q)+ (a+b)z\right)\, u(z)\, +\,
(q-z)\, u(z/q) \, = 0$$ for $a,b,c$ non-zero complex numbers. We consider as the *basic hypergeometric $q$-difference equation*. Note that if $u$ is a solution to , and $C$ is a $q$-periodic function, i.e. $C(qz)=C(z)$, then $Cu$ is also a solution to .
\[prop:solofBqDEat0atinfty\] The functions $$\begin{split}
u_1(z) \, &= \, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a, b}{c}
\ ;q, z \right)}, \quad c\not= q^{-n}, \ n=0,1,2\cdots \\
u_2(z) \, &= \, z^{1-\log_q(c)}\, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{qa/c, qb/c}{q^2/c}
\ ;q, z \right)},
\quad c\not= q^{n+2}, \ n=0,1,2\cdots,
\end{split}$$ and the functions $$\begin{split}
u_3(z) \, &= \, z^{-\log_q(a)}\, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a, qa/c}{qa/b}
\ ;q, \frac{qc}{abz} \right)}, \quad a\not= bq^{-n-1}, \ n=0,1,2\cdots, \\
u_4(z) \, &= \, z^{-\log_q(b)}\, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{b, qb/c}{qb/a}
\ ;q, \frac{qc}{abz} \right)}, \quad b\not= aq^{-n-1}, \ n=0,1,2\cdots.
\end{split}$$ are solutions of the basic hypergeometric $q$-difference equation .
Since the map $z\mapsto qz$ has two fixed points on the Riemann sphere, namely $z=0$ and $z=\infty$, it is natural to consider power series expansion solutions of at $z=0$ and $z=\infty$ using the Frobenius method.
We make the Ansatz $$u(z) = \sum_{n=0}^\infty a_n z^{n+\mu}, \qquad a_n\in \C, \ a_0\not= 0,\ \mu\in \C$$ Plugging such a solution into and collecting the coefficients of $z^{n+\mu}$, we require $$\begin{gathered}
0 = a_0 z^\mu \bigl( cq^\mu -(c+q) + q^{1-\mu}\bigr) + \\
\sum_{n=1}^\infty z^{\mu+n} \Bigl( a_n \bigl( cq^{\mu+n} -(c+q) + q^{1-\mu-n}\bigr) + a_{n-1} \bigl( -abq^{\mu+n-1}
+ a+b - q^{1-\mu-n}\bigr) \Bigr)\end{gathered}$$ So the coefficient of $z^\mu$ has to be zero, and this gives the *indicial equation* $$0 = cq^\mu -(c+q) + q^{1-\mu} = (q^\mu-1)(c-q^{1-\mu}).$$ So we find $q^\mu=1$ or $q^{\mu} = q/c$.
In case $q^\mu=1$ we find the recurrence relation $$\begin{gathered}
a_n \bigl( cq^{n} -(c+q) + q^{1-n}\bigr) = a_{n-1} \bigl( abq^{n-1}
- (a+b) + q^{1-n}\bigr) \quad \Longrightarrow \\
a_n = a_{n-1} \frac{(aq^{n-1}-1)(b-q^{1-n})}{(cq^n-q)(1-q^{-n})}
= a_{n-1} \frac{(1-aq^{n-1})(1-bq^{n-1})}{(1-cq^n)(1-q^{n+1})} = a_0 \frac{(a,b;q)_n}{(c,q;q)_n}, \end{gathered}$$ so we find the solution $u_1$ for $\mu=0$. If we take more generally $\mu = \frac{2\pi i}{\log q}k$, $k\in \Z$, then we multiply $u_1$ by the $q$-periodic function $z\mapsto z^{\frac{2\pi i}{\log q}k}$.
In case $q^{\mu} = q/c$ we find the recurrence relation $$\begin{gathered}
a_n \bigl( q^{1+n} -(c+q) + cq^{-n}\bigr) = a_{n-1} \bigl( \frac{ab}{c}q^{n}
- (a+b) + cq^{-n}\bigr) \quad \Longrightarrow \\
a_n = a_{n-1} \frac{(q^n\frac{a}{c}-1)(b-cq^{-n})}{(q^n-1)(q-cq^{-n})}
= a_{n-1} \frac{(1-q^na/c)(1-bq^n/c)}{(1-q^n)(1-q^{n+1}/c)} = a_0 \frac{(qa/c,qb/c;q)_n}{(q,q^2/c;q)_n}, \end{gathered}$$ so we find the solution $u_2$ for $\mu = 1-\log_q(c)$, and again if we add an integer multiple of $\frac{2\pi i}{\log q}$, we multiply by a $q$-periodic function.
Similarly we obtain the solutions $u_3$, $u_4$ by replacing the Ansatz by $u(z) = \sum_{n=0}^\infty a_n z^{-n-\mu}$, $a_n\in \C$, $a_0\not= 0$, $\mu\in \C$. We leave this as Exercise \[eq:prop:solofBqDEat0atinfty\].
\[rmk:linindepsolBHDE\] Note that for generic parameters the solutions $u_1$ and $u_2$, respectively $u_3$ and $u_4$, are linearly independent (over $q$-periodic functions). So we expect that these solutions satisfy relations amongst each other. In particular, gives $$\begin{gathered}
u_1(z) = C_3(z) u_3(z) + C_4(z) u_4(z) \\
C_3(z) = C_3(z;a,b;c) =\frac{(b,c/a;q)_\infty (az,q/az;q)_\infty}
{(c,b/a;q)_\infty(z,q/z;q)_\infty} z^{\log_q(a)},
\qquad C_4(z) = C_3(z;b,a;c)\end{gathered}$$ Note that indeed, $C_3(qz)=C_3(z)$ using and so $C_3$ and $C_4$ are $q$-periodic functions.
We rewrite the basic hypergeometric equation as $$\label{eq:BHqDiff-rewrite}
(c-abz)\, \frac{u(qz)-u(z)}{z} + (1-z/q) \, \frac{u(z/q)-u(z)}{z/q}
= (1-a)(1-b) u(z).$$ We consider the left hand side as an operator acting on functions $u$, so we put $$\label{eq:Labc-def}
\bigl(L u\bigr)(z) = \bigl(L^{a,b,c} u\bigr)(z)
= (c-abz)\, \frac{u(qz)-u(z)}{z} + (1-z/q) \, \frac{u(z/q)-u(z)}{z/q},$$ so that upon using the normalized $q$-difference operator, cf. , $$\bigl( \tilde{D}_qu\bigr) (z) = \frac{u(qz)-u(z)}{z}$$ we can rewrite as $$\label{eq:BHqDiff-rewrite2}
\bigl(L u\bigr)(z) =
(c-abz)\, \bigl( \tilde{D}_qu\bigr) (z) - (1-z/q) \, \bigl( \tilde{D}_qu\bigr) (z/q)
= (1-a)(1-b) u(z),$$ so that the operator $L$ in left hand side of , can be written as the composition $L=S\circ \tilde{D}_q$, where $$\label{eq:defSabc}
S = S^{a,b,c}, \qquad \bigl( Sf\bigr)(z) = (c-abz)f(z) - (1-z/q)f(z/q).$$ Note that when acting on polynomials, $\tilde{D}_q$ maps polynomials of degree $n$ to polynomials of degree $n-1$ and $S$ maps polynomials of degree $n$ to polynomials of degree $n+1$. So we see that $L= S\circ\tilde{D}_q$ gives a factorisation in terms of a lowering operator and a raising operator. It is then common to consider the reversed composition and consider it as the Darboux transform of $L$. Lemma \[lem:Darboux-L\] shows that the Darboux transform is again of the same class.
\[lem:Darboux-L\] We have $$\bigl( \tilde{D}_q \circ S^{a,b,c} f\bigr)(z)
= \frac{1}{q} \bigl(L^{aq,bq,cq} f\bigr)(z) + (1-q)(ab-q^{-1}) f(z),$$ and if $L^{a,b,c}u = (1-a)(1-b)u$, then $f=\tilde{D}_qu$ satisfies $L^{aq,bq,cq}f = (1-aq)(1-bq)f$.
A straightforward calculation gives $$\begin{gathered}
\bigl( \tilde{D}_q \circ S^{a,b,c} f\bigr)(z) =
\frac{1}{z}\bigl( (c-abqz) f(qz) -(1-z)f(z) - (c-abz)f(z) + (1-z/q)f(z/q)\bigr) \\
= (c-abqz) \frac{f(qz)-f(z)}{z} + (1-z/q) \frac{f(z/q)-f(z)}{z/q}\frac{1}{q}
+ ab(1-q)f(z) + (1-q^{-1})f(z) \\
= \frac{1}{q} \left( (cq-abq^2z) \frac{f(qz)-f(z)}{z} + (1-z/q) \frac{f(z/q)-f(z)}{z/q}
\right) + (1-q)(ab-q^{-1}) f(z)\end{gathered}$$ and the term in brackets is precisely the operator $L^{aq,bq,cq}$ acting on $f$ by .
Apply the first result to $f=\tilde{D}_q u$, so that $$\begin{gathered}
\frac{1}{q} \bigl(L^{aq,bq,cq} f\bigr)(z) + (1-q)(ab-q^{-1}) f(z)
= \bigl( \tilde{D}_q \circ S^{a,b,c} \circ \tilde{D}_q u\bigr)(z) \\
= \bigl( \tilde{D}_q \circ L^{a,b,c} u\bigr)(z) =
(1-a)(1-b) \bigl(\tilde{D}_q u\bigr)(z) = (1-a)(1-b) f(z)\end{gathered}$$ so that $$\begin{gathered}
\bigl(L^{aq,bq,cq} f\bigr)(z) = \bigl( q(1-a)(1-b) - q(1-q)(ab-q^{-1})\bigr) f(z) \\
= \bigl( q -aq-bq +abq - abq +abq^2 +1 -q)\bigr) f(z)
= (1-aq)(1-bq) f(z).\end{gathered}$$ This proves the second statement, and it is in line with .
Exercises
---------
1. Prove .
2. \[eq:prop:solofBqDEat0atinfty\] Show the second part of Proposition \[prop:solofBqDEat0atinfty\], see also Theorem \[thm:MVBHSatinfty\].
Notes {#notes-1 .unnumbered}
-----
The solutions follow unpublished notes by Koornwinder, and the factorisation and the Darboux transform seems to be well known.
Basic hypergeometric $q$-difference equation: polynomial case {#sec:BHS-qdiff-pol}
=============================================================
We first consider the basic hypergeometric $q$-difference operator on a space which we can identify with a sequence space on $\N$. This is closely connected to the little $q$-Jacobi polynomials, and we derive its orthogonality and recurrence properties from properties of this operator.
The difference equation in a special case {#ssec:littleqJacobipols}
-----------------------------------------
Replace $z=z_0q^{k+1}$ in and put $u_k = u(z_0q^{k+1})$ then we get $$\begin{gathered}
(c-abz_0 q^{k+1})\, u_{k+1} \,+\, \left(-(c+q)+ (a+b)z_0q^{k+1}\right)\, u_k\, +\,
(q-z_0q^{k+1})\, u_{k-1} \, = 0 \quad \Longrightarrow \\
(cq^{-k-1}-abz_0)\, u_{k+1} \,-\, (c+q)q^{-k-1}\, u_k\, +\,
(q^{-k}-z_0)\, u_{k-1} \, = - (a+b)z_0\, u_k \end{gathered}$$ Note that in the special case $z_0=1$ the coefficient of $u_{-1}$ is zero. So we consider the operator $L$ on sequences $u=(u_k)_{k\in\N}$ by $$\label{eq:defDOlittleqJacobi}
\begin{split}
(L u)_k &= (cq^{-k-1}-ab)\, u_{k+1} \,-\, \bigl((cq^{-k-1}-ab) +
(q^{-k}-1)\bigr) \, u_k\, +\,
(q^{-k}-1)\, u_{k-1} \\
&= (cq^{-k-1}-ab)\, (u_{k+1}-u_k) +\,
(q^{-k}-1)\, (u_{k-1}-u_k), \qquad k\geq 1, \\
(L u)_0 &= (cq^{-1}-ab)\, (u_{1}-u_0)
\end{split}$$ Note that putting $\varphi(q^k)=u_k$, we can view $L$ as $$\begin{gathered}
(L\varphi)(x) = (cq^{-1}-abx)\, \frac{\varphi(xq)-\varphi(x)}{x} + (1-x) \frac{\varphi(x/q)-\varphi(x)}{x},
\quad x=q^k, k\geq 1 \\
(L\varphi)(1) = (cq^{-1}-ab)\, \bigr(\varphi(q)-\varphi(1)\bigr)\end{gathered}$$ so $L\varphi$ is expressible in terms of $D_q\varphi$ and $D_{q^{-1}}\varphi$. Considering $L$ as an operator acting on functions by $$\label{eq:lqJacpol-defL}
(L\varphi)(x) = (cq^{-1}-abx)\, \frac{\varphi(xq)-\varphi(x)}{x} + (1-x) \frac{\varphi(x/q)-\varphi(x)}{x},$$ we see that $L$ preserves the polynomials and moreover that $L$ also preserves the degree. Note that $L$ of is slightly different from the $L$ in since there is a $q$-shift in the argument.
Letting $\C_N[x]$ be the polynomials of degree less than or equal to $N$, we see that $L\colon \C_N[x]\to \C_N[x]$ and moreover that $L$ is lower-triangular with respect to the basis $\{x^k \mid 0\leq k\leq N\}$. Then $$Lx^n = \bigr(-ab (q^n-1) + (1-q^{-n})\bigr) x^n + \text{l.o.t.} =
(1-abq^{n}) (1-q^{-n}) x^n + \text{l.o.t.}$$ where $\text{l.o.t.}$ means ‘lower order terms’.
\[lem:lqJac-poleigfunctions\] For each degree $n$, the operator $L$ as in has a polynomial eigenfunction of degree $n$ with eigenvalue $(1-abq^{n}) (1-q^{-n})$.
A lower triangular operator has its eigenvalues on the diagonal. Since the eigenvalues are different for different $n$, all eigenvalues have algebraic and geometric multiplicity equal to $1$.
From the basic hypergeometric difference equation and Proposition \[prop:solofBqDEat0atinfty\] we see that the polynomial eigenfunction is $${\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{q^{-n}, abq^n}{c}
\ ;q,qx \right)}.$$
Consider the Hilbert space $\ell^2(\N,w)$ of weighted $\ell^2$-sequences with inner product $$\langle u, v\rangle = \sum_{k=0}^\infty u_k \overline{v_k} w_k$$ for some positive sequence $w=(w_k)_{k\in \N}$.
\[prop:Lsaonell2w\] Take $0<q<1$, $ab, c\in\R$, then with $$w_k = c^k \frac{(abq/c;q)_k}{(q;q)_k}, \qquad c>0, abq<c,$$ the operator $L$ with $D(L) = \{ u=(u_k)_{k\in\N} \mid u_k\not=0 \text{ for at most finitely many } k\in\N\}$ is symmetric; $$\langle Lu, v\rangle = \langle u, Lv\rangle, \qquad \forall \, u, v\in D(L).$$
We consider for finite sequences $(u)_k$, $(v)_k$ the difference of the inner products. Note that in particular all sums are finite, so that absolute convergence of all series involved is automatic.
We do the calculation slightly more general by not making any assumptions on the sequences $(u)_k$, $(v)_k$, but by chopping off the inner product. Denote $\langle u, v\rangle_N = \sum_{k=0}^N u_k \overline{v_k} w_k$ then obviously $\lim_{N\to\infty} \langle u, v\rangle_N = \langle u, v\rangle$ for any two sequences $(u)_k$, $(v)_k \in \ell^2(\N,w)$. Now $$\begin{gathered}
\langle Lu, v\rangle_N
- \langle u, Lv\rangle_N = (cq^{-1}-ab)(u_1-u_0)\overline{v_0}w_0 -
u_0 \overline{(cq^{-1}-ab)(v_1-v_0)}w_0 +
\\
\sum_{k=1}^N \Bigl( (cq^{-k-1}-ab)\, (u_{k+1}-u_k) +\,
(q^{-k}-1)\, (u_{k-1}-u_k)\Bigr) \overline{v_k} w_k - \\
\sum_{k=1}^N u_k \overline{\Bigl( (cq^{-k-1}-ab)\, (v_{k+1}-v_k) +\,
(q^{-k}-1)\, (v_{k-1}-v_k)\Bigr)} w_k = \\
(cq^{-1}-ab)w_0\bigl( u_1\overline{v_0} - u_0\overline{v_1}\bigr) +
\sum_{k=1}^N \Bigl( (cq^{-k-1}-ab)\, u_{k+1} +\,
(q^{-k}-1)\, u_{k-1}\Bigr) \overline{v_k} w_k - \\
\sum_{k=1}^N u_k \overline{\Bigl( (cq^{-k-1}-ab)\, v_{k+1} +
(q^{-k}-1)\, v_{k-1}\Bigr)} w_k\end{gathered}$$ since the coefficients are real. Relabeling gives $$\begin{gathered}
\langle Lu, v\rangle_N
- \langle u, Lv\rangle_N = (cq^{-1}-ab)w_0\bigl( u_1\overline{v_0} - u_0\overline{v_1}\bigr) + \\
\sum_{k=2}^{N+1} u_k (cq^{-k}-ab)\overline{v_{k-1}} \frac{w_{k-1}}{w_k} w_k -
\sum_{k=1}^N u_k
(q^{-k}-1)\, \overline{v_{k-1}} w_k \\
+ \sum_{k=0}^{N-1}
u_{k} (q^{-1-k}-1)\, \overline{v_{k+1}} \frac{w_{k+1}}{w_k} w_k -
\sum_{k=1}^N u_k(cq^{-k-1}-ab)\, \overline{v_{k+1}} w_k.\end{gathered}$$ Since we want $L$ to be symmetric, most of the terms have to cancel for all sequences $(u)_k$, $(v)_k$. So we need to impose $$\label{eq:littleqJac-weight}
\begin{split}
(cq^{-k}-ab) \frac{w_{k-1}}{w_k} =
(q^{-k}-1),\qquad
(q^{-1-k}-1) \frac{w_{k+1}}{w_k} =
(cq^{-k-1}-ab)
\end{split}$$ which give the same recurrence relation for $w_k$; $$w_k = \frac{(cq^{-k}-ab)}{(q^{-k}-1)} w_{k-1} =
c \frac{(1-abq^k/c)}{(1-q^k)} w_{k-1} = c^k \frac{(abq/c;q)_k}{(q;q)_k} w_0.
$$ Since we need $w_k>0$, we require $c>0$ and $abq/c<1$.
Taking this value for $w_k$ we see that most of the terms in the sum cancels, and we get $$\begin{gathered}
\langle Lu, v\rangle_N
- \langle u, Lv\rangle_N =
(cq^{-1}-ab)w_0\bigl( u_1\overline{v_0} - u_0\overline{v_1}\bigr) + \\
u_{N+1} (q^{-N-1}-1) \overline{v_N} w_{N+1} - u_1(q^{-1}-1) \overline{v_0} w_1 \\
+ u_0(cq^{-1}-ab)\overline{v_1}w_0
- u_N(cq^{-N-1}-ab)\, \overline{v_{N+1}} w_{N} \\
= u_{N+1} (q^{-N-1}-1) \overline{v_N} w_{N+1} - u_N(cq^{-N-1}-ab)\, \overline{v_{N+1}} w_{N}.
$$ Using one obtains $$\label{eq:LuvN-uLvN-lqJac}
\langle Lu, v\rangle_N
- \langle u, Lv\rangle_N =
\sqrt{(q^{-N-1}-1)(cq^{-N-1}-ab)} \sqrt{w_Nw_{N+1}} \bigl(
u_{N+1} \overline{v_N} - u_N \, \overline{v_{N+1}}\bigr)$$ In particular, for finitely supported sequences $(u)_k$, $(v)_k$ we have $\langle Lu, v\rangle = \langle u, Lv\rangle$ by taking $N\gg0$, so that $L$ is symmetric with respect to the domain of the finitely supported sequences in $\ell^2(\N,w)$.
\[rmk:spectralLlittleqJacobi\] We do not study the spectral analysis of $L$, but see §\[ssec:AlSalamChihara-littleqJacobi\]. In the analysis the form in is closely related to the study of self-adjoint extensions of $L$.
Let us now assume that $u$ and $v$ correspond to the polynomial eigenvalues of $L$ as in Lemma \[lem:lqJac-poleigfunctions\]. So this means $u_k = A + Bq^k + \cO(q^{2k})$, $v_k = C + Dq^k + \cO(q^{2k})$ as $k\to \infty$. For these values we see that gives $$\begin{gathered}
\langle Lu, v\rangle_N
- \langle u, Lv\rangle_N = \\
\sqrt{(q^{-N-1}-1)(cq^{-N-1}-ab)} \sqrt{w_Nw_{N+1}} \Bigl(
\bigl(A + Bq^{N+1} + \cO(q^{2N})\bigr)
\bigl( \bar C + \bar Dq^N + \cO(q^{2N}) \bigr) - \\
\bigl(A + Bq^{N} + \cO(q^{2N})\bigr)
\bigl( \bar C + \bar Dq^{N+1} + \cO(q^{2N}) \bigr)
\Bigr) = \\
\sqrt{(q^{-N-1}-1)(cq^{-N-1}-ab)} \sqrt{w_Nw_{N+1}} \Bigl( (B\bar C q +A\bar D) q^N
- (B\bar C + A \bar D q) q^N + \cO(q^{2N}) \Bigr) = \\
\sqrt{(q^{-1}-q^N)(cq^{-1}-abq^N)} \sqrt{w_Nw_{N+1}}
\Bigl( (B\bar C - A\bar D) (q-1) + \cO(q^{N}) \Bigr).\end{gathered}$$ In particular, since we want to $\lim_{N\to\infty} \langle Lu, v\rangle_N
- \langle u, Lv\rangle_N = 0$ for such sequences $(u_k)$ and $(u_k)$ and since $w_N = \cO(c^N)$, we need $c<1$. Assuming $0<c<1$ and $abq<c$ we see that any polynomial $p$ gives a sequence $(p(q^k))_k \in \ell^2(\N,w)$.
The little $q$-Jacobi polynomials
---------------------------------
In order to get to the little $q$-Jacobi polynomials we relabel. We (re-)define the polynomials from Lemma \[lem:lqJac-poleigfunctions\] as the little $q$-Jacobi polynomials; $$\label{eq:deflqJacpols}
p_n(x;\al,\be;q) = {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{q^{-n},\al\be q^{n+1}}{\al q}
\ ;q, qx \right)}.$$ So we have specialized $(a,b,c)$ to $(q^{-n}, \al\be q^{n+1}, \al q)$ and the conditions $0<c<1$, $abq<c$ translate into $0<\al<q^{-1}$, $\be<q^{-1}$. By we have $$\label{eq:DqonlittleqJac}
\begin{split}
\bigl( {D}_qp_n(\cdot;\al,\be;q)\bigr)(x) &= \frac{(1-q^{-n})(1-\al\be q^{n+1})}{(1-c)(1-q)}
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{q^{1-n},\al\be q^{n+2}}{\al q^2}
\ ;q, qx \right)} \\
&=
\frac{(1-q^{-n})(1-\al\be q^{n+1})}{(1-c)(1-q)} p_{n-1}(x;\al q,\be q;q)
\end{split}$$ Let us also rename the weight of Proposition \[prop:Lsaonell2w\] to the new labeling. We get $$w(\al,\be;q)_k = (\al q)^k \frac{(\be q;q)_k}{(q;q)_k}$$ and we denote the corresponding Hilbert space $\ell^2(\N,w)$ by $\ell^2(\N;\al,\be;q)$. A polynomial sequences in $\ell^2(\N;\al,\be;q)$ is a sequence of the form $u_k= p(q^k)$ for some polynomial $p$. Note that these sequences are indeed in $\ell^2(\N;\al,\be;q)$, and we denote them by $\cP$.
\[lem:adjointDq-lqJacpol\] $\tilde{D}_q$ is an unbounded map from $\ell^2(\N;\al,\be;q)$ to $\ell^2(\N;\al q,\be q;q)$. As its domain we take the polynomial sequences $\cP$. Then we have $$\langle \tilde{D}_q p, r \rangle_{\ell^2(\N;\al q,\be q;q)}
= \langle p, S^{\al,\be} r \rangle_{\ell^2(\N;\al,\be;q)}, \qquad \forall \, p,r\in\cP$$ where $$S^{\al,\be} \colon \cP \to \cP, \qquad
\bigl( S^{\al,\be} r \bigr)_k =
(\al q)^{-1} \frac{(1-q^k)}{(1-\be q)} r(q^{k-1}) -
\frac{(1- \be q^{k+1})}{(1-\be q)}r(q^k)$$ and $-\al q (1-\be q)S^{\al,\be}$ corresponds to $S^{a,b,c}$ as in with $a=q^{-n}$, $b=\al\be q^{n+1}$, $c=\al q$ and $z=q^{k+1}$.
Note that $$\begin{gathered}
\langle \tilde{D}_q p, r \rangle_{\ell^2(\N;\al q,\be q;q)}
= \sum_{k=0}^\infty \frac{p(x^{k+1})-p(q^k)}{q^k} \overline{r(q^k)}
(\al q^2)^k \frac{(\be q^2;q)_k}{(q;q)_k} \\
= \sum_{k=1}^\infty p(x^{k}) \overline{r(q^{k-1})}
(\al q)^{k-1} \frac{(\be q^2;q)_{k-1}}{(q;q)_{k-1}} - \sum_{k=0}^\infty p(x^{k}) \overline{r(q^k)}
(\al q)^k \frac{(\be q^2;q)_k}{(q;q)_k} \\
= - p(1)\overline{r(1)} +
\sum_{k=1}^\infty p(x^{k}) \left( \overline{r(q^{k-1})}
(\al q)^{k-1} \frac{(\be q^2;q)_{k-1}}{(q;q)_{k-1}} - \overline{r(q^k)}
(\al q)^k \frac{(\be q^2;q)_k}{(q;q)_k}\right) \\
= - p(1)\overline{r(1)} +
\sum_{k=1}^\infty p(x^{k}) \left( \overline{r(q^{k-1})}
(\al q)^{-1} \frac{(1-q^k)}{(1-\be q)} - \overline{r(q^k)}
\frac{(1- \be q^{k+1})}{(1-\be q)}\right) (\al q)^k \frac{(\be q;q)_k}{(q;q)_k}.\end{gathered}$$ In this derivation we use that all sums converge absolutely, so that we can split the series and rearrange them. This calculation gives, using that $\al,\be, q\in \R$, $$\bigl( S^{\al,\be} r \bigr)_k =
(\al q)^{-1} \frac{(1-q^k)}{(1-\be q)} r(q^{k-1}) -
\frac{(1- \be q^{k+1})}{(1-\be q)}r(q^k)$$ which is again in $\cP$ since $r\in \cP$. Note that the formula is also valid for $k=0$.
We leave the fact that $\tilde{D}_q$ is unbounded to the reader.
Note that $S^{\al,\be}$ raises the degree of the polynomial by $1$. We have already observed how $\tilde{D}_q$ acts on a little $q$-Jacobi polynomial, but we also want to find out for $S^{\al,\be}$.
\[prop:adjointDq-lqJacpol2\] The little $q$-Jacobi polynomials are orthogonal in $\ell^2(\N;\al,\be;q)$. Moreover, $$\begin{gathered}
\bigl( \tilde{D}_qp_n(\cdot;\al,\be;q)\bigr)(x) =
\frac{(1-q^{-n})(1-\al\be q^{n+1})}{(1-c)} p_{n-1}(x;\al q,\be q;q), \\
\bigl( S^{\al,\be} p_{n-1}(\cdot;\al q,\be q;q)\bigr)(x)
= \frac{1}{\al q} \frac{1-\al q}{1-\be q} p_n(x;\al,\be;q)\end{gathered}$$
Note that the second order difference operator has the little $q$-Jacobi polynomial as eigenfunction, see Lemma \[lem:lqJac-poleigfunctions\]. In the relabeling the operator $L$ is given by $$\bigl(L^{\al,\be} f\bigr)(x) = \bigl(Lf\bigr)(x) = \al(1- \be q x) \frac{f(qx)-f(x)}{x}
+ (1-x) \frac{f(x/q)-f(x)}{x}$$ see with $(a,b,c,x)$ replaced by $(q^{-n}, \al\be q^{n+1}, \al q, qx)$. Since the decomposition of $L^{\al,\be}$ of corresponds, up to a scalar, with the operator $S^{\al,\be}$ we see that $\bigl( S^{\al,\be} p_{n-1}(\cdot;\al q,\be q;q)\bigr)(x)$ has to be a multiple of $p_n(x;\al,\be;q)$. By considering the evaluation at $0$, and using $p_n(0;\al,\be;q)=1$, we see that the multiple is $$\frac{(1/\al q-1)}{1-\be q} = \frac{1}{\al q} \frac{1-\al q}{1-\be q}$$ since we can write for a polynomial $r$ $$\bigl( S^{\al,\be} r\bigr) (x) = \frac{1}{\al q} \frac{(1-x)}{(1-\be q)}r(x/q) -
\frac{(1-\be q x)}{(1-\be q)}r(x).$$
In order to show the orthogonality, we consider the following inner product for $k\leq n$, using the raising and lowering operators $S^{\al,\be}$ and $\tilde{D}_q$; $$\begin{gathered}
\langle x^k, p_n(\cdot;\al,\be;q) \rangle_{\ell^2(\N;\al,\be;q)} = \al q \frac{(1-\be q)}{(1-\al q)}
\langle x^k, S^{\al,\be} p_{n-1}(\cdot;\al q,\be q;q) \rangle_{\ell^2(\N;\al q,\be q;q)} \\
= (q^k-1) \al q \frac{(1-\be q)}{(1-\al q)}
\langle x^{k-1}, p_{n-1}(\cdot;\al q,\be q;q) \rangle_{\ell^2(\N;\al q,\be q;q)} \end{gathered}$$ since $\tilde{D}_q x^k = (q^k-1) x^{k-1}$. In particular, we get $0$ for the inner product in case $k=0$. By iterating the procedure we get $$\begin{gathered}
\langle x^k, p_n(\cdot;\al,\be;q) \rangle_{\ell^2(\N;\al,\be;q)} = \al q \frac{(1-\be q)}{(1-\al q)}
\langle x^k, S^{\al,\be} p_{n-1}(\cdot;\al q,\be q;q) \rangle_{\ell^2(\N;\al q,\be q;q)} \\
= (-1)^p (q^{k-p+1};q)_p
\al^p q^{\frac12 p(p+1)}
\frac{(\be q;q)_p}{(\al q;q)_p}
\langle x^{k-p}, p_{n-p}(\cdot;\al q^p,\be q^p;q) \rangle_{\ell^2(\N;\al q^p,\be q^p;q)} \end{gathered}$$ and for $k<n$ this gives zero for $p=k+1\leq n$. Hence, the little $q$-Jacobi polynomial $p_n(\cdot;\al,\be;q)$ of degree $n$ is orthogonal to all monomials of degree $<n$. So the the little $q$-Jacobi polynomials are orthogonal.
Note that as an immediate corollary to the proof, we also obtain $$\label{eq:squarednormlqJacopols}
\begin{split}
&\langle p_n(\cdot;\al,\be;q), p_n(\cdot;\al,\be;q) \rangle_{\ell^2(\N;\al,\be;q)} =
\text{lc}\bigl(p_n(\cdot;\al,\be;q)\bigr)
\langle x^n , p_n(\cdot;\al,\be;q) \rangle_{\ell^2(\N;\al,\be;q)} \\
= &
\text{lc}\bigl(p_n(\cdot;\al,\be;q)\bigr)
(-1)^n (q^;q)_n
\al^n q^{\frac12 n(n+1)}
\frac{(\be q;q)_n}{(\al q;q)_n}
\langle 1, 1 \rangle_{\ell^2(\N;\al q^n,\be q^n;q)},
\end{split}$$ where $\text{lc}(p)$ denotes the leading coefficient of the polynomial $p$. So the shift operators can be used to find the squared norm of the orthogonal polynomials in terms of the squared norm of the constant function $1$.
The leading coefficient can be calculated directly from the definition ; $$\begin{gathered}
\text{lc}\bigl(p_n(\cdot;\al,\be;q)\bigr) =
\frac{ (q^{-n}, \al \be q^{n+1};q)_n}{(q,\al q;q)_n}q^n =
(-1)^n q^{-\frac12 n(n-1)} \frac{ (\al \be q^{n+1};q)_n}{(\al q;q)_n}. \end{gathered}$$ The squared norm of the constant function $1$ follows from the $q$-binomial sum ; $$\begin{gathered}
\langle 1, 1 \rangle_{\ell^2(\N;\al,\be;q)} =
\sum_{k=0}^\infty (\al q)^k \frac{(\be q;q)_k}{(q;q)_k} =
{\,_{1}\vp_{0} \left( \genfrac{.}{.}{0pt}{}{\be q}{-}
\ ;q, \al q \right)} = \frac{(\al \be q^2;q)_\infty}{(\al q;q)_\infty}.\end{gathered}$$
This gives the following orthogonality relations for the little $q$-Jacobi polynomials from the analysis of the hypergeometric $q$-difference operator on a very specific set of points.
\[thm:orthlittleqJac\] Let $0<\al<q^{-1}$, $\be<q^{-1}$ and consider the little $q$-Jacobi polynomials $$p_n(x;\al,\be;q) = {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{q^{-n},\al\be q^{n+1}}{\al q}
\ ;q, qx \right)}$$ and the inner product space $$\langle f, g \rangle_{\ell^2(\N;\al,\be;q)} = \sum_{k=0}^\infty f(q^k)\overline{g(q^k)}
(\al q)^k \frac{(\be q;q)_k}{(q;q)_k}$$ then the little $q$-Jacobi polynomials satisfy $$\begin{gathered}
\langle p_n(\cdot;\al,\be;q) , p_m(\cdot;\al,\be;q) \rangle_{\ell^2(\N;\al,\be;q)} =
\de_{m,n} h_n(\al,\be;q) \\
h_n(\al,\be;q) = (\al q)^n \frac{(q, \be q;q)_n}{(\al q, \al\be q;q)_n}
\frac{1-\al\be q}{1-\al\be q^{2n+1}}
\frac{(\al\be q^2;q)_\infty}{(\al q;q)_\infty}.\end{gathered}$$
This is a combination of the results in this section, and we are left with calculating the squared norm. Now $$\begin{gathered}
h_n(\al,\be;q) = \text{lc}\bigl(p_n(\cdot;\al,\be;q)\bigr)
(-1)^n (q^;q)_n
\al^n q^{\frac12 n(n+1)}
\frac{(\be q;q)_n}{(\al q;q)_n}
\langle 1, 1 \rangle_{\ell^2(\N;\al q^n,\be q^n;q)} \\
= (-1)^n q^{-\frac12 n(n-1)} \frac{ (\al \be q^{n+1};q)_n}{(\al q;q)_n}
(-1)^n (q^;q)_n
\al^n q^{\frac12 n(n+1)}
\frac{(\be q;q)_n}{(\al q;q)_n}
\frac{(\al \be q^{2n+2};q)_\infty}{(\al q^{n+1};q)_\infty} \\
= \frac{(q, \be q;q)_n}{(\al q, \al\be q;q)_n}
\frac{(\al q)^n}{1-\al\be q^{2n+1}}
\frac{(\al\be q;q)_\infty}{(\al q;q)_\infty} \end{gathered}$$ which is equal to the stated value.
The three-term recurrence relation for the little $q$-Jacobi polynomials
------------------------------------------------------------------------
The shift operators $\tilde{D}_q$ and $S^{\al,\be}$ play an essential role in the derivation of the orthogonality relations for the little $q$-Jacobi polynomials in Theorem \[thm:orthlittleqJac\]. As a final application we show how one can obtain the coefficients in the three-term recurrence relation for the monic little $q$-Jacobi polynomials. Let $\tilde{p}_n(x;\al,\be;q)$ be the monic little $q$-Jacobi polynomials. Because of the monicity we have a three-term recurrence of the form $$x\, \tilde{p}_n(x;\al,\be;q) = \tilde{p}_{n+1}(x;\al,\be;q) + b_n
\tilde{p}_n(x;\al,\be;q) + c_n \tilde{p}_{n-1}(x;\al,\be;q).$$ The value of $c_n$ can then be calculated from the knowledge we already have obtained; $$\begin{gathered}
c_n \langle \tilde{p}_{n-1}(x;\al,\be;q), \tilde{p}_{n-1}(x;\al,\be;q)\rangle_{\ell^2(\N;\al,\be;q)}
= \langle x\, \tilde{p}_n(x;\al,\be;q), \tilde{p}_{n-1}(x;\al,\be;q)\rangle_{\ell^2(\N;\al,\be;q)} \\
= \langle \tilde{p}_n(x;\al,\be;q), x \tilde{p}_{n-1}(x;\al,\be;q)\rangle_{\ell^2(\N;\al,\be;q)}
= \langle \tilde{p}_n(x;\al,\be;q), \tilde{p}_{n}(x;\al,\be;q)\rangle_{\ell^2(\N;\al,\be;q)}\end{gathered}$$ using that multiplication by $x$ is self-adjoint. So we find $$\begin{gathered}
c_n = \frac{\langle \tilde{p}_n(x;\al,\be;q), \tilde{p}_{n}(x;\al,\be;q)\rangle_{\ell^2(\N;\al,\be;q)}}
{\langle \tilde{p}_{n-1}(x;\al,\be;q), \tilde{p}_{n-1}(x;\al,\be;q)\rangle_{\ell^2(\N;\al,\be;q)}}
= \left( \frac{\text{lc}\bigl(p_{n-1}(\cdot;\al,\be;q)\bigr)}{\text{lc}\bigl(p_{n}(\cdot;\al,\be;q)\bigr)}
\right)^2 \frac{h_n(\al,\be;q)}{h_{n-1}(\al,\be;q)}\\
= \left( -q^{n-1} \frac{(1-\al\be q^n)(1-\al q^n)}{(1-\al\be q^{2n-1})(1-\al\be q^{2n})}\right)^2
\al q \frac{(1-q^n)(1-\be q^n)(1-\al\be q^{2n-1})}{(1-\al q^n)(1-\al\be q^n)(1-\al\be q^{2n+1})} \\
= \al q^{2n-1} \frac{(1-q^n)(1-\al q^n)(1-\be q^n)(1-\al\be q^n)}{(1-\al\be q^{2n-1})
(1-\al\be q^{2n})^2(1-\al\be q^{2n+1})}.
\end{gathered}$$ Writing $\tilde{p}_n(x;\al,\be;q) = x^n + r_n(\al,\be)x^{n-1} + \text{l.o.t.}$, it follows that upon comparing coefficients of $x^{n}$ in the three-term recurrence relation that $$r_n(\al,\be) = r_{n+1}(\al,\be) + b_n \qquad \Longrightarrow \qquad b_n =
r_n(\al,\be) - r_{n+1}(\al,\be).$$ We could read it off from the explicit expression , but we use the shift operators to find the values. Indeed, since, by Proposition \[prop:adjointDq-lqJacpol2\], $$\begin{gathered}
\bigl( \tilde{D}_q \tilde{p}_n(\cdot;\al,\be;q)\bigr)(x)
= (q^n-1) \, \tilde{p}_{n-1}(x;\al q,\be q;q) \quad \Longrightarrow \\
(q^{n-1}-1) r_n(\al,\be) = (q^n-1) r_{n-1}(\al q,\be q)
\quad \Longrightarrow \quad r_n(\al,\be) = \frac{(1-q^n)}{(1-q^{n-1})} r_{n-1}(\al q,\be q)
\quad \Longrightarrow \\
r_n(\al,\be) = \frac{(1-q^n)}{(1-q^{n-p})} r_{n-p}(\al q^p,\be q^p) =
\frac{(1-q^n)}{(1-q)} r_{1}(\al q^{n-1},\be q^{n-1}).\end{gathered}$$ In order to determine $r_1(\al,\be)$ we use the shift operator as well. By Proposition \[prop:adjointDq-lqJacpol2\] we have that $p_1(x;\al,\be;q)$ is a multiple of $$\begin{gathered}
\bigr( S^{\al,\be}1\bigr)(x) = \frac{1}{\al q} \frac{(1-x)}{(1-\be q)} -
\frac{(1-\be q x)}{(1-\be q)} = \frac{\be q -1/\al q}{(1-\be q)} x
+ \frac{1/\al q -1}{(1-\be q)} =
\frac{\be q -1/\al q}{(1-\be q)} \left( x + \frac{1/\al q -1}{\be q -1/\al q}\right) \\
\Longrightarrow \qquad r_1(\al,\be) = \frac{1/\al q -1}{\be q -1/\al q}
= - \frac{1-\al q}{1-\al\be q^2}\end{gathered}$$
\[prop:3trmoniclqJacpol\] The monic little $q$-Jacobi polynomials $\tilde{p}_n(x;\al,\be;q)$ satisfy the three-term recurrence relation $$\begin{gathered}
x\, \tilde{p}_n(x;\al,\be;q) = \tilde{p}_{n+1}(x;\al,\be;q) + b_n
\tilde{p}_n(x;\al,\be;q) + c_n \tilde{p}_{n-1}(x;\al,\be;q), \\
b_n = \frac{q^n\left( (1+\al) -\al(1+\be)(1+q)q^n + \al\be q(1+\al)q^{2n}\right)}
{(1-\al\be q^{2n})(1-\al\be q^{2n+2})}
\\
c_n= \al q^{2n-1} \frac{(1-q^n)(1-\al q^n)(1-\be q^n)(1-\al\be q^n)}{(1-\al\be q^{2n-1})
(1-\al\be q^{2n})^2(1-\al\be q^{2n+1})}.\end{gathered}$$
The value for $b_n$ is seemingly different from the classical value given in e.g. [@KoekS (3.12.4)]. We leave this to Exercise \[ex:bnrightvalue\].
We have already established the value for $c_n$. It remains to finish the calculation of $b_n$. This is done as follows; $$\begin{gathered}
b_n = r_n(\al,\be) - r_{n+1}(\al,\be) =
\frac{(1-q^n)}{(1-q)} r_{1}(\al q^{n-1},\be q^{n-1}) -
\frac{(1-q^{n+1})}{(1-q)} r_{1}(\al q^{n},\be q^{n}) \\
= - \frac{(1-q^n)(1-\al q^n)}{(1-q)(1-\al\be q^{2n})} +
\frac{(1-q^{n+1})(1-\al q^{n+1})}{(1-q)(1-\al\be q^{2n+2})}\end{gathered}$$ By working this out we get $$\begin{gathered}
b_n
= \frac{q^n}{(1-\al\be q^{2n})(1-\al\be q^{2n+2})}
\left( (1+\al) -\al(1+\be)(1+q)q^n + \al\be q(1+\al)q^{2n}\right)\end{gathered}$$
Note that the value $$r_n(\al,\be) = - \frac{(1-q^n)(1-\al q^n)}{(1-q)(1-\al\be q^{2n})}$$ also corresponds with taking into account division by the leading coefficient.
Let us view the operator $L=L^{\al,\be}$ as an operator acting on polynomials as well as the operator $M$ which is acting by multiplication. They can be viewed as generators (up to an affine scaling) of a limit of the Zhedanov algebra, also known as the Askey-Wilson algebra. We refer to [@KoorM] for the precise formulation and related references. Moreover, the Zhedanov algebra as well as its degenerations in [@KoorM] have relations that can be interpreted as non-homogeneous Serre relations in quantum algebras, and this type of relations hold for generators of quantum symmetric pairs as studied by Gail Letzter and coworkers, see e.g. [@Kolb §5.3]. It is not clear what the connection entails.
Relation to Al-Salam–Chihara polynomials {#ssec:AlSalamChihara-littleqJacobi}
----------------------------------------
We have circumvented the precise analytic study of the basic $q$-difference equation for the little $q$-Jacobi polynomials or the basic hypergeometric series. The reason is that this analytic study is somewhat complicated since the self-adjoint extension of the symmetric operator as in Lemma \[lem:adjointDq-lqJacpol\] depend in general on parameters. Indeed, $(L, D((L))$ as in Lemma \[lem:adjointDq-lqJacpol\] is not essentially self-adjoint in general. We explain this in this section by relating to a non-determinate moment problem.
We can also relate the eigenvalue equation to orthogonal polynomials. Indeed, rewriting $Lu=\la u$ gives $$\begin{gathered}
\la u_k(\la) = (cq^{-k-1}-ab)\, (u_{k+1}(\la)-u_k(\la)) +\,
(q^{-k}-1)\, (u_{k-1}(\la)-u_k(\la))\end{gathered}$$ which we can consider as a three-term recurrence for orthogonal polynomials with initial values $u_0(\la)=0$ (and $u_{-1}(\la)=0$). In order to determine these polynomials, we first look at the monic version. So we put $u_k(\la) = \al_k r_k(\la)$ with $$\frac{\al_{k+1}}{\al_k} (cq^{-k-1}-ab) = 1$$ so that the recurrence relation becomes $$\begin{gathered}
\la r_k(\la) = \, r_{k+1}(\la)- \bigl((cq^{-k-1}-ab)+ (q^{-k}-1)\bigr) r_k(\la) +\,
(q^{-k}-1)(cq^{-k}-ab)\, r_{k-1}(\la)\end{gathered}$$ and putting $2\mu=\al(\la-ab-1)$, $p_k(\mu) = \al^{-k} r_k(\al(2\mu-ab-1))$ we get $$\begin{gathered}
2\mu p_k(\mu) = p_{k+1}(\mu)- \frac{1}{\al} (cq^{-k-1}+ q^{-k}) p_k(\mu) +\,
\frac{ab}{\al^2}(1-q^{-k})(1-cq^{-k}/ab)\, p_{k-1}(\mu)\end{gathered}$$ and finally taking $\al = - \sqrt{ab}$ we find $$\label{eq:relAlSalChih}
2\mu p_k(\mu) = p_{k+1}(\mu) + q^{-k}(c/q\sqrt{ab}+ 1/\sqrt{ab}) p_k(\mu) +\,
(1-q^{-k})(1-cq^{-k}/ab)\, p_{k-1}(\mu).$$ Now can be matched to [@KoekS §3.8], so that $p_k(\mu)$ can be identified with the Al-Salam–Chihara polynomials $Q_k(\mu; c/q\sqrt{ab}, 1/\sqrt{ab}|q^{-1})$ in base $q^{-1}>1$.
Theorem \[thm:AlSalChih-indeterminate\] gives a characterization of the (in-)determinacy of the Al-Salam–Chihara polynomials in case the base is bigger than $1$, and it is due to Askey and Ismail [@AskeI Thm. 3.2, p. 36].
\[thm:AlSalChih-indeterminate\] Consider the sequence of monic polynomials $$xv_n(x) = v_{n+1}(x) + A q^{-n} + (1-q^{-n}) (C-Bq^{-n}) v_{n-1}(x)$$ with $0<q<1$, $B\geq 0$, $B> C$ and initial conditions $v_{-1}(x)=0$, $v_0(x)=1$. Then the corresponding moment problem is indeterminate if and only if $$A^2>4B, \quad \text{and} \quad q\geq |\be^2 B|$$ where $(1-At+Bt^2) = (1-t/\al)(1-t/\be)$ with $|\al|\geq |\be|$.
The conditions $0<q<1$, $B\geq 0$, $B> C$ in Theorem \[thm:AlSalChih-indeterminate\] ensure that the conditions of Favard’s theorem, see e.g. [@Chih], [@Deif], [@DunfS], [@Isma], [@Koel-Laredo], are met. So there is an orthogonality measure for which the polynomials $v_n(x)$ are orthogonal. In the determinate case this measure is uniquely determined by the polynomials, whereas in the indeterminate case there are infinitely many orthogonality measures for these polynomials. In the indeterminate case this means that the operator $L$ with domain $D(L)$ the finite linear combinations as in Proposition \[prop:Lsaonell2w\] is not essentially self-adjoint, see e.g. [@DunfS], [@Koel-Laredo], [@Simo].
The proof of Theorem \[thm:AlSalChih-indeterminate\] follows by observing that a moment problem is indeterminate if and only if $\sum_{n=0}^\infty |p_n(i)|^2<\infty$ for the corresponding orthonormal polynomials $p_n(x)$, see e.g. [@Akhi]. Askey and Ismail then determine the asymptotic behaviour of the Al-Salam–Chihara polynomials by applying Darboux’s method, see e.g. [@Olve], to the generating function for the Al-Salam–Chihara polynomials.
Comparing Theorem \[thm:AlSalChih-indeterminate\] with we see that we can apply Theorem \[thm:AlSalChih-indeterminate\] with $(A,B,C)=(c/q\sqrt{ab} + 1/\sqrt{ab}, c/abq,1)$ and the same base $q$. So the requirement for Favard’s theorem translates to $c/abq>1$ or $c>abq$, which we now assume. Then the first condition $A^2>4B$ translates to $$\left( \frac{c}{q\sqrt{ab}} + \frac{1}{\sqrt{ab}}\right)^2 > \frac{4c}{abq}
\ \Longleftrightarrow \ \left( \frac{c}{q\sqrt{ab}} - \frac{1}{\sqrt{ab}}\right)^2 >0,$$ which is always true unless $c=q$. For the second condition we factorise $$1-At+Bt^2= 1- (c/q\sqrt{ab} + 1/\sqrt{ab})t + \frac{c}{abq}t^2 =
(1-\frac{ct}{q\sqrt{ab}})(1-\frac{t}{\sqrt{ab}})$$ so that $\{\al,\be\} =\{\frac{q\sqrt{ab}}{c}, \sqrt{ab}\}$. So $\be= \sqrt{ab}$ if $c\leq q$, and then $q\geq |\be^2B|$ is equivalent to $c\leq q^2$. Next $\be=\frac{q\sqrt{ab}}{c}$ if $c\geq q$ and then $q\geq |\be^2B|$ is equivalent to $q\geq 1$. We conclude that $L$ is not essentially self-adjoint if $0<c<q$.
Exercises
---------
1. Show that $\tilde{D}_q$ as in Lemma \[lem:adjointDq-lqJacpol\] is unbounded.
2. \[ex:bnrightvalue\] Look up the standard value for $b_n$ as in Proposition \[prop:3trmoniclqJacpol\] and establish the equality with the value as given in Proposition \[prop:3trmoniclqJacpol\].
3. Show that the three-term recurrence relation for the litle $q$-Jacobi polynomials as in Proposition \[prop:3trmoniclqJacpol\] gives a relation for the Al-Salam–Chihara polynomials which is related to the $q$-difference operator for the Al-Salam–Chihara polynomials.
Notes {#notes-2 .unnumbered}
-----
The little $q$-Jacobi polynomials were introduced by Andrews and Askey [@AndrA] in 1977. The link to the quantum $SU(2)$-group as matrix elements of unitary representations by Vaksman & Soibelman, Koornwinder and Masuda, Mimachi, Nakagami, Noumi, and Ueno at the end of the 1980s has led to many results on (subclasses of) little $q$-Jacobi polynomials, see the references in the lecture notes [@Koor-CQGSF] by Koornwinder. The usage of the shift operators to obtain the explicit results is a technique that can be useful in other applications, such as multivariable setting or in the matrix-valued case, see also [@Koor-CQGSF] for this approach for little and big $q$-Jacobi polynomials. The duality between little $q$-Jacobi polynomials and Al-Salam–Chihara polynomials is observed by Rosengren [@Rose] and it is also observed by Groenevelt [@Groe-Ramanujan]. This duality –but in a dual way– also plays an important role in the study of the quantum analogue of the Laplace-Beltrami operator on bounded quantum symmetric domain, see Vaksman [@Vaks]. For the corresponding Zhedanov algebra, the duality is described in [@KoorM]. The duality can also be extended to big $q$-Jacobi polynomials and continuous dual $q^{-1}$-Hahn polynomials, see [@KoelS-CA]. In general, this duality is related to explicit solutions of explicit indeterminate moment problems, and several examples are known. A vector-valued analogue of [@KoelS-CA] is given by Groenevelt [@Groe-CA2009].
Basic hypergeometric $q$-difference equation: non-polynomial case {#sec:BHS-qdiff-nonpol}
=================================================================
We now consider the basic hypergeometric $q$-difference equation in a more general version. In the general version we cannot restrict naturally to a simple domain. We have to take all the general $q$-line $zq^\Z$ into account.
Doubly infinite Jacobi operators {#ssec:doublyinfiniteJacobioperators}
--------------------------------
In this section we briefly review the spectral analysis of a doubly infinite Jacobi operator, i.e. a three-term recurrence on the Hilbert space $\ell^2(\Z)$. This section requires some knowledge from functional analysis, in particular of symmetric, unbounded, and self-adjoint operators and the spectral theorem.
We consider an operator on the Hilbert space $\ell^2(\Z)$ of the form $$\label{eq:defLnonpolcase}
L\, e_k = a_k \, e_{k+1} + b_k\, e_k +
a_{k-1}\, e_{k-1}, \qquad
a_k> 0, \ b_k\in\R,$$ where $\{ e_k\}_{k\in\Z}$ is the standard orthonormal basis of $\ell^2(\Z)$. If $a_i=0$ for some $i\in\Z$, then $L$ splits as the direct sum of two Jacobi operators, so that we are essentially back to two three-term recurrence operators related to two sets of orthogonal polynomials. Recall that a three-term recurrence operator on $\ell^2(\N)$ is a Jacobi operator. The spectral analysis is closely related to the orthogonality of the corresponding orthogonal polynomials, and is essentially a proof of Favard’s theorem, see [@DunfS], [@Koel-Laredo], [@Simo]. So we will assume that $a_i\not=0$ for all $i\in \Z$. We call $L$ a Jacobi operator on $\ell^2(\Z)$ or a doubly infinite Jacobi operator.
The domain $\cD(L)$ of $L$ is the dense subspace $\cD(\Z)$ of finite linear combinations of the basis elements $e_k$, $k\in \Z$. This makes $L$ a densely defined symmetric operator.
We extend the action of $L$ to an arbitrary vector $v=\sum_{k=-\infty}^\infty v_ke_k\in\ell^2(\Z)$ by $$L^\ast \, v = \sum_{k=-\infty}^\infty (a_k \, v_{k+1}
+ b_k\, v_k + a_{k-1}\, v_{k-1})\, e_k,$$ which is not an element of $\ell^2(\Z)$ in general. Define $${\cD}^\ast = \{ v\in\ell^2(\Z)\mid L^\ast v\in \ell^2(\Z)\}.$$
\[lem:maxdomdiJacobi\] $(L^\ast, \cD^\ast)$ is the adjoint of $(L,\cD(\Z))$.
The proof of Lemma \[lem:maxdomdiJacobi\] requires a bit of Hilbert space theory, and we leave it to the Exercise \[ex:lem:maxdomdiJacobi\].
In particular, $L^\ast$ commutes with complex conjugation, so its deficiency indices are equal. Here the deficiency indices $(n_+,n_-)$ are defined as $$\begin{gathered}
n_+= \dim \ker (L^\ast - z) = \dim \ker (L^\ast - i), \qquad \Im z>0 \\
n_-= \dim \ker (L^\ast - z) = \dim \ker (L^\ast + i), \qquad \Im z<0\end{gathered}$$ since the dimension is constant in the upper and lower half plane. The solution space of $L^\ast v=z\, v$ is two-dimensional, since $v$ is completely determined by any initial data $(v_{n-1},v_n)$ for any fixed $n\in\Z$. So the deficiency indices are equal to $(i,i)$ with $i\in \{ 0,1,2\}$. From the general theory of self-adjoint operators, see [@DunfS], we have that $(L, \cD(L))$ has self-adjoint extensions since the deficiency indices $n_-=n_+$. In case $n_-=n_+=0$, the operator $(L^\ast, \cD^\ast)$ is self-adjoint, and this case will be generally assumed in this section.
### Relation to Jacobi operators
To the operator $L$ we associate two Jacobi operators $J^+$ and $J^-$ acting on $\ell^2(\N)$ with orthonormal basis denoted by $\{f_k\}_{k\in\N}$ in order to avoid confusion. Define $$\begin{split}
J^+\, f_k &= \begin{cases}
a_k \, f_{k+1} + b_k\, f_k + a_{k-1}\, f_{k-1},&
\text{for $k\geq 1$,} \\
a_0 \, f_1 + b_0\, f_0, & \text{for $k=0$,}\end{cases} \\
J^-\, f_k &= \begin{cases}
a_{-k-2}\, f_{k+1} + b_{-k-1}\, f_k + a_{-k-1}\, f_{k-1},&
\text{for $k\geq 1$,} \\
a_{-2} \, f_1 + b_{-1}\, f_0, & \text{for $k=0$,}\end{cases}
\end{split}$$ and extend by linearity to $\cD(\N)$, the space of finite linear combinations of the basis vectors $\{f_k\}_{k=0}^\infty$ of $\ell^2(\N)$. Then $J^{\pm}$ are densely defined symmetric operators with deficiency indices $(0,0)$ or $(1,1)$ corresponding to whether the associated Hamburger moment problems is determinate or indeterminate, see [@Akhi], [@BuchC], [@DunfS], [@Koel-Laredo], [@Simo]. The following theorem, due to Masson and Repka [@MassR], relates the deficiency indices of $L$ and $J^\pm$.
\[thm:MassonRepka\] The deficiency indices of $L$ are obtained by summing the deficiency indices of $J^+$ and the deficiency indices of $J^-$.
For the proof of Theorem \[thm:MassonRepka\] we refer to [@MassR], [@Koel-Laredo].
We define the Wronskian $[u,v]_k = a_k(u_{k+1}v_k-u_kv_{k+1})$. The Wronskian is also known as the Casorati determinant.
\[lem:WronskianLonl2Z\] The Wronskian $[u,v]=[u,v]_k$ is independent of $k$ for $L^\ast u=z\, u$, $L^\ast v=z\, v$. Moreover, $[u,v]\not=0$ if and only if $u$ and $v$ are linearly independent solutions.
The proof is straightforward, see Exercise \[ex:lem:WronskianLonl2Z\].
Now using Lemma \[lem:WronskianLonl2Z\] $$\begin{split}
&\sum_{k=M}^N (L^\ast u)_k \bar v_k - u_k \overline{(L^\ast v)_k}
\\ &=
\sum_{k=M}^N (a_ku_{k+1}+b_ku_k+a_{k-1}u_{k-1})\bar v_k
-u_k(a_k\bar v_{k+1}+b_k\bar v_k+a_{k-1}\bar v_{k-1}) \\ &=
\sum_{k=M}^N [u,\bar v]_k - [u,\bar v]_{k-1} =
[u,\bar v]_N-[u,\bar v]_{M-1},
\end{split}$$ so that, cf. Proposition \[prop:Lsaonell2w\], $$B(u,v) = \lim_{N\to\infty} [u,\bar v]_N
-\lim_{M\to-\infty}[u,\bar v]_M, \qquad u,v \in \cD^\ast.$$ In particular, if $J^-$ and $J^+$ are essentially self-adjoint, $(L^\ast, \cD^\ast)$ is self-adjoint, and then $$\lim_{M\to-\infty}[u,\bar v]_M=0 \quad \text{and} \quad \lim_{N\to\infty}[u,\bar v]_N=0.$$
### The Green kernel and the resolvent operator
From on we assume that $J^-$ and $J^-$ have deficiency indices $(0,0)$, so that $J^-$ and $J^+$ are essentially self-adjoint and by Theorem \[thm:MassonRepka\] the deficiency indices of $L$ are $(0,0)$. We refer to e.g. [@Koel-Laredo], [@MassR], for the case that one of the operators has deficiency indices $(0,0)$ and the other on $(1,1)$. This can also be analysed in this framework. In case $L$ has deficiency indices $(2,2)$ the restriction of the domain of a self-adjoint extension of $L$ to the Jacobi operator $J^\pm$ does not in general correspond to a self-adjoint extension of $J^\pm$, cf. [@DunfS Thm. XII.4.31], so that this is the most difficult situation. We restrict ourselves to the case of essentially self-adjoint $L$ or equivalently that $J^\pm$ have both deficiency indices $(0,0)$, i.e. the adjoint of $L$ is self-adjoint.
Let $z\in\C\backslash\R$, so that we know that $L^\ast -z\Id$ has an inverse in $B(\ell^2(\Z))$, the bounded linear operators on $\ell^2(\Z)$. The inverse is denoted by $R(z)$, and is called the *resolvent operator*. Introduce the spaces $$\label{eq:Keq409}
\begin{split}
S^-_z &= \{ \{f_k\}_{k=-\infty}^\infty \mid
L^\ast f = z\, f\text{\ and\ } \sum_{k=-\infty}^{-1}
|f_k|^2<\infty \}, \\
S^+_z &= \{ \{f_k\}_{k=-\infty}^\infty \mid
L^\ast f = z\, f\text{\ and\ } \sum_{k=0}^\infty
|f_k|^2<\infty \}.
\end{split}$$ Since the solution of a three-term recurrence operator is completely determined by two starting values $v_0$, $v_1$, we find $\dim S^\pm_z \leq 2$. The deficiency index $n_+$, respectively $n_-$, for $L^\ast$ is precisely $\dim(S^+_z\cap S^-_z)$ for $\Im z>0$, respectively $\Im z<0$. From the general theory of orthogonal polynomials we know that $\dim(S^\pm_z)\geq 1$, and in case of deficiency indices $(0,0)$ of $J^\pm$ we actually have $\dim(S^\pm_z)= 1$. Consequently, in the case of a self-adjoint $(L^\ast, \cD^\ast)$ we have $\dim(S^\pm_z)= 1$ and $\dim(S^+_z\cap S^-_z)=0$.
Choose $\Phi_z\in S^-_z$, so that $\Phi_z$ is determined up to a constant. We assume $\overline{(\Phi_z)_k} =
(\Phi_{\bar z})_k$, which we can do since $L^\ast$ commutes with complex conjugation. Let $\varphi_z\in S^+_z$, such that $\overline{(\varphi_z)_k}=(\varphi_{\bar z})_k$. We may assume
1. $[\varphi_z,\Phi_z]\not= 0$,
2. $\tilde\varphi_z$, defined by $(\tilde\varphi_z)_k=0$ for $k<0$ and $(\tilde\varphi_z)_k=(\varphi_z)_k$ for $k\geq 0$, is contained in the domain $\cD^\ast$ of the self-adjoint $L^\ast$.
Let $(L^\ast, \cD^\ast)$ be the self-adjoint extension of $L$, assuming, as before, that $J^\pm$ have deficiency indices $(0,0)$. Let $\varphi_z\in S^+_z$, $\Phi_z\in S^-_z$ as before. We define the Green kernel for $z\in\C\backslash\R$ by $$G_{k,l}(z) = \frac{1}{[\varphi_z,\Phi_z]}\begin{cases}
(\Phi_z)_k\, (\varphi_z)_l, & k\leq l, \\
(\Phi_z)_l\, (\varphi_z)_k, & k>l.
\end{cases}$$ So $\{ G_{k,l}(z)\}_{k=-\infty}^\infty,
\{ G_{k,l}(z)\}_{l=-\infty}^\infty \in \ell^2(\Z)$ and $\ell^2(\Z)\ni v\mapsto G(z)v$ given by $$(G(z)v)_k = \sum_{l=-\infty}^\infty v_l G(z)_{k,l} =
\langle v, \overline{G_{k,\cdot}(z)}\rangle$$ is well-defined. For $v\in\cD(\Z)$ we have $G(z)v\in \cD^\ast$.
\[prop:GreenkerLonl2Z\] The resolvent of $(L^\ast, \cD^\ast)$ is given by $R(z)=G(z)$ for $z\in\C\backslash\R$.
For the proof of Proposition \[prop:GreenkerLonl2Z\] we refer to [@Koel-Laredo], and we give here the basic calculation. For $v\in \cD(\Z)$ $$\begin{split}
[\varphi_z,\Phi_z]&\bigl( (L^\ast -z) G(z)v\bigr)_k =
\sum_{l=-\infty}^{k-1} v_l \bigl( a_k
(\varphi_z)_{k+1}+(b_k-z)(\varphi_z)_k +a_{k-1}(\varphi_z)_{k-1}\bigr)
(\Phi_z)_l \\
& + \sum_{l=k+1}^\infty v_l \bigl( a_k
(\Phi_z)_{k+1}+(b_k-z)(\Phi_z)_k +a_{k-1}(\Phi_z)_{k-1}\bigr)
(\varphi_z)_l \\
& + v_k\bigl( a_k(\Phi_z)_k(\varphi_z)_{k+1} +
(b_k-z)(\Phi_z)_k(\varphi_z)_k +
a_{k-1}(\Phi_z)_{k-1}(\varphi_z)_k\bigr) \\
= & v_k a_k\bigl((\Phi_z)_k(\varphi_z)_{k+1} -(\Phi_z)_{k+1}
(\varphi_z)_k\bigr) = v_k [\varphi_z,\varphi_z]
\end{split}$$ and canceling the Wronskian gives the result.
With Proposition \[prop:GreenkerLonl2Z\] we can calculate $$\langle G(z)u, v\rangle = \sum_{k,l=-\infty}^\infty
G_{k,l}(z)u_l\bar v_k =
\frac{1}{[\varphi_z,\Phi_z]} \sum_{k\leq l}
(\Phi_z)_k(\varphi_z)_l\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l}),$$
Now the spectral theorem, see [@DunfS §XII.4], [@Rudi Ch. 13], can be stated as follows. In particular, one sees that the resolvent in terms of the Green kernel gives the spectral decomposition by the Stieltjes-Perron inversion formula.
\[thmspectralthm\] Let $T\colon\cD(T)\to \cH$ be an unbounded self-adjoint linear map with dense domain $\cD(T)$ in the Hilbert space $\cH$, then there exists a unique spectral measure $E$ such that $T=\int_\R t \, dE(t)$, i.e. $\langle Tu,v\rangle =\int_\R t \,
dE_{u,v}(t)$ for $u\in\cD(T)$, $v\in {\mathcal H}$. Moreover, $E$ is supported on the spectrum $\si(T)$, which is contained in $\R$. Moreover, the Stieltjes-Perron inversion formula is valid; $$E_{u,v}\bigl( (a,b)\bigr) = \lim_{\de\downarrow 0}
\lim_{\ep\downarrow 0} \frac{1}{2\pi i}
\int_{a+\de}^{b-\de}
\langle R(x+i\ep)u,v\rangle - \langle R(x-i\ep)u,v\rangle \, dx.$$
Recall that a spectral measure $E$ is a self-adjoint orthogonal projection-valued measure on the Borel sets of $\R$ such that $E(\R)=\Id$, $E(\emptyset)=0$, $E(A\cap B) = E(A)E(B)$ for Borel sets $A$ and $B$ and that $\si$-finite additivity with respect to the strong operator topology holds, i.e. for all $x\in\cH$ and any sequence $(A_i)_{i\in\N}$ of mutually disjoint Borel sets we have $$E\Bigl( \bigcup_{i\in \N} A_i\Bigr)x = \sum_{i\in\N} E(A_i)x.$$ In particular $E_{x,y}(B)= \langle E(B)x,y\rangle$ for $x,y\in\cH$ and $B$ a Borel set gives a complex Borel measure on $\R$, which is positive in case $x=y$.
The basic hypergeometric difference equation
--------------------------------------------
This example is based on Appendix A in [@KoelS-PublRIMS], which was greatly motivated by Kakehi [@Kake] and unpublished notes by Koornwinder. On a formal level the result can be obtained as a limit case of the orthogonality of the Askey-Wilson polynomials, see [@KoelS-NATO] for a precise formulation. We take the coefficients as $$a_k = \hf \sqrt{
(1-\frac{q^{-k}}{r})(1-\frac{cq^{-k}}{d^2r})}, \qquad
b_k =\frac{q^{-k}(c+q)}{2dr},$$ where we assume $0<q<1$, and $r<0$, $c>0$, $d\in\R$. This assumption is made in order to get the expression under the square root sign positive. There are more possible choices in order to achieve this, see [@KoelS-PublRIMS App. A]. Note that $a_k$ and $b_k$ are bounded for $k<0$, so that $J^-$ is self-adjoint. Hence, the deficiency indices of $L$ are $(0,0)$ or $(1,1)$ by Theorem \[thm:MassonRepka\].
\[lem:lemma4.5.2\] Put $$\begin{split}
w_k^2 &= d^{2k} \frac{(cq^{1-k}/d^2r;q)_\infty}
{(q^{1-k}/r;q)_\infty}, \\
f_k(\mu(y)) &=
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{dy,d/y}{c}
\ ;q,rq^k \right)},
\qquad c\not\in q^{-\N},\quad \mu(y)=\hf(y+y^{-1}), \\
g_k(\mu(y)) &= q^kc^{-k} \,
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{qdy/c, qd/cy}{q^2/c}
\ ;q, rq^k \right)}
\quad \mu(y)=\hf(y+y^{-1}),\ c\notin q^{2+\Z} \\
F_k(y) &= (dy)^{-k} \,
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{dy,qdy/c}{qy^2}
\ ;q,\frac{q^{1-k}c}{d^2r} \right)}
\qquad y^2\not\in q^{-\N},
\end{split}$$ then, with $z=\mu(y)$, we have that $u_k(z)= w_kf_k(\mu(y))$, $u_k(z)= w_kg_k(\mu(y))$, $u_k(z)=w_k F_k(y)$ and $u_k(z)=w_kF_k(y^{-1})$ define solutions to $$z\, u_k(z) = a_k \, u_{k+1}(z) + b_k \, u_k(z) +
a_{k-1}\, u_{k-1}(z).$$
Put $u_k(z)=w_kv_k(z)$, then $v_k(z)$ satisfies $$2z\, v_k(z) = (d-\frac{cq^{-k}}{dr})\, v_{k+1}(z)
+q^{-k}\frac{c+q}{dr}\, v_k(z)
+ (d^{-1}-\frac{q^{1-k}}{dr})\, v_{k-1}(z)
$$ and this is precisely the second order $q$-difference equation that has the solutions given, see Proposition \[prop:solofBqDEat0atinfty\] and Section \[sec:BHS-qdiff\].
The asymptotics of the solutions of Lemma \[lem:lemma4.5.2\] can be given as follows. First observe that $w_{-k} ={\mathcal O}(d^{-k})$ as $k\to\infty$, and using $$w_k^2=c^k
\frac{(r q^k, d^2r/c, cq/d^2r;q)_\infty}
{(d^2rq^k/c, r, q/ r;q)_\infty} \Rightarrow
w_k={\mathcal O}(c^{\hf k}), \ k\to\infty.$$ Now $f_k(\mu(y))= {\mathcal O}(1)$ as $k\to\infty$, and $g_k(\mu(y)) = {\mathcal O}((q/c)^k)$ as $k\to \infty$. Similarly, $F_{-k}(y)={\mathcal O}((dy)^k)$ as $k\to\infty$.
\[prop:4.5.3\] The operator $L$ is essentially self-adjoint for $0<c\leq q^2$, and $L$ has deficiency indices $(1,1)$ for $q^2<c<1$. Moreover, for $z\in\C\backslash\R$ the one-dimensional space $S^-_z$ is spanned by $wF(y)$ with $\mu(y)=z$ and $|y|<1$. For $0<c\leq q^2$ the one-dimensional space $S^+_z$ is spanned by $wf(z)$, and for $q^2<c<1$ the two-dimensional space $S^+_z$ is spanned by $wf(z)$ and $wg(z)$.
The proof of Proposition \[prop:4.5.3\] relies on criteria establishing the defect indices of Jacobi operators. We refer to [@Koel-Laredo p. 79] for the application of these criteria leading to Proposition \[prop:4.5.3\]. Since we restrict ourselves to the self-adjoint setting, we assume from now on that $0<c\leq q^2$.
The Wronskian $$[wF(y), wF(y^{-1})] = \lim_{k\to-\infty}
a_k w_{k+1}w_k\bigl(
F_{k+1}(y)F_k(y^{-1})-F_k(y)F_{k+1}(y^{-1})\bigr)
=\hf (y^{-1}-y)$$ using $a_k\to \hf$ as $k\to-\infty$ and the asymptotics of $F_k$ and $w_k$ as $k\to-\infty$. Note that the Wronskian is non-zero for $y\not= \pm 1$ or $z\not=\pm1$. Since $wF(y)$ and $wF(y^{-1})$ are linearly independent solutions to the recurrence fo $L^\ast f=zf$ for $z\in\C\backslash\R$, we see that we can express $f_k(\mu(y))$ in terms of $F_k(y)$ and $F_k(y^{-1})$. These solutions are related by the expansion $$\label{eq:Keq4191}
\begin{split}
f_k(\mu(y))
&= c(y) F_k(y) + c(y^{-1})F_k(y^{-1}), \\
c(y) &= \frac{(c/dy,d/y,dry,q/dry;q)_\infty}
{(y^{-2},c,r,q/r;q)_\infty},
\end{split}$$ for $c\not\in q^{-\N}$, $y^2\not\in q^\Z$, which is a reformulation of . This shows that we have $$\label{eq:Wronskianexpl}
[wf(\mu(y)),wF(y)] =\hf c(y^{-1})(y-y^{-1}).$$
Since we assume that $0<c\leq q^2$, $L$ is essentially self-adjoint, or $L^\ast$ is self-adjoint. Then for $z\in\C\backslash\R$ we have $\varphi_z = wf(z)$ and $\Phi_z=wF(y)$, where $z=\mu(y)$ and $|y|<1$. In particular, it follows that $\varphi_{x\pm i\ep}\to
\varphi_x$ as $\ep\downarrow 0$. For the asymptotic solution $\Phi_z$ we have to be more careful in computing the limit of $z$ to the real axis. For $x\in \R$ satisfying $|x|>1$ we have $\varphi_{x\pm i\ep} \rightarrow wF_y$ as $\ep\downarrow 0$, where $y\in (-1,1)\backslash \{0\}$ is such that $\mu(y)=x$. If $x\in [-1,1]$, then we put $x=\cos\chi=\mu(e^{i\chi})$ with $\chi\in [0,\pi]$, and then $\Phi_{x-i\ep}\rightarrow wF_{e^{i\chi}}$ and $\Phi_{x+i\ep}\rightarrow wF_{e^{-i\chi}}$ as $\ep\downarrow 0$.
We calculate the integrand in the Stieltjes-Perron inversion formula of Theorem \[thmspectralthm\] using Lemma \[lem:lemma4.5.2\] and Proposition \[prop:4.5.3\] in the case $|x|<1$, where $x=\cos\chi=\mu(e^{i\chi})$. For $u,v\in {\mathcal D}(\Z)$ we have $$\begin{split}
&\lim_{\ep\downarrow 0}\langle G(x+i\ep)u,v\rangle -
\langle G(x-i\ep)u,v\rangle = \\ & \lim_{\ep\downarrow 0}
\sum_{k\leq l}
\Bigl( \frac{(\Phi_{x+i\ep})_k(\varphi_{x+i\ep})_l}
{[\varphi_{x+i\ep},\Phi_{x+i\ep}]} -
\frac{(\Phi_{x-i\ep})_k(\varphi_{x-i\ep})_l}
{[\varphi_{x-i\ep},\Phi_{x-i\ep}]} \Bigr)
\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l})
= \\ &2\sum_{k\leq l}
\Bigl( \frac{w_kF_k(e^{-i\chi})w_lf_l(\cos\chi)}
{c(e^{i\chi})(e^{-i\chi}-e^{i\chi})} -
\frac{w_kF_k(e^{i\chi})w_lf_l(\cos\chi)}
{c(e^{-i\chi})(e^{i\chi}-e^{-i\chi})}\Bigr)
\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l})
= \\ &2\sum_{k\leq l}
\Bigl( w_kw_lf_l(\cos\chi)
\frac{c(e^{-i\chi})F_k(e^{-i\chi})+c(e^{i\chi})F_k(e^{i\chi})}
{c(e^{i\chi})c(e^{-i\chi})(e^{-i\chi}-e^{i\chi})}
\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l}) = \\
&2\sum_{k\leq l}
\Bigl( \frac{w_kw_lf_l(\cos\chi)f_k(\cos\chi)}
{c(e^{i\chi})c(e^{-i\chi})(e^{-i\chi}-e^{i\chi})}
\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l}) = \\
&\frac{2}{c(e^{i\chi})c(e^{-i\chi})(e^{-i\chi}-e^{i\chi})}
\sum_{l=-\infty}^\infty w_lf_l(\cos\chi)u_l
\sum_{k=-\infty}^\infty w_kf_k(\cos\chi)\bar v_k
\end{split}$$ using the expansion and the Wronskian in . Now integrate over the interval $(a,b)$ with $-1\leq a<b\leq 1$ and replacing $x$ by $\cos\chi$, so that $\frac{1}{2\pi i}dx = (e^{i\chi}-e^{-i\chi})d\chi/4\pi$. We obtain, with $a=\cos \chi_a$, $b=\cos\chi_b$, and $0\leq \chi_b<\chi_a\leq\pi$, $$\begin{split}
E_{u,v}\bigl( (a,b)\bigr) &= \frac{1}{2\pi}
\int_{\chi_b}^{\chi_a} \bigl({\mathcal F}u\bigr)(\cos\chi)
\overline{\bigl({\mathcal F}v\bigr)(\cos\chi)}
\frac{d\chi}{|c(e^{i\chi})|^2}, \\
\bigl({\mathcal F}u\bigr)(x) &= \langle u, \varphi_x\rangle =
\sum_{l=-\infty}^\infty w_lf_l(\cos\chi)u_l.
\end{split}$$ This shows that $[-1,1]$ is contained in the continuous spectrum of $L$.
For $|x|>1$ we can calculate as above the integrand in the Stieltjes-Perron inversion formula, but now we have to use that $x=\mu(y)$ with $|y|<1$. This gives $$\lim_{\ep\downarrow 0}\langle G(x+i\ep)u,v\rangle
= 2\sum_{k\leq l} \frac{w_kF_k(y)w_lf_l(y)}
{c(y^{-1})(y-y^{-1})}
\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l}).$$ The limit $\lim_{\ep\downarrow 0}\langle G(x+i\ep)u,v\rangle$ gives the same result, so we can only have discrete mass points for $|x|>1$ in the spectral measure at the zeroes of the Wronskian, i.e. at the zeroes of $y\mapsto c(y^{-1})$ with $|y|<1$ or at $y=\pm 1$. Let us assume that all zeroes of the $c$-function are simple, so that the spectral measure at these points can be easily calculated.
The zeroes of the $c$-function can be read off from the expressions in , and they are $$\{ cq^k/d\mid k\in\N\}, \quad \{ dq^k\mid k\in\N\}, \quad \{ q^k/dr\mid k\in\Z\}.$$ Assuming that $|c/d|<1$ and $|d|<1$, we see that the first two sets do not contribute. (In the more general case we have that the product is less than $1$, since the product equals $c<1$. We leave this extra case to the reader.) The last set, labeled by $\Z$ always contributes to the spectral measure. Now for $u,v\in{\mathcal D}(\Z)$ we let $x_p=\mu(y_p)$, $y_p=q^p/dr$, $p\in\Z$, with $|q^p/dr|>1$, so that by the Stieltjes-Perron inversion formula and Cauchy’s residue theorem we find $$E_{u,v}(\{ x_p\}) = \text{Res}_{y=y_p^{-1}}
\Bigl( \frac{-1}{c(y^{-1})y}\Bigr)
w_k F_k(y_p^{-1}) w_l f_l(x_p)
\bigl( u_l\bar v_k+u_k\bar v_l\bigr) (1-\hf
\de_{k,l})$$ after substituting $x=\mu(y)$. Now from we find $f_k(x_p)=c(y_p)F_k(y_p^{-1})$, since $c(y_p^{-1})=0$ and we assume here that $c(y_p)\not=0$. Hence, we can symmetrise the sum again and find $$E_{u,v}(\{ x_p\}) = \Bigl(\text{Res}_{y=y_p}
\frac{1}{c(y^{-1})c(y)y}\Bigr)
\bigl({\mathcal F}u\bigr)(x_p) \overline{\bigl({\mathcal F}v\bigr)(x_p)}$$ switching to the residue at $y_p$.
We can combine the calculations in the following theorem. Note that most of the regularity conditions can be removed by continuity after calculating explicitly all the residues. The case of an extra set of finite mass points is left to the reader, as stated above. Of course, there are also other possibilities for choices of the parameters $c$, $d$ and $r$ for which the expression under the square root sign in $a_k$ in is positive. See [@KoelS-PublRIMS App. A] for details.
\[thm:orthorel2phi1\] Assume $r<0$, $0<c\leq q^2$, $d\in\R$ with $|d|<1$ and $|c/d|<1$ such that the zeroes of $y\mapsto c(y)$ are simple and $c(y)=0$ implies $c(y^{-1})\not= 0$. Then the spectral measure for the Jacobi operator on $\ell^2(\Z)$ defined by is given by, $A\subset\R$ a Borel set, $$\begin{split}
&\langle E(A)u,v\rangle =
\int_{\cos\chi\in [-1,1]\cap A}\bigl({\mathcal F}u\bigr)(\cos\chi)
\overline{\bigl({\mathcal F}v\bigr)(\cos\chi)} \frac{d\chi}
{|c(e^{i\chi})|^2} \\ &+
\sum_{p\in\Z, |q^p/dr|>1, \mu(q^p/dr)\in A}
\Bigl(\text{{\rm Res}}_{y=q^p/dr}
\frac{1}{c(y^{-1})c(y)y}\Bigr)
\bigl({\mathcal F}u\bigr)(\mu(q^p/dr))
\overline{\bigl({\mathcal F}v\bigr)(\mu(q^p/dr))}.
\end{split}$$
It only remains to prove that $\pm 1$ is not contained in the point spectrum. These are precisely the points for which $F(y)$ and $F(y^{-1})$ are not linearly independent solutions. We have to show that $\varphi_{\pm 1}\not\in\ell^2(\Z)$, and this can be done by determining its asymptotic behaviour as $k\to-\infty$, see [@Kake], [@KoelS-CA] for more information.
Take $A=\R$ and $u=e_k$ and $v=e_l$, then we find the following orthogonality relations for the ${}_2\vp_1$-series as in Lemma \[lem:lemma4.5.2\].
\[cor:thm:orthorel2phi1\] With the notation and assumptions as in Theorem \[thm:orthorel2phi1\] we have $$\begin{split}
&\int_0^\pi f_k(\cos\chi) f_l(\cos\chi)
\frac{d\chi}{|c(e^{i\chi})|^2} +
\\ &\sum_{p\in\Z, |q^p/dr|>1}
\Bigl({\text{\rm Res}}_{y=q^p/dr}
\frac{1}{c(y^{-1})c(y)y}\Bigr)
f_k(\mu(\frac{q^p}{dr}))f_l(\mu(\frac{q^p}{dr}))
= \frac{\de_{k,l}}{w_k^2}.
\end{split}$$
\[rmk:cor:thm:orthorel2phi1\] Theorem \[thm:orthorel2phi1\] and Corollary \[cor:thm:orthorel2phi1\] have been obtained under the condition that the operator $L^\ast,\cD^\ast)$ is self-adjoint, or that $0<c\leq q^2$. In [@Koel-Laredo] it is shown that in case $q^2<c<1$, there exists a self-adjoint extension of $(L,\cD(\Z))$ such that the same decomposition in Theorem \[thm:orthorel2phi1\] and Corollary \[cor:thm:orthorel2phi1\] remain valis. The case $c=q$ is a bit more intricate and requires a limiting process, since, see Proposition \[prop:solofBqDEat0atinfty\], $u_1$ and $u_2$ are the same, see [@GroeKK App. C] for the details.
\[rmk:thm:orthorel2phi1\] The result in Corollary \[cor:thm:orthorel2phi1\] can be viewed as $q$-analogue of the integral transform pair of the Jacobi functions, see for the definition. The Jacobi function transform is an integral transform pair with a ${}_2F_1$-series, the Jacobi function, as integral kernel, see [@Koor-Jacobi] for details.
Another possible option is to obtain the result of Corollary \[cor:thm:orthorel2phi1\] as a limiting case of the orthogonality relation of the Askey-Wilson polynomials, which is comparable to the limit transition of the Jacobi polynomials to the Bessel functions, see [@KoelS-NATO]. $q$-Analogues of the Bessel functions in terms of ${}_2\varphi_1$-series have also been studied in [@Koel-ITSF93]. Taking a similar limit in the little $q$-Jacobi polynomials leads to the little $q$-Bessel functions (or ${}_1\varphi_1$-$q$-Bessel function or as the Hahn-Exton $q$-Bessel function) studied by Koornwinder and Swarttouw [@KoorS-TAMS]. These $q$-Bessel functions have been studied intensively, see e.g. [@FitoD2], [@FitoD], [@KoelSw] as well as other references.
Exercises
---------
1. \[ex:lem:maxdomdiJacobi\] Prove Lemma \[lem:maxdomdiJacobi\].
2. \[ex:lem:WronskianLonl2Z\] Prove Lemma \[lem:WronskianLonl2Z\].
Notes {#notes-3 .unnumbered}
-----
The results of this section have been motivated by the paper by Kakehi [@Kake] and unpublished notes by Koornwinder. The results and techniques have been very useful in the study of various problems related to harmonic analysis on the non-compact quantum group analogue of $SU(1,1)$. In particular, we have used [@KoelS-PublRIMS App. A], where more general sets of parameters have been studied, see also [@Koel-Laredo]. A special case is studied in [@GroeKK App. C]. In [@Koel-IM] another case related to a non-selfadjoint operator is studied in detail. There is also an approach to doubly infinite Jacobi operators as in §\[ssec:doublyinfiniteJacobioperators\], due to Krein, and this is to relate it to a $2\times 2$-matrix valued three term recurrence on $\N$, see e.g. Berezanskiĭ [@Bere Ch. VII]. This leads to the theory of matrix-valued orthogonal polynomials.
See [@Koel-Laredo] for the solution to Exercise \[ex:lem:maxdomdiJacobi\] and Exercise \[ex:lem:WronskianLonl2Z\].
Transmutation properties for the basic $q$-difference equation {#sec:transmutation}
==============================================================
In §\[sec:BHS-qdiff\] we have discussed the factorisation of the basic $q$-difference operator. The Darboux factorisation in §\[sec:BHS-qdiff\] is related to a $q$-shift in both parameters. Here we discuss a related shift operator, but we use a relabeling of the parameters. Moreover, the shift is more general and leads to a $q$-analogue of fractional integral operators and other type of factorisations of the basic $q$-difference operator.
We rewrite the second order hypergeometric $q$-difference operator as studied in §\[sec:BHS-qdiff\] as $$\label{eq:KR1.1}
L=L^{(a,b)}= a^2(1+\frac{1}{x})\bigl( T_q-\text{Id}\bigr) +
(1+\frac{aq}{bx})\bigl( T^{-1}_q-\text{Id}\bigl),$$ where $T_qf(x)=f(qx)$ for suitable functions $f$ in a suitable Hilbert space. So we have eigenfunctions to $L$ in terms of basic hypergeometric series, see Proposition \[prop:solofBqDEat0atinfty\]. The little $q$-Jacobi function is defined as $$\label{eq:KR1.2}
\varphi_\la(x;a,b;q) =
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a\si,a/\si}{ab}
\ ;q,-\frac{bx}{a} \right)}, \quad \la=\hf(\si+\si^{-1})=\mu(\si)$$
The little $q$-Jacobi function satisfies $$L\varphi_\la(\cdot;a,b;q) =
(-1-a^2+2a\la)\varphi_\la(\cdot;a,b;q).$$ We note that the little $q$-Jacobi functions are eigenfunctions for the eigenvalue $\la$ of $$\label{eq:KR1.4}
\cL^{(a,b)} = \frac{1}{2a} L^{(a,b)} + \hf(a+a^{-1})
= \frac{a}{2}(1+\frac{1}{x})T_q
- \bigl( \frac{a}{2x} + \frac{q}{2bx}\bigr)\text{Id}
+ \frac{1}{2a}(1+\frac{aq}{bx})T_q^{-1}.$$ For simplicity we assume that $a,b>0$, $ab<1$ and $y>0$, but the results hold, mutatis mutandis, for the more general range of the parameters as discussed in [@KoelS-PublRIMS App. A]. Then the operator $L$ is an unbounded symmetric operator on the Hilbert space $\cH(a,b;y)$ of square integrable sequences $u=(u_k)_{k\in\Z}$ with respect to the weights $$\label{eq:KR1.5}
\sum_{k=-\infty}^\infty |u_k|^2 (ab)^k
\frac{(-byq^k/a;q)_\infty}{(-yq^k;q)_\infty},
$$ where the operator $L$ is initially defined on the sequences with finitely many non-zero entries, see §\[sec:BHS-qdiff-nonpol\], and where $x=yq^k$.
The goal is to give a general factorisation property in Theorem \[thm:KRTheorem2.3\] and Theorem \[thm:KRTheorem2.3ii\]. As a motivation we start by giving a Darboux factorisation of the second order $q$-difference operator $L^{(a,b)}$ or $\cL^{(a,b)}$, related to the one in §\[sec:BHS-qdiff\].
The backward $q$-derivative operator is $B_q=M_{1/x}(1-T_q^{-1})$, where $M_g$ is the operator of multiplication by $g$; $\bigl( M_gf\bigr)(x)=g(x)f(x)$, and $T_qf(x)=f(qx)$. Then $B_q$ is closely related to $\tilde{D}_q$ of §\[sec:BHS-qdiff\] with inverted base $q\leftrightarrow q^{-1}$. Now we check that $$\label{eq:KR5.1}
\bigl( B_q\varphi_\la(\cdot;a,b;q)\bigr)(x)
= \frac{b(1-a\si)(1-a/\si)}{qa(1-ab)}
\, \varphi_\la(x;aq,b;q).
$$ Considering $\cH(a,b;y)$ as an $L^2$-space with discrete weights $(ab)^k (-byq^k/a;q)_\infty/(-yq^k;q)_\infty$ at the point $yq^k$, $k\in\Z$, we look at $B_q$ as a (densely defined unbounded) operator from $\cH(a,b;y)$ to $\cH(aq,b;y)$. Its adjoint, up to a constant depending only on $y$, is given by $$\label{eq:KR5.2}
A(a,b) = M_{1+bx/aq} - ab M_{1+x}T_q,
$$ and it is a straightforward calculation to show that $$\label{eq:KR5.3}
\bigl( A(a,b) \varphi_\la(\cdot;aq,b;q)\bigr)(x) =
(1-ab)\, \varphi_\la(x;a,b;q)
$$ and that $-b L^{(a,b)} = aq A(a,b) \circ B_q$, with the notation as in . This calculation is essentially the same as done in §\[sec:BHS-qdiff\].
Since $B_q$ and $A(a,b)$ are triangular with respect to the standard orthogonal basis of Dirac delta’s at $yq^k$ of $\cH(a,b;y)$, this means that we have a Darboux factorisation of $L^{(a,b)}$. Also, $$-b(L^{(aq,b)}+(1-q)(1-qa^2))=aq^2 B_q\circ A(a,b),$$ from which we deduce $$B_q\circ L^{(a,b)}=L^{(aq,b)}\circ B_q \quad \text{and} \quad
L^{(a,b)}\circ A(a,b)=A(a,b)\circ L^{(aq,b)}.$$ It is the purpose of this section to generalize these intertwining properties to arbitrary powers of $B_q$.
Introduce the operator $W_\nu$, $\nu\in\C$, acting on functions defined on $[0,\infty)$ by $$\label{eq:KR5.4}
\bigl( W_\nu f\bigr)(x) = x^\nu \sum_{l=0}^\infty
f(xq^{-l}) q^{-l\nu} \frac{(q^\nu;q)_l}
{(q;q)_l}, \qquad x\in[0,\infty),
$$ assuming that the infinite sum is absolutely convergent if $\nu\not\in -\N$. So we want $f$ sufficiently decreasing on a $q$-grid tending to infinity, e.g. $f(xq^{-l}) =\cO(q^{l(\nu+\ep)})$ for some $\ep>0$. Note that for $\nu\in -\N$ the sum in is finite and $W_0=\text{Id}$ and $W_{-1}=B_q$.
This operator is a $q$-analogue of the Weyl fractional integral operator as used in [@Koor-ArkMat75 §3], [@Koor-Jacobi §5.3] for the Abel transform. With the notation $$\int_a^\infty f(t)\, d_qt = a\sum_{k=0}^\infty f(xq^{-k})q^{-k}$$ for the $q$-integral we see that for $n\in\N$ the operator $W_n$ is an iterated $q$-integral; $$\label{eq:KR5.5}
\bigl( W_n f\bigr)(x) = \int_x^\infty \int_{x_1}^\infty \ldots
\int_{x_{n-1}}^\infty f(x_n)\, d_qx_nd_qx_{n-1}\ldots d_qx_1.
$$
In the following lemma we collect some results on $W_\nu$, where we use the function space $$\label{eq:KR5.6}
\cF_\rho = \{ f\colon [0,\infty)\to \C \mid\,
|f(xq^{-l})| = \cO(q^{l\rho})
,\ l\to\infty,\ \forall x\in (q,1]\},\qquad \rho>0.
$$ Recall that $\cL^{(a,b)}$ is defined in .
\[lem:KRLemma5.1\] Let $\nu,\mu\in\C\backslash(-\N)$.
- $W_\nu$ preserves the space of compactly supported functions,
- $W_\nu\colon
\cF_\rho\to \cF_{\rho-\Re\nu}$ for $\rho>\Re\nu>0$,
- $W_\nu\circ W_\mu = W_{\nu+\mu}$ on $\cF_\rho$ for $\rho>\Re(\mu+\nu)>0$,
- $W_\nu\circ B_q = B_q\circ W_\nu = W_{\nu-1}$ on $\cF_\rho$ for $\rho>\Re\nu-1>0$, and $B_q^n\circ W_n = \Id$ for $n\in\N$ on $\cF_\rho$ for $\rho>n$,
- $\cL^{(aq^{-\nu},b)}\circ W_\nu =
W_\nu \circ \cL^{(a,b)}$, valid for compactly supported functions.
It follows from (iii) that $W_{-n}=B_q^n$, $n\in\N$, and $W_0=\Id$.
The first statement is immediate from . For (ii) we use that for $f\in\cF_\rho$ and $x\in(q,1]$ we have $$| W_\nu f(xq^{-k})| \leq M\sum_{l=0}^\infty q^{(k+l)\rho}
q^{-(k+l)\Re\nu} \frac{(q^{\Re\nu};q)_l}{(q;q)_l} =
M q^{k(\rho-\Re\nu)}
\frac{(q^\rho;q)_\infty}{(q^{\rho-\Re\nu};q)_\infty}$$ by the $q$-binomial theorem for $\rho>\Re\nu$. The third statement is a consequence of interchanging summations, valid for $f\in\cF_\rho$, $\rho>\Re(\mu+\nu)$, and $$\sum_{k+l=p} \frac{(q^\mu;q)_k (q^\nu;q)_l}
{(q;q)_k(q;q)_l} q^{-(l+k)\mu-l\nu} =
q^{-p(\mu+\nu)} \frac{(q^{\mu+\nu};q)_p}
{(q;q)_p},$$ which is the $q$-Chu-Vandermonde summation formula . For (iv) we note that $B_q\colon \cF_\rho\to \cF_{\rho+1}$, then the first statement of (iv) is a simple calculation involving $q$-shifted factorials, which reduces the second statement of (iv) to verifying the easy case $n=1$. For (v) recall , so that $\cL^{(aq^{-\nu},b)}(W_\nu f)(x)$ and $W_\nu (\cL^{(a,b)}f)(x)$ involve the values $f(xq^{-k})$, $k+1\in \N$. A straightforward calculation using $q$-shifted factorials shows that the coefficients of $f(xq^{-k})$ in $\cL^{(aq^{-\nu},b)}(W_\nu f)(x)$ and $W_\nu (\cL^{(a,b)}f)(x)$ are equal.
The asymptotically free solution $\Phi_\si(yq^k;a,b;q)$ is defined by $$\label{eq:KR1.10AFSol}
\Phi_\si(yq^k;a,b;q) = (a\si)^{-k}
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a\si,q\si/b}{q\si^2}
\ ;q, -\frac{q^{1-k}}{y} \right)}.$$ so that, see , $$\begin{split}
\varphi_\la(yq^k;a,b;q) &= c(\si;a,b;q) \Phi_\si(yq^k;a,b;q)
+ c(\si^{-1};a,b;q) \Phi_{\si^{-1}}(yq^k;a,b;q), \\
c(\si;a,b,y;q) &= \frac{(b/\si,a/\si;q)_\infty}
{(\si^{-2},ab;q)_\infty}\frac{(-by\si, -q/by\si;q)_\infty}
{(-by/a,-qa/by;q)_\infty},
\end{split}$$ valid for $\si^2\notin q^\Z$. Then $\Phi_\si$ is the asymptotically free solution; $$L\Phi_\si(\cdot;a,b;q)=(-1-a^2+2a\la)\Phi_\si(\cdot;a,b;q)$$ on $yq^\Z$ with, as before, $\la=\mu(\si)=\hf(\si+\si^{-1})$.
The asymptotically free solution $\Phi_\si(yq^k;a,b;q)
\in\cF_\rho$ for $q^\rho>|a\si|$ as follows from . A calculation using the $q$-binomial formula gives, cf. , $$\label{eq:KR5.7}
\bigl(W_\nu \Phi_\si(\cdot;a,b;q)\bigr)(yq^k)=
y^\nu \frac{(a\si;q)_\infty}
{(aq^{-\nu}\si;q)_\infty}
\Phi_\si(yq^k;aq^{-\nu},b;q),
$$ for $|a\si|<q^\nu$ in accordance with Lemma 5.1(v). Note that is a $q$-analogue of Bateman’s formula, cf. [@Gasp], [@Koor-ArkMat75].
\[lem:KRLemma5.2\] Define the operator $$S(a,b) = M_{\frac{(-x;q)_\infty}{(-bx/a;q)_\infty}}
\circ T_{b/a},\qquad T_{b/a}f(x) = f(\frac{b}{a}x),$$ then $S(a,b)^{-1}\circ \cL^{(a,b)} \circ
S(a,b) = \cL^{(b,a)}$. In particular, $\tilde W_\nu^{(a,b)} = S(a,bq^{-\nu})\circ
W_\nu\circ S(a,b)^{-1}$ satisfies the intertwining property $\cL^{(a,bq^{-\nu})}\circ \tilde W_\nu^{(a,b)} =
\tilde W_\nu^{(a,b)} \circ \cL^{(a,b)}$.
Note that $S(a,b)^{-1}=S(b,a)$ and that $S(a,b)\colon
\cH(b,a;yb/a)\to \cH(a,b;y)$ is an isometric isomorphism. For $f\in\cF_\rho$ we see that $\bigl(S(a,b)f\bigr)(xq^{-l}) = \cO( |a/b|^lq^{l\rho})$, so that $S(a,b)f\in \cF_{\rho + \ln(|a/b|)/\ln q}$.
It follows from that $$\begin{gathered}
\cL^{(a,b)}\bigl(x\mapsto \frac{(-x;q)_\infty}
{(-bx/a;q)_\infty}f(x)\bigr)(x) = \\ \frac{(-x;q)_\infty}
{(-bx/a;q)_\infty} \Bigl( \frac{b}{2}(1+\frac{a}{bx})f(qx)
+ \frac{1}{2b}(1+\frac{q}{x})f(xq^{-1}) - \hf
(\frac{a}{x}+\frac{q}{bx})f(x)\Bigr)\end{gathered}$$ and the term in parentheses can be written as $T_{b/a}\circ \cL^{(b,a)}\circ T_{a/b}$ applied to $f$. The second statement then follows from Lemma 5.1(v).
It follows directly from , and the last equation of , $$\label{eq:KR5.8}
\aligned
\bigl( S(a,b)\varphi_\la(\cdot;b,a;q)\bigr)(x)&= \varphi_\la(x;a,b;q), \\
\bigl( S(a,b)\Phi_\si(\cdot;b,a;q)\bigr)(x)&= \Phi_\si(x;a,b;q).
\endaligned
$$
\[thm:KRTheorem2.3\] Let $a,b\in\C\backslash\{0\}$, $\nu,\mu\in\C$ with $|q^{\nu-\mu}b/a|<1$. Define the operator $$\begin{gathered}
\bigl(W_{\nu,\mu}(a,b)f\bigr)(x) =
\frac{(-x;q)_\infty}{(-xq^{-\mu};q)_\infty}
q^{-\mu^2}\bigl( \frac{b}{a}\bigr)^\mu x^{\mu+\nu}
\\ \times \sum_{p=0}^\infty
f(xq^{-\mu-p})\, q^{-p\nu}\frac{(q^\nu;q)_p}{(q;q)_p}
\, {\,_{3}\vp_{2} \left( \genfrac{.}{.}{0pt}{}{q^{-p}, q^{-\mu},-q^{1+\mu-\nu}a/bx}{q^{1-p-\nu},-q^{\mu+1}/x}
\ ;q,q^{1-\mu}\frac{b}{a} \right)}\end{gathered}$$ for any function $f$ with $|f(xq^{-p})|={\cO}(q^{p(\ep+\nu)})$ for some $\ep>0$. Then $$W_{\nu,\mu}(a,b)\circ
{\cL}^{(a,b)}= {\cL}^{(aq^{-\nu},bq^{-\mu})}
\circ W_{\nu,\mu}(a,b)$$ on the space of compactly supported functions and for $|a\si|<q^\nu$ $$\bigl(W_{\nu,\mu}(a,b)\Phi_\si(\cdot;a,b;q)\bigr)(yq^k) =
y^{\mu+\nu}
\frac{(a\si,b\si;q)_\infty}
{(aq^{-\nu}\si,bq^{-\mu}\si;q)_\infty}
\Phi_\si(yq^k;aq^{-\nu},bq^{-\mu};q).$$
It follows from Lemma \[lem:KRLemma5.1\](v) and Lemma \[lem:KRLemma5.2\] that the operator $$\begin{aligned}
W_{\nu,\mu}(a,b) &= \tilde W_\mu^{(aq^{-\nu},b)}\circ W_\nu
= S(aq^{-\nu},bq^{-\mu})\circ W_\mu\circ
S(b,aq^{-\nu})\circ W_\nu\end{aligned}$$ satisfies the required interwining property. For $f\in {\cF}_\rho$ with $\rho>\Re\nu$ we can interchange summations, which leads to the sum with a terminating ${}_3\vp_2$ as kernel. Note that the ${}_3\vp_2$-series in the kernel of $W_{\nu,\mu}(a,b)$ behaves as $${\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{q^{-\mu},-q^{1+\mu-\nu}a/bx}{-q^{1+\mu}/x}
\ ;q, q^{\nu-\mu}\frac{b}{a} \right)}$$ as $p\to\infty$.
The statement for the action on $\Phi_\si(\cdot;a,b;q)$ follows immediately from and .
The results in Theorem \[thm:KRTheorem2.3\] deal with the fractional $q$-derivative $W_\nu$ related to the point at $\infty$, and these operators act nicely on the eigenfunctions $\Phi_\si$ at $\infty$ of the operator $\cL^{(a,b)}$. We want to have similar statements on a suitable intertwining operator that acts nicely on the eigenfunctions $\vp_\la$ at $0$ of the operator $\cL^{(a,b)}$. In order to get results in this direction, see Theorem \[thm:KRTheorem2.3ii\], we take appropriate adjoints of the previous construction. Consider $W_\nu$, $\nu\in\C\setminus (-\N)$, as a densely defined unbounded operator from $\cH(aq^\nu,b;y)$ to $\cH(a,b;y)$ and define $R_\nu^{(a,b)}$ as its adjoint, so $$\label{eq:KR5.9}
\langle R^{(a,b)}_\nu f, g\rangle_{\cH(aq^\nu,b;y)} =
\langle f, W_\nu g\rangle_{\cH(a,b;y)}
$$ for all compactly supported functions $g$, cf. Lemma 5.1(i). Here we use the identification of $\cH(a,b;y)$ as a weighted $L^2$-space on a discrete set, see §1. A $q$-integration by parts shows $$\label{eq:KR5.10}
\bigl(R_\nu^{(a,b)} f\bigr)(yq^p)= y^\nu
\frac{(-byq^p/a;q)_\infty}{(-byq^{p-\nu}/a;q)_\infty}
\sum_{l=0}^\infty f(yq^{p+l}) (ab)^l
\frac{(q^\nu,-yq^p;q)_l}{(q,-byq^p/a;q)_l}.
$$ Now define, for functions $f$, the operator $$\label{eq:KR5.11}
\bigl(A_\nu^{(a,b)} f\bigr)(x)=
\frac{(-bx/a;q)_\infty}{(-bxq^{-\nu}/a;q)_\infty}
\sum_{l=0}^\infty f(xq^l) (ab)^l
\frac{(q^\nu,-x;q)_l}{(q,-bx/a;q)_l},
$$ so that $A_\nu^{(a,b)}\Big\vert_{{\cH}(a,b;y)}=y^{-\nu}R_\nu^{(a,b)}$. Note that $A_\nu^{(a,b)}$ is well-defined for bounded functions assuming $|ab|<1$. Recall that the dense domain of finite linear combinations of the basis vectors for $\cL^{(a,b)}$ corresponds to the functions compactly supported in $(0,\infty)$.
\[lem:KRLemma 5.3\] $\cL^{(aq^{\nu},b)}\circ A_\nu^{(a,b)} =
A_\nu^{(a,b)} \circ \cL^{(a,b)}$ on the space of functions compactly supported in $(0,\infty)$. Moreover, $$\bigl(A_\nu^{(a,b)}\varphi_\la(\cdot;a,b;q)\bigr)(x) =
\frac{(abq^\nu;q)_\infty}{(ab;q)_\infty}
\varphi_\la(x;aq^\nu,b;q).$$ Defining $\tilde A_\nu^{(a,b)} = S(a,bq^\nu)\circ
A_\nu^{(b,a)}\circ S(b,a)$ we have $\cL^{(a,bq^\nu)}\circ \tilde A_\nu^{(a,b)} =
\tilde A_\nu^{(a,b)} \circ \cL^{(a,b)}$, and $$\bigl(\tilde A_\nu^{(a,b)}\varphi_\la(\cdot;a,b;q)\bigr)(x) =
\frac{(abq^\nu;q)_\infty}{(ab;q)_\infty}
\varphi_\la(x;a,bq^\nu;q).$$
Note that and show that the operators $R_\nu^{(a,b)}$ and $A_\nu^{(a,b)}$ preserve the space of functions compactly supported in $(0,\infty)$. The intertwining property for $R_\nu^{(a,b)}$ follows from and Lemma 5.1, and hence for $A_\nu^{(a,b)}$.
To calculate the action of $A_\nu^{(a,b)}$ on the little $q$-Jacobi function we use the last equation of to write $$\label{eq:KR5.12}
\varphi_\la(x;a,b;q) = \frac{(-x;q)_\infty}{(-bx/a;q)_\infty}
\, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{b\si,b/\si}{ab}
\ ;q,-x \right)}
$$ Using this in , interchanging summations, which is easily justified for $|x|<1$, and using the $q$-binomial theorem gives $$\bigl(A_\nu^{(a,b)}\varphi_\la(\cdot;a,b;q)\bigr)(x) =
\frac{(abq^\nu,-x;q)_\infty}
{(ab,-bxq^{-\nu}/a;q)_\infty}
\, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{b\si,b/\si}{abq^\nu}
\ ;q, -x \right)}
$$ and using again gives the result for $|x|<1$. The general case follows by analytic continuation in $x$, see , since the convergence in for $f$ the little $q$-Jacobi function is uniform on compact sets for $x$.
The statements for $\tilde A_\nu^{(a,b)}$ follow from the corresponding statements for $A_\nu^{(a,b)}$ and Lemma 5.2 and .
\[thm:KRTheorem2.3ii\] Let $a,b>0$, $ab<1$, $\nu>0$ and $\mu\in\C\setminus\Z_{\leq 0}$. Define the operator $$\begin{gathered}
\bigl( A_{\nu,\mu}(a,b)f\bigr)(x) =
\frac{(-bxq^\mu/a;q)_\infty}{(-bxq^{\mu-\nu}/a;q)_\infty}
\\ \times \sum_{k=0}^\infty f(xq^{\mu+k})\,
(ab)^k \frac{(q^\nu,-xq^\mu;q)_k}{(q,-bxq^\mu/;q)_k}
\, {\,_{3}\vp_{2} \left( \genfrac{.}{.}{0pt}{}{q^{-k}, q^\mu,-bxq^{\mu-\nu}/a}{q^{1-\nu-k},-xq^\mu}
\ ;q,q \right)}
$$ for any bounded function. Then ${\cL}^{(aq^\nu,bq^\mu)}\circ A_{\nu,\mu}(a,b) =
A_{\nu,\mu}(a,b)\circ {\cL}^{(a,b)}$ on the space of functions compactly supported in $(0,\infty)$. Moreover, $$\bigl(A_{\nu,\mu}(a,b) \varphi_\la(\cdot;a,b;q)\bigr)(x) =
\frac{(abq^{\nu+\mu};q)_\infty}{(ab;q)_\infty}
\, \varphi_\la(x;aq^\nu,bq^\mu;q).
$$
Define $$\begin{aligned}
A_{\nu,\mu}(a,b) &= \tilde A^{(aq^\nu,b)}_\mu\circ A_\nu^{(a,b)}\\
&= S(aq^\nu,bq^\mu)\circ A_\mu^{(b,aq^\nu)}\circ
S(b,aq^\nu)\circ A_\nu^{(a,b)}\end{aligned}$$ then it follows from Lemma 5.3 that the intertwining property is valid. The action on a function $f$ can be calculated and for $f$ compactly suppported in $(0,\infty)$ we find the explicit result with the ${}_3\vp_2$-series as kernel. We can extend the result to bounded $f$ if we require $\nu>0$.
The action of $A_{\nu,\mu}(a,b)$ on the little $q$-Jacobi function follows from Lemma 5.3.
These results can be used to obtain several identities involving the kernels of the transforms $A_\nu^{(a,b)}$ and $W_{\nu,\mu}(a,b)$, involving the transform of §\[sec:BHS-qdiff-nonpol\]. We refer to [@KoelR-RMJM02] for examples.
Exercises
---------
1. Gasper’s $q$-analogue [@Gasp (1.8)] of Erdélyi’s fractional integral is $$\begin{gathered}
{\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{ar\si,ar/\si}{abrs}
\ ;q,
-\frac{byq^l}{ar} \right)} =
\frac{(ab,rs;q)_\infty}{(q,abrs;q)_\infty}\sum_{k=0}^\infty
(ab)^k \frac{(q^{k+1},-byq^{k+l}/a;q)_\infty}
{(rsq^k,-byq^{k+l}/ar;q)_\infty} \\ \times
{\,_{3}\vp_{2} \left( \genfrac{.}{.}{0pt}{}{q^{-k},r,ar/b}{rs, -arq^{1-l-k}/by}
\ ;q,q \right)}
\, {\,_{2}\vp_{1} \left( \genfrac{.}{.}{0pt}{}{a\si,a/\si}{ab}
\ ;q,
-\frac{byq^{l+k}}{a} \right)}\end{gathered}$$ for $|rs|<1$, $|ab|<1$. Derive this from Theorem \[thm:KRTheorem2.3ii\].
Notes {#notes-4 .unnumbered}
-----
The result of this section are based on [@KoelR-RMJM02], and they focus on the fractional analogues of the $q$-derivative for the asymptotically free solution of the second order $q$-difference equation. Several other results related to this factorisation of the $q$-difference equation are presented in [@KoelR-RMJM02]. For the classical situation this is related to factoring the Jacobi function transform as a product of the Abel transform followed by the (standard) Fourier transform, see [@Koor-Jacobi] for details.
Askey-Wilson level {#sec:AW-level}
==================
At the level of the Askey-Wilson polynomials and Askey-Wilson functions one considers the second-order $q$-difference operator
Askey-Wilson polynomials {#ssec:AWpols}
------------------------
In this section we briefly recall the basic properties of the Askey-Wilson polynomials. We formulate these properties using the concept of duality.
The Askey-Wilson polynomials $p_n(x)=p_n(x;a,b,c,d;q)$, $n\in \N$, are defined by $$\label{eq:KSpn}
p_n(x) = p_n(x;a,b,c,d\mid q)= {\,_{4}\vp_{3} \left( \genfrac{.}{.}{0pt}{}{q^{-n}, q^{n-1}abcd, ax, ax^{-1}}{ab, ac, ad}
\ ;q,q \right)}$$ see [@AskeW]. Note that $p_n$ is a polynomial in $z=\mu(x) = \frac12(x+x^{-1})$, but we consider it as Laurent polynomial in $x$. Usually, see [@AskeW], [@KoekLS], [@KoekS], the normalization is chosen differently in order to make the Askey-Wilson polynomials symmetric in $a$, $b$, $c$ and $d$. The Askey-Wilson polynomials $\{p_n\}_{n\in\N}$ form a basis of the polynomial algebra $\C[z]=\C[x+x^{-1}]$ consisting of eigenfunctions of the Askey-Wilson second order $q$-difference operator $$\label{eq:KSLAW}
L=\alpha(x)(T_q-1)+\alpha(x^{-1})(T_q^{-1}-1),\qquad
\alpha(x)=\frac{(1-ax)(1-bx)(1-cx)(1-dx)}{(1-x^2)(1-qx^2)},$$ where $(T_q^{\pm 1}f)(x)=f(q^{\pm 1}x)$.
The eigenvalue of $L$ corresponding to the Askey-Wilson polynomial $p_n$ is $\mu(\gamma_n)$, where $\gamma_n=\tilde{a}q^n$, $\tilde{a}=\sqrt{q^{-1}abcd}$, and $$\label{eq:KSeigenvalue}
\bigl(Lp_n\bigr)(x) = \mu(\gamma_n) p_n(x), \qquad
\mu(\ga)=-1-\tilde{a}^2+\tilde{a}(\ga+\ga^{-1}).$$
In order to describe the orthogonality relations concisely we recall the dual parameters to $(a,b,c,d)$. We extend the definition of $\tilde{a}$ to $$\label{eq:KSdual}
\tilde{a}=\sqrt{q^{-1}abcd},\qquad \tilde{b}=ab/\tilde{a}=q\tilde{a}/cd,\qquad
\tilde{c}=ac/\tilde{a}=q\tilde{a}/bd,\qquad \tilde{d}=ad/\tilde{a}=
q\tilde{a}/bc.$$
\[lem:KSdualdomain\] The assignment $(a,b,c,d,t)\mapsto (\tilde{a},\tilde{b},\tilde{c},\tilde{d},
\tilde{t})$ defined by is an involution.
Lemma \[lem:KSdualdomain\] follows by calculation.
The orthogonality relations for the Askey-Wilson polynomials hold quite generally, see [@AskeW], [@GaspR], [@KoekS], but we assume $0<q<1$ as usual and moreover that $a,b,c$ and $d$ are positive and less than one. Then Askey and Wilson [@AskeW] proved the orthogonality relations $$\label{eq:KSorthoAW}
\frac{1}{2\pi i C_0}\int_{x\in {\mathbb{T}}}p_n(x)p_m(x)\Delta(x)\frac{dx}{x}
=\delta_{m,n}\frac{\underset{x=\gamma_0}{\hbox{Res}}\left(
\frac{\widetilde{\Delta}(x)}{x}\right)}
{\underset{x=\gamma_n}{\hbox{Res}}\left(\frac{\widetilde{\Delta}(x)}{x}\right)}$$ where $\de_{m,n}$ is the Kronecker delta and $\T$ is the counterclockwise oriented unit circle in the complex plane, with the weight function given by $$\label{eq:KSDelta}
\De(x)=\frac{\bigl(x^2,1/x^{2};q\bigr)_{\infty}}
{\bigl(ax,a/x,bx,b/x,cx,c/x,dx,d/x;q\bigr)_{\infty}},$$ and with $\widetilde{\De}(x)$ the weight function $\De(x)$ with respect to dual parameters. Here the positive normalization constant $C_0$ is given by the Askey-Wilson integral $$C_0=\frac{1}{2\pi i}\int_{x\in {\mathbb{T}}}\Delta(x)\frac{dx}{x}=
\frac{2\bigl(abcd;q\bigr)_{\infty}}
{\bigl(q,ab,ac,ad,bc,bd,cd;q\bigr)_{\infty}}.$$ Various different proofs of the Askey-Wilson integral exist, see e.g. references in [@Isma-LN]. The original proof follows by an elaborate residue calculus, and now there are many different approaches to the Askey-Wilson integral as well as to its various extensions, see [@GaspR] for references.
Having the dual parameters , the explicit expression for the Askey-Wilson polynomials show that the duality relation $$\label{eq:KSduality}
p_n(aq^m;a,b,c,d;q)=p_m(\tilde{a}q^n;\tilde{a},\tilde{b},\tilde{c},
\tilde{d};q),\qquad m,n\in \N$$ holds. The deeper understanding of the duality stems from affine Hecke algebraic considerations, see [@NoumS]. This duality takes an even nicer form in the case of the Askey-Wilson functions, see \[ssec:AWfnctiont\]. The duality also shows that the three-term recurrence relations for the Askey-Wilson polynomials $p_m(\cdot;\tilde{a},\tilde{b},\tilde{c},\tilde{d};q)$, $m\in \N$, follows from the eigenvalue equations $Lp_n=\mu(\ga_n)p_n$, $n\in\N$, by applying the duality , see e.g. [@AskeW] and [@NoumS].
The orthogonality relations written in the form exhibit the duality of the Askey-Wilson polynomials on the level of the orthogonality relations, since it expresses the quadratic norms explicitly in terms of the dual weight function $\widetilde{\Delta}$. This description of the quadratic norms was proved in [@NoumS].
Askey-Wilson function transform {#ssec:AWfnctiont}
-------------------------------
We define the Askey-Wilson function transform and we state the main result concerning the Askey-Wilson function transform. For this we more generally need to consider general, i.e. non-polynomial, eigenfunctions to $$\label{eq:KSeigenvalueequation}
\bigl(Lf\bigr)(x)=\mu(\gamma)f(x),$$ which reduces to the Askey-Wilson polynomial for $\gamma=\gamma_n$, $n\in\N$, and enjoys the same duality properties. The solutions of have been studied by Ismail and Rahman [@IsmaR-TAMS].
Two linearly independent solutions of the eigenvalue equation can be derived from Ismail’s and Rahman’s [@IsmaR-TAMS (1.11)–(1.16)] solutions for the three term recurrence relation of the associated Askey-Wilson polynomials. The solutions are given in terms of very well poised ${}_8\varphi_7$ series, in particular $$\label{eq:KSphi}
\begin{split}
\phi_{\gamma}(x)=&\phi_\ga(x;a,b,c; d\mid q) =\frac{\bigl(qax\gamma/\tilde{d}, qa\gamma/\tilde{d}x;
q\bigr)_{\infty}}
{\bigl(\tilde{a}\tilde{b}\tilde{c}\gamma, q\gamma/\tilde{d},
q\tilde{a}/\tilde{d}, qx/d, q/dx;q\bigr)_{\infty}}\\
&\qquad\times{}_8W_7\bigl(\tilde{a}\tilde{b}\tilde{c}\gamma/q;
ax, a/x, \tilde{a}\gamma,
\tilde{b}\gamma, \tilde{c}\gamma;q,q/\tilde{d}\gamma\bigr),\qquad
|q/\tilde{d}\gamma|<1
\end{split}$$ is a solution to . This solution is called the *Askey-Wilson function*. Lemma \[lem:KSreductionAWfunction\] shows that the Askey-Wilson function satisfies the same duality, and moreover extends the Askey-Wilson polynomials of .
\[lem:KSreductionAWfunction\] The Askey-Wilson function satisfies the duality and reduction formulas $$\begin{gathered}
\phi_{\gamma}(x;a;b,c;d\mid q)=\phi_x(\gamma;\tilde{a};\tilde{b},
\tilde{c};\tilde{d}\mid q) \\
\phi_{\gamma_n}(x)=\frac{1}
{\bigl(bc,qa/d,q/ad;q\bigr)_{\infty}}
p_n(x),\qquad n\in \N.\end{gathered}$$ and $\phi_{\gamma^{\pm 1}}(x^{\pm 1})
=\phi_{\gamma}(x)$ for all possible choices.
Note that duality for the Askey-Wilson polynomials is a special case of the duality of $\phi_{\gamma}$ in Lemma \[lem:KSreductionAWfunction\].
The proof rests on a formula expressing a very-well poised ${}_8\varphi_7$-series as a sum of two balanced ${}_4\varphi_3$-series given by Bailey’s formula [@GaspR (III.36)]. This gives $$\label{eq:KS43presentation}
\begin{split}
\phi_{\gamma}(x)=&\frac{1}
{\bigl(bc,qa/d,q/ad;q\bigr)_{\infty}}
{}_4\phi_3\left(\begin{matrix} ax, a/x, \tilde{a}\gamma, \tilde{a}/\gamma\\
ab, ac, ad \end{matrix}; q,q\right)\\
+&\frac{\bigl(ax, a/x, \tilde{a}\gamma, \tilde{a}/\gamma, qb/d,
qc/d;q\bigr)_{\infty}}
{\bigl(qx/d, q/dx, q\gamma/\tilde{d}, q/\tilde{d}\gamma,
ab,ac,bc,qa/d,ad/q;q\bigr)_{\infty}}\\
&\qquad\qquad\times
{}_4\phi_3\left(\begin{matrix} qx/d, q/dx, q\gamma/\tilde{d},
q/\tilde{d}\gamma\\
qb/d, qc/d, q^2/ad \end{matrix}; q,q\right),
\end{split}$$ hence $\phi_{\gamma}(x)$ extends to a meromorphic function in $x$ and $\gamma$ for generic parameters $a,b,c$ and $d$, with possible poles at $x^{\pm 1}=q^{1+k}/d$, $k\in\N$, and $\gamma^{\pm 1}=q^{1+k}/\tilde{d}$, $k\in \N$. It follows from that $\phi_{\gamma^{\pm 1}}(x^{\pm 1})
=\phi_{\gamma}(x)$ (all sign combinations possible), and that $\phi_{\gamma}$ satisfies the duality relation by inspection. Finaly, observe that the meromorphic continuation of $\phi_{\gamma}(x)$ implies that $$\label{redpol}
\phi_{\gamma_n}(x)=\frac{1}
{\bigl(bc,qa/d,q/ad;q\bigr)_{\infty}}
p_n(x),\qquad n\in\N.$$
Indeed, the factor $\bigl(\tilde{a}/\gamma;q\bigr)_{\infty}$ in front of the second ${}_4\varphi_3$ in vanishes for $\gamma=\gamma_n=
\tilde{a}q^n$ for $n\in\N$.
At this stage we need to specify a particular parameter domain for the five parameters $(a,b,c,d,t)$ in order to ensure positivity of measures.
\[def:AWsetV\] Let $V$ be the set of parameters $(a,b,c,d,t)\in \R^{5}$ satisfying $$t<0,\qquad0<b,c\leq a<d/q,\qquad bd,cd\geq q,\qquad ab,ac<1.$$
Observe that $b,c<1$ and $d>q$ for all $(a,b,c,d,t)\in V$. We extend the duality of to $$\label{eq:KSdualt}
\tilde{t}=1/qadt.$$
The domain $V$ is self-dual extending Lemma \[lem:KSdualdomain\].
\[lem:KS2dualdomain\] The assignment $(a,b,c,d,t)\mapsto (\tilde{a},\tilde{b},\tilde{c},\tilde{d},
\tilde{t})$ defined by and , is an involution on $V$.
Again by direct verification.
From now on we consider $(a,b,c,d,t)\in V$ fixed.
In order to motivate the measure which we will introduce we look at other solutions for the eigenvalue equation for $L$. Observe that the eigenvalue equation is asymptotically of the form $$\label{asymptequation}
\tilde{a}^2\bigl(f(qx)-f(x)\bigr)+\bigl(f(q^{-1}x)-f(x)\bigr)=\mu(\gamma)f(x)$$ when $|x|\rightarrow \infty$. For generic $\gamma$, the asymptotic eigenvalue equation has a basis $\{\Phi_{\gamma}^{\free}, \Phi_{\gamma^{-1}}^{\free}\}$ of solutions on the $q$-line $I =\{ dtq^k\}_{k\in\Z}$, where $$\Phi_{\ga}^{\free}(dtq^k)=\bigl(\tilde{a}\gamma)^{-k},\qquad
k\in\Z.$$ Furthermore, for generic $\ga$ there exists a unique solution $\Phi_{\ga}(x)$ of the eigenvalue equation on $I$ of the form $\Phi_{\gamma}(x)=\Phi_{\gamma}^{\free}(x)g(x)$, where $g$ has a convergent power series expansion around $\infty$ with constant coefficient equal to one. The solution $\Phi_{\gamma}$ is the *asymptotically free solution* of the eigenvalue equation .
Actually, an explicit expression for $\Phi_{\gamma}$ can be obtained from the study of Ismail and Rahman on the associated Askey-Wilson polynomials, in which they study solutions of the eigenvalue equation . Starting with [@IsmaR-TAMS (1.13)] and applying the transformation formula [@GaspR (III.23)] for very well poised ${}_8\varphi_7$’s we obtain $$\begin{split}
\Phi_{\gamma}(x)=
&\frac{\bigl(qa\gamma/\tilde{a}x,qb\gamma/\tilde{a}x,qc\gamma/\tilde{a}x,
q\tilde{a}\gamma/dx,d/x;q\bigr)_{\infty}}
{\bigl(q/ax,q/bx,q/cx,q/dx,q^2\gamma^2/dx;q\bigr)_{\infty}}\\
&\times{}_8W_7\bigl(q\gamma^2/dx;q\gamma/\tilde{a},q\gamma/\tilde{d},
\tilde{b}\gamma,\tilde{c}\gamma,q/dx;q,d/x\bigr)\Phi_{\gamma}^{free}(x)
\end{split}$$ for $x\in I$ with $|x|\gg 0$. We now expand the Askey-Wilson function $\phi_{\ga}(x)$ as a linear combination of the asymptotically free solutions $\Phi_{\ga}(x)$ and $\Phi_{\ga^{-1}}(x)$ for $x\in I$ with $|x|\gg 0$. Since these are all solutions to the same eigenvalue equation, we can expect a relation with coefficients being constants or $q$-periodic functions.
\[prop:KScprop\] Let $x\in I$ with $|x|\gg 0$. Then we have the $c$-function expansion $$\phi_{\gamma}(x)=\widetilde{c}(\gamma)\Phi_{\gamma}(x)+
\widetilde{c}(\gamma^{-1})
\Phi_{\gamma^{-1}}(x)$$ for generic $\ga$, where the $c$-function is given by $$c(\ga)= c(\ga;a;b,c;d;q;t) = \frac{1}{\bigl(ab,ac,bc,qa/d;q\bigr)_{\infty}\theta(qadt)}
\frac{\bigl(a/\gamma,b/\gamma,
c/\gamma;q\bigr)_{\infty}\theta(\gamma/dt)}
{\bigl(q\gamma/d,1/\gamma^2;q\bigr)_{\infty}}.$$ using the notation and $\widetilde{c}(\ga)=c(\ga;\tilde{a};\tilde{b},\tilde{c}; \tilde{d};q;\tilde{t})$.
We call $\widetilde{c}(\gamma)=
c(\gamma;\tilde{a};\tilde{b},\tilde{c};\tilde{d};q;\tilde{t})$ the dual $c$-function, with the dual parameter $\tilde{t}$ defined by .
The proof requires some calculation. The essential ingredients are as follows. First apply Bailey’s three term recurrence relation [@GaspR (III.37)] with its parameters specialized as $$a\rightarrow q\gamma^2/dx,\quad
b\rightarrow q/dx,\quad
c\rightarrow q\gamma/\tilde{a},\quad
d\rightarrow q\gamma/\tilde{d},\quad
e\rightarrow \tilde{b}\gamma,\quad
f\rightarrow \tilde{c}\gamma.$$ This gives an expansion of the required form with explicit coefficients $\widetilde{c}(\gamma)$ and $\widetilde{c}(\gamma^{-1})$, which at a first glance still depend on $x\in I$. Using the theta function and its functional equation we see that the coefficients are independent of $x$.
For the moment we furthermore assume that $x\mapsto 1/c(x)c(x^{-1})$ only has simple poles. This imposes certain generic conditions on the parameters $(a,b,c,d,t)$, which can be removed at a later stage by a continuity argument.
It is convenient to renormalize the function $1/c(x)c(x^{-1})$ as follows, $$\label{eq:KSWeight}
W(x)=\frac{1}{c(x)c(x^{-1})c_0}
=\frac{\bigl(qx/d,q/dx,x^2,1/x^{2};q\bigr)_{\infty}}
{\bigl(ax,a/x,bx,b/x,cx,c/x;q\bigr)_{\infty}\theta(dtx)\theta(dt/x)},$$ where $c_0$ is the positive constant $$c_0=\frac{\bigl(ab,ac,bc,qa/d;q\bigr)_{\infty}^2\theta(adt)^2}{a^2}.$$ It follows from and that $$\label{eq:KSdifference}
W(x)=\frac{\theta(dx)\theta(d/x)}{\theta(dtx)\theta(dt/x)}\Delta(x).$$ By , the quotient of theta functions in is a $q$-periodic function. In particular, the weight function $W(x)$ differs from $\Delta(x)$ only by a $q$-periodic function, but this factor introduces additional poles which arise in the orthogonality (or spectral) measure.
Let $S$ be the discrete subset $$\label{eq:KSS}
\begin{split}
S&=\{x\in {\mathbb{C}} \,\, | \,\,
|x|>1, c(x)=0\}=S_+\cup S_-,\\
S_+&=\{aq^k \,\, | \,\, k\in\N,\,\, aq^k>1\},\\
S_-&=\{dtq^k \,\, | \,\, k\in \Z,\,\, dtq^k<-1\}.
\end{split}$$ By $\widetilde{S}$ and $\widetilde{S}_{\pm}$ we denote the subsets $S$ and $S_{\pm}$ with respect to dual parameters. We define a measure $\nu=\nu(\cdot;a;b,c;d;t;q)$ by $$\label{eq:KSnu}
\begin{split}
\int f(x)&d\nu(x)=\frac{K}{4\pi i}\int_{x\in {\mathbb{T}}}f(x)
W(x)\frac{dx}{x}\\
&+\frac{K}{2}\sum_{x\in S}f(x)\underset{y=x}{\hbox{Res}}
\left(\frac{W(y)}{y}\right)
-\frac{K}{2}\sum_{x\in S^{-1}}f(x)
\underset{y=x}{\hbox{Res}}
\left(\frac{W(y)}{y}\right),
\end{split}$$ where the positive constant $K$ is given by $$\label{eq:KSK}
K=\bigl(ab,ac,bc,qa/d,q;q\bigr)_{\infty}\sqrt{\frac{\theta(qt)\theta(adt)
\theta(bdt)\theta(cdt)}{qabcdt^2}}.$$ This particular choice of normalization constant for the measure $\nu$ is justified in Theorem \[thm:KSmain\], since the corresponding transform is made an isometry.
In view of , we can relate the discrete masses $\nu\bigl(\{x\}\bigr)(=-\nu\bigl(\{x^{-1}\}\bigr))$ for $x\in S_+$ to residues of the weight function $\Delta(\cdot)$, which were written down explicitly in [@AskeW], see also [@GaspR (7.5.22)] in order to avoid a small misprint in [@AskeW]. Explicitly, we obtain for $x=aq^k\in S_+$ with $k\in\N$ the expression $$\label{eq:KSweightplus}
\nu\bigl(\{aq^k\}\bigr)=\frac{\bigl(qa/d,q/ad,1/a^2;q\bigr)_{\infty}}
{\bigl(q,ab,b/a,ac,c/a;q\bigr)_{\infty}\theta(adt)\theta(dt/a)}
\frac{(1-a^2q^{2k})}{(1-a^2)}\frac{K}{2\tilde{a}^{2k}}$$ for the corresponding discrete weight. For fixed $k\in\N$, the right hand side of gives the unique continuous extension of the discrete weight $\nu\bigl(\{aq^k\}\bigr)$ and $-\nu\bigl(\{a^{-1}q^{-k}\}\bigr)$ to all parameters $(a,b,c,d,t)\in V$ satisfying $aq^k>1$. Furthermore, the (continuously extended) discrete weight $\nu\bigl(\{aq^k\}\bigr)$ is strictly positive for these parameter values. Note that $S_+$ gives a finite number of discrete mass points in the measure $\nu$.
A similar argument can be applied for the discrete weights $\nu\bigl(\{x\}\bigr)(=-\nu\bigl(\{x^{-1}\}\bigr))$ with $x\in S_-$. Explicitly we obtain for $x=dtq^k\in S_-$ with $k\in\Z$, $$\label{eq:KSweighttoinfty}
\begin{split}
\nu\bigl(\{dtq^k\}\bigr)=&\frac{\bigl(qt,q/d^2t;q\bigr)_{\infty}}
{\bigl(q,q,a/dt,b/dt,c/dt,adt,bdt,cdt;q\bigr)_{\infty}}
\\
&\times
\frac{\bigl(1/t,a/dt,b/dt,c/dt;q\bigr)_{-k}}
{\bigl(q/adt,q/bdt,q/cdt,q/d^2t;q\bigr)_{-k}}
\left(1-\frac{1}{d^2t^2q^{2k}}\right)\frac{K\tilde{a}^{2k}}{2}.
\end{split}$$ As for $\nu\bigl(\{x\}\bigr)$ with $x\in S_+$, we use the right hand side of to define the strictly positive weight $\nu\bigl(\{dtq^k\}\bigr)(=-\nu\bigl(\{d^{-1}t^{-1}q^{-k}\}\bigr))$ for all $(a,b,c,d,t)\in V$ satisfying $dtq^k<-1$. Note that $S_-$ gives an infinite number of discrete mass points in the measure $\nu$.
We see that the definition of the measure $\nu$ can be extended to arbitrary parameters $(a,b,c,d,t)\in V$ using the continuous extensions of its discrete weights in , . The resulting measure $\nu$ is a positive measure for all $(a,b,c,d,t)\in V$.
\[def:KSdefhilbert\] Let $\cH=\cH(a;b,c;d;t;q)$ be the Hilbert space consisting of $L^2$-functions $f$ with respect to $\nu$ which satisfy $f(x)=f(x^{-1})$ $\nu$-almost everywhere.
We write $\widetilde{\nu}$ for the measure $\nu$ with respect to dual parameters $(\tilde{a},\tilde{b},\tilde{c},\tilde{d},\tilde{t})$, and $\widetilde{\cH}$ for the associated Hilbert space ${\cH}$.
Let ${\cD}\subset \cH$ be the dense subspace of functions $f$ with compact support, i.e. $${\cD}=\{f\in \cH \,\, | \,\, f(dtq^{-k})=0,\ k\gg 0\},$$ and define $$\label{eq:KSF}
\bigl({\cF}f\bigr)(\ga)=\int f(x){\overline{\phi_{\gamma}(x)}}
d\nu(x),\qquad f\in \cD$$ for generic $\ga\in \C\setminus \{0\}$.
We write ${\widetilde{\mathcal{D}}}\subset \widetilde{{\mathcal{H}}}$ (respectively $\widetilde{\mathcal{F}}$) for the dense subspace ${\mathcal{D}}$ (respectively the function transform $\mathcal{F}$) with respect to dual parameters $(\tilde{a},\tilde{b},\tilde{c},\tilde{d},\tilde{t})$.
\[thm:KSmain\] Let $(a,b,c,d,t)\in V$. The transform ${\cF}$ extends to an isometric isomorphism ${\cF} \colon {\cH}\to \widetilde{\cH}$ by continuity. The inverse of ${\cF}$ is given by ${\widetilde{\cF}} \colon \widetilde{{\cH}} \to {\cH}$.
The isometric isomorphism $\cF\colon {\cH}\rightarrow
\widetilde{\cH}$ is called the *Askey-Wilson function transform*.
We will not prove Theorem \[thm:KSmain\] in these notes, but we refer to the original proof in [@KoelS-IMRN2001].
Exercises
---------
1. \[ex:AW1\] Give a factorisation for the second order $q$-difference operator in terms of a lowering and raising operator.
Notes {#notes-5 .unnumbered}
-----
The Askey-Wilson polynomials have been introduced by Askey and Wilson in [@AskeW], and these polynomials are on top of the continuous part of the $q$-analogue of the Askey scheme, see [@AskeW], [@KoekLS], [@KoekS]. The discrete counterpart, the $q$-Racah polynomials are on top of the discrete part of the $q$-analogue of the Askey scheme. The solutions of the second order $q$-difference equation as considered here were studied by Ismail and Rahman [@IsmaR-TAMS], where they studied the associated Askey-Wilson polynomials. The corresponding Askey-Wilson function transform as in Theorem \[thm:KSmain\] is due to [@KoelS-IMRN2001]. It is remarkable that the $q=1$ analogue of this transform, the Wilson function transform of Groenevelt [@Groe-WT], is only established after the $q$-analogue of the Askey-Wilson function transform. Moreover, Groenevelt’s Wilson transform comes into two versions, which also map Wilson polynomials to Wilson polynomials (with dual parameters). The $q$-analogue of this statement is related to the expansion of the Askey-Wilson function in terms of a series involving a product of the Askey-Wilson polynomials and the Askey-Wilson polynomials with dual parameters, see Stokman [@Stok-JAT]. Other approaches to the same or closely related functions can be found in e.g. Haine and Iliev [@HainI], Ruijsenaars [@Ruij-AW] and Suslov [@Susl]. As to multivariable extensions, Stokman [@Stok-AoM] established the analogue of Proposition \[prop:KScprop\]. For Exercise \[ex:AW1\] one can e.g. consult [@GaspR], [@KoekS]. There seems not to be an appropriate analogue of the Frobenius method for the Askey-Wilson difference equation, see [@IsmaS] for related Taylor expansions.
Matrix-valued extensions {#sec:MVextensions}
========================
The hypergeometric differential equation has a matrix-valued analogue, studied by Tirao [@Tira]. This matrix-valued differential equation plays an important role in the study of matrix-valued spherical functions on (especially compact) symmetric spaces and extensions of these result to more general matrix-valued orthogonal polynmials, see e.g. [@GrunPT] [@KoelvPR1], [@KoelvPR2], [@KoeldlRR]. See also [@DamaPS] and the lecture notes [@Koel-OPSFA2016], as well as references given there, for more information on matrix-valued orthogonal polynomials. This section is of a preliminary nature and based on the Bachelor thesis of Nikki Jaspers [@Jasp] and finds its origin in the paper [@AldeKR]. The results can be considered as $q$-analogues of the solutions of the matrix-valued hypergeometric series at $0$ and $\infty$, see [@Tira] and [@RomaS].
Vector-valued basic hypergeometric $q$-difference equation
----------------------------------------------------------
Recall the basic hypergeometric equation , which we now adapt to $$\label{eq:vectorvBHqDE}
(q-z)\Id f(q^{-1}z)+\bigl((A+B)z-C-q\Id\bigr)f(z)+(C-ABz)f(qz) = 0$$ where $A,B,C\in \End(\C^N)$, i.e. linear maps from $\C^N$ to $\C^N$, and $f\colon \C\to \C^N$ is the unknown vector-valued function, which we want to satisfy . Note that in particular in the identity $\Id\in \End(\C^N)$ and $0\in \C^N$.
Note that $N=1$ brings us back to and the results presented in these lecture notes, but also the case of commuting diagonalizable $A$, $B$ and $C$ bring us back to .
Generically the dimension of the solution space of the vector-valued basic $q$-difference equation is $2N$.
\[rmk:eq:vectorvBHqDE\] More generally, we can consider with $A+B$ and $AB$ replaced by more generally $U$ and $V$. In the case $N=1$ this is equivalent, but in the vector-valued case this more general. In the case of the hypergeometric differential operator this is dicussed by Tirao [@Tira]. We will not discuss this case, see [@Jasp] for the $q$-case.
Note that we can generalize to $$\label{eq:matrixvBHqDE}
(q-z)\Id F(q^{-1}z)+\bigl((A+B)z-C-q\Id\bigr)F(z)+(C-ABz)F(qz) = 0$$ where $A,B,C\in \End(\C^N)$ and $F\colon \C\to \End(\C^N)$ is the unknown matrix-valued function, which we want to satisfy . Note that in now $0\in\End(\C^N)$ is the zero-matrix. Moreover, if $F(z)$ is any solution to , then $f(z) = F(z)f_0$ for a fixed vector $f_0\in \C^N$ satisfies .
Solutions of matrix-valued q-hypergeometric equation
----------------------------------------------------
The Frobenius method can be extended to case of . We first consider expansions around $z=0$. For a matrix $C\in \End(\C^N)$ we let $\si(C)$ denote its spectrum, i.e. the zeros of the characteristic polynomial of $C$. For $A,B,C\in \End(\C^N)$ with $q^{-\N}\cap \si(C)=\emptyset$ we define for $n\in \N$ the product $$\big(A,B;C;q\big)_n = \displaystyle\prod_{k=0}^{\substack{n-1\\ \gets}}(I-q^kC)^{-1}(I-q^kA)(I-q^kB)$$ where $$\displaystyle\prod_{k=0}^{\substack{n-1\\ \gets}} a_k = a_{n-1}\cdots a_0$$ for non-commuting elements $a_k$. In case $A,B,C$ are $1\times 1$-matrices, this reduces to $\frac{(A;q)_n(B(;q)_n}{(C;q)_n}$.
For $\al\in \C$ we define the matrix-valued basic hypergeometric series $$\label{eq:MVbasichypseries}
{}_2\Phi_1^\al(A,B;C;q,z)= \displaystyle\sum_{n=0}^\infty\big(\al A, \al B; \al C;q\big)_n
\frac{z^n}{(\al q;q)_n}$$ assuming $q^{-\N}\cap \si(\al C)=\emptyset$. In case $\al=1$, we drop it from the notation, i.e. ${}_2\Phi_1(A,B;C;q,z)={}_2\Phi_1^1(A,B;C;q,z)$.
Note that obvious symmetry $A\leftrightarrow B$ of the scalar case no longer holds, since $AB\not= BA$ in general.
We can now describe the solutions to in the generic case, i.e. when the eigenvalues of $C$ are sufficiently generic.
\[thm:MVBHSatzero\] Assume $C$ is diagonalizable, so that $\si(C)=\{c_1,\cdots ,c_N\}$ with $c_i\not=c_j$ for $i\not=j$. Assume $c_i\not=0$ for all $i$. Let $f_i\not=0$ be the corresponding eigenvectors of $C$; $Cf_i=c_i f_i$, $1\leq i \leq N$. Assume furthermore that $\si(C)\cap q^\Z=\emptyset$ and that $c_i/c_j \notin q^\Z$ for $i\not= j$. Then the vector-valued functions $${}_2\Phi_1(A,B;C;q,z) e_i$$ for any basis $\{e_1,\cdots,e_N\}$ of $\C^N$ and the vector-valued functions $$z^{1-\log_q(c_i)}\, {}_2\Phi_1^{q/c_i}(A,B;C;q,z)f_i, \qquad 1\leq i \leq N$$ span the solution space of (over the $q$-periodic functions).
The proof follows the Frobenius method for the basic hypergeometric $q$-difference equation. So we assume that we have a solution of the form $\sum_{n=0}^\infty f_n z^{n+\mu}$, $f_n\in \C^N$, where $f_0\not=0$. Plugging this Ansatz into we get $$\begin{gathered}
0= (q-z)\Id \sum_{n=0}^\infty f_n(q^{-1}z)^{n+\mu} +\bigl((A+B)z-(C+q\Id)\bigr)
\sum_{n=0}^\infty f_n z^{n+\mu} + (C-ABz)\sum_{n=0}^\infty f_n (qz)^{n+\mu} \\
= \bigl( f_0q^{1-\mu} - \bigl(C+q) f_0 + q^\mu C f_0 \bigr) z^{\mu}
+ \\
\sum_{n=1}^\infty z^{\mu+n}
\Bigl( \bigl( Cq^{\mu+n} -(C+q) + q^{1-\mu-n}\bigr)f_n + \bigl( -ABq^{\mu+n-1}
+ A+B - q^{1-\mu-n}\bigr)f_{n-1} \Bigr)\end{gathered}$$ So in particular, the vector-valued coefficient needs to vanish, and this gives the indicial equation $$\bigl( (q^{1-\mu} -q) - (1-q^\mu) C\bigr) f_0 =(1-q^\mu)(q^{1-\mu} -C)f_0 = 0$$ This gives $2N$ for $q^\mu$, namely $q^\mu=1$ and $f_0\in \C^N$ arbitrary and $Cf_0=q^{1-\mu} f_0$, i.e. $f_0=f_i$ and $q^{1-\mu}=c_i$ for $1\leq i \leq N$.
With each of these solutions we then need to solve recursively $$\begin{gathered}
\bigl( Cq^{\mu+n} -(C+q) + q^{1-\mu-n}\bigr)f_n = \bigl( ABq^{\mu+n-1}
-(A+B) + q^{1-\mu-n}\bigr)f_{n-1} \quad \Longrightarrow \\
-(1-q^{\mu+n})(C-q^{1-\mu-n})f_n = (Aq^{\mu+n-1}-1)(B-q^{1-n-\mu})f_{n-1} \quad \Longrightarrow \\
(1- q^{\mu+n-1}C)f_n = \frac{1}{(1-q^{\mu+n})}(1-q^{\mu+n-1}A)(1-q^{n+\mu-1}B)f_{n-1}\end{gathered}$$ since $q^{\mu+n}\not= 1$ under the assumptions on $C$ and $q^{1-\mu}\in \si(C)$.
In case $q^\mu=1$, we find $$\begin{gathered}
f_n = \frac{1}{(1-q^{n})}(1- q^{n-1}C)^{-1}(1-q^{n-1}A)(1-q^{n-1}B)f_{n-1}
= \frac{1}{(q;q)_n} (A,B;C;q)_n f_0\end{gathered}$$ without condition on $f_0$. This gives the first set of solutions by taking $\mu=0$. All other solutions of $q^\mu=1$ lead to the same solution up to $q$-periodic functions, cf. the proof of Proposition \[prop:solofBqDEat0atinfty\].
In the other case, we have $f_0=f_i$ and $q^{1-\mu}=c_i$ for some $1\leq i \leq N$. Take $\mu=1-\log_q(c_i)$, so $q^\mu=q/c_i$. Then the recurrence is $$\begin{gathered}
f_n = \frac{1}{(1-q^{1+n}/c_i)}(1- q^{n}c_i^{-1}C)^{-1}(1-q^{n}c_i^{-1}A)(1-q^{n}c_i^{-1}B)f_{n-1} \\
= \frac{1}{(q/c_i;q)_n} (qc_i^{-1}A,qc_i^{-1}B;qc_i^{-1}C;q)_n f_i\end{gathered}$$ and this gives the other set of solutions. Again, choosing a different solution of $q^\mu=q/c_i$ leads to the same solution up to a $q$-periodic function.
Since the space of solutions is $2N$ dimensional, and the set of solutions are linearly independent we have obtained all solutions. The linear independence follows since the first set is linearly independent as analytic solutions with linearly independent values at $z=0$, and the other solutions all have different behaviour as $z\to 0$.
In order to describe the solutions at $\infty$ we introduce the notation $$\label{eq:MVsolastinfty}
\begin{split}
[A,B;C;\al;q]_n & = \displaystyle\prod_{k=0}^{\substack{n-1\\ \gets}}
(A-\al q^k)^{-1}(B-\al q^k)^{-1} (C-\al q^k) \\
\Theta^\al(A,B;C;q,z) &= \sum_{n=0}^\infty (\al/q;q)_n [A,B;C;\al;q]_n z^n
\end{split}$$ where we assume that all inverses of the matrices involved exist.
\[thm:MVBHSatinfty\] Assume that $A$ and $B$ are diagonalizable with non-zero eigenvalues and such that the following genericity conditions on the spectra $\si(A)$, $\si(B)$ hold; $$\begin{gathered}
\si(A)\cap \si(B)=\emptyset, \quad \si(A)\cap \si(B)q^{1+\N}=\emptyset,
\quad \si(B)\cap \si(B)q^{1+\N}=\emptyset,\end{gathered}$$ Moreover, let $\si(A)=\{a_1,\cdots, a_N\}$, $\si(B)=\{b_1,\cdots, b_N\}$, with $Af_i^A=a_i f_i^A$, $Bf_i^A=b_i f_i^B$ then the solutions $$\begin{gathered}
z^{-\log_q(b_i)} \Theta^{qb_i}(B,A;C;q,qz^{-1}) f_i^B, \qquad 1\leq i \leq N, \\
z^{-\log_q(a_i)} \Theta^{qa_i}(B,A;C;q,qz^{-1}) (a_i-B)^{-1} f_i^A, \qquad 1\leq i \leq N, \\\end{gathered}$$ are linearly independent solutions of .
Note that in case $N=1$, this leads to the solutions $u_3$ and $u_4$ of Proposition \[prop:solofBqDEat0atinfty\].
Now assume that a solution at $\infty$ has the expansion $\sum_{n=0}^\infty f_n z^{-n-\mu}$ for $f_n\in \C^N$. Plugging this expression in in and rearranging terms we find $$\begin{gathered}
0 = z^{-1-\mu} \bigl( -q^\mu + (A+B) -q^{-\mu} AB\bigr) f_0
+ \\
\sum_{n=0}^\infty
z^{-n-\mu} \Bigl( \bigl( q^{n+\mu+1} -(C+q) + q^{-n-\mu}C\bigr) f_n
+ \bigl( -q^{n+\mu+1} + (A+B) - q^{-1-n-\mu} AB\bigr) f_{n+1} \Bigr).\end{gathered}$$ In order to have a solution, the coefficients of the powers of $z$ have to be zero. The first equation is the indicial equation (for $z=\infty$) $$-q^{-\mu}(q^\mu-A)(q^{\mu}-B) f_0 = \bigl( -q^\mu + (A+B) -q^{-\mu} AB\bigr) f_0 = 0$$ Since $A$ and $B$ are diagonalizable with $N$ different non-zero eigenvalues and $\si(A)\cap \si(B) =\emptyset$, we have $2N$ solutions for the indicial equation. In the first case, $f_0$ is an eigenvector for $B$ with eigenvalue $b_i=q^\mu$, say $f_0=f_i^B$. So $$q^\mu = b_i, \qquad f_0 = f_i^B, \qquad 1\leq i \leq N.$$ In the second case, we find that $(q^\mu-B)f_0$ is an eigenvector of $A$ for the eigenvalue $a_i=q^\mu$, i.e. $$q^\mu = a_i, \qquad f_0 = (a_i-B)^{-1} f_i^A, \qquad 1\leq i \leq N,$$ where $A f_i^A= a_i f_i^A$.
In the first case we find the recursion $$\begin{gathered}
\bigl( b_iq^{n+1} - (A+B) + b_i^{-1}q^{-1-n} AB\bigr) f_{n+1} =
\bigl( b_iq^{n+1} -(C+q) + q^{-n}b_i^{-1}C\bigr) f_n \quad \Longrightarrow\\
(b_iq^{n+1}-A)(1-b_i^{-1}q^{-1-n}B) f_{n+1} = (b_iq^{n+1}-C)(1-b_i^{-1}q^{-n})
f_n \quad \Longrightarrow\\
(A- b_iq^{n+1}) (B- b_iq^{n+1}) \, f_{n+1} =
q\, (1-q^nb_i) (C-b_iq^{n+1}) f_n \quad \Longrightarrow\\
f_{n+1} = q\, (1-q^nb_i)
(B- b_iq^{n+1})^{-1} (A- b_iq^{n+1})^{-1} (C-b_iq^{n+1}) f_n \\
= q^{n+1}\, (b_i;q)_{n+1} [B,A;C; qb_i;q]_{n+1} f_0 =
q^{n+1}\, (b_i;q)_{n+1} [B,A;C; qb_i;q]_{n+1} f_i^B,\end{gathered}$$ and this gives the first set of solutions. Other choices of $\mu$ lead to the same solution up to a $q$-constant function.
In the second case we find the recursion $$\begin{gathered}
\bigl( a_iq^{n+1} -(C+q) + a_i^{-1}q^{-n}C\bigr) f_n
= \bigl( a_iq^{n+1} - (A+B) + a_i^{-1}q^{-1-n} AB\bigr) f_{n+1} \quad \Longrightarrow\\
f_{n+1} = q (1-a_iq^n) (B-a_iq^{n+1})^{-1}(A-a_iq^{n+1})^{-1} (C-a_iq^{n+1}) f_n = \\
q^{n+1}\, (a_i;q)_{n+1} [B,A;C; qa_i;q]_{n+1} (a_i-B)^{-1} f_i^A\end{gathered}$$ which gives the second solution.
Since the singularities of the solutions are all different by the genericity assumptions on the eigenvalues, linear independence follows.
It is now a natural question to ask if one can develop an analoguous theory for matrix-valued little $q$-Jacobi polynomials using the solutions developed in this section. A first attempt is in [@AldeKR]. For this one needs to study when a matrix-valued basic hypergeometric series terminates, and we can directly see that $${}_2\Phi_1^1(A,B;C;q,z) f$$ is a polynomial of degree $l$ if $f\not\in \Ker \bigl( (1-q^kA)(1-q^kB)\bigr)$, $1\leq k<l$ and $f\in \Ker \bigl( (1-q^lA)(1-q^lB)\bigr)$.
For an analogous theory of the little $q$-Jacobi function, we need to connect the solutions of Theorem \[thm:MVBHSatzero\] with the solutions of Theorem \[thm:MVBHSatinfty\], i.e. we need the matrix-valued analogue of Watson’s formulas , . First results on this approach can be found in [@Jasp].
Exercises
---------
1. Prove that the series in converges in $\End(\C^N)$ (equipped with the operator norm) for $|z|<1$, so that it defines an analytic function. Similarly for the series in .
2. Show that the series in and can be written in terms of standard basic hypergeometric series if we assume that the matrices $A$, $B$ and $C$ pairwise commute.
3. Determine more generally solutions in power series at $0$ and $\infty$ for the equation $$(q-z)\Id f(q^{-1}z)+\bigl(Uz-C-q\Id\bigr)f(z)+(C-Vz)f(qz) = 0$$ which reduces to in case $V=AB$ and $U=A+B$.
Notes {#notes-6 .unnumbered}
-----
A slightly more general situation is considered in [@AldeKR §4], but then only the analytic solutions are considered. Conflitti and Schlosser [@ConfS] consider also matrix-valued basic hypergeometric $q$-difference equations and hypergeometric differential equation analogues of Tirao [@Tira], but the approach in [@ConfS] is different to ours. In [@AldeKR] a family of $2\times 2$-matrix valued little $q$-Jacobi polynomials is considered. In [@AldeKR] matrix-valued analogues of the Askey-Wilson polynomials (of the subclass of the Chebyshev polynomials of the 2nd kind) are constructed using representation of quantum symmetric pairs as a $q$-analogue of [@KoelvPR1], [@KoelvPR2]. Using non-symmetric Askey-Wilson polynomials a $2\times 2$-matrix-valued orthogonality is constructed by Koornwinder and Mazzoco [@KoorM]. It is not clear if these two approaches can be combined to study the matrix-valued Askey-Wilson polynomials more generally. A suitable spectral analysis of the matrix-valued $q$-difference operator has not been developed.
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|
---
abstract: 'One of the major hurdles toward automatic semantic understanding of computer programs is the lack of knowledge about what constitutes functional equivalence of code segments. We postulate that a sound knowledgebase can be used to deductively understand code segments in a hierarchical fashion by first de-constructing a code and then reconstructing it from elementary knowledge and equivalence rules of elementary code segments. The approach can also be engineered to produce computable programs from conceptual and abstract algorithms as an inverse function. In this paper, we introduce the core idea behind the MindReader online assessment system that is able to understand a wide variety of elementary algorithms students learn in their entry level programming classes such as Java, C++ and Python. The MindReader system is able to assess student assignments and guide them how to develop correct and better code in real time without human assistance.'
author:
-
title: '**Smart Assessment of and Tutoring for Computational Thinking MOOC Assignments using MindReader**'
---
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
Authentic assessment; computational thinking; automated assessment; computer programming; program equivalence; semantic similarity
Introduction
============
A significant demand is known to exist for computer science (CS) graduates, and the US government has responded with the passing of the America Competes Act of 2007 [@Kuenzi2008] and subsequent refunding in 2011 to help train the much needed workforce. Additionally, the National Science Foundation has introduced the “CS for All" program with the goal that “all students should have the opportunity to learn CS in school." This imperative requires that CS education move into K-12 poorly funded schools with woefully under prepared staff to provide education in this field. There are few teachers with the skills necessary to teach CS courses. In rural schools the situation is exacerbated by the fact that there may only be one teacher with math or science skills for the entire school.
Technological advances and economic realities are also prompting a shift in the way we learn, teach and deliver instructions to train our labor force. Tech savvy younger generation today find personalized online systems engaging and useful and are welcoming online and digital learning in all three settings – formal or institutional, blended, and self-paced and non-formal learning spaces. The expectation is that online systems will overcome much of the hurdles we face in formal education systems and will complement it in a larger way. Although some skepticism exists [@DominguezESBH16; @Sayapin13s], the excitement around Massive Open Online Courses (MOOC) and more institutional approach to digital learning using BBLearn[^1] that are at the two ends of a spectrum, are fueled by these promises. All online universities, academies and institutes such as Coursera[^2], KhanAcademy[^3], and MITOpenCourseware[^4] are then immediately faced with problems in three axes – content delivery, teaching and tutoring and assessment, much like the traditional systems do. They also grapple with enrollment and coverage, retention, cost, teaching effectiveness, and so on much like their formal counterparts. To combat these problems, new learning environments such as immersive, game-based, blended, personalized, self-regulated and self-paced, social, peer, and pair learning have been proposed, the effectiveness studies of which are ongoing [@EdwardsSF10; @Farag12s; @Sharp16s].
But what we anecdotally know already are of significant concern. For example, the retention rate in first year programming classes is extremely low nationally. A recent online study [@Holton2016s] found that about 60% STEM subject students drop out or transfer and about 55% never graduate in state and community colleges. MOOCs and other online institutes’ retention rates are even worse – about 90% enrollees never complete their courses [@Holton2016s; @Tauber2013s]. We believe the environment that currently exists within the online education community does not support many of the recommendations of experienced educationists summarized in reports such as [@Holton2016s; @Jazzar2012s], and appear to retain the drawbacks of the traditional systems, and offer a mixed mode hodgepodge or a “succeed on your own" online setting.
However, the encouraging fact is that there have been significant progress in several areas of computer science that we believe can be leveraged and assembled together to build effective and smart cyber systems for online teaching, tutoring and assessment of entry level computing classes, and other STEM subjects. In our vision, such a system will complement a human instructor or mentor, and take on the role of a human observer to monitor students in real time and detect where she is making a mistake in her coding exercise, and immediately offer assistance by providing diagnostic comments and helpful pointers that most likely will cure the error [@MartinPSR17s].
In this paper, we introduce a novel prototype online system for tutoring and assessment, called the [*MindReader*]{}, for high school and freshman college students to aid learning programming languages. We develop necessary computational technologies to advance the science of computer program understanding needed for digital tutoring, and online real time assessment of programming assignments. This system is aimed at complementing human instructors at a more massive scale fully automatically. For the want of space and brevity, however, we highlight only the salient features of MindReader and refer readers to [@MindReader-Tech-2017] for a more detailed discussion.
Related Research
================
While some progress has been made in online instruction delivery, as well as in creating exciting learning environments, real time assessment and tutoring of STEM subjects online still remain at its infancy. Most often than not, these two areas rely largely on human interaction or MCQ tests, effectiveness of which are still being debated in general [@Simonova14s], and in STEM settings [@Azevedo15s] and for computer programming classes [@ShuhidanHD10s] in particular. The general consensus appear to be that MCQ tests are great tools for formative and diagnostic assessments but for summative assessment, tests such as authentic assessment are more appropriate, particularly in computing courses.
The challenges in designing smart cyber systems for tutoring and summative assessment are manifold. Ideally, a tutoring or assessment system should not rely on a specific procedure for establishing correctness of a proof in mathematics, for example. Rather the logical argument in any order must be the basis. Such an assumption rules out most of the current approaches to establishing correspondence of a student response to a known solution. A few automated systems have attempted to capture this spirit in subjects such as mathematics [@GluzPMGV14s] and physics [@Mehta01s] education with extremely narrow success. In computer science, the success has been mixed [@NavratT14s; @SorvaS15s].
To assess programming assignments, usually understanding the code semantically is required, and for a machine it would essentially mean determining the functional equivalence of a reference solution and the student solution, which is theoretically hard – deciding functional equivalence of two programs in general is NP-complete [@HuntCS80], and only in limited instances and for special classes of programs we are able to do so [@EiterFTW07s; @ChaudhuriV94s]. Undeterred by this weakness, researchers took a different route and tried to assess correctness of programs by various means so that the method can be used in learning exercises and online settings [@DrabentM05] but faced complexity barriers of a different nature [@Hungar91s]. Other approaches used test data to assess correctness [@TangSRW16s; @LiYS16s] to match with known outcomes and “assume" correctness. We, however, are not aware of a system capable of tutoring or assessing computer programs fully automatically and comprehensively.
CDGs: Hierarchical Concept Structure
====================================
The [*program dependence graph*]{} (PDG) [@Gorg16] based matching approach to determine code equivalence for the purpose of grading programming assignments is too simplistic although such approaches have been narrowly effective in detecting code clones [@LiKKL16s; @WagnerABOR16s] and plagiarized codes [@LiuCHY06s; @ZhangW0Z14s]. In particular, such techniques call for a complete enumeration of all possible solutions for every assignment, a largely daunting task, if not impossible. For example, consider an assignment that involves writing a code segment to swap the values of two variables. As shown in figure \[swap\], a student cannot be penalized if she offered the code segment as a possible solution even though a PDG based matching approach will most likely fail to accept it as a possible solution. If a student offers a more sophisticated but unanticipated solution instead as shown below, she should be assigned higher credits, not less, though a PDG based grading will certainly be ineffective.
> void swap(int *i, int *j){
> int t = *i;
> *i = *j;
> *j = t; }
[|p[4cm]{}|p[4cm]{}|]{}
#include <iostream>
using std::cout;
int main(){
int a=27, b=43, t;
cout << "Before " << a
<< " " << b << endl;
t = a;
a = b;
b = t;
cout << "After " << a <<
" " << b << endl;
return 0; }
&
#include <iostream>
using std::cout;
void swap(int& i, int& j){
int t = i;
i = j;
j = t; }
int main(){
int a=27, b=43, t;
cout << "Before " << a
<< " " << b << endl;
swap(a, b);
cout << "After " << a <<
" " << b << endl;
return 0; }
\
Reference Solution & Student Solution\
In this paper, we propose a novel and a more effective approach to matching solutions based on the idea of [*concept dependance graphs*]{} (CDGs) in which nodes are matched semantically as opposed to syntactic matching using PDGs. In a CDG, each node represents a hierarchically defined concept, and the graph represents the precedence relationship among the concepts. Thus, the matching of two CDGs have a much higher likelihood of determining functional and semantic equivalence of two code segments necessary for grading assignments, and offering tutoring help. We illustrate the idea using a simple problem of averaging a list of values in C++.
For a list of $n$ values $x_i$, their average is the simple mathematical formula $a = \frac{\sum_{i=1}^n x_i}{n}$, represented as the CDG in figure \[avg-cdg\] in which rectangles are [*declarations*]{}, ellipses are [*computable*]{} concepts, solid [*arrows*]{} represent precedence, and dashed [*arrows*]{} represent possible replacements. In the concept symbols, there are four quadrants which represents name (upper left), contextual concept parameters (lower left), node ID (upper right), and node membership (lower right). In CDGs, concepts are defined hierarchically, and terminal nodes are either declaration, or base computable concepts such as [*print, decide, for loop, while loop*]{}, and so on such that all variables needed for the base concepts are also in the CDG. CDGs can be simple or complex. A simple CDG is a connected and directed acyclic graph of base concepts and declarations, while a complex CDG is a forest of simple CDGs and CDGs involving concept nodes connected using dashed arrows.
Technically, a CDG is a graph $\langle N, \prec\rangle$ of a set of nodes $N=\cup_i n_i\subseteq V$ and a precedence relation $\prec=\cup_j e_j\subseteq E$, where $V$ is the set of all possible concepts, and $E$ is all pairs $2^{V\times V}$. In figure \[avg-cdg\], the CDG $\langle \{u_1, u_4, v_6\}, \{u_1 \prec v_6, u_4\prec v_6\}\rangle$ is a simple, while $\langle \{u_1, u_4, v_3, v_7, v_8, v_9\}, \{u_1 \prec v_3,u_4 \prec v_3,v_3 \prec v_7,v_7 \prec v_8,v_8 \prec v_9\}\rangle$ is a complex. In figure \[avg-cdg\], the concept [*counter loop*]{} is replaceable with the sequence $v_7, v_8, v_9$, or with node $v_6$ alone. In other words, the CDG in figure \[avg-cdg\] assumes a [*counter loop*]{} can be implemented in two possible ways, and thus defines an equivalence relation. Note that the concepts are also hierarchically defined. The concept [*average*]{} is defined as an aggregation of a list of values, followed by a division by the size of the list. An aggregate on the other hand is defined as the summation of the elements of the list inside a counter loop (note node $v_4$ is part of node $v_3$, the counter loop). Finally a counter loop is defined as a for loop or a while loop. For the student program $P_s$ below, we can transform it to construct a corresponding CDG, and match it with the conceptual solution in the knowledgebase even if the student solution is implemented using a for loop.
> 1: #include <iostream>
> 2: void main() {
> 3: int k=0, total, size=9, mean, elements[10];
> 4: while (k<=size) {
> 5: total=total + elements[k];
> 6: k++; }
> 7: std::cout << total/(size+1); }
Formal Model
============
Let $L$ be a programming language, $\mu_L$ be a function that can parse a program $P$ in $L$ and convert each sentence into either a [*declaration*]{} concept or a [*computational*]{} concept in the language $\cal L$ of MindReader. $\cal L$ consists of two types of expressions – [*abstract statements*]{} and [*precedence relations*]{}. Abstract statements are of two types: [*declaration*]{} and [*computational*]{} type. Declaration type expressions are tuples of the form $[N, V, T, C]$, where $N$ is the statement number in $P$, $V$ is the variable name, $T$ is the class of variable such as individual variable, boolean or a list, and $C$ is the statement or program in which the statement is included. For example, statement number 5 and 6 in the program $P_s$ above are contained in statement number 4, while the statement number 4 is contained is statement number 2. Likewise, statements 3 and 7 are contained in statement 2.
Similarly, computable expressions are tuples of the form $\langle N, E, P, C\rangle$, where $N$ is the statement number in $P$, $E$ is the type of executable statement such as assignment, loop or decision statement, $P$ is a list of context sensitive parameters, and $C$ is the statement number of which the statement is a part of. For example, statement 4 in program $P_s$ is a while loop, represented as the expression $\langle 4, whileLoop, param(cond(i<=n)), 2\rangle$, and the expression $\langle 6, tran, param(k, k+1), 4\rangle$ represents statement 6. Finally, precedence relation is a set of expressions of the form $n_1 \prec n_2$, where $n_1$ and $n_2$ are statement numbers such that $n_1$ precedes $n_2$.
The language $\cal L$ of MindReader is a tuple $\langle \mu_L, {\cal C}, \Sigma, \Gamma, \Psi\rangle$ of a concept extractor $\mu_L$, concept hierarchy $\cal C$ (e.g., figure \[cgh\]), concept mapper $\Sigma$, concept dependence graph $\Gamma$, and a subgraph isomorph function $\Psi$. The concept hierarchy organizes higher level concepts from computable expressions. For example, a counter loop can be a composite of an [*assignment*]{}, a [*while*]{} and an [*increment*]{} statement as discussed earlier in the context of figure \[avg-cdg\]. The $\Sigma$ function transforms the CDG created by $\mu_L$ into higher level concepts using the concept hierarchy $\cal C$ and the CDG $\Gamma$, iteratively. Therefore, given a program $P$, the least fixpoint $lfp(\Sigma(\mu_L(P),{\cal C},\Gamma))$ is the final CDG of a program $P$. Observe that the concept hierarchy $\cal C$, the CDG $\Gamma$, and the summarization function $\Sigma$ help abstract programs into CDGs and increases the matching likelihood with high level abstract algorithms stored as a reference CDG independent of their lower level implementations.
![Example concept hierarchy.[]{data-label="cgh"}](swap-loop-branch-cg.eps){height="1.5in" width=".49\textwidth"}
Finally, the subgraph matching function $\Psi$ ensures proper matching of CDGs independent of the variable declarations, typing and naming. It ensures proper substitutions throughout the code segment. Therefore, for any given program $P$, if $\Psi(C_a, lfp(\Sigma(\mu_L(P),{\cal C},\Gamma))\approx 1$, where $C_a$ is the conceptual CDG of any algorithm $a$, then we assume that the submitted code is acceptable and correct. Conversely, if for any “unknown" program $P$, $\Psi(C_a, lfp(\Sigma(\mu_L(P),{\cal C},\Gamma)))\approx 1$, then we can be confident that the unknown program is a candidate implementation of the abstract and conceptual algorithm $a$. This a significant and powerful method to determine functional equivalence of unknown codes which is extremely difficult, if not impossible, using PDG based approach due to its inability to summarize codes functionally.
Assessment and Tutoring using MindReader
========================================
The high level architecture in figure \[archi\] depicts MindReader’s two broad subsystems for two distinct but complementary functions – tutoring and grading. In MindReader, all learners have a profile which includes background, past lessons, tests and tutoring activities, known problem areas, and their peer groups. MindReader generates tutorials based on students’ profile and level of programming competence expected along the lines of the systems such as [@BoumizaSB16s] keeping in mind that for computation thinking classes, the challenges primarily involve the difficulties in learning the syntax and understanding the semantics and use of constructs such as loops, conditional statements, and simple algorithms [@HullsNKPB05s]. For the purpose of both grading and tutoring, MindReader assembles the statement structures written by the student into possible CDGs using the concept structures in the [*Concept Database*]{} according to the rules in [*Concept Construction*]{} rule base with the aim of matching the CDG with one of the known templates in the [*Algorithm Templates*]{}. Failure to match CDGs of the student code and the reference template results into a dataflow pattern match using known and random test data of the compiled codes. Failure to match flow patterns forces a diagnostic feedback, but a success indicates a new way of solving a problem unknown to MindReader, and the new CDG is included in the knowledgebase after proper curation.
{height="1.7in" width=".9\textwidth"}
Learning Complex Concepts
=========================
Building concepts hierarchically and generating corresponding CDGs though intuitive, learning new concepts could be challenging. In MindReader, we assume that it is impossible to enumerate all reference solutions regardless of the complexity. We thus adopt an incremental learning approach with the assistance of a panel of curators or experts in MindReader’s architecture in figure \[archi\]. To understand how MindReader learns new concepts, consider an abstract algorithm for bubble sort as shown in algorithm \[alg:bubble\], its C++ implementation $Q$ as shown below, and its CDG representation $C_b$ shown in figure \[bubble-cdg\] as a reference solution. Obviously, the $lfp(\Sigma(\mu_L(Q),{\cal C},\Gamma))$ will not match with $C_b$, i.e., $\Psi(C_b, lfp(\Sigma(\mu_L(Q),{\cal C},\Gamma))) << 1$, since the loop in statement 1 is not a sentinel loop. But a dataflow analysis and random data test comparison will show a match, prompting a curation step and learning the rule that bubble sort can also be performed with an outer counter loop, and a reverse inner counter loop. Note that the blue starred nodes in the CDG in figure \[bubble-cdg\] will also need to be implemented.
set $sorted=false$
0: void bubbleSort(int ar[]) {
1: for (int i = (n - 1);
i >= 0; i--) {
2: for (int j = 1; j = i; j++) {
3: if (ar[j-1] > ar[j]) {
4: int temp = ar[j-1];
5: ar[j-1] = ar[j];
6: ar[j] = temp;
} } } }
Summary and Future Research
===========================
In this paper, we reported a late breaking result of a research focusing on semantic understanding of student codes in an online learning environment. We have demonstrated that CDG based matching code fragments have a higher likelihood of detecting semantic and functional equivalence of two programs. The process is complemented by a dataflow analysis and random testing regime to identify possible valid solutions and learn new rules. We have also demonstrated that detecting code clones and plagiarized codes based on PDGs is fundamentally different from matching two codes functionally using CDGs. In CDGs we substitute equivalent nodes under the guidance of a template CDG, and concept hierarchy to determine semantic similarity essential for grading tasks of MOOC student assignments. It should be evident that summarization of concepts in the concept graphs allows for abstract algorithm development, and it should be possible to actually write codes in various languages as an inverse function and develop new languages such as Scratch.
Initial evaluations of MindReader was encouraging and a more serious performance analysis and comparison with existing systems is being planned. Once deployed, and students use it for a period of time, we plan to collect a large number of coding examples and investigate students learning behavior, and effectiveness and do comparative analysis with the traditional classroom teaching. Identification of problem areas of learning where a significant number of students are having difficulty manifested by their inability to solve problems could imply gaps in instruction delivery, course content design or learning habits warranting a revision, and could help develop personalized teaching, tutoring and assessment regimes, and measured for continuous improvement.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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|
---
abstract: 'Icosahedral quasicrystals spontaneously form from the melt in simulations of Al–Cu–Fe alloys. We model the interatomic interactions using oscillating pair potentials tuned to the specific alloy system based on a database of density functional theory (DFT)-derived energies and forces. Favored interatomic separations align with the geometry of icosahedral motifs that overlap to create face-centered icosahedral order on a hierarchy of length scales. Molecular dynamics simulations, supplemented with Monte Carlo steps to swap chemical species, efficiently sample the configuration space of our models, which reach up to 9846 atoms. Exchanging temperatures of independent trajectories (replica exchange) allows us to achieve thermal equilibrium at low temperatures. By optimizing structure and composition we create structures whose DFT energies reach to within $\sim$2 meV/atom of the energies of competing crystal phases. Free energies obtained by adding contributions due to harmonic and anharmonic vibrations, chemical substitution disorder, phasons, and electronic excitations, show that the quasicrystal becomes stable against competing phases at temperatures above 600K. The average structure can be described succinctly as a cut through atomic surfaces in six-dimensional space that reveal specific patterns of preferred chemical occupancy. Atomic surface regions of mixed chemical occupation demonstrate the proliferation of phason fluctuations, which can be observed in real space through the formation, dissolution and reformation of large scale icosahedral motifs – a picture that is hidden from diffraction refinements due to averaging over the disorder and consequent loss of information concerning occupancy correlations.'
author:
- Marek Mihalkovič
- Michael Widom
bibliography:
- 'acf.bib'
title: 'Spontaneous formation of thermodynamically stable Al–Cu–Fe icosahedral quasicrystal from realistic atomistic simulations'
---
Introduction
============
Since the discovery of quasicrystals as a distinct phase of matter [@Shechtman], and recognition of their quasiperiodicity [@LevineSteinhardt], two fundamental questions remain to be definitively answered: where are the atoms [@Bak]? What stabilizes their quasiperiodic order? Excellent descriptions of their [*average*]{} structures are possible in terms of cuts through higher dimensional periodic lattices obtained by single-crystal diffraction refinements [@cdyb]. However, quasicrystalline structures can only be reliably equilibrated at high temperatures; consequently, these models contain ambiguous atomic positions with uncertain occupation and chemistry. They omit important [*correlations*]{} in the case of mixed or partial occupation, and they omit atomic vibrations and diffusion. As regards their thermodynamic stability, local icosahedral motifs are clearly favored energetically, but this need not force long-range quasiperiodicity, as is clearly illustrated by the prevalence of periodic “approximants”, which mimic quasiperiodic order locally within an ordinary crystalline unit cell that in turn repeats periodically.
An intriguing puzzle that has eluded researchers for three decades is identifying the mechanism that selects an ordered yet non-periodic state. One possible explanation is that quasicrystal are energy minimizing structures, whose structure is forced by specific interatomic interactions [@SocolarMatchingRules], by maximizing the density of some favorable motif [@Gummelt; @Steinhardt], or by creating a deep pseudogap in the electronic density of states [@Pseudogap; @ebands-almnpd; @ebands-alconi]. Another possibility is that the structural ambiguity is an [*intrinsic*]{} characteristic of quasicrystals [@ElserRT; @HenleyRT; @WidomEntropic]. In this view, the entropy to be gained from chemical or positional fluctuations serves to reduce the free energy relative to competing phases whose energies (without entropy) are lower. Quasiperiodicity might arise spontaneously because it allows icosahedral symmetry, and this high symmetry maximizes the entropy.
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The icosahedral phase of Al–Cu–Fe is an excellent place to seek theoretical insight. Experimentally, the $i$-phase of Al–Cu–Fe was the first well-ordered thermodynamically stable quasicrystal to be discovered [@Tsai_1987]. It exhibits a particular symmetry classified as [*face-centered*]{}-icosahedral, and exhibits strong pseudogap in electronic states density near Fermi energy [@pgap-barman]. Recently, these quasicrystals attracted renewed attention following their discovery in a meteorite [@bindi-alcufe].
In this paper, we report computer simulations leading to spontaneous formation of icosahedral quasicrystals from the melt. Such simulations have been previously reported [@BinaryLJ; @Dzugutov; @Engel2015], but only for artificial models that do not describe actual chemical species and hence yield no direct insight into specific experimentally studied compounds. Here we model Al–Cu–Fe ternary alloys using a combination of chemically accurate density functional theory (DFT) calculations and accurate interatomic pair potentials. Based on structure models and thermodynamic data provided by our simulation, we calculate a temperature-dependent phase diagram for the Al-rich region of the alloy system showing that energy and entropy conspire in the emergence of the quasicrystal phase as thermodynamically stable at elevated temperatures. Figure \[fig:diffract\] illustrates the diffraction patterns of our simulated structures and verifies that we indeed obtain a quasicrystal with the expected face-centered icosahedral symmetry.
Simulation Methods
==================
Three important ingredients enable the success of our atomistic simulations: realistic DFT-derived interatomic interactions; appropriately sized simulation cells with periodic boundary conditions; efficient hybrid Monte Carlo/molecular dynamics augmented by replica exchange. We exploit the data generated by our simulation to present optimized structure models revealing icosahedral order on a hierarchy of length scales. Combining DFT-calculated formation enthalpies with entropies derived from fluctuations, we calculate the absolute free energies of the quasicrystal phase and competing crystalline phases. Then, from the convex hull of the set of free energies we predict a temperature-dependent phase diagram that shares characteristics with the experimentally assessed behavior, including the emergence of the quasicrystal as a high temperature stable phase.
Oscillating interatomic pair potentials accurately describe elemental metals and alloys characterized by weakly-bound $s$ electrons, even in the presence of $s$-$pd$ hybridization. While these can be derived analytically within electronic density functional theory [@Phillips1994; @ico-almn; @Moriarty1997], we instead employ a parametrized empirical form known as EOPP [@EOPP] that we fit to a database of DFT energies and forces (Appendix \[app:dft\]). This form has found success modeling many binary and ternary alloys [@mmclh-alir; @ScZn; @mmclh-mgzndy]. EOPP also lead to spontaneous formation of single-component icosahedral quasicrystals [@Engel2015]. Henley [@clh-ideas] connected the oscillating potentials with pseudogap near $E_F$ via the Hume-Rothery scenario, and argued that their second minima participate in formation of fundamental clusters. Indeed, under some circumstances the second minima can even create local matching rules favoring quasiperiodicity [@Lim2008]. Figure \[fig:ppgofr\] illustrates our fitted potentials and the comparison with interatomic separations in our simulated structures. Details of our fitting procedure are provided in Appendix \[app:eopp\].
![ \[fig:ppgofr\] Empirical oscillating pair potentials $V_{\alpha\beta}(r)$ (thick solid lines, energy axis on the left) and partial pair correlation functions $g_{\alpha\beta}(r)$ (dashed lines, vertical axis on the right) for Al–Cu–Fe quasicrystals. The EOPP are fit to a database of DFT energies and forces (see Appendix \[app:eopp\]). The pair correlation functions are obtained from single snapshot of the 9846-atom 8/5 approximant EOPP MD simulation. ](fig2.pdf){width="3.25in"}
Because the quasicrystal is aperiodic it cannot be precisely represented in a finite size system. Fortunately, a series of “rational approximants” are known that capture the local quasicrystal structure and minimize the deviation caused by application of periodic boundary conditions. Reflecting the hierarchical nature of quasiperiodic order, these special sizes grow geometrically as $a_{cub}=2 a_q\tau^n/\sqrt{\tau+2}$ where $\tau=(1+\sqrt{5})/2\approx1.61803...$ is the Golden Mean and $a_q$ is “quasilattice parameter” or Penrose Rhombohedron edge length [@eh85]. The volume grows rapidly as $a_{cub}^3\sim \tau^{3n}$. Strictly, face-centered icosahedral symmetry implies $\tau^3$ scaling for self similarity (volume $\sim\tau^9$) along 5–fold and 3–fold directions, but $\tau^1$ scaling along 2–fold icosahedral directions. Here, we label the approximants with a ratio of successive Fibonacci numbers $F_{n+1}/F_n$, with 128 atoms for “2/1” up to 9846 for “8/5”. Our naming convention is drawn from the definition of Henley’s canonical cells designed for packing icosahedral clusters [@clhcct]. Since the fundamental clusters in AlCuFe are $\tau$-times smaller than the proper Mackay clusters decorating the B.C.C. lattice in $\alpha$-AlMnSi [@eh85; @elser-model; @fujita-double32], our “2/1” approximant has about the same edge length $a_{cub}\sim$12.3Å as $\alpha$-AlMnSi.
For small cell sizes (2/1 and 3/2-2/1-2/1), imposing this special length encourages nucleation of the quasicrystal from the melt [@MeltQuench]. In larger cells (3/2, 5/3 and 8/5), the entropic barrier to nucleation is hard to overcome; instead we [*seed*]{} the structure using the previous approximant size. Because of the discrete cell size scaling, a single unit cell of the seed occupies less than 24% of the cell volume. Thus we take a supercell of the smaller approximant and enforce the periodic boundary of the bigger approximant. Near the large approximant boundary we let the small approximant overlap itself in a region that is 10% of the large approximant cell size. This introduces a 25% excess of atoms in the large approximant placed at unphysically short separations. We remove the excess atoms through a fixed-site lattice gas annealing [@HBSD2] to reach the desired atomic density.
To efficiently anneal both chemical and positional order we utilize a hybrid Monte Carlo/molecular dynamics (MC/MD [@MCMD]) method that samples continuous evolution of atomic positions through molecular dynamics while enabling interchange of chemical species through Monte Carlo swaps. Species swaps are accepted with the Boltzmann probability $\exp{(-\Delta E/{k_{\text{B}}}T)}$ with $\Delta E$ the change in total energy for the swap. Our simulations are performed in the canonical ensemble with constant temperature, volume and numbers of atoms of each species. To achieve equilibration at low temperatures and enhance sampling of the configurational ensemble at all temperatures, we supplement our MC/MD simulation with replica exchange [@Swendsen86]. MC/MD simulations are performed in parallel at many temperatures $T_i$. At fixed intervals we suspend the simulations and consider swapping configurations at adjacent temperatures $T_i$ and $T_{i+1}$. The swap is accepted with a Boltzmann-like probability based on the energy difference between the configurations. Although the temperature of a configuration jumps during the swap, the configuration remains a properly weighted member of the equilibrium ensemble at its instantaneous temperature. Further details are in Appendix \[app:replica\].
Results {#sec:results}
=======
We analyze the simulated structures to demonstrate their quasiperiodicity. The clearest demonstration is their diffraction pattern (Fig. \[fig:diffract\]) which shows characteristic 2x, 3x and 5x patterns with sharp peaks near the characteristic positions for face-centered icosahedral symmetry (minor deviations occur due to the finite size approximant). The face-centering causes certain diffraction peak positions to occur in ratios of $\tau^3$, while the remainder show $\tau^1$ scaling.
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Examining the structure in real space, we observe that small approximants solidify into well-ordered structures that can be conveniently described as packings of two cluster types. One is a small 13-atom (Al$_{12-x}$Cu$_x$)Cu icosahedron that we denote as [[*I *]{}]{}(see Fig. \[fig:clus\]a). The other is a larger pseudo-Mackay icosahedron ([[*pMI *]{}]{}) (Fig. \[fig:clus\]b) with a partially occupied Al$_{12-x}$Fe inner shell ($x\sim 1-3$ due to Al-Al repulsion), and a second shell made up of two subshells: a (Cu,Fe)$_{12}$ “unit-icosahedron” on the 5–fold axes at 4.45Å from the center, and an Al-rich (Al,Cu)$_{30}$ icosidodecahedron on the 2–fold axes. In the example shown in Fig. \[fig:clus\]b, the Cu and Fe atoms on the unit-icosahedron segregate to break the symmetry from icosahedral to 5–fold. [[*I *]{}]{}-clusters connect along 2–fold icosahedral directions with a spacing of $b=$7.55 Å, and alternate with [[*pMI *]{}]{}along 3–fold directions ($c=$6.54 Å spacing). This even/odd alternation implements the face-centered icosahedral order. For the 2/1 approximant this packing is a unique structure producing A, B and C-type canonical cells [@clhcct] (CCT), while the 3/2-2/1-2/1 approximant also contains a symmetry-broken D-cell.
Larger approximants avoid the bulkiest canonical cell D by introducing a new three-shell cluster extending to $\sim$7.7Å (2–fold radius, see Fig. \[fig:clus\]c). This cluster is entirely bounded by CCT $Y$-faces, hence it extends, rather than violates, the CCT concept. Its innermost shell is Al$_{12-x}$Fe as in usual [[*pMI*]{}s ]{}, but the second shell, albeit topologically similar to [[*pMI *]{}]{}, is Cu-rich with only 25% Al and no Fe atoms. Finally, the third shell is made up from three icosahedral subshells: Al$_{60}$ (at 6.4Å), a Fe$_{30-x}$Cu$_x$ ($x\sim$ 6) $\tau$-Icosidodecahedron (at 2–fold radius 7.7 Å) and a Cu$_{12}$ $\tau^2$-Icosahedron (at 5–fold radius 7.2Å). These 12 Cu atoms are in fact all centers of the small [[*I *]{}]{}clusters; upon including them the whole cluster has 282 atoms.
The three clusters provide a simple, highly economical zero-order description of the structure, since they cover practically all atoms in the structure (99% in 3/2, 98% in 5/3 and 97% in 8/5 approximant). A typical example of these clusters as they appear in our simulations is shown in Fig. \[fig:clus\]d, taken from the equilibrium ensemble at T=1200K, followed by relaxation. Mixed chemical occupation breaks the cluster symmetry and can serve as a stabilizing source of entropy, while, together with the cluster covering, it is potentially a means of forcing quasiperiodicity [@Gummelt; @Steinhardt].
The ensemble of structures can be represented using the 6D cut and project scheme. This is an efficient way to represent the average structure in a manner that automatically enforces perfect quasiperiodicity. Details are presented in Appendix \[app:6D\]. Approximately 80% of atoms match projected 6D positions with an accuracy of 0.45 Å or better. The exceptions are primarily the frustrated Al atoms of the [[*pMI *]{}]{} inner shell: their positions are dictated by Al-Al repulsion in the tight inner shell of 9-10 Al atoms around the central Fe, rather than the wells of the Al–TM potential. Three atomic surfaces emerge (see Fig. \[fig:perp\]), two large ones (AS$_1$ and AS$_2$) sit at hypercubic lattice sites (“nodes”), one even and the other odd. The remaining small one (B$_1$) sits at the hypercubic body center. Each atomic surface has a particular pattern of chemical occupation. AS$_1$ is primarily Fe, concentrated at the center, with Cu surrounding and finally traces of Al. AS$_2$ is primarily Al, with a small concentration of Fe at the center and Cu separating the Al from the Fe. The remaining surface B$_1$, at the body center, is primarily Cu. The contrast between AS$_1$ and AS$_2$, along with absence of B$_2$, reflects the strong symmetry breaking from simple icosahedral to face-centered lattice.
There is a unique connection between the three fundamental clusters (Fig. \[fig:clus\]) and the three atomic surfaces. The B$_1$ surface Cu atoms are centers of the [[*I *]{}]{}clusters, the central Cu/Fe part of the AS$_2$ surfaces occupy [[*pMI *]{}]{}clusters, and the Fe core atoms of the AS$_1$ surface are centers of the large [[*$\tau$-pMI*]{} ]{}clusters.
Notice the smooth variations in color (i.e. chemical occupancy). Cu separates Al from Fe on the atomic surfaces while blending continuously into each. Curiously, the location of Cu at the boundary of Fe and Al on the atomic surface is consistent with the position of Cu in the periodic table between transition ($d$-band) metals and nearly free electron ($sp$-band) metals. Mixed occupation implies swaps in chemical occupation in real space, and low atomic surface densities correspond to partial site occupation. Because these lead to spreading of the occupation domains in perpendicular space, we may identify these chemical species swaps and fractional occupation as types of phason fluctuations.
![ \[fig:perp\] (top) Atomic surface occupation averaged over ensemble of 3000 configurations from 9846-atom “8/5” approximant at T=1242K; (bottom) color bars for chemical species occupancy. Mixed chemical occupation is represented by adding the RGB color values.](AS-with-ColorBars-KG-axes.png){width="3.5in"}
Our simulated atomic surface occupation largely agrees with the popular Katz-Gratias model [@kg]. Specifically, the node vertex surfaces (AS$_1$ and AS$_2$) both transition from Fe at the center, through Cu, to Al around the edges. Only a single body center (B$_1$) is occupied, solely by Cu. Short-distance constraints determined specific shapes of the KG atomic surfaces. In our model these constraints are obeyed through positional correlations so our surfaces have outer shapes that differ from the KG model. Note that the outer shapes are defined by regions of low occupation probability.
After annealing down to low temperatures using the interatomic potentials, we apply DFT to relax the structures to T=0K and compute their enthalpies of formation, $\Delta H$ relative to pure elements, and energetic instabilities, $\Delta E$ relative to the tie-plane of competing crystal structure enthalpies. Table \[tab:ene\] summarizes the compositions and formation enthalpies of several competing phases.
-------------------------- ------------ ------------ ---------- ------ ------ ------
name $\Delta E$ $\Delta H$ N$_{at}$ Al Cu Fe
meV/at meV/at per cell % % %
$\omega$ (tP40) S -280.0 40 70.0 20.0 10.0
$\lambda$ (mC102) S -361.5 102 72.5 3.9 23.5
$\alpha$/$\tau_1$ (oC28) S -377.4 60 66.7 6.7 26.7
$\beta$ (tP16) S -344.3 16 50.0 18.2 31.2
$\phi$’ (cP50)[^1] S -267.0 50 46.0 46.0 8.0
[*i*]{}-(2/1) +1.8 -285.4 128 64.1 25.8 10.9
[*i*]{}-(3/2) +4.3 -292.5 552 63.8 23.9 12.3
[*i*]{}-(5/3) +4.0 -292.6 2324 65.0 22.5 12.5
-------------------------- ------------ ------------ ---------- ------ ------ ------
: \[tab:ene\]Chemistry, instability $\Delta E$ (letter “S” for stable) and formation enthalpy $\Delta H$ for quasicrystal approximants and competing ternary phases at T=0$K$. The two variants of the B2–type $\beta$–phase, resulting from annealing simulations, are a 2$\times$2$\times$2 Fe-richer supercell with Pearson symbol [*tP16*]{}, and a 3$\times$3$\times$3 supercell with ordered vacancies and Pearson symbol [*cP50*]{}).
The small approximants exhibit an unusual electronic density of states (eDOS), with a wide and deep pseudogap as is usual in Al-based quasicrystals, and in addition a deeper and very narrow second pseudogap [*inside*]{} the broad pseudogap (see Appendix \[app:nrg\]). For optimal density and composition, which we achieve by replacing Cu with a combination of Al and Fe, the Fermi level lies inside this second pseudogap. Specifically, we find the effective valence rules of Al=+3, Cu=+1 and Fe=-2 apply, so we can raise or lower $E_F$ by 1 electron without altering the number of atoms through targeted chemical substitutions such as 2Cu$\leftrightarrow$Al$+$Fe. Composition can be shifted at constant effective valence through substitutions such as 3Al+2Fe$\leftrightarrow$5Cu. This rule matches the slope of the quasicrystal and approximant phase fields in the ternary composition space [@alcufe-approximants]. We discovered that neighboring Fe-Fe pairs lead to states in the pseudogap that can be removed by avoiding these pairs. These optimizations can substantially lower the total energy. However, these structures remain unstable by 2-4 meV/atom relative to competing ordinary crystal phases in the triangle $\eta_2(AlCu)$–$\lambda(Al_3Fe)$–$\omega(Al_7Cu_2Fe)$ as described in Table \[tab:ene\]. Simulated larger approximants, and supercells of the 2/1 approximant, exhibit only the broad pseudogap. Apparently the higher entropy available in larger simulation cells introduces disorder that washes out the detailed structure leading to the narrow second pseudogap, trading a gain in energy for a compensating gain in entropy. It is conceivable that the EOPP potentials are not sufficiently accurate to capture the interactions responsible for the narrow pseudogap feature.
Notice the sequences of enthalpies of formation, $\Delta H$, that decreases monotonically with increasing approximant size. This suggests a possible energetic mechanism favoring quasiperiodicity. However, this is not yet clear, as we have not demonstrated that the energetically optimized structures are more perfect in their quasiperiodicity than representative high temperature structures. Indeed, the decreasing enthalpy is primarily a reflection of increasing Fe content. From the energies relative to the convex hull, $\Delta E$, which are positive and not systematically decreasing, it is likely that any quasicrystal model will be unstable relative to competing crystal phases at low temperatures. Hence, to explain the formation of the quasicrystals from the melt at high temperatures we must seek either a kinetic or a thermodynamic argument.
We consider the larger approximants, 3/2 (552 atoms) and 5/3 (2324 atoms), as representative of the true quasicrystal. They exhibit interesting structures and dynamics at elevated temperatures. Because clusters are difficult to identify in individual snapshots due to chemical disorder and atomic displacements, it is best to examine [*time averages*]{} of the structure. In Fig. \[fig:mcmd\] the inner shells of the [[*pMI*]{}s ]{}are clearly visible as smeared circles due to the high mobility of the Al atoms, whose positions are frustrated by the incompatible length scale of the icosahedral potential produced by the outer shells with the short-range repulsion of the Al-Al potential. An azimuth of the three-shell [[*$\tau$-pMI*]{} ]{}clusters with Cu-rich interior is indicated by large black circles. Cu-centered small icosahedra are also clearly visible. As time evolves, the identity of these clusters shifts, with some becoming more distinct while others dissolve. Occasionally the structures take pleasing hexagon-boat-star-decagon (HBSD [@HBSD1; @HBSD2]) tiling forms as in Fig. \[fig:mcmd\], but these structures, in turn, further evolve. We provide a video illustrating the evolving structure in our Supplemental Online Material.
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We find a curious behavior at very high temperatures as the quasicrystals melt. One aspect is that the melting point grows as the approximant size increases. The 2/1 approximant (in 2x2x2 supercell) melts at T=1669 K, the 3/2 at T=1683 K, the 5/3 at T=1723 K, and the 8/5 at T=1788 K. Additionally, the 2/1 and 3/2 melts in a single step, while the 5/3 melts in two(the first broad heat-capacity maximum at 1590 K), and the 8/5 have three additional smaller but sharp heat capacity peaks at 1266 K, 1471 K and 1680 K that precede melting. Melting of the larger approximants begins with small Al-Cu-rich regions that leave behind an Fe-richer (and hence more mechanically stable) solid quasicrystal phase. Because we perform our simulations at fixed volumes, the actual melting occurs at high pressure - we estimate P=7GPa at the melting point of 8/5. Note that this is consistent with recent experiments of Bindi [@bindi-alcufe-curich]. A video showing liquid-quasicrystal phase coexistence is provided in our Online Supplemental Material. In this video the quasicrystal melts and resolidifies as the liquid interface advances into the solid and then recedes.
Finally, we seek to explain the thermodynamic stability. As shown in Table \[tab:ene\] our lowest energy quasicrystal models remain above the convex hull of energy by 2–5 meV/atom. Thus at low temperatures we anticipate phase separation into the competing phases, $\eta_2(AlCu)$–$\lambda(Al_3Fe)$–$\omega(Al_7Cu_2Fe)$. Indeed, this is what is shown in the standard phase diagram for the Al–Cu–Fe system [@pdiag-alcufe]. Finite temperature stability is given by the convex hull of [*free*]{} energy. As discussed in Appendix \[app:thermo\] the free energy includes harmonic vibrational free energy $F_h$ derived from phonons, electronic free energy $F_e$, and other contributions such as anharmonic phonons, chemical substitution and tiling degrees of freedom, $F_a$.
Comparing free energies of the competing ordinary crystals, small quasicrystalline approximants and the quasicrystal (considered as a large approximant) we predict the stable phases at various temperatures by evaluating the convex hull of free energies over the composition space. Details are presented in Appendices \[app:ternary\] and \[app:nrg\]. Notably, we observe that the 2/1 approximant gains stability at T=0K when quantum zero point vibrational energy is included, and the 5/3 approximant, which we take as a proxy for the quasicrystal emerges as a stable phase at temperatures above T=600K owing to the excitations contained in $F_a$.
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Discussion
==========
We have provided four pieces of evidence demonstrating the appearance of a thermodynamically stable quasicrystal state in our model of Al–Cu–Fe. First is the spontaneous formation of quasicrystal approximants, directly from the melt in the case of small approximants, and with the assistance of smaller approximant seeds in the case of large approximants. Second is the appearance of progressively larger clusters and superclusters with increasing approximant size. Third is the decreasing total energy of highly optimized structures with increasing approximant size, suggesting that energy could favor quasiperiodicity. Finally, we have calculated free energies for quasicrystal approximants and competing ordinary crystalline phases and foud that the 5/3 approximant, taken as a proxy for the true quasicrystal, acquires thermodynamic stability above T=600K.
Our findings shed light on the underlying reasons for quasicrystal formation and suggest there is not a single explanation but rather a coincidence of favorable conditions. Although the formation enthalpies decrease ([[*i.e.*]{}]{} become more negative, see Table \[tab:ene\]) with increasing approximant size, they actually [*lose*]{} stability relative to competing crystal states owing to the slope of the convex hull with increasing Fe content. The largest cluster for which we have DFT energies, namely 5/3, is the [*smallest*]{} approximant that can accomodate the [[*$\tau$-pMI*]{} ]{}supercluster including its complete surrounding I clusters. If we postulate that this is an energetically favorable motif then it may be that DFT energies are needed for yet larger approximants before we can claim existence of an energetic preference for the quasicrystal as compared with finite approximants. Further, the appearance of superclusters is a consequence of quasiperiodicity, not a cause. Indeed both matching rule and random tiling models share this feature provided that these are viewed in a time-averaged sense as in the present case. While our explanation for thermodynamic stability at high temperatures is an example of entropic stabilization, the entropy includes phonon anharmonicity in addition to phason-related chemical species swaps and tile flips.
In conclusion, although our explanation for quasicrystal stability is not simple, it illuminates the complex interplay of multiple factors. These include the composition-dependence of competing phase energies, as well as multiple sources of entropy. Chemical preferences for site classes in cluster motifs and on atomic surfaces are favored by our EOPP interatomic potentials. We remark that cluster overlap together with cluster symmetry breaking provides a possible mechanism to force quasiperiodicity, but this appears insufficient to stabilize the Al-Cu-Fe quasicrystal against competing ordinary crystalline phases at low temperature. In our model, the quasicrystal is a high temperature phase.
This work was supported by the Department of Energy under grant DE-SC0014506. Assistance with videos was provided by the Pittsburgh Supercomputer Center under XSEDE grant DMR160149. M. M. also acknowledges support from Slovak Grant Agency VEGA (No. 2/0082/17), and from APVV (No. 15-0621). Most of the calculations were performed in the Computing Center of the Slovak Academy of Sciences using the supercomputing infrastructure acquired under projects ITMS 26230120002 and 26210120002.
DFT {#app:dft}
===
First principles calculations within the approximations of electronic density functional theory (DFT) lie at the foundation of our structural and thermodynamic models. We employ projector augmented wave potentials [@PAW] in the PW91 generalized gradient approximation [@PW91] as implemented in the plane-wave code VASP [@Kresse96]. Our $k$-point meshes are increased to achieve convergence to better than 1 meV/atom with tetrahedron integration. We employ the default energy cutoffs. For T=0K enthalpies, all internal coordinates and lattice parameters are fully relaxed. Finite difference methods are applied to calculate interatomic force constants that we need for low temperature vibrational free energies. Ab-initio molecular dynamics (AIMD) was performed to generate additional energy and force data for fitting interatomic pair potentials. AIMD calculations contained 544 atoms in cubic cell and utilized only a single $k$-point. For accurate cohesive energies, approximant $k$-meshes were converged to 10$^3$ $k$-points/BZ for 128-atom 2/1, 6$^3$ for 552-atom 3/2, and 2$^3$ for 2324-atom 5/3.
Empirical Oscillating Pair Potentials {#app:eopp}
=====================================
We choose a 6-parameter empirical oscillating pair potentials (EOPP [@EOPP]) of the form $$\label{eq:oscil6}
V(r) = \frac{C_1}{r^{\eta_1}} + \frac{C_2} {r^{\eta_2}} \cos(k_* r + \phi_*)$$ to fit a DFT-derived database of force components and energies. The database contains binary compounds Al$_2$Cu (both tetragonal and cubic), Al$_3$Fe, Al$_5$Fe$_2$, ternary tetragonal Al$_7$Cu$_2$Fe, orthorhombic Al$_{23}$CuFe$_4$, as well as the ternary extension of Al$_3$Fe, and a number of approximant structures. The database contains a significant portion of AIMD data at elevated temperatures, ranging from T=300K up to 2000K, covering both solid and liquid configurations. In total we use 13176 force-component data points and 63 energy differences (see Fig. \[fig:ppfit\]).
$C_1$ $\eta_1$ $C_2$ $\eta_2$ $k_*$ $\phi_*$
------- --------------------- ---------- --------- ---------- -------- ----------
Al–Al 4337 10.416 -0.1300 2.2838 4.1702 0.8327
Al–Fe 1.03$\times$10$^5$ 17.511 4.8643 3.3527 3.0862 1.6611
Al–Cu 482 8.899 -2.8297 3.7479 3.2019 4.3551
Fe–Fe 1.233$\times$10$^6$ 13.622 5.0695 2.5591 2.5215 3.8725
Fe–Cu 461 7.363 -3.7766 3.1410 2.9191 5.7241
Cu–Cu 1069 9.321 -2.3005 3.2640 2.8665 0.0586
: \[tab:eopp\] Fitted parameters for Al–Cu–Fe EOPP potentials.
We initialized the fit from parameter values that fit GPT potentials [@HBSD2] for a similar system (Al-Co-Ni). The fit quickly converged, with RMS deviation 0.16 eV/Å for forces, and 9.4 meV/atom for energy differences. Since Fe-Fe and Fe-Cu potentials are prone to softening at nearest-neighbor distances due to the lack of data in Al-rich systems, we increased repulsion term coefficients $C_1$ manually for Fe-Fe and Fe-Cu. Final parameters of our potentials are listed in Tab. \[tab:eopp\]. Our cutoff radius is taken as 7 Å.
![ \[fig:ppfit\] EOPP potentials fitted to ab–initio (VASP) force and energy data. Parity plots on left are $\Delta E$ (units meV/atom) and $F$ (units eV/Å). The resulting potentials are shown at right. Parameters are summarized in Table \[tab:eopp\]. ](ppfit-combine.pdf){width="3.25in"}
As a demonstration of the accuracy of our pair potentials, we computed the vibrational densities of states (VDOS) for a 208-atom “3/2-2/1-2/1” approximant (see Fig. \[fig:pvdos\]). The three partials computed by EOPP semiquantitatively match the DFT result, with accuracy comparable to the Sc-Zn case [@ScZn].
![ \[fig:pvdos\] Partial vibrational DOS for a 208-atom orthorhombic approximant of icosahedral quasicrystal, calculated directly by DFT (solid lines), or by EOPP (broken lines). ](pdos.pdf){width="3.25in"}
It should be noted that the potentials are valid only for a particular density of the free-electron sea, and they should be used exclusively at constant volume; or in constant–pressure simulations, with additional external pressure set to a value leading to the same equilibrium volume.
Replica Exchange Simulations {#app:replica}
============================
To enhance the efficiency with which our simulations explore the configurational ensemble, we employ a replica exchange mechanism [@Swendsen86], also known as parallel tempering, in which we perform many runs simultaneously at different temperatures. The probability for a configuration $i$ of energy $E_i$ to occur at temperature $T_i$ is $$P_i =\Omega(E_i) e^{-\beta_i E_i}/Z_i$$ where $\Omega(E)$ is the configurational density of states, $Z_i$ is the partition function at temperature $T_i$, and $\beta_i=1/{k_{\text{B}}}T_i$. The joint probability for configuration $i$ at $T_i$ and configuration $j$ at $T_j$ is $$P = P_i P_j = (\Omega(E_i)\Omega(E_j) e^{-(\beta_i E_i+\beta_j E_j)}/Z_iZ_j$$ Now consider swapping temperature between configurations $i$ and $j$. The joint probability for configuration $i$ to occur in the equilibrium ensemble at temperature $T_j$ and configuration $j$ to occur at temperature $T_i$ is $$\begin{aligned}
P' &= (\Omega(E_i)\Omega(E_j) e^{-(\beta_j E_i+\beta_i E_j)}/Z_iZ_j \\
&= P e^{(\beta_i-\beta_j)E_i-(\beta_i-\beta_j)E_j}\end{aligned}$$ Hence, if the swap is performed with probability $P'/P=e^{\Delta\beta\Delta E}$, equilibrium is preserved following the swap.
This process works most efficiently if energy fluctuations are sufficiently large that the energy distributions $H(E)$ at adjacent temperatures overlap, so that swaps occur frequently. In this case, a given run (sequence of consecutive configurations) will diffuse between low and high temperatures. Rapid structural evolution at high temperatures thus provides an ongoing source of independent configurations for low temperatures where structural change is intrinsically slow.
We perform isochoric replica-exchange atomistic simulations in the temperature range of 200-1800 K, for several sizes of approximants: 2/1 (128-129 atoms), 3/2 (548 atoms), 5/3 (2324 atoms) and 8/5 (9844 atoms). The basic cycle of the simulation is a species-swap fixed-lattice Metropolis Monte Carlo (MC) stage, consisting of $\sim$30-100 swap attempts per pair, followed by 100-200 MD steps starting directly from the final MC configuration, and finally attempted replica swaps between adjacent temperatures. The temperatures spacing increases linearly with temperature following $$\Delta T = \frac{\alpha}{\sqrt{N_{a}}},$$ where $\alpha$ is a multiplicative coefficient and $N_a$ number of atoms; such spacing guarantees uniform replica exchange acceptance rates assuming constant heat capacity. Consequently, for large systems an added load is due to the increasingly finer temperature grid. The parameters of most important simulations are summarized in Table \[tab:repex\].
L/S supercell T-range $N_a$ MCS MDS cycles/10$^3$
----- ---------------------------------- ---------- ------- ----- ----- ---------------
8/5 1 650-1824 9844 25 100 33
5/3 1 200-1800 2324 30 500 91
3/2 $\sqrt{2}\times\sqrt{2}\times 1$ 400-1800 1096 25 200 80
2/1 $2\times 2\times 2$ 200-1811 1032 25 200 110
: \[tab:repex\]Cooling simulations for sequence of quasicrystal approximants. All sizes have the same composition Al$_{65.0}$Cu$_{22.5}$Fe$_{12.5}$. Column MCS is number of Monte Carlo attempts per atom for the lattice-gas stage of the simulation, MDS number of MD steps (time step $dt$=4fs, per one cycle of the simulation. Whole simulation
Hyperspace reconstruction {#app:6D}
=========================
Lifting a raw atomic structure to hyperspace amounts to associating the position of each atom to an ideal point in a 6D hypercubic crystal. We require that the projection of the ideal position into physical space match the actual atomic position to within a tolerance $r_{core}$. Additionally we require that the ideal position lie close in 6D space to the physical 3D space within “atomic surfaces” of definite positions, shapes and sizes. Specifically, we choose $\tau^2$-triacontahedra at 6D nodes, and unit triacontahedra at 6D body-centers, of radius $r_{6D}$=11.7Å in the 5-fold direction. In real space, the projected sites are separated by at least 1.05Å. We obtain unambiguous registration with $r_{core}=$0.45Å.
We register each structure by: ($i$) identifying all Al$_{12}$Cu icosahedra in the actual atomic structure (centers of perfectly icosahedral clusters will have the smallest positional deviation from ideal sites); ($ii$) mapping these to the even-only [*body-center*]{} sublattice (hence resolving the even/odd ambiguity); ($iii$) Given the relative shift obtained from step ($ii$), we attempt to map the entire set of atoms positions to the ideal 6D sites. In practice about 80% of sites map, with the exceptions being primarily the disordered inner shells of the [[*pMI *]{}]{}.
Working with periodically bounded boxes in real space results in finite resolution of the perp-space, namely $d_{res}^\perp =
a_q/\sqrt{F_{n}^2+F_{n-1}^2}$, $F_n$ are Fibonacci numbers and $a_q$=4.462 Å. For our 8/5 approximant, $d_{res}^\perp =
a_q/\sqrt{5^2+3^2}\sim$0.765 Å .
The atomic surfaces that result from the registration for our largest 8/5 approximant, after averaging over 3000 configurations, are shown in Fig. \[fig:perp\].
Ternary phase diagrams {#app:ternary}
======================
We model phase stability by exploring the full composition space. In addition to the quasicrystal phase, we include the pure species in their favored structures, all known binary Al–Cu and Al–Fe phases, and all known ternary phases: $\lambda$-Al$_3$Fe.mC102, $\beta$-AlCuFe, Al$_6$Mn structure type $\tau_1$-Al$_{23}$CuFe$_4$.oC28, $\omega$-Al$_7$Cu$_2$Fe, $\phi$-Al$_{10}$Cu$_{10}$Fe. All of the phases have known structures with the exception of $\beta$ and $\phi$, exhibiting vacancy and chemical disorder; the $\phi$ phase was claimed to be a superstructure of Al$_3$Ni$_2$ structure [@zhang] but belongs to the same $B2$–type family.
To represent the $\beta$ phase family, we evaluated DFT cohesive energies for every member of the 2x2x2-supercell ensemble of the cubic $B2$ structure type, under the constraint of fixing the cube-vertex occupation as Al. For a given Cu content, we created a list of all symmetry-independent Cu/Fe orderings on the [ *body–center*]{} sublattice. Ground states with 1-5 Cu atoms per 16-atom supercell revealed that 2-4 Cu atom range yields stable structures, while 1/16 or 5/16 Cu atom compositions (with 7/16 and 3/16 Fe atom content respectively) are unstable. By examining the electronic DOS we concluded that the ternary $\beta$ phase is electronically stabilized at low T by pushing $E_F$ beyond the steep Fe-$d$-band shoulder, optimally by substituting 3Fe$\rightarrow$3Cu (per 16-atoms). This ternary $\beta$-phase is predicted to be stable at T=0K in the composition range 12.5-25% of Cu.
At the Cu-rich composition, the $\beta$ phase takes a vacancy-ordered form with experimental composition Al$_{10}$Cu$_{10}$Fe. Since we could not find any promising T=0K structure in the 2x2x2 supercell, and larger supercells ensembles are inaccessible to our direct DFT method, we proceeded with EOPP potentials and fixed-site lattice-gas annealing [@HBSD2], in which atoms are constrained to occupy fixed lattice of sites, but pairs of species with different chemistry are allowed to swap their positions. Using this method we discovered a T=0K stable state in a 3x3x3 supercell, whose composition can be described by a single parameter $x$=4/(2$\times$3$^2$)$\sim$0.074 and composition (Al$_{0.5-x}$Cu$_x$)(Cu$_{0.5-2x}$Fe$_x$Vac$_x$) where the parentheses separate cube-vertex/body-center sublattices respectively.
The quasicrystal family of structures is represented by 2/1 and 5/3 approximants. The former turns out to be the most stable T=0K structure within the family ($\delta E=+1.8$ meV/atom), while the 5/3 model at ($\delta E=+4.0$ meV/atom is our best representation of the quasicrystal phase.
To predict the phase diagram at finite temperature, T$>$0, we add the free energy $F_{\rm Tot}$ in Eq. (\[eq:Ftot\]) to the DFT-calculated enthalpy for each structure considered. Because full phonon calculations for large QC approximants are prohibitive, we assume that they all share the same $F_h$ as the 2/1 approximant. We then compute the convex hull of the set of free energies (see Fig. \[fig:pdiag\]). Vertices of the convex hull are predicted to be stable. We also include a few structures whose free energies lie slightly above the convex hull by up to 4 meV/atom. Structures are labeled using their phase name followed by their Pearson symbol.
All known structures lie on or near the convex hull, both at T=0K and at T=600K, with the exception of Al$_2$Fe in its observed structure of Pearson type aP19. Instead we find the hypothetical Al$_2$Fe structure of Pearson type tI6 that is believed to be more energetically stable [@MihalkovicWidom2012]. Some binaries extend into the ternary composition space, for example Al(Cu,Fe).cP2. The two quasicrystal approximants, iQC-2/1 and iQC-5/3 (reference numbers 21 and 20 respectively), swap stability between low and high temperature. This occurs because the smaller 2/1 approximant has an optimized structure and composition at which a deep pseudogap appears and the energy reaches the convex hull, whereas the larger 5/3 approximant lacks a unique optimized structure and composition but instead enjoys a large anharmonic contribution to its entropy. The 2/1 approximant appears on the convex hull at $T=0$ despite having $E>0$ according to DFT because of the quantum zero point vibrational (or competing phases) that is contained in $F_h$. The 5/3 approximant is the largest for which we can reliably obtain its low temperature enthalpy by DFT, and hence we take it as a proxy for the true quasicrystal state.
Thus we predict that the 2/1 approximant should be stable at low temperatures but transform to the icosahedral quasicrystal at elevated temperatures of 600K and above. Correspondingly, the quasicrystal loses stability at low temperature and transforms into the 2/1 approximant. Many actual or implicit transformations have been reported for Al–Cu–Fe quasicrystals [@Bancel1989; @Audier1991; @Bancel1993; @Menguy1993]. Fine detail of the phase diagram at 600 C [@alcufe-approximants] revealed that except for small window of single-phase stable quasicrystal composition around Al$_{62}$Cu$_{25.5}$Fe$_{12.5}$, the quasicrystal phase transforms into a rhombohedral approximant ($R$-5/3 in our notation), and this transformation is reversible [@menguy-R-alcufe]. The quasicrystal phase field shrinks with decreasing temperature [@pdiag-alcufe] but remains finite at T=560C, a temperature above our predicted transformation. At low temperatures the kinetics becomes slow and the transformation will be inhibited, so the quasicrystal remains metastable at low temperatures.
According to the experimental phase diagram, our energy minimizing, electronically optimized composition (Al$_{65}$Cu$_{22.5}$Fe$_{12.5}$) with $E_F$ in the center of the pseudogap lies in a coexistence region of three phases: $\lambda$, $\omega$ and $R$, and the latter should have composition Al$_{63.5}$Cu$_{24}$Fe$_{12.5}$ in agreement with Ref. [@menguy-R-alcufe]. However, the center of the $R$-approximant phase field is around $x_{Cu}=$0.26 and $x_{Fe}=$0.12, not far from the composition of our low-temperature winner, the 2/1 approximant Al$_{63.3}$Cu$_{25.8}$Fe$_{10.9}$, which lies just outside the $R$-phase stability range.
We simulated the $R$-phase structure in a cell of 718 atoms at our optimized composition of Al$_{65}$Cu$_{22.5}$Fe$_{12.5}$. Although the optimal structure places $E_F$ at the center of a pseudogap, the energy remains higher than the cubic 5/3 approximant by $\sim$ 3 meV/atom. Due to kinetic barriers at low temperatures, existence of the 2/1 approximant cannot be ruled out. Adequate evaluation of all competing phases around 800-1000 K would require systematic variation of composition and density of all competing phases.
Energetic optimization of quasicrystal approximants {#app:nrg}
===================================================
The most direct comparison between approximants is achieved under the constraint of equal density/composition revealing the impact of structure alone. Also, these are the conditions under which the EOPP energies are most meaningful (see Appendix \[app:eopp\]). Expressing atomic density per $b^3$ volume, where $b\sim$12.2 Å is the side of the 2/1 cubic cell, we chose density 129.51 atoms/Å$^3$ and composition Al$_{65}$Cu$_{22.5}$Fe$_{12.5}$. In order to satisfy this constraint accurately for all approximants, we worked with the 2/1 approximant in a $2\times 2\times 2$ supercell (1032 atoms), the 3/2 approximant in a $\sqrt{2}\times\sqrt{2}\times 1$ supercell with 1096 atoms, and the 5/3 in its unit cell with 2324 atoms. Supercells were also required in order to counter the size effect when measuring anharmonic heat capacity, $F_a$. The resulting optimized energies are presented in Table \[tab:appene\]. The sequence of formation enthalpies, both for EOPP and full DFT calculations, appears to favor larger approximants that minimize the phason strain and accommodate larger superclusters.
Since the optimal density and composition could vary between approximants, we varied these individually for each approximant within its conventional unit cell, with the results presented previously in Table \[tab:ene\]. Again the enthalpy is found to be a decreasing function of approximant size, both for EOPP and for DFT. However, the energy relative to the convex hull is minimized for the smallest approximant. This is because the larger approximants favor greater Fe content, and the strong bonding of Fe (see Fig. \[fig:ppgofr\]) causes a strong slope of the convex hull facets in the direction of increasing Fe.
The 2/1 approximant system size (128-129 atoms) allowed for full exploration of all degrees of freedom. Under EOPP the icosahedral structure forms easily from the melt, and we can apply full DFT refinement to simultaneously explore compositional and density variation for EOPP pre-optimized models. The structure with lowest $\Delta E$ occured at increased Cu content, and density of 128 atoms/cell (identical with the Katz-Gratias model prediction). Starting from this model, we then performed AIMD for 5000 steps (5fs time step) of MD annealing at 1100, 900 and 700 K, and quenched several snapshots from each temperature. The lowest energy snapshot was from the 900K annealing batch, and yielded the best atomic structure. This optimal structure exhibits a deep pseudogap (0.015 states/eV/atom according to tetrahedron method calculation, see Fig. \[fig:edos\]) centered on the Fermi energy. The pseudogap becomes shallower and broader for structures in the equlibrium ensemble at higher temperatures.
In the 3/2 approximant cell, we explored several densities: 544 (KG model density), 552 and 448 atoms/unit cell. The composition was refined by seeking deepening of the pseudogap and controlling the Fermi energy assuming a rigid-band picture and simple valence rules. The best structure was a 552-atom model, greater by 1.5% than the KG-model density. This structure was then annealed under AIMD for 5000 steps at 1250K; we observed strong atomic diffusion and some Al atoms moved as far as 6Å. Subsequently, we cooled gradually from 1250K to 800K in another 5000 steps, and finally from 800K down to 300 K in 1000 steps. At the end, we found that maximal displacement was 5.1Å for Al, 2.9Å for Cu and 0.5Å for Fe atoms; 20 Cu and 100 Al atoms displaced by more than 0.5Å. Despite that, the energy of the final annealed configuration was nearly identical to the initial configuration. Our best 3/2 approximant has a narrow true gap (according to the tetrahedron method) at E$_F$.
The 5/3 approximant system size did not allow for systematic variations, but we did explore several densities (2304, 2324, 2338 atoms) and compositions, using pseudogap depth and Fermi level position as guides. The final structure was optimized under ab-initio relaxation in $\sim$70 ionic steps. AIMD annealing was not feasible. In contrast to the smaller approximants, all 5/3 models considered had broader and less deep pseudogaps. Nonetheless, the 5/3 approximant achieves the lowest formation enthalpy of the approximants, most likely as a result of its greatest Fe content.
$N_{at}$ $\Delta$E$_{eopp}$ H$_{DFT}$ $\Delta$H$_{DFT}$ $\Delta$E$_{DFT}$
----- ---------- -------------------- ----------- ------------------- -------------------
5/3 2324 ref. -293.0 ref +4.0
3/2 1096 +0.9 -290.1 +2.9 +7.1 (+3.1)
2/1 1032 +5.3 -284.6 +8.4 +12.50 (+8.5)
8/5 9846 +0.3 – –
: \[tab:appene\] Energetic competition (in meV/atom) within cubic approximant family at equal composition Al$_{65.0}$Cu$_{22.5}$Fe$_{12.5}$, placing $E_F$ exactly at the center of the pseudogap (see Fig. \[fig:edos-fixch\]). 2/1 and 3/2 approximants are modeled in supercells of sizes $2\times 2\times 2$ and $\sqrt{2}\times\sqrt{2}\times 1$, respectively. The largest 8/5 approximant (last row of the Table) is inaccessible to DFT evaluation. Notice the enthalpy H$_{DFT}$ decreases with system size. Column $\Delta$E$_{DFT}$ from convex hull evaluation would be equal to the $\Delta$H$_{DFT}$ if the approximant compositions were strictly equal.
![ \[fig:edos-fixch\] Electronic density of states for 2/1, 3/2 and 5/3 approximants at fixed composition Al$_{65.0}$Cu$_{22.5}$Fe$_{12.5}$. 2/1 and 3/2 approximants represented by supercells (see Table \[tab:appene\]) contain amount of frozen disorder comparable to the 5/3 approximant. Resolution with Gaussian $\sigma$=0.02 eV ($a$), 0.01 eV ($b$) and 0.006 eV ($c$). ](edos_fixch.pdf){width="3.25in"}
![ \[fig:edos\] Electronic density of states for 2/1, 3/2 and 5/3 approximants at their optimal compositions (see Table \[tab:ene\]), eigenenergies density smeared using Gaussian $\sigma$=0.02 eV ($a$). Zoom into fine-accuracy tetrahedron method DOS calculation using 10$\times$10$\times$10 and 6$\times$6$\times$6 meshes respectively for 2/1 and 3/2 approximants in ($b$) and ($c$). 5/3 approximant (2$\times$2$\times$2 calculation) with resolutions $\sigma$=0.006 eV ($b$) and 0.002 eV ($c$). ](edos.pdf){width="3.25in"}
Thermodynamics {#app:thermo}
==============
The Helmholtz free energy $F(N,V,T)$ can be approximately decomposed into a relaxed T=0K energy $E_0$, plus corrections due to harmonic vibrational free energy, $F_h$, anharmonic positional disorder $F_a$, and electronic excitations $F_{\rm elect}$. Thus we write $$\label{eq:Ftot}
F_{\rm Tot} = E_0+F_h+F_a+F_e.$$
The harmonic vibrational free energy of a single phonon mode of frequency $\omega$ is $$\label{eq:Fmode}
f_h(\omega)={k_{\text{B}}}T \ln{[2\sinh{(\hbar\omega/2{k_{\text{B}}}T)}]}.$$ Notice that as ${k_{\text{B}}}T\ll \hbar\omega$, $f_h(\omega)\rightarrow \hbar\omega/2$, which is the zero point vibrational energy. As ${k_{\text{B}}}T\gg\hbar\omega$, $f_h(\omega)\rightarrow {k_{\text{B}}}T\ln{(\hbar\omega/{k_{\text{B}}}T)}$, which is the classical limit of the free energy. The full harmonic free energy $$\label{eq:Fh}
F_h(T)=\sum_i f_h(\omega_i).$$
The anharmonic contribution includes corrections due to shifts in phonon frequency with large amplitude of oscillation and additional discrete degrees of freedom connected to chemical substitution and possible additional tiling flips. At low temperature these contributions can be neglected, so we only include them beyond a temperature, $T_0$, which we set at 200K. At these elevated temperatures positional degrees of freedom behave nearly classically, so we will evaluate $F_a$ from our classical MC/MD simulations using EOPP.
In the canonical $NVT$ ensemble, the Helmholtz free energy $F(N,V,T)=U-TS$ has the differential $$\label{eq:dF}
dF(N,V,T)=-S {{\rm d}}T -p {{\rm d}}V +\mu {{\rm d}}N.$$ In particular, $S=-\partial F/\partial T$. The entropy is also related to the heat capacity $C=\partial U/\partial T$ through $C/T = \partial S/\partial T$. Hence, $$\label{eq:C}
C=-T \frac{\partial^2 F}{\partial T^2}.$$ Conveniently, $C$ can be obtained at high temperatures from classical MC/MD simulation through fluctuations of the energy, $$\label{eq:Cfluct}
C=\frac{1}{{k_{\text{B}}}T^2}\left(\avg{E^2}-\avg{E}^2\right).$$ $C$, thus obtained, includes contributions from both harmonic and anharmonic atomic vibrations, and potentially also from chemical substitution and tile flipping.
Because we seek the anharmonic component of the free energy, we define $C_a\equiv C-3N{k_{\text{B}}}$ for $T>T_0$, and $C_a\equiv 0$ for $T<T_0$. We now integrate $C_a$ once to obtain $$\label{eq:SofT}
S_a(T)=\int_0^T \frac{C_a(T')}{T'}{{\rm d}}T',$$ and then integrate once more to obtain $$\label{eq:Fint}
F_a(T)=-\int_{T_0}^T S_a(T') {{\rm d}}T'.$$ By definition, $C_a$, $S_a$ and $F_a$ all vanish below $T_0$, then grow continuously at high $T$.
The electronic free energy is obtained from the superposition of single state energies and entropies. An electronic state of energy $E$ is occupied with probability given by the Fermi-Dirac occupation function, $f_\mu(E)=1/(\exp{((E-\mu)/{k_{\text{B}}}T})+1)$. For $N$ electrons, the chemical potential is defined by the requirement that $N=\sum_i f_\mu(E_i)=N$. Fractional state occupation leads to electronic entropy $$\label{eq:Se}
S_e(E)=-{k_{\text{B}}}[f_\mu(E)\ln{f_\mu(E)}+(1-f_\mu(E))\ln{(1-f_\mu(E))}].$$ Summing over electronic states, we obtain free energy $$\label{eq:Fe}
F_e=\sum_i \left(E_i f_\mu(E_i)-T S_e(E_i)\right).$$
The overlapping energy distributions at different temperatures created by replica exchange provide an opportunity for accurate calculation of heat capacity, entropy and free energy through the method of histogram analysis. At a single temperature $T$, the frequency distribution of simulated energies $E$ is proportional to the Boltzmann probability $$P_T(E)=\Omega(E) e^{-E/{k_{\text{B}}}T}/Z(T)$$ This equation can be inverted to obtain the density of states $\Omega(E)\sim H_T(E) e^{+E/{k_{\text{B}}}T}$, where $H_T(E)$ is a normalized histogram of energies obtained from a simulation at fixed temperature $T$. Given the density of states, the partition function may be calculated (up to an undetermined multiplicative factor) by integrating, $$\label{eq:Zhisto}
Z=\int{\rm d}E\Omega(E)e^{-E/{k_{\text{B}}}T}.$$ The free energy is determined (up to an additive linear function of T) from $$\label{eq:Fhisto}
F=-{k_{\text{B}}}T\log{Z},$$ and all other thermodynamic functions can be obtained by differentiation. Notice that $Z(T)$ and $F(T)$ are obtained as continuous functions of temperature $T$ over a range of temperatures surrounding the original simulation temperature.
The same approach interpolates between the fixed simulation temperatures by consistently merging densities of states obtained from each temperature [@Swendsen89]. Up to an unknown multiplicative constant, we have $$\label{eq:Whisto}
\Omega(E)=\frac{\sum_T H_T(E)}{\sum_T e^{(F_T-E)/{k_{\text{B}}}T}}$$ where the free energies $F_T$ must be obtained self-consistently with $\Omega(E)$ from Eqs. (\[eq:Whisto\]) and (\[eq:Fhisto\]). By setting the value of $F$ and its derivative $S=-\partial F/\partial T$ to values determined from first principles methods at the lowest simulation temperature, we obtain absolute free energy across the entire simulated temperature range.
[^1]: similar to experimentally known Al$_{10}$Cu$_{10}$Fe $\phi$-phase
|
---
abstract: 'Searching for black hole echo signals with gravitational waves provides a means of probing the near-horizon regime of these objects. We demonstrate a pipeline to efficiently search for these signals in gravitational wave data and calculate model selection probabilities between signal and no-signal hypotheses. As an example of its use we calculate Bayes factors for the Abedi-Dykaar-Afshordi (ADA) model on events in LIGO’s first observing run and compare to existing results in the literature. We discuss the benefits of using a full likelihood exploration over existing search methods that used template banks and calculated p-values. We use the waveforms of ADA, although the method is easily extendable to other waveforms. With these waveforms we are able to demonstrate a range of echo amplitudes that is already is ruled out by the data.'
author:
- 'Alex B. Nielsen'
- 'Collin D. Capano'
- Ofek Birnholtz
- Julian Westerweck
title: Parameter estimation for black hole echo signals and their statistical significance
---
Introduction
============
Black holes are defined by their horizons [@Hawking:1973uf]. Although a large amount of astrophysical data is compatible with the existence of black holes [@Narayan:2013gca], a number of theoretical models still predict dark compact objects without horizons or for which the horizon structure is significantly modified from classical vacuum general relativity [@Visser:2009pw; @Cardoso:2016oxy; @Cardoso:2017njb; @Cardoso:2017cfl; @Holdom:2016nek]. These models are typically motivated by quantum effects or attempts to address issues related to black hole information and evaporation [@Polchinski:2016hrw]. One possible observational signature of such structure is that infalling waves would not be entirely absorbed by the horizon as is generally expected in general relativity, but instead some amount of the infalling wave would be reflected.
Recent observations of gravitational waves from coalescences of binary black holes [@Abbott:2016blz; @Abbott:2016nmj; @TheLIGOScientific:2016pea; @Abbott:2017vtc; @Abbott:2017gyy; @Abbott:2017oio] by the LIGO [@TheLIGOScientific:2014jea] and Virgo [@TheVirgo:2014hva] detectors have allowed for a number of new tests of the near horizon structure of black holes [@TheLIGOScientific:2016src; @Cabero:2017avf; @Nielsen:2017lpd]. One such test involves searching for echo signals that could potentially be caused by reflective structure forming at or near the location of the black-hole horizon. A number of groups have searched for such signals in gravitational wave data with contrasting conclusions [@Abedi:2017; @LowSignificance; @Conklin:2017lwb]. Here we propose a new method to search for these echo signals that provides an explicit probability for the compatibility of the data with the echoes hypothesis relative to Gaussian noise. We demonstrate this method on the binary black hole events detected during the first observing run of the Advanced LIGO detectors; these events were the subject of previous studies [@Abedi:2017; @LowSignificance; @Conklin:2017lwb].
The general physical picture of echoes is that infalling radiation is reflected due to some mechanism near the putative horizon location. This radiation is then partially trapped between the near-horizon structure and the angular momentum light-ring barrier [@Cardoso:2016rao]. Some of the energy is transmitted away from the system by successive bounces, thereby forming a series of echoes. Generic parameters in the physical models are the amount of wave reflected by the boundary and the effective location where this reflection occurs. These in turn are related to the amplitude of the reflected echo signals and the time separation between the successive echoes. Bounds on the amplitude and time separation of echo signals derived from the data can be translated into bounds on the reflectivity and location of the near-horizon structure.
For illustrative purposes here, we focus on the explicit model of Abedi-Dykaar-Afshordi (ADA) [@Abedi:2017], which has been the subject of discussion in the literature [@EchoComments; @Abedi:2017isz; @LowSignificance; @Abedi:2018pst]. However, we note that our methodology can just as well be applied to other, more detailed models with explicit waveforms, including those recently proposed in the literature [@Mark:2017dnq; @Nakano:2017fvh]. Efforts to search for echo templates using Bayesian model selection have been developed with LALInference [@lalinf] in parallel to our own work, and published concurrently with our own [@Lo:2018sep]. Other, model-agnostic searches [@Tsang:2018uie], have also been ongoing, along with different techniques to constrain horizonless objects through their impact on the stochastic background [@Barausse:2018vdb].
The primary result of [@Abedi:2017] is a p-value, calculated as the probability of observing a signal-to-noise ratio (SNR) in noise (assumed to be free of signal) at least as significant as that observed in the on-source data that potentially contains the signal. This by itself does not indicate the probability that the on-source data contains a signal. A probability that the data contains a signal can however be obtained using Bayes’ theorem: . It is most convenient to compare this probability to an alternative hypothesis, for example that the data contains only Gaussian noise: . In the above, the first factor on the right hand side is the likelihood ratio and the second factor is the prior odds. Evaluating the prior odds is difficult without prior data (and in the case of a signal model that violates expected physics, might well be a very small factor) but the likelihood ratio can be calculated by exploring the likelihood function over the model parameters using a stochastic sampling algorithm, such as a Markov chain Monte Carlo (MCMC). To obtain a final Bayes factor, the model parameters must be marginalized over using their respective prior distributions.
The example we consider here is based on the hypothesis of ADA [@Abedi:2017]; we refer the reader to that work for more detail on the model and the meaning of the various model parameters. The most important of these parameters are the overall amplitude of the echoes relative to the original signal’s peak $A$, the relative amplitude between successive echoes $\gamma$, and the time separation between successive echoes [$\Delta t_{\mathrm{echo}}$]{}; these and the other parameters [$t_{\mathrm{echo}}$]{} and [$t_{0}$ trunc.]{} are explained more fully in [@Abedi:2017]. Table \[tab:table1\] gives the prior ranges we use for the relevant parameters. These are adapted for our purposes from the template bank search performed in [@Abedi:2017].
\[tab:table1\]
-------------------------------- --------------------------------------------------------------------- -------------------- --------------
**Echo** **Prior** **GW150914** **Injected**
**param.** **range** **range** **value**
[$\Delta t_{\mathrm{echo}}$]{} inferred 0.2825 to 0.3025 s 0.2925 s
[$t_{\mathrm{echo}}$]{} ${\ensuremath{\Delta t_{\mathrm{echo}}}}\pm 1\%$ 0.2795 to 0.3055 s 0.2925 s
[$t_{0}$ trunc.]{} $(-0.1 \mathrm{\; to \;} 0){\ensuremath{\Delta t_{\mathrm{echo}}}}$ -0.02925 to 0 s -0.02457 s
$\gamma$ 0.1 to 0.9 0.1 to 0.9 0.8
$A$ unconstrained 0.00001 to 0.9 varying
-------------------------------- --------------------------------------------------------------------- -------------------- --------------
: Table of prior ranges and values used for injection studies. The ranges are adopted from [@Abedi:2017] and the injected values are chosen to lie close to the parameter values found in that work, except for $\gamma$ and [$t_{0}$ trunc.]{} which are chosen to lie within the prior range rather than at the boundary.
In the ADA model the range for [$\Delta t_{\mathrm{echo}}$]{} is inferred from the published parameters of GW150914 [@TheLIGOScientific:2016pea], using $50\%$ ranges, and assuming Gaussian distributions. The Kerr metric formula is used for the light travel time between the light ring and a perfectly reflecting surface. This surface is assumed to be at a proper distance one Planck length along Boyer-Lindquist time slices from the Kerr metric event horizon. The parameter $\gamma$ was chosen to reflect the physical expectation that the amplitude of successive echoes should decrease due to energy loss through one or both of the boundaries. We allow the parameter [$t_{\mathrm{echo}}$]{} to vary independently from [$\Delta t_{\mathrm{echo}}$]{} within $1\%$ of its maximum values, and choose an explicit prior for the relative amplitude.
![Posterior on the echo parameters for a loud (SNR $\sim 17$) simulated signal. The signal has GW150914-like parameters at a fiducial distance of $400\,$Mpc. An amplitude factor of 0.4 is used for the echoes. Off-diagonal plots show 2D marginal posteriors; the white contours show the $50\%$ and $90\%$ credible regions. Each point represents a random draw from the posterior, colored by the SNR ($\rho$) at those parameters. The diagonal plots show the 1D marginal posteriors, with the median and $90\%$ credible intervals indicated by the dashed lines. The reported values are the median of the 1D marginal posterior plus/minus the $5/95$ percentiles. We see that the injected parameter values, shown by the red lines, are all within the $90\%$ credible intervals. The log Bayes factor for this signal is $140.57$.[]{data-label="fig:dist400_corner"}](injection100_posterior.png){width="\columnwidth"}
![Posterior on the echo parameters for a quiet simulated signal. The signal has GW150914-like parameters at a fiducial distance of $400\,$Mpc. An amplitude factor of 0.0125 is used for the echoes. Again, the injected values are shown by the red lines, while points are colored by the SNR at that point in the parameter space. The log Bayes factor for this injection is -1.55, thus indicating what we would expect when the signal is not distinguishable from noise. The prior ranges are largely saturated and lines appear in [$t_{\mathrm{echo}}$]{}.[]{data-label="fig:dist3200_corner"}](injection3200_posterior.png){width="\columnwidth"}
Since the value of the amplitude will have a direct influence on the signal strength, and hence the signal likelihood, its prior range is of central importance to our results. In the template bank search of [@Abedi:2017] a prior for the amplitude is not explicitly given. Instead, it is maximised over the template bank. To replicate as closely as possible the method of [@Abedi:2017] we choose a flat amplitude prior from $10^{-5}$ to $0.9$. This ensures we are sensitive to relatively quiet amplitude signals, although not arbitrarily quiet, and implements the reasonable assumption that the first echo should not be louder than the main signal.
For simplicity we choose to fix the number of echoes to 30. In principle this could be allowed to vary, but for values of $\gamma$ less than $0.9$, 30 echoes capture the main part of the signal that influences the SNR. In testing, we found that varying this number did not change the results substantially.
To establish that our method can correctly identify echo signals in the data, we first test it on simulated echo signals with a variety of different amplitudes. These simulations are added to real detector data, which is made available by the Gravitational Wave Open Science Center (GWOSC) [@LOSC; @Vallisneri:2014vxa], $100$ seconds after GW150914. The $100$-second delay makes it unlikely that the data at that time is contaminated by a real astrophysical signal [@Nielsen:2018bhc]. We then apply our method directly to the three binary black hole events in O1: GW150914, LVT151012 and GW151226. Finally, we show how these results can be used to place bounds on the reflectivity of structure that has formed a given distance from the location of the would-be horizon.
![The $90\%$ credible regions of the 2D marginal posteriors of ${\ensuremath{\Delta t_{\mathrm{echo}}}}$ and $\gamma$ for GW150914-like simulated signals. Shown are a range of echo amplitudes (relative to the peak amplitude of the original signal) $A$. The injected values are given by the horizontal and vertical red lines. For small values of $A$, the $90\%$ contour covers most of the prior range, whereas for larger amplitudes the contours narrow down onto the injected values.[]{data-label="fig:inj_rec_gamma_amplitude"}](injection_posterior_compare.png){width="\columnwidth"}
Methodology and analysis pipeline
=================================
The pipeline we use is based on [@Biwer:2018osg]. It employs a parallel-tempered MCMC algorithm, [@ForemanMackey:2012ig; @Vousden:2016], to sample the likelihood function for a hypothesis based on the existence of a signal in the data. The likelihood function is chosen to be compatible with the assumption that the underlying noise is Gaussian with a given power spectral density. Once the likelihood has been mapped, the marginalization over the model parameters is performed using thermodynamic integration to obtain a probability for the hypothesis given the data. Although it is known that LIGO data is not Gaussian over long periods of time, over shorter periods it is approximately Gaussian [@Nielsen:2018bhc]. To account for the non-Gaussianities without a model hypothesis for them, it is possible to sample the Gaussian Bayes factor over many realisations of the true detector noise.
In the results presented here we used 100 Markov chains to sample the likelihood. We require that each chain run for at least five auto-correlation lengths (ACL) beyond 1000 iterations of the sampler. The ACL is measured by averaging parameter samples over all chains, then taking the maximum ACL over all parameters. For the thermodynamic integration of the likelihood function, care has to be taken that it is sufficiently sampled both near its peak, but also at lower values of the likelihood. In tests we found that using 16 different temperatures, each placed by inspection, was sufficient to guarantee a consistent value of the Bayes factor. Convergence of this result was checked by running with double the number of temperatures and ensuring that the results were consistent. The posterior distributions are constructed from the coldest temperature chain.
Injections based on GW150914 {#sec:injections}
============================
To test our method we choose to examine simulated echo signals based on GW150914. This is, to date, the loudest binary black hole signal that has been observed via gravitational waves, and should play a central role in constraints derived from the data. Following ADA for simplicity, we choose to fix the base inspiral-merger-ringdown (IMR) waveform to be echoed for both injections and for the search templates. The parameters for these base IMR waveforms are given in the appendix and are obtained from the maximum likelihood results of [@Biwer:2018osg]. The waveforms are constructed using the phenomenological IMR waveform family IMRPhenomPv2 [@Khan:2015jqa; @Hannam:2013oca] which is freely available as part of LALSuite [@LALSuite]. These IMR signals are then used to produce echo signals with echo parameters given in Table \[tab:table1\]. The simulated echo injections are added linearly at varying amplitudes to real detector noise (chosen to be 100 seconds after GW150914, far enough away to be uncontaminated by echo signals or any pre-merger signal). We then attempt to recover them with our analysis pipeline. Example results are shown in Figs \[fig:dist400\_corner\] and \[fig:dist3200\_corner\].
![Values of the maximum likelihood SNR and log Bayes factors for GW150914-based injections with amplitudes from 0.025 to 0.4 at a distance of 400Mpc. A linear fit is possible through the SNR points down to an amplitude around 0.1. The log Bayes factor is negative for amplitude values below $\sim0.07$ (indicating formal preference for Gaussian noise over the echoes hypothesis, although at low absolute values of the log Bayes the data is uninformative). For amplitudes larger than 0.08, the log Bayes factor is greater than 1, indicating positive preference for echoes (the injected signal) over Gaussian noise, by the nomenclature of [@Kass].[]{data-label="fig:inj_snr_logbf"}](fixed_injection_distance_snr_and_logbf_versus_amplitude.png){width="\columnwidth"}
Figure \[fig:dist400\_corner\] shows a very loud injection with a relative amplitude of 0.4 and a maximum likelihood SNR of $\sim 17.7$. The log Bayes factor for this injection is 140.57, showing a strong preference for the echoes hypothesis over the pure Gaussian noise hypothesis. In this case the echo parameters are well recovered, with the injected values lying within the 90% credible intervals of the marginalised one-dimensional posterior distributions.
Figure \[fig:dist3200\_corner\] shows a much quieter injection with a relative amplitude of $0.0125$ and a maximum likelihood SNR of only $3.8$. The log Bayes factor for this injection is $-1.55$ showing a preference for the pure Gaussian noise hypothesis. In this case most echo parameters are not well recovered and their posterior distributions are close to the original prior distributions.
Figure \[fig:inj\_rec\_gamma\_amplitude\] shows the recovery of $\gamma$ and ${\ensuremath{\Delta t_{\mathrm{echo}}}}$ for a range of different injection amplitudes. As the amplitude is increased, the recovered value is increasingly constrained to the injected value.
The recovery of signals with different amplitudes is shown in Fig. \[fig:inj\_snr\_logbf\]. This figure can be compared with Fig. 4 of [@LowSignificance], which shows the recovery of amplitudes relative to the injected amplitudes. In that work it was found that below a certain injection strength, the recovered echo amplitude was no longer reliable using the template bank method. Our results here are consistent with that finding. Here we find that below an amplitude of $\sim0.1$ the recovered maximum likelihood SNR no longer falls off linearly and flattens out to an approximately constant value of $\sim4$. At amplitudes below $\sim0.07$ the log Bayes factor becomes negative.
Events in the first observing run {#sec:events}
=================================
![Corner plot for ADA echoes templates in data just after the merger of GW150914. The log Bayes factor for this data is $-1.81$, indicating a preference the Gaussian noise hypothesis over the Echoes hypothesis. Lines are visible in the t\_echo subplots, but the SNR associated with these is still not high. These lines are also seen in tests of the pipeline on simulated Gaussian noise.[]{data-label="fig:GW150914_corner"}](GW150914_posterior.png){width="\columnwidth"}
The developed pipeline can be run directly on data immediately after the observed GW events (without injections). We show results for the three events of the first LIGO observing run in Table \[tab:table2\]. This shows that Gaussian noise is favoured over the echoes hypothesis for GW150914 with a log Bayes factor of $-1.81$. GW150914 is the loudest binary black hole merger yet detected. A corner plot of the posterior distributions for the echo parameters for GW150914 is shown in Fig. \[fig:GW150914\_corner\]. The 90% credible interval for the marginalised posterior of the parameter $\gamma$ is almost as wide as the prior range. The posterior of the amplitude, $A$, prefers lower values of the amplitude. The posterior for [$t_{\mathrm{echo}}$]{} shows distinct lines at certain values of time. These lines are unlikely to be associated with an astrophysical signal and are also seen in tests on simulated Gaussian noise with the same pipeline.
As seen in Table \[tab:table2\], both GW151226 and LVT151012 prefer the echoes hypothesis over Gaussian noise, but only marginally. These two events have lower amplitudes for the main signal than for GW150914 and thus echoes signals with the same relative amplitude would have a lower absolute amplitude relative to the ambient noise [@LowSignificance]. The detector noise is known not to be truly Gaussian for the LIGO detectors [@TheLIGOScientific:2016zmo]. We performed 20 background tests on off-source data that lies before or after the time of LVT151012 at intervals of 50 seconds. Each of these tests is sufficiently separated in time from the others that it will not be contaminated by a common signal. In these background tests, two examples were found with a Bayes factor larger than the result for LVT151012 shown in Table \[tab:table2\]. A total of four intervals returned Bayes factors that favoured the echo hypothesis over Gaussian noise. Backgrounds for similar (but not identical) echoes hypotheses were also studied in [@Lo:2018sep] who found evidence for significant tails in the distribution of Bayes factors in real detector noise versus simulated Gaussian noise.
While it is interesting to speculate whether a signal model could be developed that postdicts echo signals for certain events, such as LVT151012, but not for others, such as GW150914, we do not pursue that here. The argument that LVT151012 should be accepted as a genuine binary black hole merger was given recently in [@Nitz:2018imz], however we do not feel that the echoes data for LVT151012 is sufficiently strong to seriously entertain a model where LVT15012-like events display echoes, but GW150914-like events do not.
The SNR values found for the maximum likelihood templates in Table \[tab:table2\] are comparable, although not identical to those found in [@Abedi:2017] and [@LowSignificance]. The values computed here use a slightly modified echo waveform and the finite template spacing in the template banks of [@Abedi:2017] and [@LowSignificance] also causes a minor difference. The main differences are the different base IMR waveform employed and the different power spectral density (PSD) used to calculate the matches. The work of [@Abedi:2017] and [@LowSignificance] used a PSD directly from [@LOSC] whereas here we have used a PSD computed in [@Canton:2014ena; @Usman:2015kfa] using Welch’s method. We estimate the PSD by taking the median value over 64 8 second-long segments (each overlapped by 4 seconds), centered on the main event.
With the simplistic hypothesis that all three binary black hole events should show evidence for echo signals in the range of parameters assumed, we can simply add the log Bayes factor together to obtain an overall log Bayes factor for this model relative to Gaussian noise of $-1.81 + 1.25 + 0.42 = -0.14$. This is negative, indicating a preference for Gaussian noise, but not by much. It is worth noting that this simplistic combination assumes that the values for the echo parameters can lie anywhere in their prior ranges for any of the three events. This is slightly different from the hypothesis of [@Abedi:2017] that assumes certain echo parameters should have the same value in all three events. With a hypothesis that fixes the values of certain echo parameters to be the same in all cases, it is possible that the overall Bayes factor would be different from our result. But this issue also raises the question of how these common parameters should be fixed; a simple maximization of the sum of the squares of the template SNRs as in [@Abedi:2017], or as a maximization or marginalization of the likelihood function introduced here. We defer investigation of these subtle issues to future work.
**Event** **Log Bayes factor** **Max SNR**
----------- ---------------------- ------------- --
GW150914 -1.8056 2.86
LVT151012 1.2499 5.5741
GW151226 0.4186 4.07
: Table of Bayes factor results. Negative values indicate that the Gaussian noise hypothesis is preferred. Positive values indicate that the echoes hypothesis is preferred after marginalization over parameters. Log Bayes values with magnitude $<1$ are “not worth more than a bare mention” in the nomenclature of [@Kass]. []{data-label="tab:table2"}
Discussion and conclusions {#sec:bounds}
==========================
With knowledge of how sensitive our pipeline is from the injection test runs of Sec. \[sec:injections\] we can determine the amplitude of echoes that would have been detectable had they been present in the data. This allows us to place a bound on the amplitude of echoes emitted from the events considered here. We remind the reader that bounds from our search only relate to the family of echo waveforms considered here. These are based on the model proposed in [@Abedi:2017] and adopting the prior ranges of Table \[tab:table1\].
As shown in Fig. \[fig:GW150914\_corner\], the posterior amplitude recovery has a 90% confidence interval from $0.0583 + 0.1206 = 0.1789$ to $0.0583-0.0532 =
0.0051$. For this realization of the noise, amplitudes above $0.1789$ are ruled out at 90% confidence. This is consistent with the injection studies depicted in Fig. \[fig:inj\_snr\_logbf\] which show that (for noise at a different time, 100 seconds after the main event) echo signals with amplitudes $\gtrsim0.15$ would have been unambiguously identified in the data.
Echo signals of amplitudes $0.1$ relative to GW150914 would correspond to approximately 0.1 solar masses of energy being reflected from near the black horizon [@EchoComments]. Although this value of the amplitude is not conclusively ruled out with the current data, an amplitude as high as 0.2 is conclusively ruled out by our results.
For numerical simulations of systems similar to GW150914 within general relativity, it is estimated that approximately 4 solar masses of gravitational energy flows across the horizon [@Gupta:2018znn]. Our constraints here on the amplitude of echoes within the model of [@Abedi:2017] suggest that no more than $5\%$ of this energy is being reflected by near-horizon structure and re-emitted as echoes.
We have seen that there is little evidence of ADA echo-like signals in the data of GW150914. Although there is some evidence of echoes in LVT151012 and GW151226, as both show positive log Bayes factors, this evidence is not very strong. Sampling the true detector noise by running over off-source times, shows that the log Bayes factor found for LVT151012 is not unusual. A number of improved echo waveform models have been proposed; we defer running with these on further events to future work.
Acknowledgments
===============
O.B. acknowledges the National Science Foundation (NSF) for financial support from Grant No. PHY-1607520. This work was supported by the Max Planck Gesellschaft and we thank the Atlas cluster computing team at AEI Hanover. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.
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Fiducial IMR waveform parameters
================================
We list here the parameters of the base IMR waveforms used to construct the echo templates both for injections and for the searches. These values are obtained from the maximum likelihood values of [@Biwer:2018osg].
\[tab:table3\]
**Parameter** **GW150914** **LVT151012** **GW151226**
------------------------- --------------- --------------- ---------------
mass1 39.03 22.87 18.80
mass2 32.06 18.67 6.92
spin1x -0.87 0.12 0.44
spin1y -0.43 0.19 0.59
spin1z -0.06 -0.20 0.33
spin2x -0.11 0.018 0.00
spin2y -0.03 -0.019 -0.017
spin2z -0.15 0.062 0.0033
distance 477 751 315
ra 1.57 0.65 2.23
dec -1.27 0.069 0.98
tc 1126259462.42 1128678900.46 1135136350.66
polarization 5.99 5.64 1.43
inclination 2.91 2.32 0.68
$\mathrm{coa\_phase}$ 0.69 4.44 1.64
$\mathrm{phase\_shift}$ -0.92 -0.91 1.86
|
---
author:
- Aurélien Philippe Bouvier
bibliography:
- 'bibliography/mybib.bib'
title: 'Gamma-Ray Burst Observations at High-Energy with the $Fermi$ Large Area Telescope'
---
|
---
author:
- 'E. Spitoni [^1]'
- 'G. Cescutti'
- 'I. Minchev'
- 'F. Matteucci'
- 'V. Silva Aguirre'
- |
\
M. Martig
- 'G. Bono'
- 'C. Chiappini'
date: 'Received xxxx / Accepted xxxx'
title: '2D chemical evolution model: the impact of galactic disc asymmetries on azimuthal chemical abundance variations'
---
Introduction
============
In recent years, integral field spectrographs (IFSs) have largely substituted long-slit spectrographs in studies designed to characterize the abundance distribution of chemical elements in external galaxies. IFSs have permitted for the first time to measure abundances throughout the entire two-dimensional extent of a galaxy (or a large part thereof) and, thus, to detect azimuthal and radial trends (Vogt et al. 2017).
In the last years, several observational works have been found evidence of significant azimuthal variations in the abundance gradients in external galaxies. S[á]{}nchez et al. (2015) and S[á]{}nchez-Menguiano et al. (2016) analyzed in detail the chemical inhomogeneities of the external galaxy NGC 6754 with the Multi Unit Spectroscopic Explorer (MUSE), concluding that the azimuthal variations of the oxygen abundances are more evident in the external part of the considered galaxy.
Vogt et al. (2017) studied the galaxy HCG 91c with MUSE and arrived to the conclusion that the enrichment of the interstellar medium has proceeded preferentially along spiral structures, and less efficiently across them.
Azimuthal variations have been detected in the oxygen abundance also in the external galaxy M101 by Li et al. (2013). Ho et al. (2017) presented the spatial distribution of oxygen in the nearby spiral galaxy NGC 1365. This galaxy is characterized by a negative abundance gradient for oxygen along the disc, but systematic azimuthal variations of $\sim$ 0.2 dex occur over a wide radial range of galactic radii and peak at the two spiral arms in NGC 1365. In the same work, the authors presented a simple chemical evolution model to reproduce the observations. Azimuthal variations can be explained by two physical processes: after a local self enrichment phase in the inter-arm region, a consequent mixing and dilution phase si dominant on larger scale (kpc scale) when the spiral density waves pass through.
Probing azimuthal inhomogeneities of chemical abundances has been attempted in the Milky Way system too. Balser et al. (2011), measuring H II region oxygen abundances, found that the slopes of the gradients differ by a factor of two in their three Galactic azimuth angle bins. Moreover, significant local iron abundance inhomogeneities have also been observed with Galactic Cepheids (Pedicelli et al. 2009; Genovali et al. 2014).
Balser et al. (2015) underlined the importance of azimuthal metallicity structure in the Milky Way disc making for the first time radio recombination line and continuum measurements of 21 HII regions located between Galactic azimuth $\phi$=90$^{\circ}$- 130$^{\circ}$. The radial gradient in \[O/H\] is -0.082 $\pm$ 0.014 dex kpc$^{-1}$ for $\phi$=90$^{\circ}$- 130$^{\circ}$, about a factor of 2 higher than the average value between $\phi$=0$^{\circ}$- 60$^{\circ}$. It was suggested that this may be due to radial mixing from the Galactic Bar.
Analyzing the Scutum Red-Supergiant (RSG) clusters at the end of the Galactic Bar, Davies et al. (2009) concluded that a simple one-dimensional parameterisation of the Galaxy abundance patterns is insufficient at low Galactocentric distances, as large azimuthal variations may be present. Combining these results with other data in the literature points towards large-scale ( $\sim$ kpc) azimuthal variations in abundances at Galactocentric distances of 3-5 kpc. It thus appears that the usual approximation of chemical evolution models assuming instantaneous mixing of metallicity in the azimuthal direction is unsubstantiated.
Azimuthal abundance gradients due to radial migration in the vicinity of spiral arms in a cosmological context have been studied in detail by Grand et al. (2012, 2014, 2016), and S[á]{}nchez-Menguiano et al. (2016).
Alternatively, Khoperskov et al. (2018) investigated the formation of azimuthal metallicity variations in the discs of spiral galaxies in the absence of initial radial metallicity gradients . Using high-resolution N -body simulations, they modeled composite stellar discs, made of kinematically cold and hot stellar populations, and study their response to spiral arm perturbations. They found that azimuthal variations in the mean metallicity of stars across a spiral galaxy are not necessarily a consequence of the reshaping, by radial migration, of an initial radial metallicity gradient. They indeed arise naturally also in stellar discs which have initially only a negative vertical metallicity gradient.
The aim of this paper is to develop a detailed 2D galactic disc chemical evolution model, able to follow the evolution of several chemical elements as in previous 1D models, but also taking into account azimuthal surface density variations. In this the paper when we refer to the thin and thick discs we mean the low- and high-\[$\alpha$/Fe\] sequences in the \[$\alpha$/Fe\]-\[Fe/H\] plane. Defining the thin and thick discs morphologically, rather than chemically, identifies a mixture of stars from both the low- and high-\[$\alpha$/Fe\] sequences, and vise versa (Minchev et al. 2015, Martig et al. 2016). It is, therefore, very important to make this distinction and avoid confusion. We follow the chemical evolution of the thin disk component, i.e. the low-$\alpha$ population. We assume that the oldest stars of that low-$\alpha$ component are associated with ages of $\sim$ 11 Gyr, in agreement with asteroseismic age estimates (Silva Aguirre et al. 2018).
Starting from the classical 1D Matteucci & Fran[ç]{}ois (1989) approach (the Galactic disc is assumed to be formed by an infall of primordial gas) we included 2D surface density fluctuation in the Milky Way disc chemo-dynamical model by Minchev et al. (2013) (hereafter MCM13), as well as using analytical spiral arm prescriptions.
Our paper is organized as follows. In Section 2, we describe the framework used for the new model. In Section 2.1 the adopted nucleosynthesis prescriptions are reported. In Section 2.2 the density fluctuation from the chemo-dynamical model by MCM13 are indicated. In Section 2.3 we present the analytical expressions for the density perturbations due to Galactic spiral arm. In Section 3 we presents our results with the density fluctuation from chemo-dynamical models and with an analytical spiral arm prescription are reported. Finally, our conclusions are drawn in Section 4.
A 2D galactic disc chemical evolution model
===========================================
The basis for the 2D chemical evolution model we develop in this section is the classical 1D Matteucci & Fran[ç]{}ois (1989) approach, in which the Galactic disc is assumed to be formed by an infall of primordial gas. The infall rate for the thin disc (the low-$\alpha$ sequence) of a certain element $i$ at the time $t$ and Galactocentric distance $R$ is defined as: $$\label{infall}
B(R,t,i)= X_{A_i} \, b(R) \, e^{-\frac{t}{\tau_D(R)}},$$ where $X_{A_i}$ is the abundance by mass of the element $i$ of the infall gas that here is assumed to be primordial, while the quantity $\tau_D(R)$ is the time-scale of gas accretion. The coefficient $b(R)$ is constrained by imposing a fit to the observed current total surface mass density $\Sigma_{D}$ in the thin disc as a function of the Galactocentric distance given by: $$\Sigma_D(R,t_G)=\Sigma_{D,0}e^{-R/R_{D}},
\label{mass}$$ where $t_G$ is the present time, $\Sigma_{D,0}$ is the central total surface mass density and $R_{D}$ is the disc scale length. The fit of the $\Sigma_D(R)$ quantity using the infall rate law of eq. (\[infall\]) is given by:
$$\sum_i \int_0^{t_G} X_{A_i} b(R) e^{-\frac{t}{\tau_D(R)}} dt = \Sigma_D (R,t_G),$$
The observed total disc surface mass density in the solar neighbourhood is $\Sigma_D (8 \mbox{ kpc}, t_G)=$ 54 M$_{\odot}$ pc$^{-2}$ (see Romano et al. 2000 for a discussion of the choice of this surface density). The infall rate of gas that follows an exponential law is a fundamental assumption adopted in most of the detailed numerical chemical evolution models in which the instantaneous recycling approximation (IRA) is relaxed.
An important ingredient to reproduce the observed radial abundance gradients along the Galactic disc is the inside-out formation on the disc (Spitoni & Matteucci 2011, Cescutti et al. 2007, Mott et al. 2013). The timescale $\tau_D(R)$ for the mass accretion is assumed to increase with the Galactic radius following a linear relation given by (see Chiappini et al. 2001):
$$\tau_{D}(R) = 1.033 R(\mbox{kpc}) - 1.27 \mbox{ Gyr}
\label{tau}$$
for Galactocentric distances $\geq$ 4 kpc. For the star formation rate (SFR) we adopt a Kennicutt (1998) law proportional to the gas surface density: $$\Psi(R,t) = \nu \Sigma_g^k(R,t),
\label{SFR}$$ where $\nu$ is the star formation efficiency (SFE) process and $\Sigma_g(R,t)$ is the gas surface density at a given position and time. The exponent $k$ is fixed to 1.5 (see Kennicutt 1998).
We divide the disc into concentric shells 1 kpc wide in the radial direction. Each shell is itself divided into 36 segments of width $\ang{10}$. Therefore at a fixed Galactocentric distance 36 zones have been created.
With this new configuration we can take into account variations of the SFR along the annular region, produced by density perturbations driven by spiral arms or bars. Therefore, an azimuthal dependence appears in eq. (\[SFR\]) and, which can be written as follows: $$\Psi(R,t,\phi) = \nu \Sigma_g^k(R,t,\phi).
\label{SFR2}$$
In this paper we will show results related to the effects of density fluctuations of the chemo-dynamical model of MCM13 and we will test the effects of an analytical formulation for the density perturbations created by spiral arm waves. The reference model without any density azimuthal perturbation is similar to the one by Cescutti et al. (2007), which as been shown to be quite successful in reproducing the most recent abundance gradients observed in Cepheids (Genovali et al. 2015).
Nucleosynthesis prescriptions
-----------------------------
In this work we present results for the azimuthal variations of abundance gradients for oxygen and iron. As done in a number of chemical evolution models in the past (e.g. Cescutti et al. 2006, Spitoni et al. 2015, 2019, Vincenzo et al. 2019), we adopt the nucleosynthesis prescriptions by Fran[ç]{}ois et al. (2004) who provided theoretical predictions of \[element/Fe\]-\[Fe/H\] trends in the solar neighbourhood for 12 chemical elements.
Fran[ç]{}ois et al. (2004) selected the best sets of yields required to best fit the data (details related to the observational data collection are in Fran[ç]{}ois et al. 2004). In particular, for the yields of Type II SNe they found that the Woosley & Weaver (1995) ones provide the best fit to the data: no modifications are required for the yields of iron, as computed for solar chemical composition, whereas for oxygen, the best results are given by yields computed as functions of the metallicity. The theoretical yields by Iwamoto et al. (1999) are adopted for the Type SNeIa, and the prescription for single low-intermediate mass stars is by van den Hoek & Groenewegen (1997).
Although Fran[ç]{}ois et al. (2004) prescriptions still provide reliable yields for several elements, we must be cautious about oxygen. Recent results have shown that rotation can influence the oxygen nucleosynthesis in massive stars (Meynet & Meader 2002) and therefore chemical evolution (Cescutti & Chiappini 2010), in particular at low metallicity. However, this does not affects our results being the data shown in this project relatively metal rich. Moreover, we are mostly interested in differential effects, rather than absolute values.
2D disc surface density fluctuations from the MCM13 model
---------------------------------------------------------
We consider the gas density fluctuations present in the Milky Way like simulation obtained by Martig et al. (2012) and chosen in MCM13 for their chemodynamical model. The simulated galaxy has a number of properties consistent with the Milky Way, including a central bar. MCM13 followed the disc evolution for a time period of about 11 Gyr, which is close to the age of the oldest low-$\alpha$ disc stars in the Milky Way. The classical 1D chemical evolution model is quite successful in reproducing abundance gradient along the Galactic disc (Cescutti et al. 2007).
The chemical evolution model used by MCM13 was very similar to the one adopted here; a comparison between its star formation history and that of the simulation was presented in Fig. A.1 by Minchev et al. (2014), showing good agreement. To extracted the gas density variations we binned the disk into 18 1-kpc-wide radial bins and 10$^{\circ}$-wide azimuthal bins at $|z|<$ 1 kpc. The time resolution is 37.5 Myr for 11 Gyr of evolution. All of the above is used for our new model described below.
With the aim of preserving the general trend of the 1D chemical evolution model, we introduce a density contrast function $f$ related to the perturbations originated by the MCM13 model. At a fixed Galactocentric distance $R$, time $t$ and azimuthal coordinate $\phi$, the new surface mass density is: $$\Sigma_D(R,t,\phi)=\Sigma_D(R,t) f(\phi,R,t).$$ We impose that the average value of the density contrast $f$ is 1, i.e.:
$$\langle f(\phi,R,t) \rangle_{\phi}=1.$$
This guarantees that, at a fixed Galactocentric distance $R$ and a time $t$, the average surface mass density is the one predicted by the 1 D chemical evolution model.
ISM density fluctuations from analytical spiral structure
---------------------------------------------------------
Here we investigate the effect of an analytical spiral arm formulation on the azimuthal variations of the abundance gradients.
In particular, we analyse steady wave spiral patterns. As suggested by Bertin et al (1989) and Lin & Shu (1966) when the number of important spiral modes of oscillation is small, the spiral structure is expected to have a highly regular grand design and to evolve in time in a quasi- stationary manner.
In this work, we consider the model presented by Cox & G[ó]{}mez (2002). The expression for the time evolution of the density perturbation created by spiral arms, referred to an inertial reference frame not corotating with the Galactic disk, in terms of the surface mass density is:
$$\Sigma_S(R,\phi,t)= \chi(R,t_G) M(\gamma),$$
where $\chi(R,t_G)$ is the present day amplitude of the spiral density: $$\chi(R,t_G)=\Sigma_{S,0} e^{-\frac{R-R_0}{R_{S}}},$$ while $M(\gamma)$ is the modulation function for the “concentrated arms” given by Cox & G[ó]{}mez (2002). The $M(\gamma)$ function can be expressed as follows:
$$M(\gamma)= \left(\frac {8}{3 \pi} \cos(\gamma)+\frac {1}{2} \cos(2\gamma) +\frac{8}{15 \pi} \cos(3\gamma) \right),
\label{MGAMMA}$$
$$\gamma(R,\phi,t)= m\left[\phi +\Omega_s t -\phi_p(R_0) -\frac{\ln(R/R_0)}{\tan(\alpha)} \right].
\label{gamma}$$
In eq. (\[gamma\]), $m$ is the number of spiral arms, $\alpha$ is the pitch angle, $R_S$ is the radial scale-length of the drop-off in density amplitude of the arms, $\Sigma_{0}$ is the surface arm density at fiducial radius $R_0$, $\Omega_s$ is the pattern angular velocity, with the azimuthal coordinate $\phi$ increasing counter-clockwise and a clockwise rotation, $\phi_p(R_0)$ is the coordinate $\phi$ computed at $t$=0 Gyr and $R_0$. An important feature of such a perturbation is that its average density at a fixed Galactocentric distance $R$ and time $t$ is zero,
$$\langle \Sigma_S \rangle_{\phi}= \Sigma_{S, 0} e^{-\frac{R-R_0}{R_{S}}} \langle M(\gamma) \rangle_{\phi}=0.$$
In Fig. \[MGAMMAF\] we show the modulation function $M(\gamma)$ of “concentrated arms” on the Galactic plane using the model parameters suggested by Cox & G[ó]{}mez (2002): $R_0=8$ kpc, $\alpha=\ang{15}$, $R_S=7$ kpc. The modulation function is computed at 5 Gyr assuming the angular velocity value of $\Omega_s$ = 20 km s$^{-1}$ kpc$^{-1}$ and $\phi_p(R_0)=0$. In this work we aim to investigate the effects of spiral arm density perturbations on the chemical enrichment by ejecta from stellar populations perfectly corotating with the Galactic disk. Our purpose here is the study of regular gas density perturbation linked to simple but reliable spiral arm descriptions.
To properly describe the temporal evolution of local density perturbations, the relative spiral arm speed pattern compared to the Galactic disk motion must be computed (further details will be provided in Section 3.2, in the Result discussion).
Cox & G[ó]{}mez (2002) provided a value for the spiral arm perturbation density at 8 kpc equal to $\rho_0= \frac{14}{11}$ m$_H$ cm$^{-3}$. Our implementation requires the surface density $\Sigma_{S,0}$, which can be recovered from the $z$ direction amplitude provided by Cox & G[ó]{}mez (2002) (their eq. 1), with the following relation:
$$\Sigma_{S,0}=2 \rho_0 \int_0^\infty \mbox{sech}^2\left(\frac{x}{H}\right)dx=2 H \rho_0,$$
where $H$ is the disc scale-height. Adopting $H$=180 pc (chosen to match the scale-height of the thin stellar disc proposed by Dehnen & Binney 1998, Model2; and in agreement with Spitoni et al. 2008) we obtain
$$\Sigma_{S,0}=21.16 \mbox{ M}_{\odot} \mbox{ pc}^{-2}.$$
It is important to underline that in our approach the time dependence of the density perturbation by the spiral arms is only in the modulation function $M(\gamma)$ through the term $\Omega_s t$ (see eqs. \[MGAMMA\] and \[gamma\]). Currently, there are no analytical prescriptions for the time evolution of both the amplitude of the spiral arm perturbation and its radial profile in the Galactic evolution context (spiral arm redshift evolution). Therefore, we make the reasonable assumption that during the Galactic evolution the ratio between the amplitude of the spiral density perturbation $\chi(R,t)$ and the total surface density $\Sigma_D(R,t)$ computed at the same Galactic distance $R$ remains constant in time, i.e. $ \frac{d}{dt} \, \left[
\chi(R,t)/\Sigma_D(R,t) \right]$=0, assuming a coeval evolution of these two structures in time. We define the dimensionless quantity $\delta_S(R,\phi,t)$ as the following ratio:
$$\delta_S(R,\phi,t)= \frac{ \Sigma_S(R,\phi,t)+ \Sigma_D(R,t)}{\Sigma_D(R,t)}=1 + \frac{ \Sigma_S(R,\phi,t)}{\Sigma_D(R,t)}.
\label{delta}$$
With the assumption that the ratio $\chi(R,t)/\Sigma_D(R,t)$ is constant in time, eq. (\[delta\]) becomes:
$$\delta_S(R,\phi,t) =1 + M(\gamma)\frac{
\chi(R,t_G)}{\Sigma_D(R,t_G)}.
\label{delta2}$$
If we include the contribution of the perturbation originated by spiral arm in the SFR driven by a linear Schmidt (1959) law (i.e. $\Psi= \nu \Sigma_g(R,t)$) we have that:
$$\Psi(R,t,\phi)_{d+s} = \nu \Sigma_g(R,t) \delta_S(R,\phi,t).
\label{SFR_D}$$
We are aware that this is a simplification to the more complex behavior seen in N-body simulations (Quillen et al. 2011, Minchev et al. 2012b, Sellwood and Carlberg 2014) and external galaxies (Elmegreen et al. 1992; Rix & Zaritsky 1995; Meidt et al. 2009), where multiple spiral patterns have been found. We will make use of this description in Section 3.2.2, where we will consider the simultaneous perturbation by a number of spiral patterns moving at different angular velocities.
![The modulation function $M(\gamma)$ of eq. (\[MGAMMA\]) for concentrated arms by Cox & G[ó]{}mez (2002) with $N=2$ spiral arms, fiducial radius $R_0=8$ kpc, pitch angle $\alpha=\ang{15}$, and $\phi_p(R_0)=0$.[]{data-label="MGAMMAF"}](Figure_1_5gyr.png)
As stated in the previous Section, the average modulation function over the azimuth $\phi$ at a fixed time $t$ and Galactocentric distance $R$ is null ($\langle M(\gamma) \rangle_{\phi}=0$). Therefore, in presence of a linear Schmidt (1959) law at a fixed Galactocentric distance the average value of $\Psi(R,t,\phi)_{d+s}$ over $\phi$ of the SFR defined in eq. (\[SFR\_D\]) is equal to the unperturbed SFR by the following expression:
$$\langle\Psi(R,t,\phi)_{d+s}\rangle_{\phi} =\Psi(R,t) \langle 1 + M(\gamma)\frac{ \chi(R)}{\Sigma_D(R,t_G)}\rangle_{\phi}=$$
$$=\Psi(R,t)\left(1+\langle M(\gamma) \rangle_{\phi}\frac{ \chi(R)}{\Sigma_D(R,t_G)}\right)=\Psi(R,t).$$
Here, we do not adopt a linear Schmidt (1959) law, and we use the SFR proposed by Kennicutt (1998) which exhibits the exponent $k$=1.5. Hence, the SFR in the Galactic disc in presence of spiral arm density perturbations becomes: $$\Psi_k(R,t,\phi)_{d+s} = \nu \Sigma_g(R,t)^k \delta_S(R,\phi,t)^{k}.
\label{SFR_k}$$
Roberts (1969) provided the exact shape of the steady gas distribution in spiral arms, finding an offset between the maximum of the stellar spiral arm and the maximum of the gas distribution driven by galactic shocks. In his Figure 7, it is shown that the regions of newly born luminous stars and the HII regions lie on the inner side of the observable gaseous spiral arm of HI. The presence of a small but noticeable offset between the gas and stellar spiral arms has been also found in the study of interactions between disc galaxies and perturbing companions in 3D N-body/smoothed hydrodynamical numerical simulations by Pettitt (2006).
Because of uncertainties related to the real magnitude of this offset (small offsets are predicted by Pettitt 2006), in our work we do not consider it, and the SFR is more enhanced in correspondence of the total density perturbation peak (see eq. \[delta2\] and the modulation function in Figure 1). We are aware that is true only near the corotation radius, however with our simpler approach we provide an upper limit estimate for the azimuthal abundance variations generated by steady spiral arms density perturbations. In presence of an off-set the density perturbation should be less “concentrated” and more smeared.
Our model in the presence of analytical spiral arms must be considered as a first attempt to include spiral structure in a classical chemical evolution model. As stated in Section 2.2, we will also present results for the azimuthal abundance variations originated by chemodynamical Milky Way like simulation in the presence of spiral arms and bar in a self consistent way. Our analytical spiral arms model is meant to break down the problem to understand the reason for the causes of azimuthal variations. Assuming that modes add linearly, we can approximate a realistic galactic disk by adding several spiral sets with different pattern speeds, as seen in observations (e.g., Meidt et al. 2009) and simulations (e.g., Masset & Tagger 1997, Quillen et al. 2011, Minchev et al. 2012a).
 
![Results for the chemical evolution model in which we consider the density fluctuation by the chemo-dynamical model by MCM13. Time evolution of the oxygen abundance gradient at $\phi$=0$^{\circ}$. []{data-label="minchev_grad_ev"}](ev_gradient_ivan.png)
![ [*Upper Panel*]{}: We present the average Fe abundances of Galactic Cepheids presented by Genovali et al. (2014) in bin of 15$^{\circ}$ for the azimuthal coordinate $\phi$ at different Galactocentric distances. [*Lower left Panel*]{}: Fe abundances as functions of the azimuthal coordinates computed at 6, 8, 10, 12 kpc predicted by the chemical evolution model in which we implemented the density fluctuation by the MCM13 model. [*Lower right Panel*]{}: residual of the Fe abundances predicted by our model computed after subtracting the average radial gradient. []{data-label="CEP1"}](cep1_large.png "fig:") ![ [*Upper Panel*]{}: We present the average Fe abundances of Galactic Cepheids presented by Genovali et al. (2014) in bin of 15$^{\circ}$ for the azimuthal coordinate $\phi$ at different Galactocentric distances. [*Lower left Panel*]{}: Fe abundances as functions of the azimuthal coordinates computed at 6, 8, 10, 12 kpc predicted by the chemical evolution model in which we implemented the density fluctuation by the MCM13 model. [*Lower right Panel*]{}: residual of the Fe abundances predicted by our model computed after subtracting the average radial gradient. []{data-label="CEP1"}](ivan_cep_new.png "fig:")
Results
=======
In this section we apply our 2D model by using surface density fluctuations from the MCM13 chemo-dynamical model and from an analytical prescription.
Density fluctuation from the MCM13 chemo-dynamical model
--------------------------------------------------------
In this section we present our results based on the new 2D chemical evolution model including the density mass fluctuation extracted from the chemo-dynamical model by MCM13.
Fig. \[mSFR\] shows the galactic disc SFR computed at 0.1, 0.7, 6, 11 Gyr after the start of disc formation, for the chemical evolution model in which we tested the effects of the density fluctuation by MCM13 in units of M$_{\odot}$ pc$^{-2}$ Gyr$^{-1}$. We notice that at early times (i.e the “1 Gyr” case reported in the upper left panel), the SFR is more concentrated in the inner Galactic regions, the SFR in the innermost regions decreases and the outer parts become more star forming active because of the “inside-out” prescription coupled with the inclusion of the density fluctuation. At the Galactic epoch of 1 Gyr after the start of disc formation, regions with the same Galactocentric distances have approximately the same SFR. Already after 0.7 Gyr of Galactic evolution, azimuthal star formation inhomogeneities are not negligible. Concerning the panel with the model results at 6 Gyr, azimuthal inhomogeneities are evident, in particular at 8 kpc the ratio between the maximum and the minimum values assumed by the SFR is SFR$_{max}$ /SFR$_{min}$=6.72.
In Fig. \[mSFR\] the bar and spiral arms features do not show up clearly, especially in early times. This is caused by the adopted inside out prescription (eq. \[tau\]) which leads to huge differences between the SFRs computed in inner and outer regions. In Fig. \[1SFR\], the galactic disc SFR($R$, $\phi$) is normalized to the maximum value SFR$_{R, \ max}$ of the annular region located at the Galactocentric distance $R$, i.e SFR($R$, $\phi$)/SFR$_{R, max}$, computed at 0.1, 0.7, 6, 11 Gyr after the start of disc formation, respectively. Here, different features related to density perturbations originated by spiral arms and bar can be noted.
In Fig. \[minchev\_av\] the main results related to the present day oxygen abundance azimuthal variation are presented. The top panel shows the azimuthal distribution of the residual of the oxygen abundances computed with our chemical evolution model at 4, 8, 12, 16, and 18 kpc after subtracting the average radial gradient (i.e. the one obtained with the reference model without any density perturbation). Throughout this paper we adopt the photospheric values of Asplund et al. (2009) as our solar reference. We see that the behavior is in excellent agreement with the observations by S[á]{}nchez et al. (2015); indeed, data show that outer regions display larger azimuthal variations, and the amplitude of the risidual variations are of the order of 0.1 dex (see Figure 7 by S[á]{}nchez et al. 2015) . In our model the maximum variations are $\sim$ 0.12 dex for the chemical evolution models computed at 18 kpc. Our results appears to have a bit less scatter.
In the lower panel of Fig. \[minchev\_av\] we present our “mock” observations. We draw oxygen abundances of different ISM regions at different Galactocentric distance at random azimuthal coordinates $\phi$. Hence, we add an error of $\sigma_{\phi}$=5$^{\circ}$ to alleviate the fact that our model presents a resolution of 10$^{\circ}$ in the azimuthal component $\phi$. Moreover, the average observational uncertainty associated to the oxygen abundances of $\sigma_{[O/H]}$ = 0.05 dex provided by S[á]{}nchez et al. (2015) has been considered. We define the “new” oxygen abundance including these uncertainties as follows: $$[\mbox{O/H}]_{new} = [\mbox{O/H}]+ U([-\sigma_{[O/H]}, \sigma_{[O/H]}]),
\label{Err1}$$ where $U$ is the random generator function. Similarly, we implement the uncertainty in the azimuthal component through the following relation: $$\phi_{new} = \phi + U([-\sigma_{\phi}, \sigma_{\phi}]).
\label{Err2}$$
Here, it is clearly visible the similarity between the S[á]{}nchez et al. (2015) observations and our results. To summarize, the inclusions of density perturbations taken from a self-consistent dynamical model at different Galactic times, leads to significant variations in chemical abundances in the outer Galactic regions.
In Fig. \[minchev\_grad\] we show results for the present day abundance gradient (after 11 Gyr of evolution) for iron computed for six azimuthal slices (as indicated) of width 10$^{\circ}$ at different azimuthal coordinates. In the same plot is indicated with a shaded grey area the maximum spread in the abundance ratio \[Fe/H\] obtained by the azimuthal coordinates we considered (0$^{\circ}$, 60$^{\circ}$, 120$^{\circ}$, 180$^{\circ}$, 240$^{\circ}$, and 300$^{\circ}$). As a consequence of the results presented above, the shaded area is larger towards external regions. We also overplot the data from Genovali et al. (2014) in order to compare to our model predictions. We notice that the predicted gradient is slightly steeper than the observed one in the external Galactic regions. However, we notice that the model lines pass within in the data standard deviation computed dividing the data by Genovali et al. (2014) in six radial bins.
In Fig. \[minchev\_grad\_ev\] we tested the effects of chemo-dynamical fluctuations on the time evolution of the oxygen abundance gradient at a fixed azimuth ($\phi$=0$^{\circ}$). In agreement with Minchev et al. (2018) the abundance gradient flattens in time, because of the chemical evolution model assumptions. As shown by Spitoni et al. (2015) and Grisoni et al. (2018), the inclusion of radial gas flows can in lead to even steeper gradients in time during the whole Galactic history.
![ Spiral pattern speed $\Omega_s$ and disk angular velocity $\Omega_d$ computed by Roca-F[à]{}brega et al. (2014) are indicated with light blue and violette lines, respectively. With the vertical long dashed red line we show the position of the corotation radius located at the Galactocentric distance $R=8.31$ kpc. Outer and Inner Lindblad resonances extracted by Roca-F[à]{}brega et al. (2014) simulation are also drawn with dotted magenta and dotted purple lines, respectively. []{data-label="omega"}](omega.png)
[c|cccc]{}\
Models &$m$ &$\alpha$& $\Omega_{s}$\
& & & \[km s$^{-1}$ kpc$^{-1}$\]\
\
S2A & 2 & 15$^{\circ}$ & 20\
S2B & 2 & 15$^{\circ}$ & 17.5\
S2C & 2 & 15$^{\circ}$ & 15\
S2D & 2 & 15$^{\circ}$ & 13.75\
S2E & 2 & 15$^{\circ}$ & 12.5\
S2F& 2 & 15$^{\circ}$ & 25\
S2G & 2 & 7$^{\circ}$ & 20\
S2H & 2 & 30$^{\circ}$ & 20\
S1A & 1 & 15$^{\circ}$ & 20\
S1B & 1 & 15$^{\circ}$ & 17.5\
S1C &1 & 15$^{\circ}$ & 15\
S1D & 1 & 15$^{\circ}$ & 13.75\
S1E & 1 & 15$^{\circ}$ & 12.5\
S1F& 1 & 15$^{\circ}$ & 25\
![ Results for the chemical evolution model in which we consider the density fluctuation associated with the analytical spiral arm formulation. [*Upper Panel*]{}: The azimuthal distribution of the residual of the oxygen abundances computed with our chemical evolution model at 4, 8, 12, 16, and 20 kpc (after subtracting the average radial gradient for a model with $R_S$=7, $R_D$=3.5, $\Sigma_0$=20, $\nu$=1.1, $\Omega_s$=20 km s$^{-1}$ kpc$^{-1}$, and $m$=2 spiral arm (model S2A in Table 1). [*Lower Panel*]{}: the time evolution of the \[O/H\] abundance as a function of the azimuthal coordinate computed at 8 kpc. []{data-label="SA"}](R3.png "fig:") ![ Results for the chemical evolution model in which we consider the density fluctuation associated with the analytical spiral arm formulation. [*Upper Panel*]{}: The azimuthal distribution of the residual of the oxygen abundances computed with our chemical evolution model at 4, 8, 12, 16, and 20 kpc (after subtracting the average radial gradient for a model with $R_S$=7, $R_D$=3.5, $\Sigma_0$=20, $\nu$=1.1, $\Omega_s$=20 km s$^{-1}$ kpc$^{-1}$, and $m$=2 spiral arm (model S2A in Table 1). [*Lower Panel*]{}: the time evolution of the \[O/H\] abundance as a function of the azimuthal coordinate computed at 8 kpc. []{data-label="SA"}](R3_t.png "fig:")
![ [*Upper Panel*]{}: Galactic disc SFR resulting from model S2A after 1 Gyr of evolution (see Table 1 and text for model details). The color code indicates the SFR in units of M$_{\odot}$ pc$^{-2}$ Gyr$^{-1}$. [*Lower Panel*]{}: same but computed at 11 Gyr.[]{data-label="SASFR"}](R3_SFR_1Gyr.png "fig:") ![ [*Upper Panel*]{}: Galactic disc SFR resulting from model S2A after 1 Gyr of evolution (see Table 1 and text for model details). The color code indicates the SFR in units of M$_{\odot}$ pc$^{-2}$ Gyr$^{-1}$. [*Lower Panel*]{}: same but computed at 11 Gyr.[]{data-label="SASFR"}](R3_SFR_11Gyr.png "fig:")
![ Disk angular velocity $\Omega_d$ computed by Roca-F[à]{}brega et al. (2014) is indicated with light blue and violette lines. With different horizontal solid lines are indicated the spiral pattern speed $\Omega_s$ adopted in our models (see text and Table 1 for model details). The vertical long dashed lines show the positions of the corotation radii assuming different $\Omega_s$ values.[]{data-label="DOS"}](DIF_OMEGA_S.png)
![ Present day residual azimuthal variations in oxygen abundance for the corotation regions (as indicated) of the different pattern speeds shown in Fig. \[S2\_grad\]. An increase in the effect is found as the corotation shifts to larger radius, i.e., for slower spiral patterns. Such a set of spirals with progressively slower patterns speeds as radius increases, can be a realistic representation of a galactic disk. []{data-label="CS2"}](cor_spir_2_lab.png)
![ As in Fig. \[SA\] but for model S1A, with with multiplicity $m$=1 of spiral arms.[]{data-label="S1A"}](R04.png "fig:") ![ As in Fig. \[SA\] but for model S1A, with with multiplicity $m$=1 of spiral arms.[]{data-label="S1A"}](R04_t.png "fig:")
In Fig. \[CEP1\] we compare the average iron abundance azimuthal variation in bins of $\phi$=15$^{\circ}$ presented by Genovali et al. (2014) computed at 6, 8, 10, and 12 kpc, respectively with our 2D chemical evolution model, resulting from the MCM13 density variations. We see that the observed azimuthal variations are for limited Galactocentric distances (6-12 kpc) and with a narrow range of azimuthal coordinates. Although it is evident that the observed amplitude of azimuthal variations are larger than the ones predicted by our models, more precise Galactic Cepheid data are required to make firm conclusions.
Moreover, other dynamical processes that we have not considered in this work had maybe played important roles in the evolution and in the building up of the Galactic gradients and their azimuthal variations - radial migration processes can already introduce some variations in about a Gyr (Quillen et al. 2018).
![ As in Fig. \[CS2\] but for models with $m=1$ multiplicity (see Table 1).[]{data-label="CS1"}](cor_spir_1_lab.png)
Density fluctuations from an analytical spiral arm formulation
--------------------------------------------------------------
In this Section we discuss the results of chemical evolution models with only analytical prescriptions for spiral arm density perturbations without including any density fluctuations from chemo-dynamical models. The primary purpose here is to test the effect of regular perturbations (i.e. spiral arms evolution described by an analytical formulation) on the chemical evolution of a Milky Way like galaxy. We underline that the results showed in the previous Section reflect more closely the complex behavior of the Milky Way. However, we are also interested to explore different spiral arm configurations which could characterize external galactic systems by varying the free parameters of the analytical expression of the spiral arms. In particular, we will show the effects on the azimuthal variations of abundance gradients for oxygen by varying:
i) the multiplicity $m$ of spiral arms;
ii) the spiral pattern speed, $\Omega_s$;
iii) the pitch angle $\alpha$.
For all model results that will be presented we assume the following Cox & Gomez (2002) prescriptions: the radial scale length of the drop-off in density amplitude of the arms fixed at the value of $R_S=7$ kpc, the pitch angle is assumed constant at $\alpha=\ang{15}$, and the surface arm density $\Sigma_0$ is 20 M$_{\odot}$ pc$^{-2}$ at the fiducial radius $R_0=8$ kpc; finally we assume $\phi_p(R_0)=0$.
The disk rotational velocity $\Omega_d(R)$ has been extracted from the simulation by Roca-F[à]{}brega et al. (2014) (see their left panel of Figure 1). The exponential fit of $\Omega_d(R)$ variations as a function of the Galactocentric distance $R$ (expressed in kpc) is: $$\Omega_d(R)=98.93 \, e^{-0.29 \, R}+ 11.11 \mbox{ [km } \mbox{s }
^{-1}\mbox{kpc}^{-1}].
\label{eqom}$$ We start by adopting the constant pattern angular velocity $\Omega_s$ = 20 km s$^{-1}$ kpc$^{-1}$ consistent with the Roca-F[à]{}brega et al. (2014) model. Similar value was first estimated from moving groups in the U-V plane by Quillen & Minchev (2005, 18.1 $\pm$ 0.8 km s$^{-1}$ kpc$^{-1}$) and a summary of derived values for the Milky Way can be found in Bland-Hawthorn & Gerhard (2016). In Fig. \[omega\] we show the $\Omega_s$ and $\Omega_d(R)$ quantities as well as the Outer and Inner Lindblad resonances as a function of the Galactocentric distance, the corotation radius is located at 8.31 kpc.
### Results with a single analytical spiral pattern
We begin our analysis discussing the results obtained with model S2A (see Table 1), which has a pattern speed of $\Omega_s$ = 20 km s$^{-1}$ kpc$^{-1}$, placing the corotation resonance at the solar radius.
The upper panel of Fig. \[SA\] shows the the oxygen abundance residual azimuthal variations after 11 Gyr of disc evolution for different Galactocentric distances. The average radial gradient is subtracted. As expected, larger abundance azimuthal variations are found near the corotation radius. In this region the chemical enrichment should be more efficient due to the lack of the relative gas-spiral motions. Higher SFR at the corotation radius caused by locally higher gas overdensity lasts for a longer time, therefore more massive stars can be created and more metals can be ejected into the local ISM under the spiral arm passage.
At 8 kpc we have $\Delta$\[O/H\] $\approx$ 0.05 dex. For other Galactocentric distances, away from the corotation, variations are much smaller. In the lower panel of Fig. \[SA\] we present the temporal evolution of the oxygen abundance azimuthal variations for the model S2A as a function of the azimuthal coordinate $\phi$ computed at 8 kpc. As expected, larger inhomogeneities are present at early times, decreasing in time.
As discussed in Section 2.2, we assume that during the Galactic evolution the ratio between the amplitude of the spiral density perturbation and the total surface density computed at the same radius $R$, remains constant in time. However, this analytical approach is not capable to put constraints on the temporal evolution of pattern speed.
Galactic chemical evolution is an integral process in time. The stronger spiral structure induced azimuthal variations at early times are, therefore, washed out by phase mixing.
Fig. \[SASFR\] depicts the SFR after 1 Gyr of evolution (upper panel) and at the present time (lower panel) on the galactic plane computed with the model S2A. Here, it is evident the way in which the spiral arm density perturbation affects and modulates SFR computed at the present time in unit of M$_{\odot}$ pc$^{-2}$ Gyr$^{-1}$. The shape of the two spiral arm over-densities is clearly visible in the SFR. This is in contrast to our results using the MCM13 density fluctuations (see Fig. \[mSFR\]), where multiple spiral density waves were present. Moreover, we can appreciate the inside-out disk formation: at later times the external regions become star formation active.
### The effect of different pattern speeds
In this Section we vary the spiral pattern speed, which has the effect of shifting the corotation resonance in radius. We argue that a combination of multiple spiral modes with different pattern speeds can be a realistic representation of a galactic disk. The horizontal and vertical lines in Fig. \[DOS\] show the different pattern speeds and corresponding corotation radii, respectively, used in this Section: it is clear that smaller $\Omega_{s}$ values lead to a more external corotation radius.
In Fig. \[S2\_grad\] we show the oxygen abundance gradients computed at different azimuths after 11 Gyr of disk evolution for models with spiral multiplicity $m$ = 2 and different spiral pattern speed $\Omega_{s}$ (see Table 1 for model details).
We notice that the more the corotation radius is shifted towards the external Galactic regions the more the oxygen azimuthal abundance variations are amplified near the corotation radius. This result is reasonable in the light of our previous findings presented above with our model assuming chemo-dynamical fluctuations by MCM13. We recall that larger variations in the chemical abundance of outer galactic regions have been found by observations in external galaxies (S[á]{}nchez et al. 2015).
In Fig. \[CS2\] we show the present day azimuthal residual of the oxygen abundances after subtracting the average radial gradient computed for the Galactic annular regions which include the relative corotation radius for the following models with $m=2$ multiplicity: S2A, S2B, S2C, S2D, S2F (see Table 1 for other parameter details). The model S2D computed at $R=$13 kpc has $\Delta$\[O/H\]$\approx$ 0.32 dex. Already in regions not so far from the solar neighbourd, the variations are important, i.e., model S2C whose corotation resides in the annular region centered at $R=$ 11 kpc, presents an oxygen abundance variation of $\Delta$\[O/H\] is $\approx$ 0.20 dex.
As discussed in Setion 2.3, it is well accepted that multiple patterns can be present in galactic disks (e.g., Meidt et al. 2009) including our own Milky Way (Minchev & Quillen 2006, Quillen et al. 2011), with slower patterns shifted to outer radii. This will have the effect of placing the corotation regions very similarly to what Fig. \[CS2\] presents and having corotating arms at all radii as found by Grand et al. (2012), Hunt et al. (2019). Therefore, the increasing scatter in abundance with galactic radius can be explained as the effect of multiple patterns propagating at the same time. Note that radial migration will introduce additional scatter, that can in principle be accounted for.
![ [*Upper Panel*]{}: The Galactic disc SFR related to the model S2G computed after 11 Gyr of Galactic evolution (see Table 1 and text for model details) with a pitch angle $\alpha=7^{\circ} $. The color code indicates the SFR in units of M$_{\odot}$ pc$^{-2}$ Gyr$^{-1}$. [*Lower Panel*]{}: as the upper panel but for the model S2H where the pitch angle $\alpha$ is 30$^{\circ}$.[]{data-label="SESFSFR"}](R1_SFR_11.png "fig:") ![ [*Upper Panel*]{}: The Galactic disc SFR related to the model S2G computed after 11 Gyr of Galactic evolution (see Table 1 and text for model details) with a pitch angle $\alpha=7^{\circ} $. The color code indicates the SFR in units of M$_{\odot}$ pc$^{-2}$ Gyr$^{-1}$. [*Lower Panel*]{}: as the upper panel but for the model S2H where the pitch angle $\alpha$ is 30$^{\circ}$.[]{data-label="SESFSFR"}](R2_SFR_11.png "fig:")
![ Effects of different pitch angles $\alpha$ on the azimuthal distribution of the residual of the oxygen abundances computed with our chemical evolution model at 4, 8, 12, 16, and 20 kpc In the upper panel the pitch able is set at the value of 7$^{\circ}$ (SE model in Table 1), while in the lower panel $\alpha=30 ^{\circ}$ (SF model in Table 1). []{data-label="SE_SF"}](R1.png "fig:") ![ Effects of different pitch angles $\alpha$ on the azimuthal distribution of the residual of the oxygen abundances computed with our chemical evolution model at 4, 8, 12, 16, and 20 kpc In the upper panel the pitch able is set at the value of 7$^{\circ}$ (SE model in Table 1), while in the lower panel $\alpha=30 ^{\circ}$ (SF model in Table 1). []{data-label="SE_SF"}](R2.png "fig:")
### Results with an $m$=1 spiral pattern
We want to test whether the intensity of the amplitude of the azimuthal chemical abundance variations is dependant on the number $m$ of spiral arms. In Table 1 we label as model S1A a model identical to the model S2A but with an $m$=1 spiral structure, i.e., having only one spiral arm. Such a mode arises naturally from the coupling of $m$=2 and $m$=3 modes as found by Quillen et al. (2011) and Minchev et al. (2012a) using pure N-body and SPH simulations, and is seen in external galaxies (Zaritsky & Rix 1997).
In the upper panel of Fig. \[S1A\] we notice that the abundance variations are larger than the ones obtained with the same model but $m$=2 (upper panel of Fig. \[SA\]): a fluctuation of about $\Delta$\[O/H\]=0.1 dex is seen at the corotation radius ($\sim$ 8 kpc).
In the same Figure is presented the time evolution of azimuthal abundance inhomogeneities for oxygen computed at 8 kpc with the model S1A at 2, 4, 6, 8, and 11 Gyr.
In Fig. \[S1\_grad\] we have the oxygen abundance gradients computed at different azimuths after 11 Gyr of disk evolution for models with spiral multiplicity $m$ = 1 and the same spiral pattern speeds $\Omega_{s}$ as in Fig. \[S2\_grad\] (see Table 1 for model details). We notice that around the corotation radii the azimuthal abundance variations are generally more evident for models with one spiral arm compared to ones with spiral multiplicity $m$ = 2.
In Fig. \[CS1\] we show the present day azimuthal residual of the oxygen abundances after subtracting the average radial gradient computed in annular regions which contain the corotation radii for models with $m=1$ multiplicity: S1A, S1B, S1C, S1D, S1F (see Table 1 for other parameter details). For the model S1D at the Galactic distance of 13 kpc we have $\Delta$\[O/H\]$\approx$ 0.40 dex, which is about $\approx 25$% larger than the S2D case. As found for the model with $m=2$, the oxygen abundance variations become important in regions not so far from the solar vicinity, i.e., model S1C whose corotation resides at $R=$ 11 kpc, $\Delta$\[O/H\] $\approx$ 0.23 dex.
### Results for different pitch angles
In this Section we consider different pitch angles $\alpha$ for the spiral arms in our Milky Way galaxy.
Recent work by Quillen et al. (2018) and Laporte et al. (2018) suggest that tightly wound spiral structure should be considered, based on modeling of phase-space structure found in the second Gaia data release (Gaia collaboration et al. 2018).
A smaller pitch angle gives rise to more tightly wound spiral structure. The upper panel of Fig. \[SESFSFR\] depicts the present time SFR computed with a pitch angle $\alpha=7^{\circ}$ (model S2G in Table 1), whereas the lower panel shows the case of $\alpha=30^{\circ}$ (model S2H in Table 1). For both panels the other model parameters as the same as model S2A. The spiral pattern is clearly visible in the SFR, and for the model S2G a tighter wound spiral structure is present.
In Fig \[SE\_SF\] we compare the azimuthal variations for models S2G and S2H. We see that the chemical variations are identical at the corotation radius and simply azimuthally shifted for other Galactocentric distances.
Conclusions
===========
In this paper we presented a new 2D chemical evolution model, able to trace azimuthal variations in the galactic disc density. We applied this model to (i) the density fluctuations arising in a disc formation simulation by Martig et al. (2012), used for the MCM13 Milky Way chemo-dynamical model, and (ii) the density perturbations originating from an analytical spiral arm formulation.
The main conclusions for density perturbation from Milky Way chemo-dynamical model by MCM13 can be summarized as follows:
- We found that the density fluctuations produce significant oxygen azimuthal variations in the abundance gradients of the order of 0.1 dex.
- The azimuthal variations are more evident in the external galactic regions, in agreement with the recent observations of the galaxy NGC 6754, using MUSE data (S[á]{}nchez et al. 2015).
In an effort to understand the above findings, we constructed simple analytical spiral arm models, for which we varied the pattern speed, multiplicity and pitch angle with the following main findings:
- The larger fluctuations in the azimuthal abundance gradients are found near the corotation radius, where the relative velocity with respect to the disk is close to zero.
- Larger azimuthal variations are found at corotation radii shifted to larger radii, i.e., slower pattern speeds.
- The variation is more enhanced for the model with only one spiral arm, which is expected to result from the combination of an $m$=2 and $m$=3 spiral structure.
- We found that the more significant azimuthal abundance variations seen at early times in presence of a regular, periodic perturbation tend to quench at later times. This is expected, as galactic chemical evolution is cumulative process and phase-mixing and radial migration tends to wipe structure with time.
Combining the effect of corotaton radii by assuming the simultaneous propagation of multiple spiral modes through galactic disks, we can obtain a realistic picture of azimuthal variations induced at stellar birth found in self-consistent models, such as the MCM13. Material spiral arms propagating near the corotation at all galactic radii have been described by a number of recent numerical work with different interpretations (see Grand et al. 2012, Comparetta & Quillen 2012, Hunt et al. 2019).
In future work we will improve the new 2D chemical evolution model introduced here by taking into account stellar radial migration of long-lived stars and the pollution to the ISM abundance introduced by them at radii and azimuths different than their birth places. We will also use this model to update the Galactic habitable zone results presented by Spitoni et al. (2014, 2017) and study the effect of spiral structure and the Galactic bar.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank the anonymous referee for various suggestions that improved the paper. E. Spitoni and V. Silva Aguirre acknowledge support from the Independent Research Fund Denmark (Research grant 7027-00096B). V. Silva Aguirre acknowledges support from VILLUM FONDEN (Research Grant 10118). G. Cescutti acknowledges financial support from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 664931. This work has been partially supported by the EU COST Action CA16117 (ChETEC). I. Minchev acknowledges support by the Deutsche Forschungsgemeinschaft under the grant MI 2009/1-1. F. Matteucci acknowledges research funds from the University of Trieste (FRA2016).
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[^1]: email to: [email protected]
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abstract: 'In the helium case of the classical Coulomb three-body problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. A sequence of periodic orbits are predicted and are actually found numerically. The results obtained here will be a cornerstone for finding the remaining periodic orbits, which needed for semiclassical applications such as periodic orbit quantization.'
author:
- 'Mitsusada M. Sano'
- Kiyotaka Tanikawa
title: 'Periodic Orbits and Binary Collisions in the Classical Coulomb Three-Body Problem'
---
The microscopic three-body problem, for example, the helium atom, attracts both theoreticians and experimentalists because of rich structure of its spectrum. In particular, the excited states below the double ionization threshold ($E=0$) show the most complicated spectral structure, and contain various information on electron-electron correlations. This spectral region is not only most interesting but also most difficult in both theoretical and experimental aspects. In investigating its spectrum, semiclassical methods inevitably face with chaotic nature of its classical dynamics [@TRR]. In a theoretical aspect, for the collinear [*eZe*]{} configuration, the semiclassical periodic orbit quantization was carried out by using the celebrated Gutzwiller trace formula [@Gutzwiller] within reasonable accuracy [@ERTW; @TW]. However, such successes of semiclassical and classical approaches have been restricted to one-dimensional case, namely the collinear [*eZe*]{} and [*eeZ*]{} configurations and the Wannier ridge configuration [@TRR; @ERTW; @TW; @RTW]. The helium atom in two dimensions remains as a challenging object.
Main difficulties are due to (1) the mixture of chaos and tori, (2) the high-dimensionality, and (3) the singular nature of its dynamics. To overcome these difficulties, one needs sophisticated mathematical tools. McGehee’s blow-up transformation and the concept of the triple collision manifold [@McGehee] are important examples [@BGY; @Sano1; @Sano2; @Sano3; @CLT; @LTN; @LCT]. Thanks to these tools, the dynamics near triple collision and the structure of stable and unstable manifolds were elucidated to some extent. In fact, a quantum manifestation of triple collision was demonstrated [@BCLT; @TCLCD]. But we still know less about periodic orbits in two-dimensional dynamics, which are needed for semiclassical analysis.
Quite recently, one additional tool, i.e., the symbolic dynamics for the planar gravitational problem is developed [@TM]. In this method, binary collisions are naturally obtained as boundaries of different symbol sequences. We already have one-dimensional symbolic dynamics [@Sano1]. Then, we can systematically search periodic orbits in the dynamics with zero initial velocities (DZIV) using one-dimensional and two-dimensional symbolic dynamics. We do this in the present report.
As is known in [@Sano3], the initial condition space of the DZIV is bounded by the collinear initial conditions (see below). First, using the two-dimensional symbolic dynamics, we numerically confirm the existence of curves of initial conditions of orbits exhibiting binary collisions. We call these the binary-collision curves (BCCs). Next, we go back to symbolic dynamics for the collinear problem and obtain symbol sequences of triple collision orbits. As predicted in [@Sano3], a sequence of BCCs is shown to have two end-points on $\alpha = \pi$. In other words, these BCCs can be uniquely specified by the end-points. Then, we informally consider a one-dimensional symbolic dynamics along BCCs and obtain self-retracing periodic orbits along them. Our procedure (Fig. \[fig3\]) will be justified by numerical integration (Fig. \[fig4\]). Finally, we characterize periodic orbits with physical quantities.
Now we introduce the system considered. In the helium atom, two electrons are denoted by particles $1$ and $2$, and the nucleus is denoted by particle $3$. Let $\boldsymbol{r}_{i}$ be the position vector of the $i$th particle, and let $\boldsymbol{r}_{ij}=\boldsymbol{r}_{i}-\boldsymbol{r}_{j}$. We assume that the nucleus has infinite mass and that initial velocities of particles are zero, which implies that the problem is planar. Then, the Hamiltonian in the hyperspherical coordinates is given by $$\begin{aligned}
H & = & \frac{1}{2} \left ( p_{r}^{2} + \frac{4p_{\chi}^{2}}{r^{2}}
+ \frac{4p_{\alpha}^{2}}{r^{2}\sin^{2}(\chi)} \right )
\nonumber \\
& &
- \frac{1}{r} \left (
\frac{1}{\cos\left ( \frac{\chi}{2} \right )}
+ \frac{1}{\sin\left ( \frac{\chi}{2} \right )}
- \frac{1}{Z[1-\sin(\chi)\cos(\alpha)]^{1/2}}
\right ),
\label{eq:hamiltonian_hyper}\end{aligned}$$ where $r=(r_{1}^{2}+r_{2}^{2})^{1/2}$ is the hyperradius with $r_{1} = r \cos(\chi/2)$, $r_{2}= r \sin(\chi/2)$, and $\chi=2 \mbox{arctan}(r_{2}/r_{1})$. $Z$($=2$ in this Letter) is the charge of the nucleus. $Z$ appears in the denominator of the potential term because of the scaling of variables. $\alpha$ is the angle between the vectors $\boldsymbol{r}_{13}$ and $\boldsymbol{r}_{23}$. The total energy is $E=H$.
Next we introduce symbolic dynamics [@TM]. Three particles from a triangle. The area of this triangle has the sign $\eta = {\bf e}_{z}\cdot (\boldsymbol{r}_{13}\times
\boldsymbol{r}_{23})/|\boldsymbol{r}_{13}\times\boldsymbol{r}_{23}|$. The length of the $i$th side of the triangle is $l_{i} = |\boldsymbol{r}_{jk}|$, where $(i,j,k)$ is a cyclic permutation of $(1,2,3)$. If $\eta$ changes its sign from plus to minus and $\mbox{max}\{l_{1},l_{2},l_{3}\}=l_{k}$, then we assign a symbol $k$. If $\eta$ changes its sign from minus to plus and $\mbox{max}\{ l_{1},l_{2},l_{3}\}=l_{k}$, then we assign a symbol $k+3$. Then we obtain a symbol sequence ${\bf s}= \bullet s_{1}s_{2}s_{3}\cdots$ for a given orbit with the symbol set ${\bf S}=\{1,2,3,4,5,6\}$.
By scaling the system, we remove the dependence on $r$ from the DZIV. Then the initial conditions are uniquely specified by $0 \leq \chi < \pi$ and $0 \leq \alpha < 2\pi$. Moreover, the quarter of the initial condition space $D_{1/4}= \{ (\chi,\alpha)| 0\leq \chi \leq \frac{\pi}{2},
0 \leq \alpha \leq \pi \}$ suffices for our purpose owing to the symmetry of the problem. We integrate orbits starting at points in $D_{1/4}$, and assign a symbol sequence to each orbit following the procedure given in [@TM]. Then, $D_{1/4}$ is partitioned into regions of different symbol sequences. One example of partitions by symbol sequences up to symbol length $8$ is shown in Fig. \[fig1\]. The BCCs are clearly seen as the boundaries of the partitioned regions. This shows a big advance compared with [@Sano3]. In that work, BCCs were obtained point by point along them, so the task was rather time consuming. In addition, orbit integrations near $\alpha=\pi$ were difficult due to the closeness to triple collision. The present method basically avoids triple collision, so is accurate and fast.
The intersections of BCCs with $\alpha = \pi$ are the initial conditions for triple-collision orbits (TCOs). We call them the triple-collision points (TCPs). We attach, in Fig. \[fig1\], the orbits of some of the TCPs with symbol sequences $0$, $210$, $20$, and $220$ where ’$0$’ represents triple collision, ’$1$’ binary collision between particles $1$ and $3$, and ’$2$’ binary collision between particles $2$ and $3$. Hereafter the symbolic dynamics means the symbolic dynamics in the collinear [*eZe*]{} configuration.
![\[fig1\] Partition in $D_{1/4}$ and triple-collision points in the collinear [*eZe*]{} configuration: The large figure represents the partition obtained from the symbol sequences of symbol length $8$ in the symbolic dynamics for two-dimensional dynamics. The boundaries between the regions of the partition are binary-collision curves. The small figures represent the triple-collision orbits in the collinear [*eZe*]{} configuration. (a) $0$. (b) $210$. (c) $20$. (d) $220$. The blank region at the lower-right corner is the forbidden region ($E>0$). ](fig1.pdf){width="8cm"}
We observe two groups of BCCs. One group, which we call the [*first kind*]{}, consists of the BCCs which start at $\alpha = \pi$ and end at $\alpha = \pi$. We denote a BCC of this group by a lobe from its shape, and express it as $w_m0$-$w_n0$ using the symbol sequences $w_m0$ and $w_n0$ of the left and right end points where $w_k$ is a word of length $k$. The other, which we call the [*second kind*]{}, consists of the BCCs which start at $\alpha =\pi$ and end at $\alpha = 0$. There are other possibilities of the terminals of the BCCs. For our present purpose, we only consider periodic orbits on BBCs of the first kind. There exist an infinite number of TCPs on $\alpha = \pi$. Correspondingly, the number of lobes is infinite (see also Fig. \[fig5\] and [@Sano1]), which immediately implies that there exist infinitely many (unstable) POs on BCCs of the first kind, as we show later.
![\[fig2\] Appearance of triple-collision points on the line $\alpha = \pi$: $l$ is the symbol length in the symbolic dynamics for the collinear [*eZe*]{} dynamics. The symbol sequences of triple-collision orbits on $\alpha = \pi$ are represented in the symbolic dynamics of the collinear [*eZe*]{} case. The solid (resp. dotted) line represents the lobe corresponding to $2$-$3$ (resp. $1$-$3$) collision. ](fig2.pdf){width="7cm"}
The appearance order of TCPs is illustrated in Fig. \[fig2\]. If the symbol length is $l$, the total number of the TCPs appearing on $\alpha = \pi$ is $2^{l-1}$ where $l$ denotes the [*generation*]{} of TCPs. The successive generations of TCPs are connected by the following rule: In the $(l+1)$th generation, new TCPs are added in between the already existing TCPs in the $l$th generation. Let $w$ be a word of length $l-1$, and $w0$ be a symbol sequence of a TCP which already exists in the $l$th generation. Then, $w10$ appears to the immediate right of $w0$, while $w20$ appears to the immediate left of $w0$. (The former rule does not apply for $l=1$.) Any orbit in BCCs with this new $w10$ or $w20$ at the right (resp. left) end experiences a 2-3 (resp. 1-3) collision.
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![\[fig3\] Tails of orbits on the lobe $w20$-$w210$ ](fig3.pdf "fig:"){width="7cm"}
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![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4a.jpg "fig:"){width="3.5cm"} ![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4b.jpg "fig:"){width="3.5cm"}
![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4c.jpg "fig:"){width="3.5cm"} ![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4d.jpg "fig:"){width="3.5cm"}
![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4e.jpg "fig:"){width="3.5cm"} ![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4f.jpg "fig:"){width="3.5cm"}
![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4g.jpg "fig:"){width="3.5cm"} ![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4h.jpg "fig:"){width="3.5cm"}
![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4i.jpg "fig:"){width="3.5cm"} ![\[fig4\] (Color online) Periodic orbits (POs) found : The nucleus is fixed at the origin. For the POs with binary collisions, names of the lobe to which they belong are indicated. In the orbit integration, we set $E=-1$. ](fig4j.jpg "fig:"){width="3.5cm"}
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Here we give an intuitive proof of the existence of a PO on each BCC of the first kind (i.e., lobe). We introduce coordinate $h$ ($0 \leq h \leq 1$) on the curve so that $h=0$ at the left end and $h=1$ at the right end. The orbit changes its topology when we move along the curve from $h=0$ to $h=1$ since the orbits at both ends have different histories of collisions. There are three types of lobes: (i) lobe $w20$-$w0$ ($w \neq 2^k$, $k \geq 0$) or lobe $w0$-$w10$; (ii) lobe $w0$-$w'0$ where symbol lengths of $w$ and $w'$ differ by 2 or more; (iii) lobe $2^{k+1}0$-$2^{k}0$ ($k\geq 0$).
First consider case (i), i.e., the case of lobe $w20$-$w210$. Let us move from $w20$ ($h=0$) to $w210$ ($h=1$). The relative position of the minimum of $r_{1}$ continuously changes as in Fig. \[fig3\]. The triple collision at $h=0$ becomes a $2$-$3$ binary collision for $ 0 < h < 1$. As $h$ increases, the next minimum of $r_1$ approaches the $2$-$3$ binary collision from the right, and finally coincides with it at $h=1$ resulting in triple collision. In this process, there exists a parameter value such that the $2$-$3$ binary collision and the local maximum of $r_{1}$ take place at the same time (see the third panel of Fig. \[fig3\]). At this moment, the angular momentum of electron 2 and the nucleus is zero with respect to their center of mass, which means that the angular momentum of electron 2 is zero, since the nucleus has infinite mass. This in turn means that the angular momentum of electron 1 is zero, which implies that electron 1 stands still at this moment. Then, both electrons retrace the path they tread, that is, a self-retracing PO is obtained. In the above proof, the persistence of the minima of $r_1$ and $r_2$ is crucial.
The proof for case (ii) may be similar to case (i). However, there is a possibility that the orbit escapes, that is, one of electrons escapes to infinity. We skip this case, since we need a special care to treat the non-persistence of minima of $r_1$ and/or $r_2$.
In case (iii), the situation is different from that of case (i). In this case, we can draw a similar orbital change as in Fig. \[fig3\]. This time, however, the third minimum of $r_2$ disappears due to the escape of the orbit when the first minimum approaches the second minimum of $r_2$. Then we do not have the simultaneous occurrence of the maximum of $r_2$ and the minimum of $r_1$. Therefore, there is no POs on the $2^{k+1}0$-$2^{k}0$ lobes.
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![\[fig5\] The Lyapunov exponents and the period for a family of periodic orbits: (a) The Lyapunov exponents. (b) The period. $n_{r}$ is the number of revolutions. ](fig5a.jpg "fig:"){width="7cm"}
![\[fig5\] The Lyapunov exponents and the period for a family of periodic orbits: (a) The Lyapunov exponents. (b) The period. $n_{r}$ is the number of revolutions. ](fig5b.jpg "fig:"){width="7cm"}
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![\[fig6\] Schematic pictures of binary-collision curves and periodic orbits (POs) in $D_{1/4}$: (a) Schematic picture of binary-collision curves (BCCs) of the first kind and the second kind. Solid (resp. dotted) lines represent the BCCs for $2$-$3$ (resp. $1$-$3$) collisions. There are infinite number of fundamental blocks. The fundamental block is numbered by $N$ from the right hand side. (b) POs in the fundamental block $N=1$: The triple collision orbits in the line $\alpha = \pi$ are indicated. The BCCs of the first kind are nested. The hexagon (resp. star) represents a position of a PO with (resp. without) binary collisions. The connected graph, which links the positions of the POs, is displayed. ](fig6a.pdf "fig:"){width="6cm"}
![\[fig6\] Schematic pictures of binary-collision curves and periodic orbits (POs) in $D_{1/4}$: (a) Schematic picture of binary-collision curves (BCCs) of the first kind and the second kind. Solid (resp. dotted) lines represent the BCCs for $2$-$3$ (resp. $1$-$3$) collisions. There are infinite number of fundamental blocks. The fundamental block is numbered by $N$ from the right hand side. (b) POs in the fundamental block $N=1$: The triple collision orbits in the line $\alpha = \pi$ are indicated. The BCCs of the first kind are nested. The hexagon (resp. star) represents a position of a PO with (resp. without) binary collisions. The connected graph, which links the positions of the POs, is displayed. ](fig6b.pdf "fig:"){width="6cm"}
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To obtain POs numerically, we look for the moment of zero velocities other than the initial moment. We actually carried out this procedure with many trials and errors. Then we find some POs using the improved Newton method.
We show example trajectories of POs in Fig. \[fig4\]. We note one remarkable feature in these POs. One electron, by rapidly revolving round the nucleus, screens (or weakens) the Coulomb field which the other electron feels. During this phase, the other electron moves slowly. Two electrons alternate in one period between a rapid motion near the nucleus and a slow screened motion at a distance from the nucleus. Interestingly, the PO, which exhibits this alternation, is unstable. We found POs without binary collision through interpolation between two POs with binary collision. Some of stable frozen planetary orbits [@RW] are among them (see Figs. \[fig4\](e-g)).
Finally, we characterize POs as follows. In Figs. \[fig4\](a-d), it is clear that one electron revolves round the nucleus more than the other. Its number of revolutions up to the turn-back point (in the half period) is characterized by a half integer or an integer, say $n_{r}$. For Figs. \[fig4\](a-d), it is easily inferred that there exists a sequence of POs from $n_{r}=1$ to $n_{r}=\infty$ (i.e., toward ionization). For these POs, the Lyapunov exponents and the period are numerically calculated (Fig. \[fig5\]). It shows a remarkable regularity of the period, $T=S/2=an_{r}+b$, where $T$ is the period and $S$ is the action.
Now taking the partial summation over $n_{r}$ in the Gutzwiller periodic orbit sum $d_{\mbox{\scriptsize osc}}(E)=\sum_{p} A_{p} e^{\frac{i}{\hbar}S_{p}}$ with a daring approximation, i.e., $A_{p}$ is factored out and is replaced by the mean value $\overline{A}$, thanks to Poisson summation formula, we obtain the Rydberg energy levels, $E_{n}=-\frac{C}{n^{2}}$ with constant $C$. This means that the regular structure found is important for the energy levels of the helium atom. The amplitude factor $A_{p}$ may play a role of determining the quantum defects for the helium atom as observed in the collinear [*eZe*]{} case [@TW]. To complete this program to obtain the energy levels, we have to seek the geometrical structure of the whole (unstable) POs. The result of the present Letter is a hint to this problem.
In Fig. \[fig6\], a schematic disposition of BCCs and POs are illustrated. As shown in Fig. \[fig6\](a), there is a fundamental block between two BCCs of the second kind. The number of fundamental blocks is infinite (Fig. \[fig6\](a)). The fractal structure of BCCs and the tree structure of POs similar to a binary Cayley tree are expected to be topologically the same in every block (Fig. \[fig6\](b)) reflecting the fractal distribution of TCPs on $\alpha = \pi$. We show in Fig. \[fig6\](b) the structure of the rightmost block. We find that there exists a PO on each lobe in Fig. \[fig6\](b), even on the lobe of case (ii). It is interesting to note that though POs are far from $\alpha = \pi$, they keep a kind of one-dimensionality along with other orbits in BCCs.
In conclusion, we have informally proved the existence of self-retracing POs with binary collisions, which form the backbone of the phase space, using the structure of BCCs and have indeed found such POs numerically. These findings would serve theoreticians with new stimuli, who would like to carry out the semiclassical treatment of the helium atom in two dimensions with zero angular momentum.
This work was supported by a Grant-in-Aid for Scientific Research (No.17740252) from the MEXT, Japan.
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---
abstract: 'The constituent parts of a quantum computer are inherently vulnerable to errors. To this end we have developed quantum error-correcting codes to protect quantum information from noise. However, discovering codes that are capable of a universal set of computational operations with the minimal cost in quantum resources remains an important and ongoing challenge. One proposal of significant recent interest is the gauge color code. Notably, this code may offer a reduced resource cost over other well-studied fault-tolerant architectures using a new method, known as gauge fixing, for performing the non-Clifford logical operations that are essential for universal quantum computation. Here we examine the gauge color code when it is subject to noise. Specifically we make use of single-shot error correction to develop a simple decoding algorithm for the gauge color code, and we numerically analyse its performance. Remarkably, we find threshold error rates comparable to those of other leading proposals. Our results thus provide encouraging preliminary data of a comparative study between the gauge color code and other promising computational architectures.'
author:
- 'Benjamin J. Brown'
- 'Naomi H. Nickerson'
- 'Dan E. Browne'
title: 'Fault-tolerant error correction with the gauge color code'
---
Introduction
============
Scalable quantum technologies require the ability to maintain and manipulate coherent quantum states over an arbitrarily long period of time. It is problematic then that the small quantum systems that we might use to realize such technologies decohere rapidly due to unavoidable interactions with the environment. To resolve this issue we have discovered quantum error-correcting codes [@Shor95; @Steane96] which make use of a redundancy of physical qubits to maintain encoded quantum states with arbitrarily high fidelity over an indefinite period.
Ideally, we will design a fault-tolerant quantum computer that requires as few physical qubits as possible to minimize the resource cost of a quantum processor, and indeed, the cost in resources of a computational architecture is very sensitive to the choice of quantum error-correcting code used by a fault-tolerant scheme. It is therefore of great interest to analyse different quantum error-correction proposals to compare and contrast their resource demands.
Color codes [@Bombin06; @Bombin07; @Bombin10; @Bombin13; @Bombin13a] are a family of topological quantum error-correcting codes [@Kitaev03; @Dennis02; @Terhal13; @LidarBrun] with impressive versatility [@Eastin; @BravyiKonig13; @Kubica15] for performing fault-tolerant logic gates [@Bombin11; @Fowler11; @Landahl14]. This is an important consideration as we search for schemes that realise fault-tolerant quantum computation with a low cost in quantum resources. In particular, a fault-tolerant quantum computer must be able to perform a non-Clifford operation, such as the $\pi / 8$-gate, to realise universal quantum computation.
In general, performing Non-Clifford gates can present a considerable resource cost over the duration of a quantum computation. As such, the resource cost of realising scalable quantum computation is sensitive to the method a fault-tolerant computational scheme uses to realise non-Clifford gates. To this end the gauge color code [@Bombin13a; @Kubica14; @Bombin14a; @Bombin14b] has attracted significant recent interest because, notably, this three-dimensional quantum error-correcting code can achieve a universal gate set via gauge fixing [@Paetznick13; @Anderson14].
In contrast to the gauge-color code, surface code quantum computation, a leading approach towards low-resource quantum computation [@Kitaev03; @Dennis02; @Raussendorf07], makes use of magic state-distillation [@BravyiKitaev05] to perform $\pi/8$-gates. Magic state distillation can be achieved with $\mathcal{O}(L^3)$ space-time resource cost [@Raussendorf07; @Fowler12a; @Bravyi12; @Meier13; @Li14]. Similarly, the gauge color code performs $\pi/8$-gates via gauge fixing in constant time [@Bombin14a], and as such has an equivalent scaling in space-time resource cost as the surface code since the gauge color code requires $\mathcal{O}(L^3)$ physical qubits. However, given that gauge fixing requires no additional offline quantum resources to perform a non-Clifford rotation, the gauge color code may reduce the quantum resources that are necessary for fault-tolerant quantum computation by a constant fraction.
It is also noteworthy that the gauge color code is local only in three dimensions and as such, unlike the surface code, cannot be realised using a two-dimensional array of locally interacting qubits. Instead, the gauge color code may be an attractive model for non-local quantum-computational architectures such as networked schemes [@Barrett05; @Fujii12; @Nickerson13; @Monroe14; @Nickerson14].
Given the significant qualitative differences between the gauge color code and the surface code, it is interesting to perform a comparative analysis of these two proposals. In this Manuscript we investigate error correction with the gauge color code.
Dealing with errors that continually occur on physical qubits is particularly difficult in the realistic setting where syndrome measurements can fail and return false readings [@Dennis02]. Attempting to correct errors using inaccurate syndrome information will introduce new physical errors to the code. However, given enough syndrome information, we can distinguish measurement errors from physical errors with enough confidence that the errors we introduce are few, and can be identified at a later round of error correction [@Bombin14a]. In the case of the toric code [@Dennis02], we accumulate sufficient error data by performing multiple rounds of syndrome measurements. Surprisingly, the structure of the gauge color code enables the acquisition of fault-tolerant syndrome data using only one round of local measurements [@Bombin14a]. This capability is known as single-shot error correction.
Here, we obtain a noise threshold for the gauge color code using a phenomenological noise model where both physical errors and measurement faults occur at rate $p$. We develop a single-shot decoder to identify the sustainable operating conditions of the code, i.e. the noise rate below which information can be maintained arbitrarily well, even after many cycles of error correction. We estimate a sustainable error rate of $p_{\text{sus}} \sim 0.31\%$ using an efficient clustering decoding scheme [@Harrington; @BravyiHaah11A; @Anwar14; @Hutter14; @Watson14; @Wootton15] that runs in time $\mathcal{O}(L^6 \log L)$, [@BravyiHaah11A], where the distance of the code is $d = L + 2$, [@Bombin13a]. Remarkably, the threshold we obtain falls within an order or magnitude of the optimal threshold for the toric code under the same error model, $\sim2.9\%$, [@Wang03]. Furthermore, we also use our decoder to estimate how the logical failure rate of the gauge color code scales below the threshold error rate by fitting to a heuristic scaling hypothesis.
Results
=======
![(a) In the primal picture qubits sit on the vertices of a four-valent lattice. The three-dimensional cells of the lattice are four colorable, i.e., every cell can be assigned one of four colors such that it touches no other cell of the same color. (b) The gauge color code of linear size $L=5$ drawn in the dual picture. Qubits lie on simplices of the lattice. The faces of the tetrahedra in the figure are given one of four colors such that no two faces of a given tetrahedron have the same color. The tetrahedra are stacked such that faces that touch always have the same color. For the gauge color code to encode a single logical qubit, the lattice must have four distinct, uniformly colored boundaries, as is shown in the Figure. We give more details on the lattice construction in the dual picture in Supplementary Note 1 together with Supplementary Figures 1-6. \[ColorCodeLattice\]](ColorCodePrimalAndDual.pdf)
The gauge color code
--------------------
The gauge color code is a subsystem code [@Poulin05] specified by its gauge group, $\mathcal{G}$. From the center of the gauge group, $Z(\mathcal{G})$, we obtain the stabilizer group for the code, $\mathcal{S} = Z(\mathcal{G}) \cap \mathcal{G}$, and its logical operators, $\mathcal{L} = Z(\mathcal{G}) \backslash \mathcal{G}$. Elements of the stabilizer group, $S \in \mathcal{S}$, satisfy the property that $S | \psi \rangle = | \psi \rangle$ for all codewords of the code $| \psi \rangle$.
The code is defined on a three-dimensional four-valent lattice of linear size $L$ with qubits on its vertices [@Bombin07; @Bombin07a]. The lattice must also be four-colorable, i.e. each cell of the lattice can be given a color, red, green, yellow, or blue, denoted by elements of the set $\mathcal{C} = \{ \mathbf{r}, \mathbf{g}, \mathbf{y}, \mathbf{b} \} $, such that no two adjacent cells are of the same color. The lattice we consider is shown in Fig. \[ColorCodeLattice\](a), where the cells are the solid colored objects in the Figure.
The cells of the lattice define stabilizer generators for the code, while the faces of cells define its gauge generators. The gauge operators, otherwise known as face operators, are measured to infer the values of stabilizer operators. The face operators have weight four, six and eight, where the weight-eight face operators lie on the boundary of the lattice. More specifically, for each cell $c$ there are two stabilizer generators, $S^X_c = \prod_{j\in \mathcal{V}(c)}X_j$ and $S^Z_c = \prod_{j\in \mathcal{V}(c)} Z_j$, where $\mathcal{V}(c)$ are the qubits on the boundary vertices of cell $c$, and $X_j$ and $Z_j$ are Pauli-X and Pauli-Z operators acting on vertex $j$. For each face $f$ there are two face operators, $G_f^X = \prod_{j\in \mathcal{V}(f)} X_j$ and $G_f^Z= \prod_{j\in \mathcal{V}(f)} Z_j$ where $\mathcal{V}(f)$ are the vertices on the boundary of face $f$. We will call the outcome of a face operator measurement a face outcome.
Given suitable boundary conditions [@Bombin07a; @Bombin13a], the code encodes one qubit, whose logical operators are $\overline{X} = \prod_{j \in \mathcal{Q}} X_j$ and $\overline{Z} = \prod_{j \in \mathcal{Q}} Z_j$, where $\mathcal{Q}$ is the set of physical qubits of the code. A lattice with correct boundaries is conveniently represented on a dual lattice of tetrahedra using the convention given in Ref. [@Bombin13a]. The lattice we consider is illustrated in Fig. \[ColorCodeLattice\](b), where we discuss its construction in detail in the Supplementary Notes 1 and 2, together with Supplementary Figures 1-7. Importantly, we require that the lattice has four boundaries, distinguished by colors from set $\mathcal{C}$.
Single-shot error correction
----------------------------
![(a) The faces of the cells of the color code are three colorable. (b) An example of a set of face measurement outcomes where measurements are reliable. Face operators that return value -1 are colored, otherwise they are left transparent. The stabilizer at the cell with thick red edges contains a stabilizer defect. This cell has one face operator of each color returning a -1 outcome. All other cells have an even parity of -1 face outcomes over their colored subsets, indicating that these cells contain no stabilizer defect. (c) A gauge syndrome where colored subsets of face measurements about the green cells do not agree, thus indicating measurement errors. \[GaugeMeasurements\]](FaultTolerantSyndromes.pdf)
A quantum error-correcting code is designed to identify and correct errors. Due to the symmetry of the gauge group, it suffices here to consider only bit-flip, i.e. Pauli-X errors. We consider a phenomenological error model consisting of physical errors and measurement errors. A physical error is a Pauli-X error on a qubit, while a measurement error returns the opposite outcome of the correct reading. Errors will be identically and independently generated with the same probability $p$.
In a stabilizer code, errors are identified by stabilizer measurements that return eigenvalue $-1$, which we call stabilizer defects. We use the stabilizer syndrome, a list of the positions of stabilizer defects, to predict the incident error. In the gauge color code we do not measure stabilizer operators directly, but instead infer their values by measuring face operators, which is possible due to the fact that $\mathcal{S} \subseteq \mathcal{G}$.
In addition to using face outcomes to infer stabilizer eigenvalues, we can also exploit the local constraints in $\mathcal{G}$ of the gauge color code to detect and account for measurement errors. Remarkably, measurement errors can be detected reliably by measuring each face operator only once, so called single-shot error correction [@Bombin14a].
The local constraints stem from the structure of the code. We first observe that the faces of a cell are necessarily three colorable, as shown in Fig. \[GaugeMeasurements\](a). It follows from the three colorability of the faces that the product of the gauge operators $G^Z_f$ of any of the differently colored subsets of faces of cell $c$ recover the stabilizer operator $S^Z_c$. By measuring all the faces of the lattice, we redundantly recover each stabilizer eigenvalue three times where the three outcomes of a given cell are constrained to agree. In Fig. \[GaugeMeasurements\](b) we show an example of a gauge measurement configuration where the outcomes are reliable. Following this observation, we can use violations of the local constraints about a cell to indicate the positions of measurement errors. In Fig. \[GaugeMeasurements\](c) we show a syndrome where the product ofthe face outcomes of different subsets of of two cells, colored with thick green edges, do not agree.
Fault-tolerant decoding with the gauge color code proceeds in two stages. The first stage, syndrome estimation, uses face outcomes that may be unreliable to estimate the locations of stabilizer defects. The second stage, stabilizer decoding, takes the estimated stabilizer defect locations and predicts a correction operator to reverse physical errors.
The topological nature of the gauge color code is such that its stabilizer defects satisfy conservation laws that enable us to employ well-studied decoding algorithms [@Duclos-CianciPoulin10; @BravyiHaah11A; @WoottonLoss12] to complete stabilizer decoding. The syndrome identified by stabilizer measurements is studied extensively in Refs. [@Bombin07; @Bombin07a]. Stabilizers that return $-1$ outcomes occur in pairs at the endpoints of strings of errors, whose endpoints take the color of its terminal cells, as shown in Fig. \[Syndrome\](a). As such, one can also regard an error string as carrying the color of the stabilizer defects at its endpoints. Error strings on the gauge color code can also branch into three strings of conjugate colors, thus creating one syndrome of each color, as shown in Fig. \[Syndrome\](b). Finally, error strings of a given color can terminate at the boundary of their respective color, as shown in Fig. \[Syndrome\](c). In our simulation we decode stabilizer defects by adapting a clustering decoder [@Harrington; @BravyiHaah11A; @Anwar14; @Hutter14; @Watson14; @Wootton15] where clusters grow linearly.
We concentrate now on syndrome estimation. Syndrome estimation uses a gauge syndrome, a list of gauge defects, to estimate a stabilizer sydrome. A lattice cell can contain as many as three gauge defects. Gauge defects are distinguished by a color pair $\mathbf{uv}$, with $\mathbf{u},\, \mathbf{v} \in \mathcal{C}$, such that $\mathbf{u} \not= \mathbf{v}$ and $\mathbf{uv} = \mathbf{vu}$. The color pair of a gauge defect relates to the coloring of the lattice faces. A face is given the color pair opposite to the colors of its adjacent cells, i.e. the face shared by two cells with colors $\mathbf{r}$ and $\mathbf{g}$ is colored $\mathbf{yb}$. A cell $c$ contains a $\mathbf{uv}$ gauge defect if the product of all the $\mathbf{uv}$ face outcomes bounding $c$ is $-1$. Following this definition, a stabilizer defect is equivalent to three distinct gauge defects in a common cell.
Studying the gauge syndrome enables the identification of measurement errors. We consider face $f$, colored $\mathbf{uv}$, that is adjacent to cell $c$. In the noiseless measurement case, where $c$ contains no stabilizer defect, by definition, cell $c$ should contain no gauge defects. However, if face $f$ returns an incorrect outcome, we identify a $\mathbf{uv}$ gauge defect at $c$. Conversely, if cell $c$ contains a stabilizer defect in the ideal case, and $f$ returns an incorrect outcome, no $\mathbf{uv}$ gauge defect will appear in $c$. With these examples we see that cells that contain either one or two gauge defects indicate incorrect face outcomes.
An incorrect face outcome affects gauge defects in both of its adjacent cells. In general, incorrect face outcomes of color $\mathbf{uv}$ form error strings on the dual lattice, whose end points are $\mathbf{uv}$ gauge defects, where individual incorrect face outcomes are segments of the string. Error strings of incorrect $\mathbf{uv}$ face outcomes changes the parity of $\mathbf{uv}$ gauge defects at both of its terminal cells.
We require an algorithm that can use gauge syndrome data to predict a likely measurement error configuration, and thus estimate the stabilizer syndrome. We adapt the clustering decoder [@Harrington; @BravyiHaah11A; @Anwar14; @Hutter14; @Watson14] for this purpose. The decoder combines nearby defects into clusters that can be contained within a small box. Clusters increase linearly in size to contain other nearby defects until they contain a set of gauge defects that can be caused by a measurement error contained within the box. Once clustering is completed, a correction supported inside the boxes is returned.
We briefly elaborate on correctable configurations of gauge defects. Pairs of $\mathbf{uv}$ gauge defects are caused by strings of incorrect $\mathbf{uv}$ face outcomes, and therefore form correctable configurations, as shown in Fig. \[Syndrome\](d). As an example, the gauge syndrome in Fig. \[Syndrome\](d) depicts the gauge defects and measurement error string shown in Fig. \[GaugeMeasurements\](c) where the measurement errors have occurred on the faces that returned $-1$ measurement outcomes. Triplets of gauge defects, colored $\mathbf{uv}$, $\mathbf{vw}$ and $\mathbf{uw}$, also form correctable configurations. Error strings that cause triplets of correctable gauge defects branch at a cell, and thus indicate a stabilizer defect, shown in Fig. \[Syndrome\](e). The stabilizer defect where the error string branches lies at a cell colored $\mathbf{x} \not= \mathbf{u}, \mathbf{v}, \mathbf{w} $. Gauge defects can also arise due to incorrect face outcomes on the lattice boundary. Specifically, the boundary colored $\mathbf{w}$ contains faces of color $\mathbf{uv}$ where $ \mathbf{u}, \mathbf{v} \not= \mathbf{w}$. With this coloring we can find correctable configurations of single $\mathbf{uv}$ gauge defects, together with a boundary of color $\mathbf{w}$, see Fig. \[Syndrome\](f). In general, a cluster can contain many correctable pairs and triplets of gauge defects.
As we have mentioned, correctable clusters of gauge defects can give rise to stabilizer defects. It is important to note that the error-correction procedure is sensitive to the positions of stabilizer defects within a correctable cluster, as discrepancies in their positions later affect the performance of the stabilizer decoding algorithm. As such we must place stabilizer defects carefully. For cases where a correctable cluster of gauge defects returns stabilizer defects, we assign their positions such that they lie at the mean position of all the gauge defects within the correctable cluster, at the nearest cell of the appropriate color. Once syndrome estimation is complete, the predicted stabilizer syndrome is passed to the stabilizer decoder, and a correction operator is evaluated.
We remark that gauge defects can be incorrectly analyzed during syndrome estimation. In which case, measurement errors sometimes masquerade as stabilizer defects, and sometimes stabilizer defects can be misplaced. We will then attempt to decode the incorrect stabilizer syndrome and mistakenly introduce errors to the code. In general, any error-correction scheme that takes noisy measurement data will introduce residual physical errors to a code. These errors can be corrected in the future, provided the remaining noise is of a form that a decoder can correct. In general however, one must worry that large correlated errors can be introduced that adversarially corrupt encoded information [@Aharonov06; @Ng09; @Preskill13; @Jouzdani14; @Fowler14; @Hutter14a]. Such errors may occur in the gauge color code if, for instance, we mistakenly predict two stabilizer defects of the same color that are separated by a large distance. We give an example of a mechanism that might cause a correlated error during syndrome estimation with the gauge color code in the Supplementary Note 3, together with Supplementary Figure 8.
A special property of the gauge color code is that measurement errors, followed by syndrome estimation, will only introduce false defects in locally correctable configurations. Therefore, residual errors remain local to the measurement error. Moreover, the code is such that the probability of obtaining configurations of face outcomes that correspond to faux stabilizer defects decays exponentially with the separation of their cells. This is because the number of measurement errors that must occur to produce a pair of false stabilizer defects is extensive with their cell separation. To this end, the errors introduced from incorrect measurements are local to the measurement error and typically small. This property, coined ‘confinement’ in Ref. [@Bombin14a], is essential for fault-tolerant error correction. Most known codes achieve confinement by performing syndrome measurements many times. We give numerical evidence showing that our error-correction protocol confines errors in the following Subsection.
 Two blue syndromes indicate the end points of a string of errors connecting the two points. (b) A string error branches to create one syndrome of each color. (c) A green string error can terminate at a green boundary, thus generating only a single green syndrome. (d) A pair of $\mathbf{yb}$ gauge defects, shown by the vertices, can be caused by a string of incorrect $\mathbf{yb}$ face outcomes on the dual lattice. The displayed gauge syndrome is equivalent to that shown in Fig. \[GaugeMeasurements\](c) where the measurement errors have occurred on the faces that returned $ - 1 $ outcomes. (e) Three gauge defects, colored $\mathbf{gy}$, $\mathbf{gb}$ and $\mathbf{yb} $, can be caused by an error string that branches at a red cell. The branching point indicates a stabilizer defect at a red cell. (f) Strings of incorrect face outcomes of colors $\mathbf{rg}$ and $\mathbf{gb}$ terminate at a yellow boundary.](GaugeSyndromes.pdf)
The Simulation
--------------
We simulate fault-tolerant error correction with encoded states $|\psi_j \rangle $ of linear size $L$ where $j$ indicates the number of error-correction cycles that have been performed, and where $|\psi_0 \rangle$ is a codeword. We seek to find a correction operator $C$ such that $CE \in \mathcal{G}$ where $E$ is the noise incident to $|\psi_0 \rangle$ after $N$ error-correction cycles.
To maintain the encoded information over long durations, we repeatedly apply error-correction cycles to keep the physical noise sufficiently benign. After a short period, the state $|\psi_{j-1} \rangle$ will accumulate physical noise $E_{j}(p) $ with error rate $p$. To correct the noise, we first estimate a stabilizer syndrome, $\mathbf{s}_{j}$, using gauge syndrome data with the syndrome estimation algorithm $M_q$, where measurement outcomes are incorrect with probability $q = p$. Specifically, we have that $\mathbf{s}_{j} = M_p(E_{j}(p) |\psi_{j-1} \rangle)$. We then use the stabilizer decoding algorithm $D$ to predict a suitable correction operator $C_{j}= D(\mathbf{s}_{j})$, such that we obtain $$|\psi_{j} \rangle =C_{j} E_{j}(p) |\psi_{j-1} \rangle.$$
It is important to note is that $ |\psi_{j} \rangle $ is not necessarily in the code subspace. For $q>0$, stabilizer syndromes will in general be incorrectly estimated, and thus the correction operator $C_{j}$ introduces some new errors to the code.
![\[ThresholdFigure\]We show logical error rates, $P_{\text{fail}}$, as a function of physical error rate, $p$, for system sizes $L = 17,\, 23,\, 29 $ and $35$ shown in blue, yellow, green and red, respectively, as is marked in the legend, where we collect data after $N = 8$ rounds of error correction during which measurements are performed unreliably. The error bars show the standard error of the mean given by the expression $\Delta p = \sqrt{ (1 - P_{\text{fail}})P_{\text{fail}} /\eta } $ where $\eta$ is the number of Monte Carlo samples we collect. The data used to determine the threshold error rate is shown in the inset, where we determine the threshold using the fitting described in the Methods Section. The fitting is also plotted in the inset. In the main Figure the solid lines show the fitted expression, Eqn. (\[Eqn:Low-pFitting\]), to demonstrate the agreement of our scaling hypothesis with numerically evaluated logical error rates where $p < p_{\text{th}}$. We remark that the fitting is made using data for all values of $N$, and not only the data shown in this plot, as we explain in the Methods Section.](threshold_inset_plot.pdf){width="\columnwidth"}
We require that after $N$ error-correction cycles we can estimate error $E$ of state $ |\psi_{N} \rangle = E | \psi_0 \rangle $ to perform a logical measurement. To perform the $\overline{Z}$ logical measurement [@Alicki10; @Bombin14a], we measure each individual qubit of the code in the Pauli-Z basis. This destructive transversal measurement gives us the eigenvalues of stabilizers $S_c^Z$ to diagnose $E$, and to thus recover the eigenvalue of $\overline{Z}$.
During readout, measurement errors and physical errors have an equivalent effect; both appear as bit flips. To simulate errors that occur during the readout process, we apply the noise operator $E(p)$ to the encoded state before decoding. We therefore calculate logical failure rates $$P_{\text{fail}}(N) = \text{prob}(C(N)E(p) E \in \mathcal{G} ), \label{Eqn:LogicalFailureCalculation}$$ where $C(N) = D(M_0(E(p)| \psi_N \rangle))$. We evaluate $P_{\text{fail}}(N)$ values using Monte Carlo simulations.
To analyze the performance of the proposed decoding scheme, we first look to find the sustainable error rate of the code, $p_{\text{sus}}$, below which we can maintain quantum information for an arbitrary number of correction cycles. The discovery of such a point suggests that the error-correlations caused by our correction protocol do not extend beyond a constant, finite, and decodable length, thus showing that we can preserve information indefinitely with arbitrarily high fidelity in the $ p < p_{\text{sus}}$ regime.
We define $p_{\text{sus}}$ as the threshold error rate, $p_{\text{th}}$, at the $N \rightarrow \infty $ limit, where the threshold is the error rate below which we can decrease $P_{\text{fail}}$ arbitrarily by increasing $L$ [@Terhal13]. In the inset of Fig. \[ThresholdFigure\] we show the near threshold data we use to evaluate a threshold, where we show the data for $N = 8$ as an example. We give the details of the fitting model we use to evaluate thresholds in the Methods Section of this Manuscript.
![\[PhaseDiagram\]Threshold error rates, $p_{\text{th}}$, are calculated with system sizes $L=23,\,29,\,35$ after $N$ error-correction cycles using $\eta \sim 10^4$ Monte Carlo samples. Error bars show the standard error of the mean which are determined using the [**NonLinearModelFit**]{} function in [**Mathematica**]{}. The solid blue line shows the fitting given in Eqn. (\[Fitting\]). The dashed red line marks $p_{\text{sus}} \sim 0.31 \%$, the sustainable noise rate of the code, the limiting value of $p_{\text{th}}$ from to the fitting as $N\rightarrow \infty$.](Decay_errorbars.pdf){width="\columnwidth"}
We next study the evaluated threshold values as a function of $N$. We show this data in the plot given in Fig. \[PhaseDiagram\]. The data shows that $p_{\text{th}}$ converges to $p_{\text{sus}} \sim 0.31\%$ where we fit for values $N \le 8$. We obtain this value with a fitting that converges to $p_{\text{sus}}$, namely $$p_{\text{th}}(N) = p_{\text{sus}} \left[1 - (1 - p_{\text{th}}(0) / p_{\text{sus}} ) \exp(-\gamma N) \right]. \label{Fitting}$$ We find $p_{\text{th}}(0) \sim 0.46 \%$ and $\gamma \sim 1.47 $. The convergent trend provides evidence that we achieve steady-state confinement in the high-$N$ limit, as is required of a practical error-correction scheme.
To verify further the threshold error rates we have determined, we next check that the logical failure rate decays as a function of system size in the regime where $p < p_{\text{th}}$. In the large $N$ limit, we fit our data to the following hypothesis $$P_{\text{fail}}(N) = (N+1)A \exp \left(\alpha \log \left( \frac{p}{ p_{\text{sus}}} \right) d^\beta \right), \label{Eqn:Low-pFitting}$$ where $A$, $\alpha$ and $ \beta $ are positive constants to be determined and $d$ is the distance of the code. We evaluate the variables in Eqn. (\[Eqn:Low-pFitting\]) as $A \sim 0.033$, $\alpha \sim 0.516$ and $\beta \sim 0.822$ using $\sim 5000$ CPU hours with data for $N \le 10$. Details on the fitting calculation are given in the Methods Section. We plot the fitted scaling hypothesis on Fig. \[ThresholdFigure\] for the case of $N = 8$ to show the agreement of Eqn. (\[Eqn:Low-pFitting\]) with the available data.
Discussion
==========
To summarize, using only a simple decoding scheme, we have obtained threshold values that lie within an order of magnitude of the optimal threshold for the toric code under the phenomenological noise model. Moreover, we can expect that higher thresholds are achievable using more sophisticated decoding strategies [@Duclos-CianciPoulin10; @Wang10; @Bombin12; @WoottonLoss12; @Sarvepalli12; @HutterWoottonLoss13; @Delfosse14; @Anwar14; @Herrold; @Hutter14; @Stephens14]. It may be possible to achieve a sufficiently high sustainable noise rate to become of practical interest, thus meriting comparison with the intensively-studied surface code [@Fowler13; @Fowler15]. To this end, further investigation is required to learn its experimental viability.
To continue such a comparative analysis, one should study the code using realistic noise models [@Fowler12a] that respect the underlying code hardware. We expect that the threshold will suffer relative to the surface code when compared using a circuit-based noise model where high-weight gauge measurements are more error prone [@Landahl11]. Fortunately, gauge color code lattices are known where face operators have weight no greater than six [@Bombin13a]. While this is not as favorable as the weight-four stabilizer measurements of the surface code, given the ability to perform single-shot error correction, and $\pi/8$-gates through gauge fixing, we argue that the gauge color code is deserved of further comparison.
To the best of our knowledge, we have obtained the first threshold using single-shot error correction. Fundamentally, our favorable threshold is achieved using redundant syndrome data to identify measurement errors. It is interesting to ask if we can make use of a more intelligent collection of measurement data to improve thresholds further. Discovering single-shot error-correction protocols with simpler codes might help to address such questions.
We gratefully acknowledge S. Bartlett, H. Bombín, E. Campbell, S. Devitt, A. Doherty, S. Flammia and M. Kastoryano for helpful discussions. Computational resources are provided by the Imperial College HPC Service. This work is supported by the EPSRC and the Villum Foundation.\
[**Author contributions:**]{} Original concept conceived by BJB and DEB, simulations designed and written by BJB and NHN, data collected and analysed by BJB and NHN, the manuscript was prepared by all the authors.\
[**Data availability:**]{} The code and data used in this Manuscript is available upon request to the corresponding author.\
[**Competing financial interests:**]{} The authors declare no competing financial interests.\
Methods
=======
Threshold calculations {#SCTN:ThresholdCalculation}
----------------------
The threshold error rate, $p_{\text{th}}$, is the physical error rate below which the logical failure rate of the code can be arbitrarily suppressed by increasing the code distance. We identify thresholds by plotting the logical failure rate as a function of physical error rate $p$ for several different system sizes, and identify the value $p = p_\text{th}$ such that $P_{\text{fail}}$ is invariant under changes in system size. In the main text we show the data used for a specific threshold calculation in Fig. \[ThresholdFigure\] where we use the $N = 8$ data as an example, and in Fig. \[PhaseDiagram\] we evaluate threshold error rates for the gauge color code as a function of the number of error correction cycles, $N$.
We evaluate threshold error rates by performing $\eta \sim 10^4$ Monte Carlo simulations for each value of $p$ close to the crossing point using codes of system sizes $L = 23, \, 29$ and $35$, except for the case that $N=0$ where we evaluate the logical failure rate with system sizes $L = 31,\,39$ and $47$. Simulating larger system sizes where $N = 0$ is possible as in this case we read out information immediately after encoding it, such that $E = 1$ as shown in Eqn. (\[Eqn:LogicalFailureCalculation\]). We therefore need not perform syndrome estimation in the $N= 0$ simulation. The threshold error rate at $N=0$ is thus the threshold error rate of the clustering decoder for the gauge color code where measurements are performed perfectly, i.e., $ q = 0$.
We identify the crossing point by fitting our data to the following formula $$P_{\text{fail}} = B_0 + B_1 x + B_2 x^2,$$ where $x = (p - p_{\text{th}})L^{1/\mu}$ and $B_j$, $p_{\text{th}}$ and $\mu$ are constants to be determined. We show an example of this fitting in the Inset of Fig. \[ThresholdFigure\].
At the threshold error rate the code produces logical failures at a rate between $0.075$ and $0.27$ depending on $N$. We expect such behavior as the number of logical failures will increase with repeated use of a decoder. We therefore obtain between $\sim 750$ and $\sim2700$ logical failures per data point close to the threshold error rate.
Overhead analysis {#SubSec:Overheads}
-----------------
Here we summarize the resource-scaling analysis we give in the $p < p_{\text{th}}$ regime. We suppose the logical failure rate in this regime scales like $$P_{\text{fail}}(N) \approx (N+1)A \exp( \alpha \log (p / p_{\text{th}}(N)) d^\beta ), \label{Eqn:LogicalFailureRate}$$ where $A$, $\alpha$ and $\beta$ are constants to be determined, $d $ is the distance of the code, $p$ is the error rate and $N$ is the number of uses of the decoder we make before readout. The value of $p_{\text{th}}(N)$ is determined by the method given in the previous Subsection. This equation is derived by assuming that a single use of the decoder will fail with probability $P_{\text{fail}} = A \exp(\alpha d^\beta \log (p / p_{\text{th}}) ) $ in the low-$p$ regime. We then calculate to first order the probability that the decoder will fail a single time in $N+1$ uses to give Eqn. (\[Eqn:LogicalFailureRate\]), where we include an additional use of the decoder to account for a possible logical failure during readout.
We manipulate Eqn. (\[Eqn:LogicalFailureRate\]) to show a method to evaluate $\alpha$ and $\beta$. We first take the logarithm of both sides of Eqn. (\[Eqn:LogicalFailureRate\]) to find the linear expression $$y = \log((N+1)A) + \alpha d^\beta u, \label{Eqn:GradientFitting}$$ where we write $ y = \log P_\text{fail} $ and $ u = \log (p / p_{\text{th}}) $. We then take the gradient, $g(d) = \dd y / \dd u$, from Eqn. (\[Eqn:GradientFitting\]) to find $$\log g(d) = \log \alpha + \beta \log d. \label{Eqn:FinalFitting}$$
![\[Fig:LogPFit\]Plot showing $ \log P_{\text{fail}}$ as a function of $ \log p $ for the case that $N = 8$ where we have plotted system sizes $L = 17,\, 23,\, 29 $ and $ 35$ shown in blue, yellow, green and red, respectively. The error bars show the standard error of the mean given by the expression $\Delta \log p = \left[ (1 - P_{\text{fail}})P_{\text{fail}} /\eta \right]^{1/2} / P_{\text{fail}} $ where $\eta$ is the number of Monte Carlo samples we collected. The logarithm of the gradients found for the linear fittings are plotted as a function of $\log d$ in Fig. \[Fig:GradientFitting\]. \[Fig:ExampleData\] ](ExampleLogpData.pdf){width="\columnwidth"}
In Fig. \[Fig:ExampleData\] we plot $\log P_{\text{fail}}$ as a function of $\log p$. For fixed system sizes we observe a linear fitting that Eqn. (\[Eqn:GradientFitting\]) predicts. We take the gradient of each of these fittings to estimate $g(d)$, as given in Eqn. (\[Eqn:FinalFitting\]). Then, to find $\alpha$ and $\beta$, we plot $\log g(d)$ as a function of $\log d$ where the gradients are taken from the fittings shown in Fig. \[Fig:ExampleData\]. This data is shown in Fig. \[Fig:GradientFitting\]. We can now determine $ \alpha$ and $\beta$ using, respectively, the $d = 1$ intersection, and the gradient of a linear fit shown in Fig. \[Eqn:GradientFitting\].
![Plot shows the available data fitted to the trend anticipated in Eqn. (\[Eqn:FinalFitting\]). The Figure shows the logarithm of the gradients found in Fig. \[Fig:LogPFit\], $ \log g(d)$, plotted as a function of the logarithm of the code distance, $\log d$, for the case where $N = 8$. Error bars show the standard error of the mean, evaluated using the [**LinearModelFit**]{} function in [**Mathematica**]{}. For this case we obtain a fitting $ \log g(d) \approx 0.81 \log d - 0.63 $, as is shown in the plot. \[Fig:GradientFitting\]](ExamplelogGLogLFitting.pdf){width="\columnwidth"}
The logical failure rates we use to find $\alpha$ and $\beta$ are found using between $10^5$ and $10^6$ Monte Carlo samples for each value of $p$ where we only take values of $p < 0.8 \times p_{\text{th}}$ for each $N$. We discard data points where we observe fewer than ten failures for a given $p$. The data was collected over 5000 CPU hours.
![\[Fig:AlphaBetaInN\] Unitless values, $\alpha$ and $\beta$, plotted as a function of error-correction cycles, $N$, shown in blue and yellow respectively. The error bars show the standard error of the mean, which are found using the [**NonLinearModelFit**]{} function included in [**Mathematica**]{}. The $\alpha$ and $\beta$ data points are fitted to Eqns. (\[Eqn:AlphaFitting\]) and (\[Eqn:BetaFitting\]), respectively. The fitted functions are shown in the plot.](AlphaBeta.pdf){width="\columnwidth"}
We plot the values we find for $\alpha$ and $\beta$ as a function of $N$ in Fig. \[Fig:AlphaBetaInN\]. Importantly, we observe convergence in the large $N$ limit. We see this using the fitting functions $$\alpha(N) = \alpha_\infty \left[ 1 - (1-\alpha_0 / \alpha_\infty) \exp(-\gamma_\alpha N) \right], \label{Eqn:AlphaFitting}$$ and $$\beta(N) = \beta_\infty \left[ 1 - (1-\beta_0 / \beta_\infty) \exp(-\gamma_\beta N) \right], \label{Eqn:BetaFitting}$$ to find the $N \rightarrow \infty$ behaviour of our protocol, where $\alpha_0$, $\beta_0$, $\alpha_\infty$, $\beta_\infty$, $\gamma_\alpha$ and $\gamma_\beta$ are constants to be determined, such that $$\alpha_\infty = \lim_{N \rightarrow \infty} \alpha(N), \quad \beta_\infty = \lim_{N \rightarrow \infty} \beta(N).$$ We fit these functions to our data to find $$\alpha_\infty = 0.516 \pm 0.005 , \quad \beta_\infty = 0.822 \pm 0.004 ,$$ together with the following values $\alpha_ 0 = 0.65 \pm 0.02$, $\gamma_\alpha = 1.9 \pm 1.5$, $\beta_0 = 0.73\pm 0.01$ and $\gamma_\beta = 1.7\pm 1.4$. The fittings are shown in Fig. \[Fig:AlphaBetaInN\].
![\[Fig:AFitting\]Figure shows unitless values of $A$ numerically determined as a function of the number of error-correction cycles we perform, $N$. The error bars show the standard error of the mean which is calculated using [**Mathematica**]{}. We fit the data to Eqn. (\[Eqn:AFitting\]) as we show in the Figure. ](A.pdf){width="\columnwidth"}
Finally, given that we have evaluated $\alpha(N)$ and $\beta(N)$ for different values of $N$, we use these values together with Eqn. (\[Eqn:LogicalFailureRate\]) to determine $A$ as a function of $N$. We show the data in Fig. \[Fig:AFitting\]. We fit the values of $A(N)$ to the following expression $$A(N) = A_\infty \left[ 1 - (1-A_0 / A_\infty) \exp(-\gamma_A N) \right], \label{Eqn:AFitting}$$ to find values $A_\infty = 0.033 \pm 0.001 $, $A_0 = 0.09 \pm 0.01 $ and $\gamma_A = 2.1\pm 0.5$, thus giving all of the variables, $A_\infty$, $\alpha_\infty$ and $\beta_\infty$, we require to estimate the $N \rightarrow \infty$ behavior of the decoding scheme in the below threshold regime.\
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Supplementary Figures
=====================
Supplementary Figure 1 {#supplementary-figure-1 .unnumbered}
----------------------
\(a) An odd and an even unit cube that each consist of five tetrahedra. The odd unit cube differs from the even unit cube by a $\pm \pi/2$ rotation about any of the canonical axes. (b) An exploded even unit cell that reveals the internal structure of a unit cube. (c) A fundamental tetrahedron of the lattice whose vertices are correctly four colored. We color each face of every tetrahedron with the color that is not the color of any of its three vertices. The face that is not visible is colored green. \[FundamentalUnits\]
Supplementary Figure 2 {#ManyUnitCubes .unnumbered}
----------------------
A cube of linear size $L=5$ formed by stacking odd and even unit cubes. The odd and even unit cubes, shown in grey and white, are stacked such that no two even unit cubes meet at a face and no two odd unit cubes meet at a face.
Supplementary Figure 3 {#supplementary-figure-3 .unnumbered}
----------------------
We remove tetrahedra from the cubic lattice shown in Supplementary Figure 2 to obtain the four-sided tetrahedral structure we require of the gauge color code lattice. The figure shows some of the tetrahedra that have been removed from the cubic lattice to the right of the Figure. The remaining tetrahedra do not define a gauge color code dual lattice without replacing some of the removed tetrahedra, as we explain in Supplementary Note 1 and show in Supplementary Figure 4.
Supplementary Figure 4 {#supplementary-figure-4 .unnumbered}
----------------------
We modify the lattice of Supplementary Figure 3 to uniformly color the boundaries by replacing some of the tetrahedra that were removed from the cubic lattice. Patches of six incorrectly colored faces are outlined with white hexagons. We add additional tetrahedra to these incorrectly colored patches such that all four boundaries are uniformly colored. The Figure shows some hexagons where the additional tetrahedra are already added. \[Boundaries\]
Supplementary Figure 5 {#supplementary-figure-5 .unnumbered}
----------------------
The $L=5$ lattice where the layers of tetrahedra are separated. Interior tetrahedra are colored white and grey. \[LatticeLayers\]
Supplementary Figure 6 {#supplementary-figure-6 .unnumbered}
----------------------
A repeating cell of the lattice that consists of eight of the fundamental unit cubes. Stabilizers of weight eight and thirty-two are represented by diamonds and circles on the vertices of the repeating cell respectively, as shown in the key at the right of the Figure. Opposite faces of the repeating cell are equivalent. \[StabilizerGeometry\]
Supplementary Figure 7 {#supplementary-figure-7 .unnumbered}
----------------------
Qubits on the exterior of the gauge color code lattice. Qubits lie on all the faces, on the edges where two differently colored boundaries meet, fattened and colored in grey in the figure, and on the vertex where three differently colored boundaries meet, marked by a large yellow vertex in the Figure. Qubits are numbered in accordance with the text in Supplementary Note 2. Qubits 7, 8, 13 and 15 lie on lattice edges and qubit 14 lies on the vertex where the red blue and green vertices meet. All other labeled qubits lie on the external faces of the lattice.
Supplementary Figure 8 {#supplementary-figure-8 .unnumbered}
----------------------
Errors introduced during fault-tolerant error correction. (a) An error string that, given perfect measurement outcomes, generates two blue stabilizer defects. (b) Measurement errors appear as strings on the dual lattice picture, which are shown as multi-colored strings in the diagram. Gauge defects are identified where strings of measurement errors terminate. The original physical error is marked in grey. (c) During syndrome estimation, we incorrectly estimate the position of the measurement errors, and thus incorrectly identify the true position of the stabilizer defect. The correction operator we apply is represented by a blue string. (d) The correction operator we apply to correct the estimated stabilizer defect introduces a new error to the system. The net error that is introduced by the initial physical error and the inaccurate correction operator effectively acts like the discrepancy between the estimated position of the stabilizer defect from its true position. This discrepancy is marked by the blue error string in the diagram. \[Fig:CorrelationsDevelop\]
Supplementary Notes
===================
Supplementary Note 1 {#SCTN:GCCLattice .unnumbered}
--------------------
Here we elaborate on the dual lattice we use to construct the gauge color code simulated in the main text. The gauge color code dual lattice is a neatly stacked structure of many tetrahedra. Conveniently though, the lattice dual to the lattice geometry shown in Figure 1(a) in the main text is composed of odd and even cubic units of five tetrahedra, shown in white and grey, respectively, in Supplementary Figure 1(a). We also show the decomposition of an even unit cell in Supplementary Figure 1(b). An odd unit cell differs from an even unit cell only by a $\pi/2$ rotation about any of the canonical axes. Unit cubes on the boundaries of the lattice are modified simply by removing subsets of their tetrahedra.
To ease the explanation of the lattice construction, it is instructive to begin with a cubic block of fundamental cubes, as shown in Supplementary Figure 2. The block must have an odd linear dimension of unit cubes. Tetrahedra are removed from this block to find an appropriate four-sided structure, and finally some of the tetrahedra that have been removed are once again replaced to find suitable boundaries that can be correctly colored.
In the dual representation we require that the vertices of the lattice are four colorable, i.e. we can consistently color every vertex with one of four colors, red, green, yellow or blue, such that no two vertices of the lattice that share an edge can have the same color. We show a single tetrahedra whose vertices are correctly colored in Supplementary Figure 1(c). The vertices of the dual lattice in Supplementary Figure 2 are indeed four colorable.
The gauge color code requires four distinct boundaries where boundaries differ by the subset of colors of the vertices that lie on their surface. Specifically, a boundary of a given color contains no vertices of that color, i.e. a green boundary contains only red, blue and yellow vertices. We therefore look for a tetrahedral structure with four boundaries of four different colors. We design the correct structure by first removing large subsets of fundamental tetrahedra to find the global shape of the lattice. We find a four-sided tetrahedral structure by removing four corners of the cubic block of unit cubes, as shown in Supplementary Figure 3.
The lattice shown in Supplementary Figure 3 does not define the gauge color code. This is because we cannot four-color the lattice such that all four boundaries of the lattice are correctly three colored, as we have explained. However, by replacing some of the tetrahedra we have removed from the lattice, we recover suitable boundaries. To find three-colored boundaries it is convenient to color the faces of the fundamental tetrahedra. We assign to each face of the tetrahedra the color that is not given to any of its adjacent vertices, as shown in Supplementary Figure 1(c). With this face coloring, we require that the lattice has four distinct, uniformly colored boundaries, as shown in the main text in Figure 1(b).
We modify the tetrahedral lattice shown in Supplementary Figure 3 to find a suitable lattice. In Supplementary Figure 4 we color the external faces of the fundamental tetrahedra to show the additional tetrahedra we must replace on the surface of the sheered lattice in Supplementary Figure 3. In particular, we observe hexagonal ‘patches’ of faces which are inconsistently colored compared with the rest of the boundary. We outline some of these hexagonal patches in white. At the centre of each of these hexagons lies a single vertex of a color that is not suitable for the given boundary. To rectify this we replace all of the removed tetrahedra to the lattice that were originally touching the central vertex of each hexagonal patch. Upon doing so we recover the dual lattice shown in the main text. In Supplementary Figure 4 we show some of the ‘bulbs’ of sixteen tetrahedra we reintroduce to the lattice. Some of the bulbs have already been reattached in the Figure. For the convenience of the reader we show the lattice separated into layers in Supplementary Figure 5.
We finally remark that the lattice we consider is particularly convenient for simulation because the stabilizers lie on the vertices of a cubic lattice. In Supplementary Figure 6 we show a repeating unit of the lattice geometry on a cubic lattice. In the Figure stabilizers are represented by circular and diamond-shaped vertices of appropriate color as explained by the key.
Supplementary Note 2 {#SCTN:GCCLatticeCharacteristics .unnumbered}
--------------------
Here we analyse the lattice by counting the total number of qubits as a function of $L$. We also count the number of stabilizers, and the number of gauge operators. We also discuss the qubit support of the stabilizers, and the gauge operators. The quantities evaluated in this Note serve as good sanity checks for readers that are reproducing the gauge color code lattice.
### Qubits
The gauge color code is a complicated system where qubits are placed on tetrahedra, and on subsets of faces, edges and vertices of the dual lattice we described in Supplementary Note 1. A qubit is placed on each of the exterior vertices where three differently colored boundaries meet. We therefore have vertex qubits $$Q_v(L) = 4.$$
A qubit is also placed on exterior edges of the lattice where two differently colored boundaries meet. We have $$Q_e(L) = 6L,$$ qubits on exterior edges of the lattice.
The lattice has qubits placed on the exterior faces of the dual lattice. We find that there are $$Q_f(L) = 9L^2 - 5,$$ face qubits.
Finally, the lattice has a qubit on each of its tetrahedra. We find $$Q_t(L) = (5L^3 + 24L^2 - 2L - 24 )/ 3,$$ tetrahedron qubits.
A gauge color code following our construction therefore has $Q(L)= Q_v(L)+Q_e(L)+Q_f(L)+Q_t(L)$ qubits, which is explicitly $$Q(L) = (5L^3 + 51L^2 + 16L - 27)/3,$$ qubits.
### Stabilizers in the Dual Lattice
Each vertex of the lattice supports two stabilizers. We find the number of vertices $$v(L) = (L^3 + 12L^2 +5L - 6)/3.$$ Each stabilizer represented by a given vertex acts on every vertex, edge, face or tetrahedron qubit that contains the respective stabilizer vertex. To make this statement more rigorous, we can specify each object on the lattice that supports a qubit by a list of vertices, $v$, that are contained by the respective object. A vertex qubit is specified by a single vertex, $v = \{v \}$, an edge qubit is specified by a pair of vertices, $e = \{ v_1, v_2 \}$, each triangular face that supports a qubit is specified by three vertices $f = \{ v_1, v_2, v_3 \}$, and a tetrahedron is specified by its four vertices $t = \{ v_1, v_2, v_3, v_4 \}$.
Given that the different objects of the lattice that support qubits, known as simplices, can be uniformly denoted by a lists of vertices of varying length, we are free to group all vertex qubits, edge qubits, face qubits and tetrahedron qubits into the set of qubits, $\mathcal{Q}$, using this simplicial description. Written as simplices, vertex, edge, face and tetrahedron qubits only differ by the length of their list of vertices. Using this notation we can conveniently write down stabilizer operators $$S^X_v = \prod_{Q\ni v} X_Q,\quad S^Z_v = \prod_{Q\ni v} Z_Q,$$ where we take the product of all qubits $Q \in \mathcal{Q}$ that contain vertex $v$. We point out that we have used notation here that is inconsistent with the main text, as here we index stabilizers by vertices, $v$, whereas in the main text we chose the index $c$ to represent cells that support stabilizers. This reflects the change from the primal to dual lattice notation.
### Gauge Operators in the Dual Lattice
We finally find the number of face operators we use in the simulation in the main text, and explicitly describe the supports of the gauge operators on the dual lattice. Gauge operators are represented on edges, and on exterior vertices of the dual lattice. Instead of counting the number of gauge operators exactly, we count distinct supports of gauge operators. The number of supports are exactly the number of faces on the primal lattice. For each support, $s$, we have two gauge operators, $G^X_s$ and $G^Z_s$. On the dual lattice, gauge supports are uniquely represented by either an edge, $e$, or a pair $(v,\mathbf{c})$ that contains a vertex, $v$, and a color $\mathbf{c} \in \mathcal{C} \equiv \left\{ \mathbf{r}, \mathbf{g}, \mathbf{y}, \mathbf{b} \right\}$, as is defined in the main text.
To count the gauge supports, we first find the number of edges contained on the lattice, $e(L)$. We find this number using the Euler characteristic $$\chi_{\text{3D}} = v(L) - e(L) + f(L) - t(L), \label{EulerChar}$$ where $\chi_{\text{3D}} = 1$ for a ‘ball-shaped’ triangulation, such as that which describes the gauge color code, and where $v(L) $, $f(L)$ and $t(L)$ are the number of vertices, faces and tetrahedra of the lattice, respectively. We already have the number of tetrahedra $t(L) = Q_t(L)$, and $v(L)$ is already counted to find the number of stabilizers. We easily find the number of faces of the lattice $f(L)$ using the fact that each interior face lies on the surface of two tetrahedra, and each tetrahedra has four faces. Following this, up to the exterior faces, we can regard each tetrahedra as contributing half to the face count per face of each tetrahedra. We therefore obtain a contribution of $\sim 2Q_t(L)$ faces for the tetrahedra of the lattice. To account for exterior faces, we must add an additional half-unit per exterior face of the lattice to count the total number of faces, giving the number of faces $f(L) = 2Q_t(L) + Q_f(L)/2$. Explicitly, we have $$f(L) = (20L^3 + 123L^2 - 8L - 111 )/ 6.$$ Using $f(L)$ and Eqn. (\[EulerChar\]) we find the number of edges $$e(L) = ( 4L^3 + 33 L^2 + 2L - 27)/2.$$
Each edge of the lattice represents one gauge support that acts on all edge, face, and tetrahedron qubits that contain the respective edge. We use once again the simplicial notation to denote gauge supports represented by edges, $e$, such that $$G^X_{e } = \prod_{Q \ni \{ v_1, v_2 \}} X_Q,\quad G^Z_{e } = \prod_{Q \ni \{ v_1, v_2 \}} Z_Q,$$ where we take the product over all qubits $Q \in \mathcal{Q}$ that contain both vertices of edge $e = \{ v_1, v_2 \}$. We point out that no vertex qubit on the lattice can contain the two vertices of an edge, and therefore vertex qubits are not found in gauge operator supports associated to edges.
Gauge supports are also represented by vertices $v$ on the exterior of the lattice. Each exterior vertex represents either one, two or three gauge supports. For exterior vertices that represent multiple gauge operator supports, we specify the supports uniquely with a vertex, $v$, and a color $\mathbf{c}$.
Specifically, a gauge support of a given exterior vertex $v$ contains a [*subset*]{} of the vertex, edge or face qubits that contain $v$. The subset of lattice objects $Q \ni v$ are taken to be all of those that do not contain any vertices of one particular color, $\mathbf{c}$.
To make this statement rigorous using the simplex notation, we define the function $C:v \rightarrow \mathcal{C}$ that returns the color of the vertex as defined by the choice of the four-coloring of the lattice.
External vertex gauge operators are written $$G_{(v,\mathbf{c})}^X = \prod_{Q \ni v \backslash \mathbf{c}} X_{Q}, \quad G_{(v,\mathbf{c})}^Z = \prod_{Q\ni v \backslash \mathbf{c}} Z_{Q},$$ where we use a shorthand notation ‘$Q \ni v \backslash \mathbf{c}$’ to denote qubits $Q$ that contain vertex $v$, but do not contain any vertices of color $\mathbf{c}$, i.e. where $C(v') \not= \mathbf{c} $ for all vertices $v' \in Q$. We point out that all tetrahedra contain one vertex of each color, and as such, no tetrahedra appear in any gauge supports represented by exterior vertices.
In some cases there are exterior vertices with multiple nonempty subsets $Q \ni v \backslash \mathbf{c}$ for different choices of $\mathbf{c}$. In what follows we explicitly consider exterior vertices that contain one, two, and three gauge operator supports. We will see that exterior vertices in the middle of a boundary will contain only one gauge operator support, exterior vertices found where two different colored boundaries meet contain two distinct gauge operator supports, and vertices that lie where three different boundaries meet contain three different gauge operator supports.
Exterior vertices in the middle of a boundary, far away from any edge or vertex qubits, denote one gauge operator support. We explicitly consider the example of the gauge support associated to the blue vertex shown on the red boundary in Supplementary Figure 7. The gauge operator support associated to this vertex act on the qubits labeled 1, 2, 3 and 4. All qubits that contain the exterior vertex of interest contain a yellow, a green, and a blue vertex, but no red vertex.
Vertices that lie at a point where two differently colored boundaries meet represent two distinct gauge operator supports. We consider the green vertex in Supplementary Figure 7 that lies where the blue boundary and the red boundary meet. One of the supports acts on the qubits labeled 3, 4, 5, 6, 7 and 8. Common to all of these qubits is that none of their faces or edges contain a red vertex. The other support acts on qubits 7, 8, 9, 10, 11 and 12. Different from the first support we discussed that is associated to the green vertex, none of the faces or edges involved in this support contain a blue vertex.
We finally consider the support of the gauge operators associated to the vertices that lie where three boundaries meet, such as at the large yellow vertex shown in Supplementary Figure 7 that contains the qubit indexed 14. This vertex denotes three supports, two of which are shown in the diagram. One support acts on qubits 6, 7, 14 and 15, the qubits that contain the yellow vertex of interest, but do not contain a red vertex. The other support acts on the qubits numbered 7, 12, 13 and 14, the lattice objects that contain the large yellow vertex, but do not contain any blue vertices. The third support associated to the large yellow vertex acts on the green face qubit that contains the large yellow vertex which cannot be seen in the diagram, together with qubits 13, 14 and 15.
We can now count the number of gauge supports we must measure to realize fault-tolerant error correction using the gauge color code. We count the number of exterior vertices $v^E(L)$ with the expression $$v^E(L) = (9L^2 - 1)/2 .$$ Then, using that the $6(L-1)$ vertices that lie where two boundaries meet represent two gauge operator supports, and four vertices that lie where three boundaries meet represent three gauge operator supports, we find the number of gauge operator supports $G(L) = e(L) + v^E(L) + 6(L-1) + 8$, giving the number of gauge operator supports $$G(L) = 2L^3 + 21L^2 + 7L - 12.$$ Twice this quantity gives the number of face operators we use to perform fault-tolerant error correction with the gauge color code to identify both Pauli-X and Pauli-Z type errors.
Supplementary Note 3 {#SCTN:CorrelatedErrorsDevelop .unnumbered}
--------------------
In the main text we identify the threshold error rate as a function of the number of rounds of error correction that are performed, $N$, to show that the threshold error rate is robust in the limit $N\rightarrow \infty$. It is important to look at the single-shot error-correction scheme after repeated applications because, when one considers noisy measurements, the error-correction protocol will leave some residual noise on the code. In general, this residual noise is not easily characterized. In fact, the noise introduced to the system is a complex function of the physical error rate, the measurement error rate, and the choice of error-correction protocol. Given that the nature of the residual noise is not well understood, it is not clear that the error-correction protocol will be able to successfully deal with the residual noise after many cycles of error correction.
In Supplementary Figure 8 we give an example of a mechanism that leads to the development of a correlated error. We show a physical error in Supplementary Figure 8(a). The error is represented by a string. Given perfect measurements, we expect to observe two stabilizer defects at the left and right ends of the error string.
For this example, some measurement errors occur in the vicinity of the right stabilizer defect when we attempted to learn the positions of stabilizer defects. The measurement errors, where face operators return incorrect measurement outcomes, are represented by multi-colored strings in Supplementary Figure 8(b). Gauge defects lie at the points where the strings of measurement errors terminate. Given these measurement errors occur, we cannot be sure of the true location of the right stabilizer defect. Instead, we can make use of the positions of the three gauge defects, and knowledge of the noise model, to attempt to determine the location of the stabilizer defect.
Our ability to predict the positions of stabilizer defects depends on our choice of syndrome estimation algorithm. In Supplementary Figure 8(c) we show where the syndrome estimation algorithm mistakenly predicts measurement errors. The measurement errors that have been predicted are represented by strings that terminate at the gauge defects, and branch at the estimated stabilizer defect. The incorrect estimation made by the algorithm leads us to believe that the stabilizer defect is not in its true location, but is displaced to some estimated location.
Next, we apply a correction operator to attempt to repair the initial physical error drawn in Supplementary Figure 8(a). However, using the estimated position of the right stabilizer defect, we apply a correction operator that connects the left stabilizer defect to the estimated right stabilizer defect, as shown by a blue string in Supplementary Figure 8(c), thus introducing additional errors to the code. The initial error, and the applied correction operator is shown in grey in Supplementary Figure 8(d).
Due to the topological nature of the string errors in the gauge color code, the net effect of the initial error and the correction operator equates to the discrepancy between the true position of the right stabilizer defect, and the position of the estimated stabilizer defect. The effective error is shown by the blue error string in Supplementary Figure 8(d). This is easily checked as we can continuously deform the grey string onto the blue string. In other words, in the gauge color code the grey error string is equivalent to the blue error string up to multiplication by gauge operators.
In general it is not clear how able a syndrome estimation algorithm is for correctly predicting the positions of stabilizer defects. A bad syndrome estimation algorithm might displace stabilizer defects over long distances compared to their true positions, in which case large correlated errors that we cannot correct may develop. Moreover, it is not clear that the character of the residual noise remains constant over many error-correction cycles. Indeed, one should be concerned that an error correction protocol might cause correlations to develop over repeated use of the error-correction procedure while information is stored. It is therefore important to study a single-shot error-correction protocol over many cycles of error correction to interrogate its performance, and to check that the noise incident to a code achieves a steady state in the long-time limit. In doing so, we are able to establish that a code has a finite threshold after an arbitrarily long time.
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abstract: 'The paper deals with state estimation of a spatially distributed system given noisy measurements from pointwise-in-time-and-space threshold sensors spread over the spatial domain of interest. A *Maximum A posteriori Probability* (MAP) approach is undertaken and a *Moving Horizon* (MH) approximation of the MAP cost-function is adopted. It is proved that, under system linearity and log-concavity of the noise probability density functions, the proposed MH-MAP state estimator amounts to the solution, at each sampling interval, of a convex optimization problem. Moreover, a suitable centralized solution for large-scale systems is proposed with a substantial decrease of the computational complexity. The latter algorithm is shown to be feasible for the state estimation of spatially-dependent dynamic fields described by *Partial Differential Equations* (PDE) via the use of the *Finite Element* (FE) spatial discretization method. A simulation case-study concerning estimation of a diffusion field is presented in order to demonstrate the effectiveness of the proposed approach. Quite remarkably, the numerical tests exhibit a *noise-assisted* behavior of the proposed approach in that the estimation accuracy results optimal in the presence of measurement noise with non-null variance.'
author:
- '\'
title: MAP moving horizon estimation for threshold measurements with application to field monitoring
---
*Keywords:* State estimation; moving-horizon estimation; threshold measurements; dynamic field estimation; spatially distributed systems.
Introduction
============
Threshold sensors, which provide a binary output just indicating whether the noisy measurement of the sensed variable falls below or above a given threshold, are widely used for monitoring and control [@Wang1]-[@SelvaratnamIEEE2017]. The motivation is that by a multitude of low-cost and low-resolution sensing devices it is possible to attain the same estimation accuracy that a fewer (even a single one) high-cost and high-resolution ones could provide, but with significant practical advantages in terms of ease of sensor deployment and minimization of communication requirements. Since a threshold measurement just conveys the least possible amount (i.e., a single bit) of information, while implying communication bandwidth savings and consequent improved energy efficiency, it becomes of paramount importance to fully exploit the little available information by means of smart estimation algorithms. Over the last two decades, interesting work has been devoted to system identification, [@Wang1; @Wang2] parameter [@Ristic]-[@Ribeiro2] and state estimation, [@Irr-sampling]-[@Capponi] with a specific focus on source localisation, [@RisticAE2016; @SelvaratnamIEEE2017] using threshold measurements according to either a deterministic [@Wang1]-[@Bai] or a probabilistic [@Ristic]-[@SelvaratnamIEEE2017] approach.
In a deterministic setting, [@Irr-sampling; @Gherardini2] the information provided by a threshold sensor is mainly associated to the switching instants, corresponding to discontinuities of the threshold signal. In fact, a switch at the current discrete time implies, by continuity, the existence of a continuous time instant, within the latest sampling interval, such that the continuous output crosses the threshold at that time. As shown in previous work, [@Gherardini2] additional information can also be exploited in the non-switching sampling instants by penalizing values of the estimated variable such that the corresponding predicted measurement is on the opposite side, with respect to a threshold sensor reading, far away from the threshold. Nevertheless, it is clear that there is no or very little information available for estimation purposes whenever no or very few threshold sensor switchings occur. Hence, a possible way to achieve high estimation accuracy is to have many threshold sensors sensing the same variable with different thresholds as this would clearly increase the number of switchings, actually emulating, when the number of sensors grows to infinity, a single continuous-valued (analog) measurement.
Conversely, following a probabilistic approach, each threshold sensor can be characterized in terms of the probability that its binary output takes the two possible outcomes, in relation to the dynamical evolution of the state of the monitored system. In this way, also a threshold sensor always provides an informative contribution. Furthermore, it is worth noting how, in presence of a sufficiently large number of threshold sensors, such a stochastic approach can also be more advantageous in case the outcomes from the sensors are noisy. Indeed, as it will be shown via simulation experiments, there is a non-null value of the measurement noise variance, for which optimal state estimation accuracy from threshold measurements is obtained. This, referred to hereafter as *noise-assisted* paradigm, [@ACC_binary] is in sharp contrast with what happens in the case of linear sensors, and depends on the strong nonlinearity of threshold sensors.
From the above discussed *noise-assisted* paradigm, this paper develops a novel approach to recursive estimation of the state of a discrete-time dynamical system given threshold measurements. The proposed approach relies on a *Moving-Horizon* (MH) approximation [@Morari]-[@SchneiderIEEE2017] of the *Maximum A-posteriori Probability* (MAP) estimation [@Delgado] and extends previous work [@Ristic; @Wong] concerning parameter estimation to recursive state estimation. A further contribution is to show that, for a linear system and log-concave probability distributions, the optimization problem arising from the MH-MAP formulation turns out to be convex and, hence, practically feasible for real-time implementation.
The present paper significantly expands our preliminary work [@ACC_binary] by considering a more general class of probability distributions, by providing a mathematical proof of the above stated convexity result and, most importantly, by proposing a novel efficient MH-MAP filter for field estimation of large-scale systems. The paper is structured as follows. Section 2 formulates *Maximum A posteriori Probability* (MAP) state estimation with threshold measurements, relying on the probabilistic description of the amount of information provided by each threshold sensor. Section 3 introduces its *moving-horizon* (MH) approximation, referred to as MH-MAP estimator, and analyzes the properties of the resulting optimization problem. Section 4 presents the finite element approximation for the estimation of diffusion fields from threshold pointwise-in-space-and-time field measurements. Due to the large-scale of the dynamical system resulting from spatial discretization of the field dynamics, a fast MH-MAP filter for field estimation of large-scale systems is proposed in Section 5. Section 6 shows simulation results relative to the dynamic field estimation case-study, while Section 7 ends the paper with some discussion and perspectives for future work.
Maximum a posteriori state estimation with threshold sensors
============================================================
Let us consider the problem of recursively estimating the state of the discrete-time nonlinear dynamical system $$\label{1}
\begin{array}{rcl}
x_{k+1} &=& f(x_{k},u_{k})+ w_{k} \\
z_{k}^{(i)} &=& h^{(i)}(x_{k})+ v_{k}^{(i)},\hspace{3mm}i=1,\ldots,l
\end{array}$$ from a set of measurements provided by threshold sensors $$\label{2}
\begin{array}{rclcl}
y_{k}^{(i)} & = & g^{(i)}(z_{k}^{(i)}) & = & \left\{
\begin{array}{ll} 1, & \mbox{if } z_{k}^{(i)} \geq \tau^{(i)} \\
0, & \mbox{if } z_{k}^{(i)} < \tau^{(i)}
\end{array}
\right.
\end{array}$$ where $x_k \in \mathbb{R}^n$ is the state to be estimated, $u_k \in \mathbb{R}^m$ is a known input, and $\tau^{(i)}$ is the threshold of the $i-$th sensor. The vector $w_{k}\in \mathbb{R}^n$ is an additive disturbance affecting the system dynamics while $v^{(i)}_k$ is the measurement noise of sensor $i$. Notice from (\[1\])-(\[2\]) that sensor $i$ produces a threshold measurement $y^{(i)}_{k}\in\{0,1\}$ depending on whether the noisy system output $z^{(i)}_{k}$ is below or above the threshold $\tau^{(i)}$. We define, for the sake of simplicity, $$z_k = {\rm col} \left ( z_k^{(1)} , \ldots, z_k^{(l)} \right ) \, , \quad y_k = {\rm col} \left ( y_k^{(1)} , \ldots, y_k^{(l)} \right ) \, , \quad
v_k = {\rm col} \left ( v_k^{(1)} , \ldots, v_k^{(l)} \right ) \, .$$
Let $\mathcal{N}(\mu,\Sigma)$ denote the normal distribution with mean $\mu$ and variance $\Sigma$. Then the statistical behavior of the system is modelled according to the following assumption.
1. The initial state $x_{0}$ and disturbance $w_{k}$ are normally-distributed random vectors $$x_{0}\sim\mathcal{N}(\overline{x}_{0},P^{-1}),\hspace{2mm}w_{k}\sim\mathcal{N}(0,G^{-1})
\label{prob}$$ where $\mathbb{E}[w_{j}w_{k}']=0$ if $j\neq k$ and $\mathbb{E}[w_{j}x_{0}']=0$ for any $j$. Further, the measurement noises $v_k^{i}$ of the sensors are mutually independent as well as independent from the initial state and disturbance.
According to the available probabilistic description, hereafter the problem of state estimation from threshold measurements is recast into a Bayesian framework exploiting a MAP estimation approach. As discussed in the introduction, each threshold measurement $y^{(i)}_k$ provides intrinsically relevant probabilistic information on the state $x_k$. Such information can be effectively exploited by introducing the likelihood functions $p(y^{(i)}_{k}|x_{k})$ of the $i-$th threshold sensor. To this end, let us observe that each threshold measurement $y^{(i)}_{k}$ is a Bernoulli random variable such that, for any threshold sensor $i$ and any time instant $k$, the likelihood function $p(y^{(i)}_{k}|x_{k})$ is given by [@Ristic; @Ribeiro1] $$p(y^{(i)}_{k}|x_{k})~=~p(y^{(i)}_{k}=1|x_{k})^{y^{(i)}_{k}} ~p(y^{(i)}_{k}=0|x_{k})^{1-y^{(i)}_{k}}$$ where $$\label{8}
p(y^{(i)}_{k}=0|x_{k}) = F^{(i)}(\tau^{(i)}-h^{(i)}(x_{k}))$$ and $p(y^{(i)}_{k}=1 | x_{k})=1-p(y^{(i)}_{k}=0|x_{k}) \triangleq 1 - F^{(i)}(\tau^{(i)}-h^{(i)}(x_{k}))$. In particular, $F^{(i)}(\tau^{(i)}-h^{(i)}(x_{k})) = prob( v_k^{(i)} \leq \tau^{(i)} - h^{(i)}(x_k))$ is the *Cumulative Distribution Function* (CDF) of the random variable $v_k^{(i)}$ evaluated at $\tau^{(i)}-h^{(i)}(x_{k})$. For example, when the measurement noise is normally distributed $v^{(i)}_{k} \sim \mathcal{N}(0,r^{(i)})$, the conditional probability $p(y^{(i)}_{k}=1|x_{k}) = 1-F^{(i)}(\tau^{(i)}-h^{(i)}(x_{k})$ can be written in terms of a $Q$-function, describing the tail probability of a standard normal probability distribution [@Wim2006]$$p(y^{(i)}_{k}=1 | x_{k}) = \frac{1}{\sqrt{2\pi r^{(i)}}}\int_{\tau^{(i)}-h^{(i)}(x)}^{\infty}\exp\left(-\frac{u^{2}}{2r^{(i)}}\right) du
= Q \left(\frac{\tau^{(i)}-h^{(i)}(x)}{\sqrt{r^{(i)}}}\right) \, . \label{eq7}$$
Let us now denote by $Y_{k} \triangleq {\rm col} (y_0,\ldots,y_{k} )$ the vector of all threshold measurements collected up to time $k$ and by $X_{k}\triangleq {\rm col} (x_{0},\ldots,x_{k} )$ the vector of the state trajectory. Further, let us denote by $\hat{X}_{k|k} \triangleq {\rm col} ( \hat{x}_{0|k},\ldots,\hat{x}_{k|k} )$ the estimate of $X_{k}$ at time $k$. Then, at each time instant $k$, given the a posteriori probability $p(X_{k}|Y_{k})$, the estimate of the state trajectory can be obtained by solving the following MAP estimation problem: $$\hat{X}_{k|k} = \text{arg}\max_{X_{k}}p(X_{k}|Y_{k})
=\text{arg} \min_{X_{k}} \left\{ - \ln p(X_{k}|Y_{k}) \right\} \label{12}.$$ From the Bayes rule $$p(X_{k}|Y_{k}) ~\propto ~p(Y_{k}|X_{k})~p(X_{k}),$$ where $p(Y_{k}|X_{k})$ is the likelihood function of the threshold measurement vector $Y_{k}$, and $$p(X_{k}) = \prod_{j=0}^{k-1}p(x_{k-j}|x_{k-j-1},\ldots,x_{0})~p(x_{0}) = \prod_{j=0}^{k-1}p(x_{k-j}|x_{k-j-1})~p(x_{0})$$ thanks to the Markov property of the system state. Furthermore, in view of assumption A1, we have $$\begin{aligned}
p(x_0) &\propto & \exp\left(-\frac{1}{2}\|x_{0}-\overline{x}_0 \|^{2}_{P}\right)\label{15} \\
p(x_{k+1}|x_{k}) &\propto& \exp\left(-\frac{1}{2}\|x_{k+1}-f(x_{k},u_{k})\|^{2}_{G}\right)\label{16},\end{aligned}$$ so that $$p(X_{k}) = \exp\left(-\frac{1}{2}\left[\|x_{0}-\overline{x}_{0}\|^{2}_{P}+\sum_{j=0}^{k}\|x_{j+1}-f(x_{j},u_{j})\|^{2}_{G}\right]\right).$$ In view of the mutual independence of the measurement noises, the likelihood function $p(Y_{k}|X_{k})$ can be written as $$p(Y_{k}|X_{k})=\displaystyle{\prod_{j=0}^{k}}p(y_{j}|x_{j})=\prod_{j=0}^{k}~\prod_{i=1}^{l}p(y_{j}^{(i)}|x_{j})
=\displaystyle{\prod_{j=0}^{k}~\prod_{i=1}^{l}} F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j}))^{1-y_{j}^{(i)}} ~ \left[ 1 - F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j})) \right]^{y_{j}^{(i)}} .$$
In conclusion, the log-likelihood function, natural logarithm of the likelihood function, is given by $$\ln p(Y_{k}|X_{k})=\displaystyle{\sum_{j=0}^{k}~\sum_{i=1}^{l}}\left\{ (1-y_{j}^{(i)}) \, \ln F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j}))
+ y_{j}^{(i)} \, \ln \left[ 1 - F^{(i)} (\tau^{(i)}-h^{(i)}(x_{j})) \right] \right\},$$ and the cost function $J_k(X_k) =-\ln p(Y_{k}|X_{k})-\ln p(X_{k})$ to be minimized in the MAP estimation problem (\[12\]), up to the constant term $p(Y_k)$, turns out to be $$\begin{aligned}
\label{18}
J_{k} (X_k) &=& \|x_{0}-\overline{x}_{0}\|^{2}_{P}+\displaystyle{\sum_{j=0}^{k}}\|x_{j+1}-f(x_{j},u_{j})\|^{2}_{G}\nonumber \\
&-&\displaystyle{\sum_{j=0}^{k}~\sum_{i=1}^{l}}\left\{ (1-y_{j}^{(i)}) \ln F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j}))+y_{j}^{(i)} \ln \left[ 1 - F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j})) \right] \right\} \, ,\end{aligned}$$ which is defined for all vectors $X_k$ such that the arguments of the logarithms are different from zero. Unfortunately, a closed-form expression for the global minimum of (\[18\]) does not exist and, hence, the optimal MAP estimate $\hat X_{k|k}$ has to be determined by resorting to some numerical optimization routine. In this respect, the main drawback is that the number of optimization variables grows linearly with time, since the vector $X_k$ has size $(k+1) \, n$. As a consequence, as time $k$ grows the solution of the full information MAP state estimation problem (\[12\]) becomes eventually unfeasible, and some approximation has to be introduced.
Moving-horizon approximation {#section_MH}
============================
In this section, we propose an approximate method, based on the MHE approach, [@Morari]-[@SchneiderIEEE2017] to solve the MAP state estimation problem. To this end, let us introduce the sliding window $\mathfrak W_k = \{k-N, k-N+1, \ldots, k\}$, so that the goal of the estimation problem becomes that to find an estimate of the partial state trajectory $X_{k-N:k} \triangleq {\rm col} ( x_{k-N},\ldots,x_{k} )$ by using the information available in $\mathfrak W_k$. Therefore, in place of the full information cost $J_k (X_k)$, at each time instant $k$ the minimization of the following *moving-horizon cost* is considered: $$\begin{aligned}
\label{MH}
J_{k}^{\rm MH} (X_{k-N:k}) &=& \Gamma_{k-N} (x_{k-N}) + \displaystyle{\sum_{j=k-N}^{k}}\|x_{j+1}-f(x_{j},u_{j})\|^{2}_{G}\nonumber \\
&-&\displaystyle{\sum_{j=k-N}^{k}~\sum_{i=1}^{l}}\left\{ (1-y_{j}^{(i)}) \ln F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j}))+ y_{j}^{(i)} \ln \left[ 1 -F^{(i)}(\tau^{(i)}-h^{(i)}(x_{j})) \right] \right\},\end{aligned}$$ where the non-negative initial penalty function $\Gamma_{k-N} (x_{k-N}) $, known in the MHE literature as *arrival cost*, [@Rao2; @CDC] is introduced so as to summarize the past data $y_0, \ldots, y_{k-N-1}$ not explicitly accounted for in the objective function. The form of the arrival cost plays an important role in the behavior and performance of the overall estimation scheme. While in principle $\Gamma_{k-N} (x_{k-N})$ could be chosen so that minimization of (\[MH\]) yields the same estimate that would be obtained by minimizing (\[18\]), an algebraic expression for such a true arrival cost seldom exists, even when the sensors provide continuous (non-threshold) measurements. [@Rao2] Hence, some approximation must be used. With this respect, a common choice, [@NLMHE; @CDC] also followed in the present work, consists of assigning to the arrival cost a fixed structure penalizing the distance of the state $x_{k-N}$ at the beginning of the sliding window from some prediction $\overline{x}_{k-N} $ computed at the previous time instant, thus making the estimation scheme recursive. A natural choice is then a quadratic arrival cost of the form $$\label{eq:arrival}
\Gamma_{k-N} (x_{k-N}) = \|x_{k-N}-\overline{x}_{k-N}\|^{2}_{\Psi} \, ,$$ which, from the Bayesian point of view, corresponds to approximating the PDF of the state $x_{k-N}$ conditioned to all the measurements collected up to time $k-1$ with a Gaussian having mean $\overline{x}_{k-N}$ and covariance $\Psi^{-1}$. As for the choice of the weight matrix $\Psi$, in the case of continuous measurements it has been shown [@NLMHE; @CDC] that stability of the estimation error dynamics can be ensured provided that $\Psi$ is not too large (so as to avoid an overconfidence on the available estimates). Recently, [@Gherardini2] similar results have been proven to hold also in the case of threshold sensors in a deterministic context. In practice, $\Psi$ can be seen as a design parameter which has to be tuned by pursuing a suitable trade-off between such stability considerations and the necessity of not neglecting the already available information (since in the limit for $\Psi$ going to zero the approach becomes a finite memory one).
It is worth noting that when the PDFs of the measurement noises are always strictly positive, like in the case of the normal distribution, then the cost function (\[MH\]) is well defined for any $X_{k-N:k} \in \mathbb R^{n (N+1)}$ because the arguments of the logarithms are always strictly positive. Otherwise, in the minimization of the cost function (\[MH\]) only the vectors $X_{k-N:k}$ for which the arguments of the logarithms are strictly positive have to be taken into account. Hereafter, we denote the set of such vectors as $\mathbb X_k \subseteq \mathbb R^{n (N+1)}$. Summing up, at any time instant $k = N,N+1,\ldots$, the following problem has to be solved.\
\
**Problem $E_{k}$:** Given the prediction $\overline{x}_{k-N}$, the input sequence $\{ u_{k-N}, \ldots, u_{k-1} \}$ and the measurement sequences $\{ y^{(i)}_{k-N}, \ldots, y^{(i)}_k ; \, i = 1, \ldots, l \}$, find the optimal estimates $\hat X_{k-N:k} = {\rm col} ( \hat{x}_{k-N|k},\ldots,\hat{x}_{k|k} )$ that minimize the cost function (\[MH\]) with arrival cost (\[eq:arrival\]) under the constraint $X_{k-N:k} \in \mathbb X_k$.\
In order to propagate the estimation procedure from Problem $E_{k-1}$ to Problem $E_{k}$, the prediction $\overline{x}_{k-N}$ is set equal to the value of the estimate of $x_{k-N}$ made at time instant $k-1$, i.e., $\overline{x}_{k-N} = \hat{x}_{k-N|k-1}$. Clearly, the recursion is initialized with the a priori expected value $\overline{x}_0$ of the initial state vector.
In general, solving Problem $E_{k}$ entails the solution of a non-trivial optimization problem. However, when both equations (\[1\]) and (\[2\]) are linear, the resulting optimization problem turns out to be convex for a large class of measurement noise distributions so that standard optimization routines can be used in order to find the global minimum. To see this, let us consider the following assumptions.
1. The functions $f(\cdot)$ and $h^{(i)}(\cdot)$, $i=1,\ldots,l$, are linear, i.e., $f(x_k,u_k) = A x_k + B u_k$ and $ h^{(i)}(x_k) = C^{(i)} x_k$, $i=1,\ldots,l$, where $A$, $B$, $C^{(i)}$ are constant matrices of suitable dimensions.
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1. The PDFs of the measurement noises are log-concave.
Concerning assumption A3, it is worth noting that the class of probability distributions having a log-concave PDF is quite general and includes, among others, normal, exponential, uniform, logistic, and Laplace distributions. Then, the following result, whose proof is in the Appendix, can be stated.
If assumption A2 holds, then the set $\mathbb X_k$ is an open convex polyhedron for any $k$. If in addition also assumptions A1 and A3 hold, then the cost function (\[MH\]) with arrival cost (\[eq:arrival\]) is convex. $\square$
Notice that the convexity of the cost function (\[MH\]) is guaranteed also when assumption A1 is replaced by the milder requirement that the PDF of the process disturbance is log-concave.
Finite element approximation and dynamic field estimation
=========================================================
In this section, we consider the problem of reconstructing a two-dimensional dynamic field, sampled with a network of threshold sensors arbitrarily deployed over the spatial domain of interest $\Omega$. The process is governed by the following parabolic PDE: $$\dfrac{\partial c}{\partial t} - \lambda \nabla^2 c ~=~ 0 \,\,\, \mbox{in} \,\, \Omega
\label{PDE}$$ which models various physical phenomena, such as the spread of a pollutant in a fluid. In (\[PDE\]): $c(\xi,\eta,t)$ represents the space-time dependent substance concentration; $\lambda$ is the constant diffusivity of the medium; $\nabla^2 = {\partial^2 } / {\partial \xi^2} + {\partial^2 } / {\partial \eta^2}$ is the Laplace operator; $(\xi,\eta) \in \Omega$ and $t \in \mathbb{R}$ denote the planar Cartesian coordinates and, respectively, the continuous time instant. Furthermore, let us assume mixed boundary conditions, i.e. (i) a non-homogeneous Dirichlet condition $$c = \psi \,\, \mbox{on} \,\, \partial \Omega_D,
\label{Dbc}$$ which specifies a constant-in-time value of concentration on the boundary $\partial \Omega_D$, and (ii) a homogeneous Neumann condition on $\partial \Omega_N = \partial \Omega \setminus \partial \Omega_D$, assumed impermeable to the contaminant, so that $$\frac{\partial c}{\partial \upsilon} = 0 \,\, \mbox{on} \,\, \partial \Omega_N,
\label{Nbc}$$ where $\upsilon$ is the outward pointing unit normal vector of $\partial \Omega_N$.
The objective is to estimate the dynamic field of interest $c(\xi,\eta,t)$, given pointwise-in-time-and-space threshold measurements of the field itself. The PDE system (\[PDE\])-(\[Nbc\]) is spatially discretized with a mesh of finite elements over $\Omega$ via the Finite Element (FE) approximation described in previous work on dynamic field estimation. [@source; @ECC15; @TAC17] Specifically, the domain $\Omega$ is subdivided into a suitable set of non overlapping regions, or elements, and a suitable set of basis functions $\phi_{j} (\xi , \eta)$, $j=1,\ldots,n_\phi$, is defined on such elements. The choices of the basis functions $\phi_{j}$ and of the elements are key points of the FE method. In the specific case under investigation, the elements are triangles in 2D and define a FE mesh with vertices $({\xi}_j, \eta_j) \in \Omega, j=1,\ldots,n_\phi $. Then each basis function $\phi_{j}$ is a piece-wise affine function which vanishes outside the FEs around $({\xi}_j, \eta_j)$ and such that $\phi_{j}({\xi}_i, \eta_i)=\delta_{ij}$, $\delta_{ij}$ denoting the Kronecker delta. In order to account for the mixed boundary conditions, the basis functions are supposed to be ordered so that the first $n$ correspond to vertices of the mesh which lie either in the interior of $\Omega$ or on $\partial \Omega_N$, while the last $n_\phi-n$ correspond to the vertices lying on $\partial \Omega_D$. Accordingly, the unknown function $c(\xi, \eta,t)$ is approximated as $$c(\xi,\eta,t) \approx \sum_{j=1}^{n} \phi_{j}(\xi, \eta) \, c_j(t) + \sum_{j=n+1}^{n_\phi} \phi_{j}(\xi, \eta) \, \psi_j
\label{EXPA}$$ where $c_j(t)$ is the unknown expansion coefficient of the function $c(\xi, \eta,t)$ with respect to time $t$ and basis function $\phi_j(\xi,\eta)$, and $\psi_j$ is the known expansion coefficient of the function $\psi(\xi, \eta)$ only relative to the basis function $\phi_j(\xi,\eta)$. Notice that the second summation in (\[EXPA\]) is needed so as to impose the non-homogeneous Dirichlet condition (\[Dbc\]) on the boundary $\partial \Omega_D$. The PDE (\[PDE\]) can be recast into the following integral form: $$\int_\Omega \frac{\partial c}{\partial t} \varphi \, d\xi d\eta \, - \,
\lambda \int_\Omega ~\nabla^2 c~ \varphi \, d\xi d\eta =0$$ where $\varphi(\xi,\eta)$ is a generic space-dependent weight function. By applying the Green’s identity, one obtains: $$\label{eq:Green}
\int_\Omega \frac{\partial c}{\partial t} \varphi \, d\xi d\eta + \lambda
\int_\Omega \nabla^T c ~\nabla \varphi \, d\xi d\eta
- \lambda
\int_{\partial \Omega} \frac{\partial c}{\partial \upsilon} \varphi \,
d\xi d\eta
= 0 \, .$$ By choosing the test function $\varphi$ belonging to the selected basis functions and exploiting the approximation (\[EXPA\]), the Galerkin weighted residual method is then applied and the following equation is obtained $$\label{eq:Galerkin}
\sum_{i=1}^{n} \int_\Omega \phi_i \phi_j \, d\xi d\eta \, \dot{c}_i(t) + \lambda \sum_{i=1}^{n} \int_\Omega \nabla^T \phi_i ~\nabla \phi_j \, d\xi d\eta \, c_i (t) + \lambda \sum_{i=n+1}^{n_\phi} \int_\Omega \nabla^T \phi_i ~\nabla \phi_j \, d\xi d\eta \, \psi_i = 0$$ for $j = 1, \ldots, n$. In the latter equation the boundary integral of equation (\[eq:Green\]) is null thanks to the homogeneous Neumann condition (\[Nbc\]) on $\partial \Omega_N$ and to the fact that, by construction, the basis functions $\phi_j$, $j = 1, \ldots, n$, vanish on $\partial \Omega_D$. The interested reader is referred to the standard literature [@Brenner96] for further details on the FEM theory, in particular on how to convert the case of inhomogeneous boundary conditions to the homogeneous one.
By defining the state vector $x \triangleq {\rm col} (c_1 , \ldots , c_n) $ and the vector of boundary conditions $\gamma \triangleq {\rm col} (\psi_{n+1}, \ldots, \psi_{n_\phi})$, equation (\[eq:Galerkin\]) can be written in the more compact form $$M \dot x (t) + S x(t) + S_D \gamma = 0$$ where $S$ is the so-called stiffness matrix, $M$ is the mass matrix, and $S_D$ captures the physical interconnections among the vertices affected by boundary condition (\[Dbc\]) and the remaining nodes of the mesh. By applying, for example, the implicit Euler method, the latter equation can be discretized in time, thus obtaining the linear discrete-time model $$x_{k+1} = A \, x_k + B \, u + w_k\,,
\label{dt-sys}$$ where $$\begin{aligned}
A &=& \left[ I + T_s ~M^{-1}S\right]^{-1} \\
B &=& \left[ I + T_s ~M^{-1}S\right]^{-1}M^{-1} T_s \\
u &=& - S_D ~\gamma ~.
\label{eq:def}\end{aligned}$$ $T_s$ is the sampling interval, and $w_k$ is the process disturbance taking into account also the space-time discretization errors. Notice that the linear system (\[dt-sys\]) has dimension $n$ equal to the number of vertices of the mesh not lying on $\partial \Omega_D$. The system (\[dt-sys\]) is assumed to be monitored by a network of $l$ threshold sensors. Each sensor, before threshold quantization is applied, directly measures the pointwise-in-time-and-space concentration of the contaminant in a point $(\xi_i,\eta_i)$ of the spatial domain $\Omega$. By exploiting (\[EXPA\]), such a concentration can be written as a linear combination of the concentrations on the grid points in that $$c(\xi_i,\eta_i, kT_s) \approx C^{(i)} x_k + D^{(i)} \gamma \,,$$ where $$\begin{aligned}
C^{(i)} &=& \left [ \phi_1(\xi_i,\eta_i) \; \cdots \; \phi_n(\xi_i,\eta_i) \right ] \\
D^{(i)} &=& \left [ \phi_{n+1}(\xi_i,\eta_i) \; \cdots \; \phi_{n_\phi}(\xi_i,\eta_i) \right ].\end{aligned}$$ Hence the resulting output function takes the form $$z_{k}^{(i)} = c(\xi_i,\eta_i,kT_s) - D^{(i)} \gamma + v_{k}^{i} = C^{(i)} x_{k} + v_{k}^{i},\hspace{3mm}i=1,\ldots,l \label{z}$$ where the constant $D^{(i)} \gamma$ is subsumed into the threshold $\tau^{(i)}$, so that assumption A2 is fulfilled.
Fast MH-MAP filter for field estimation of large-scale systems
==============================================================
In order to achieve a good approximation of the original continuous field, a large number of basis functions need to be used in the expansion (\[EXPA\]). Hence, in general the FEM-based space discretization gives rise to a large-scale system possibly characterized by thousands of state variables, equal to the number of the vertices of the mesh not lying on the boundary $\partial\Omega$. This means that a direct application of the MH-MAP filter of Section 4 to system (\[dt-sys\]) involves the solution, at each time instant, of a large-scale (albeit convex) optimization problem. Although today commercial optimization software can solve general convex programs of some thousands variables, the problem becomes intractable from a computational point of view when the number of variables (that is, the number of vertices of the FE grid) is too large. Further, even when a solution to the large-scale optimization problem can be found, the time required for finding it may not be compatible with real-time operations (recall that the MH-MAP filter requires that each optimization terminates within one sampling interval).
In this section, we propose a more computationally efficient and faster version of the MH-MAP filter for the real-time estimation of a dynamic field that is based on the idea of decomposing the original large-scale problem into simpler, small-scale, subproblems by means of a two-stage estimation procedure. The proposed method allows one to efficiently solve the problem of estimating the state (ideally infinite-dimensional) of a spatially-distributed dynamical system just by using sensors with minimal information content, such as threshold sensors. The fast version of the aforementioned MH-MAP filter, which can be suitable for large-scale systems, will split the estimation problem into two main steps.
(1) Estimation of the local concentration in correspondence of each threshold sensor by means of $l$ independent MH-MAP filters. The concentration estimates provided by each local MH-MAP filter allows one to recast the threshold (binary-valued) measurements as *linear* real-valued pseudo-measurements.
(2) Field estimation over a mesh of finite elements defined over the (spatial) domain $\Omega$ on the basis of the linear pseudo-measurements provided by the local filters in step1. For this purpose, any linear filtering technique suitable for large-scale systems can be used (see e.g.the finite-element Kalman filter [@TAC17]). In this paper, field estimation is performed by minimizing a single quadratic MH cost function for linear systems.
This solution turns out to be more computationally efficient as compared to a direct application of the MH-MAP filter to system (\[dt-sys\]) in that: (i) the number of threshold sensors spread over the domain $\Omega$ is typically much smaller than the number of vertices of the FE mesh (i.e. $l\ll n$); (ii) as it will be clarified in the following, each local MH-MAP filter in step 1 involves the solution of a convex optimization problem with a greatly reduced number of variables.
Step 1
------
Let us examine in more detail step 1 of the fast MH-MAP filter. To this end, let us denote by $\sigma_{k}^{(i)}$ the value of the concentration in correspondence of sensor $i$ at the $k$-th sampling instant, i.e. $\sigma_{k}^{(i)} = c(\xi_i,\eta_i, k T_s)$, and let $\sigma^{(i)}_{m,k}$ denote the value of the $m$-th time-derivative of such a concentration, so that $$\label{eq:taylor}
\sigma^{(i)}_{m,k} = \left . \frac{\partial^m}{\partial t^m} c(\xi_i,\eta_i,t) \right |_{t = k T_s}.$$ Under the hypothesis of a small enough sampling interval, the dynamical evolution of the propagating field in correspondence of each binary sensor can be approximated by resorting to a truncated Taylor series expansion, i.e. $$\sigma_{k+1}^{(i)} \approx \sigma_{k}^{(i)} + \sum_{m=1}^M \frac{T_s^m}{m!} \sigma^{(i)}_{m,k}$$ Then, the local dynamics of the concentration in correspondence of the sensor $i$ can be described by a linear dynamical system with state $\chi_{k}^{(i)} \triangleq {\rm col} \left( \sigma_{k}^{(i)},\sigma_{1,k}^{(i)} , \ldots, \sigma_{M,k}^{(i)}\right)$ and state equation $$\label{eq:chi}
\chi_{k+1}^{(i)} = \tilde A \, \chi_{k}^{(i)} + w_k^{(i)}\,,$$ where the matrix $\tilde A$ is obtained by time-discretization of (\[eq:taylor\]) with sampling interval $T_s$ and $w_k^{(i)}$ is the disturbance acting on the local dynamics with zero mean and inverse covariance $\tilde G$. Notice that models like (\[eq:chi\]) are widely used in the construction of filters for the estimation of time-varying quantities whose dynamics is unknown or too complex to be modeled (for instance, they are typically used in tracking as motion models of moving objects [@Bar-Shalom]). With this respect, a crucial assumption for the applicability of this kind of models is that the sampling interval is sufficiently small as compared to the time constants characterizing the variation of the quantities to be estimated. Hence, their application in the present context is justified by the fact that, in practice, (binary) concentration measurements can be taken at a high rate so that between two consecutive measurements only small variations can occur.
In model (\[eq:chi\]), the simplest choice amounts to take $M=0$ and $w^{(i)}_k$ as a Gaussian white noise, which corresponds to approximate the concentration as nearly constant (notice that, in this case, we have $\tilde A = 1$). Instead, by taking $M=1$, we obtain a nearly-constant derivative model with state transition matrix $$\tilde A = \left [ \begin{array}{cc} 1 & T_s \\ 0 & 1 \end{array} \right],$$ which is equivalent to the nearly-constant velocity model widely adopted for moving object tracking. [@Bar-Shalom] Clearly, each local model (\[eq:chi\]) is related to the $i$-th binary measurement via the measurement equation $$\begin{aligned}
z_{k}^{(i)} &=& \widetilde C \, \chi_{k}^{(i)} + v_{k}^{(i)} \\
y_{k}^{(i)} &=& g^{(i)} \left ( z_{k}^{(i)} \right )\,, \label{meas:chi}\end{aligned}$$ where $$\widetilde C = [1 \; 0 \; \cdots \; 0] \, .$$ Then, for each sensor $i$ and time instant $k$, the minimization of the following MH-MAP cost function is addressed $$\begin{aligned}
\label{eq:cost_step1}
\widetilde J_{k}^{(i)} (\chi_{k-N:k}^{(i)}) &=& \|\chi_{k-N}^{(i)} - \overline{\chi}_{k-N}^{(i)}\|^{2}_{\widetilde \Psi} + \displaystyle{\sum_{j=k-N}^{k}}\|\chi_{j+1}^{(i)} - \tilde A \, \chi_{j}^{(i)} \|^{2}_{\widetilde G}\nonumber \\
&-&\displaystyle{\sum_{j=k-N}^{k}}\left\{ (1-y_{j}^{(i)}) \ln F^{(i)} \left (\tau^{(i)}- \widetilde C \, \chi_{j}^{(i)} \right )+
y_{j}^{(i)} \ln \left[ 1 - F^{(i)} \left (\tau^{(i)}- \widetilde C \, \chi_{j}^{(i)} \right ) \right] \right\},\end{aligned}$$ where $\chi_{k-N:k}^{(i)} \triangleq {\rm col}\left( \chi_{k-N}^{(i)} , \ldots, \chi_{k}^{(i)}\right)$ and $\overline{\chi}_{k-N}^{(i)}$ is the estimate of the local state at time $k-N$ computed at the previous iteration.
In conclusion, at any time instant $k = N, N+1, \ldots$, for any threshold sensor $i$ the following problem has to be solved.\
\
**Problem $\widetilde E_k^{(i)}$:** Given the prediction $\overline{\chi}_{k-N}^{(i)}$ and the measurement sequence $\{ y^{(i)}_{k-N}, \ldots, y^{(i)}_k \}$, find the optimal estimates $\hat{\chi}_{k-N|k}^{(i)},\ldots,\hat{\chi}_{k|k}^{(i)}$ that minimize the cost function $\widetilde J_{k}^{(i)}(\chi_{k-N:k}^{(i)})$ in (\[eq:cost\_step1\]).\
\
As before, the propagation of the estimation problem from time $k-1$ to time $k$ is ensured by choosing $\overline{\chi}_{k-N}^{(i)} = \hat{\chi}_{k-N|k-1}^{(i)}$. The number of variables involved in each of such optimization problems is $(M+1)(N+1)$ and, in view of (\[eq:chi\]) and (\[meas:chi\]), the cost function $\widetilde J_{k}^{(i)}$ is *convex* according to Proposition 1. Hence, basically, step 1 amounts to solving $l$ convex optimization problems of low/moderate size.
Step 2
------
In step 2, the concentration estimates $\hat{\sigma}_{j|k}^{(i)} = \widetilde{C} \hat{\chi}_{j|k}^{(i)}$, $j= k-N, \ldots, k$, obtained by solving Problem $\widetilde E^{(i)}_k$ in correspondence of any threshold sensor $i$, are used as linear pseudo-measurements in order to estimate the whole concentration field over the spatial domain $\Omega$. By resorting again to the FE approximation of Section V, the vector of coefficients $x_k$ can be estimated, for example, by minimizing a quadratic MH cost function of the form: $$\label{eq:cost_function_2}
\overline J_{k}(X_{k-N:k}) = \|x_{k-N} - \overline{x}_{k-N}\|^{2}_{\Psi} + \sum_{j=k-N}^{k}\|x_{j+1}-Ax_{j}-Bu\|^{2}_{G}
+\sum_{j=k-N}^{k}\sum_{i=1}^{l}\|\hat{\sigma}_{j|k}^{(i)}-C^{(i)}x_{j} - D^{(i)} \gamma \|^{2}_{\Xi^{(i)}}$$ where the quantities $A$, $B$, $\gamma$, $C^{(i)}$, $D^{(i)}$, for $i=1,\ldots,l$, are obtained by means of the FE method as in Section V. Notice that each term weighted by the positive scalar $\Xi^{(i)}$ penalizes the distance of the concentration $C^{(i)} x_{j} + D^{(i)}\gamma$, estimated through the FE approximation from the concentration $\hat{\sigma}_{j|k}^{(i)}$ at step 1 on the basis of the threshold measurements. In practice, each weight $\Xi^{(i)}$ can be set approximately equal to the inverse of the variance of the estimation error in the step 1 of the procedure, computed for example by means of numerical simulations. Once these weights have been fixed, the weight matrix $\Psi$ in the arrival cost can be tuned so as to ensure the stability of the estimation error. We refer to the related literature on MHE [@receding; @NLMHE; @CDC] for a discussion on this issue. The prediction $\overline{x}_{k-N}$ is computed in a recursive way as shown in the previous sections. Accordingly, at any time instant $k = N, N+1, \ldots$, after the application of step 1 the following problem has to be addressed.\
\
**Problem $\overline E_k$:** Given the prediction $\overline{x}_{k-N}$ and the optimal estimates $\{\hat{\sigma}_{k-N|k}^{(i)},\ldots,\hat{\sigma}_{k|k}^{(i)}\}$, $i = 1, \ldots, l$, obtained by solving Problem $\widetilde E_k^{(i)}$, find the optimal estimates $\hat{x}_{k-N|k},\ldots,\hat{x}_{k|k}$ that minimize the cost function $\overline J_{k}(X_{k-N:k})$ in (\[eq:cost\_function\_2\]).\
\
Since the cost function (\[eq:cost\_function\_2\]) depends quadratically on the states $\{x_{k-N},\ldots,x_{k}\}$, the above estimation problem admits a closed-form solution at any time $k$. Hence, the computational effort needed to perform step 2 of the fast MH-MAP filter turns out to be limited. Recall also that Step 1 of the proposed approach involves the solution of $l$ nonlinear nonquadratic convex optimization problems in $(K+1)(N+1)$ variables where $l$ is the number of threshold sensors, $N$ is the moving-horizon length and $K+1$ the low-order of the assumed local dynamics. This means that the overall algorithm is computationally efficient if compared to a direct application of the MH-MAP filter on the large-scale system arising from the FE discretization, which would require the solution of a large-scale optimization problem with a number of variables equal to $(N+1)n$ where $n$ is the order of the adopted FE model. Such improvement becomes even more relevant if compared with other common solutions already present in the literature. As a matter of fact, it is worth noting that approaches relying on particle filtering [@Djuric_1; @Djuric_2] cannot be applied to solve estimation problems relative to large-scale systems, and deterministic approaches [@Irr-sampling; @Gherardini2; @Wim2006; @Bai] require the optimization of a piece-wise cost function, due to the highly discontinuous nature of the outcomes from threshold sensors. Although piece-wise optimization can be solved by means of sequential quadratic programming, the corresponding computational cost is feasible only if the number of state variables is not too large.
Numerical results
=================
Simulation experiments concerning state estimation of a spatially distributed system are used to demonstrate the effectiveness of the proposed (both standard and computationally fast) MH-MAP approach. The process under consideration is modeled by a diffusion PDE described by with known constant diffusivity $\lambda = 5 \times 10^{-8} ~ [m^2/s]$ and initial condition $x_{0} = 0_n ~ [g/m^2]$. Diffusion is a fundamental transport process in fluid dynamics which governs the movement of a substance from a region of high concentration to a region of low concentration. The discrete-time state space system (\[dt-sys\])-(\[z\]) is obtained by first using FEM for spatial discretization of the PDE with a mesh of $1695$ triangular elements ($915$ vertices) generated by the Matlab PDE Toolbox, and then via time-discretization of the resulting model with fixed sampling interval $T_s = 1 ~ [s]$.
![Diffusive field monitored by a constellation of 20 threshold sensors (cyan $\square$) randomly deployed over the 2D bounded domain $\Omega$.[]{data-label="fig:field200"}](domain3b.pdf)
![Triangular finite-element mesh (152 elements, 97 nodes) used by the proposed MH-MAP field estimators.[]{data-label="fig:mesh"}](mesh_estimator3b.pdf)
The *true* dynamic field to be estimated is defined over an L-shaped spatial domain $\Omega$ covering an area of $7.44 ~ [m^2]$ (see Fig.\[fig:field200\]). The problem is characterized by mixed boundary conditions, i.e. a condition of type (\[Dbc\]) with $\gamma = 30 ~ [g/m^2]$ on the bottom end A–B, and no-flux condition (\[Nbc\]) on the remaining five edges of boundary $\partial\Omega$. The *true* concentrations are generated by recursively using (\[dt-sys\]) on the simulated system. Then, threshold measurements are obtained by first corrupting the state variables with a Gaussian noise with variance $r^{(i)}$, and finally by applying a threshold $\tau^{(i)}$, different for each sensor $i$ of the network. Note that, in order to collect threshold measurements which are as informative as possible, $\tau^{(i)}, ~ i = 1,...,l$, are generated as uniformly distributed random numbers in the interval $[0.05,29.95]$, where $[0,30]$ is the range of nominal concentration values throughout each experiment. The duration of each simulation experiment is fixed to $1200 ~ [s]$ (120 samples). In order to account for model uncertainty, the MH-MAP estimator is taken as one order of magnitude coarser and slower than the corresponding *ground truth* simulator, by implementing a triangular mesh of ${n_\phi} = 97$ vertices (of which $n=89$ are internal) and $152$ elements (see Fig.\[fig:mesh\]), and by running at a sample rate of $0.1 ~ [Hz]$. It is worth to point out that such differences in the FE mesh resolutions and sampling rates between the ground-truth simulator and MH-MAP estimator clearly induce model mismatches in the simulation experiments, thus allowing to investigate the robustness of the proposed approach. The time evolution of the field to be estimated is represented through the $n$-dimensional linear discrete-time model , which describes the dynamics of the concentration field in correspondence of the $n$ nodes lying on the interior of $\Omega$. As discussed in Section V, based on these pointwise-in-space estimates and thanks to the FEM, it is possible to approximately reconstruct the overall state of the infinite-dimensional diffusive process . In this regard, we have set the following parameters: initial guess of the estimated field randomly generated as normally distributed with mean $\overline{x}_{0} = 5 \cdot {1}_n ~ [g/m^2]$ and variance $P= 10 ~ I_n$; measurement noise variance $r^{(i)} = 0.1$; moving window of size $N = 15$; total number of threshold sensors $l=20$; $G = 10 ~ I_n$ in ; weight matrices $\widetilde{\Psi} = 10^3 ~ I_n$ and $\widetilde{Q} = 10^{2} ~ I_n$ in (where $1_{n}$ indicates the $n-$dimensional vectors with all unit entries, while $I_n$ denotes the $n-$dimensional identity matrix). Additionally, in all the numerical experiments we assumed a nearly-constant concentration field, i.e. $\tilde A = 1$ in . Further, we have also assumed a mismatch on the measurement noise covariance by taking $r^{(i)} = 1$ in (\[eq7\]) instead of the ground truth value $0.1$.
![Comparison in terms of RMSE between the standard and fast MH-MAP state estimators as a function of time.[]{data-label="fig:fastvsstand"}](Figure-RMSE-FAST2018vsACC2016_std.pdf)
![Fast MH-MAP filter: convergence of the concentration field estimates as simulation moves forward in time. The estimation error is computed for 304 sampling points evenly spread within the domain $\Omega$ in a single simulation run.[]{data-label="fig:convergence"}](estimation_error_new.pdf)
\[tab:1\]
The performance of the proposed MH-MAP state estimators (both the standard and fast versions) is given in terms of *Root Mean Square Error* (RMSE) of the estimated concentration field, i.e. $$\label{64}
\text{RMSE}(k)=\left(\sum_{\ell=1}^{L}\frac{\|e_{k,\ell}\|^{2}}{L}\right)^{\frac{1}{2}}$$ where $\|e_{k,\ell}\|$ is the norm of the estimation error at time $k$ in the $\ell-$th simulation run (averaged over $L$ independent Monte Carlo realizations) and the estimation error is computed on the basis of the estimate $\hat{x}_{k-N|k}$. Fig.\[fig:fastvsstand\] shows the comparison in performance between the standard and fast MH-MAP filters implemented in MATLAB, in terms of the RMSE, and relative standard deviation with respect to Monte Carlo runs, of the estimated diffusive field as given by (\[64\]). To obtain the RMSEs plotted in Fig.\[fig:fastvsstand\], the estimation error $e_{k,\ell}$ at time $k$ in the $\ell-$th simulation run has been averaged over $304$ sampling points (evenly spread within $\Omega$) and $L=100$ Monte Carlo realizations. As shown in Fig.\[fig:fastvsstand\], the estimation accuracy and relative statistical variability of the standard and fast MH-MAP filters is comparable, therefore the fast implementation is preferable in large-scale problems, usually arising from discretization of PDEs, where its computational efficiency is a strong advantage. In addition, the computational times associated to state estimation using the standard or fast filter are presented in Table \[tab:1\], which highlights the effort required by the optimization step with respect to the total CPU time in the two cases. It is clearly shown that the on-line calculation times (tested on an Intel Xeon CPU @ $3.30$ GHz, $16$ GB RAM) are reduced dramatically when the fast MH-MAP filter is adopted. In particular, the computational burden can be reduced up to two orders of magnitude by running the fast two-step approach instead of directly applying the MH-MAP method. Notice also that around $96 \%$ and, respectively, $80 \%$ of the total CPU time is devoted to the optimization step, and that the considered field estimation problem can be reliably solved by the fast algorithm under the given sampling rate. Fig.\[fig:convergence\] shows how the diffusive field estimates obtained by the computationally fast MH-MAP filter tend to the true concentration values as time goes by.
Further simulations have been run in order to investigate the effect of the moving window length $N$, the measurement noise variance $r^{(i)}$, and the number of threshold sensors $l$ on the overall state estimation accuracy of the fast MH-MAP filter. Its performance under different values of moving window length $N$ is illustrated in Fig.\[fig:normalizedH\]. Specifically, Fig.\[fig:normalizedH\] highlights the increase of RMSE that occurs when reducing the size of the moving window. It shows the RMSE for $N=1,5,10$ normalized over the estimation error obtained when $N=15$, i.e.the normalized RMSE (NRMSE), that is defined as $$\text{NRMSE}_h=\frac{\text{RMSE}_{N=h}}{\text{RMSE}_{N=15}}, \quad \mbox{ for } h=1,5,10.$$ Figs.\[fig:fig1\]-\[fig:fig2\] show the effect of measurement noise variance on the RMSE. Although the performance given varying values of $r^{(i)}$ is similar (see Fig.\[fig:fig1\]), surprisingly the choice of an observation noise with higher variance can actually improve the overall quality of the field estimates. Such results are valid for both standard and fast MH-MAP filters (as shown in preliminary results [@ACC_binary] on standard MH-MAP field estimation), and they numerically demonstrate the validity of the above stated noise-assisted paradigm in problems of recursive state estimation using threshold measurements. Finally, Fig. \[fig:RMSE\_nsens\] shows the evolution of the RMSE as a function of the number of threshold sensors available for field estimation.
![Increase of the RMSE of concentration estimates for varying moving window size ($N=1,5,10$) compared to $N=15$.[]{data-label="fig:normalizedH"}](fig-ex9-comparisonRMSEvsH15-1-5-10.pdf)
![Comparison in terms of RMSE for varying measurement noise variance of the fast MH-MAP filter ($N=5$).[]{data-label="fig:fig1"}](RMSE_fast_R.pdf)
![Fast MH-MAP filter: average RMSE over time as a function of the measurement noise variance, for a random network of 20 threshold sensors ($N=15$). As shown above, operating in a noisy environment turns out to be beneficial, for certain values of $r^{(i)}$, to the state estimation problem.[]{data-label="fig:fig2"}](bathtub.pdf)
![Fast MH-MAP filter: average RMSE over time as a function of the number of sensors deployed over the monitored area.[]{data-label="fig:RMSE_nsens"}](figure_RMSE_fast_nsens5102050_C.pdf)
Conclusions
===========
State estimation with noisy threshold measurements has been addressed. The problem has been recast in a Bayesian framework in order to always ascribe to each threshold sensor a probabilistic information content encoded by the likelihood function. Accordingly, a MAP optimization approach has been undertaken by maximizing the conditional probability of the whole state trajectory given the whole sequence of threshold (binary) measurements. Since the dimension of such MAP optimization clearly grows with time, an MH approximation over a moving time-window of suitable fixed length has been then adopted in order to make the problem solution computationally feasible. Further, it has been shown how the resulting MH-MAP state estimator can be efficiently applied to the monitoring of dynamic spatial fields by means of threshold sensors deployed over the area to be monitored. Such fields are described by PDEs, which are spatially discretized with a mesh of finite elements over the spatial domain of interest by means of the FE method. In order to efficiently estimate the time-space evolving field with a network of pointwise-in-time-and-space threshold sensors, we have proposed a faster version of the aforementioned MH-MAP filter, suitable for large-scale systems. The effectiveness of the proposed approach has been demonstrated via simulation experiments. The faster strategy has been shown to be able to provide similar levels of accuracy while requiring minimal computational effort compared to the standard MH-MAP filter, which is of great advantage in practical large-scale applications. Future work on the topic will possibly concern application of the MH-MAP state estimator to target tracking with threshold proximity sensors.
Proof of Proposition 1 {#proof-of-proposition-1 .unnumbered}
======================
In order to show that $\mathbb X_k$ is an open convex polyhedron, let us consider the terms $- \ln F^{(i)}(\tau^{(i)}-h^{(i)} (x_{j}) ) = - \ln F^{(i)}(\tau^{(i)}-C^{(i)}x_{k})$ and $- \ln ( 1- F^{(i)}(\tau^{(i)}-h^{(i)} (x_{j})) ) = - \ln ( 1 - F^{(i)}(\tau^{(i)}-C^{(i)}x_{j}) )$. Since the CDF $F^{(i)} (z)$ is non-negative and monotonically non-decreasing, the set of values $z$ such that $F^{(i)}(z) > 0$ is of the form $(a , \infty)$ for some $a \in \mathbb R \cup \{- \infty\}$. Hence, the domain of $- \ln F^{(i)}(\tau^{(i)}-C^{(i)}x_{k})$ has the form $\{ x_j \in \mathbb R^n : \tau^{(i)}-C^{(i)}x_{j} > a \}$ which defines an open hyperplane in $\mathbb R^n$. Similarly, the set of values $z$ such that $1-F^{(i)}(z) > 0$ is of the form $(-\infty , b)$ for some $b \in \mathbb R \cup \{ \infty\}$. Hence, the domain of $- \ln ( 1 - F^{(i)}(\tau^{(i)}-C^{(i)}x_{j}) )$ has the form $\{ x_k \in \mathbb R^n : \tau^{(i)}-C^{(i)}x_{j} < b \}$ which also defines an open hyperplane in $\mathbb R^n$. Therefore, the set $\mathbb X_k$ is the intersection of a finite number of open hyperplanes which corresponds to an open convex polyhedron.
Concerning the convexity of the cost function, since the sum of convex functions is again convex, it is sufficient to show that each term of the cost is convex. Clearly, under assumptions A1 and A2, each term $\| x_{j+1} - f(x_j, u_j) \|^2_G = \| x_{j+1} - A x_j -B u_j \|^2_G$ is convex. Consider now the terms $ - \ln F^{(i)}(\tau^{(i)}-C^{(i)}x_{k})$ and $ - \ln ( 1 - F^{(i)}(\tau^{(i)}-C^{(i)}x_{k}) )$. Since convexity is preserved under affine transformations, these terms are convex if and only if the functions $ \ell^{(i)}_1 (z) = - \ln F^{(i)} (z)$ and $\ell^{(i)}_2 (z) = - \ln (1- F^{(i)} (z) ) $ are convex. A general proof of the convexity of these functions hinges on the observation that the functions $F^{(i)} (z) $ and $1-F^{(i)} (z)$ are obtained by integrating the log-concave PDF of the measurement noise $v^{(i)}_k$ over the convex sets $(-\infty , z]$ and $(z, \infty)$, respectively. Then, as discussed in previous work, [@Boyd] we can apply the integral property of log-concave functions \cite{} and show that both $F^{(i)} (z) $ and $1-F^{(i)} (z)$ are log-concave, thus proving the convexity of $ \ell^{(i)}_1 (z) $ and $\ell^{(i)}_2 (z) $.
Alternatively, a simple proof of the convexity of the functions $ \ell^{(i)}_1 (z) $ and $\ell^{(i)}_2 (z) $ can be given under the additional assumptions that the PDF of the measurement noise is always strictly positive and differentiable. This implies that the PDF of the measurement noise can be written as $e^{-\Phi^{(i)}(z) } $ for some convex differentiable function $\Phi^{(i)} (z)$. We discuss in detail this case because it includes, among others, the normal distribution which is the most commonly used PDF and because it provides useful insights on the form of the cost function. In the following of the proof, to simplify the notation, the dependence on the sensor index $i$ will be dropped. We start by noting that the CDF of the measurement noise can be written as $$F (z) = \int_{-\infty}^{z} e^{-\Phi (s)} ds \,$$ Then, since the differentiability of $\Phi (z)$ implies that $\ell_1 (z)$ and $\ell_1 (z)$ are twice differentiable in their domain, convexity of these functions corresponds to the second-order conditions $ \ell''_1 (z) \ge 0$ and $ \ell''_2 (z) \ge 0$, respectively. As it can be easily checked, we have $$\label{eq:logconc}
\ell''_1 (z) = \frac{e^{-\Phi (z)}}{[F(z)]^2} \left [ e^{-\Phi (z)} + F(z) \, \Phi'(z) \right ] \, .$$ We distinguish two cases. When $ \Phi'(z) \ge 0$, the second derivative is non-negative because all terms are non-negative. When instead $ \Phi'(z) < 0$, we can exploit the fact that, for a convex function $\Phi (z)$, we have $\Phi(s) \ge \Phi (z) + \Phi'(z) (s-z)$ to derive the upper bound $$F (z) = \int_{-\infty}^{z} e^{-\Phi (s)} ds \le e^{-\Phi (z)} e^{z \, \Phi' (z)} \int_{-\infty}^{z} e^{-\Phi' (z) \, s} ds = - \frac{e^{-\Phi (z)} }{\Phi' (z) } \, .$$ Then, when $ \Phi'(z) < 0$, we can lower bound the second derivative in (\[eq:logconc\]) as $$\ell''_1 (z) \ge \frac{e^{-\Phi (z)}}{[F(z)]^2} \left [ e^{-\Phi (z)} - \frac{e^{-\Phi (z)} }{\Phi' (z) } \, \Phi'(z) \right ] = 0 \, ,$$ proving the convexity of $\ell_1 (z) $. Similarly, for $\ell_2 (z)$ we have $$\label{eq:logconc2}
\ell''_2 (z) = \frac{e^{-\Phi (z)}}{[1-F(z)]^2} \left [ e^{-\Phi (z)} - (1-F(z)) \, \Phi'(z) \right ] \, .$$ Again we distinguish two cases. When $ \Phi'(z) \le 0$, the second derivative is non-negative because all terms are non-negative. When instead $ \Phi'(z) > 0$, we can exploit again the convexity of $\Phi (z)$ and derive the upper bound $$1- F(z) = \int_{z}^{\infty} e^{-\Phi (s)} ds \le e^{-\Phi (z)} e^{z \, \Phi' (z)} \int_{z}^{\infty} e^{-\Phi' (z) \, s} ds = \frac{e^{-\Phi (z)} }{\Phi' (z) } \, .$$ Then, when $ \Phi'(z) > 0$, we can lower bound the second derivative in (\[eq:logconc2\]) as $$\ell''_2 (z) \ge \frac{e^{-\Phi (z)}}{[1-F(z)]^2} \left [ e^{-\Phi (z)} - \frac{e^{-\Phi (z)} }{\Phi' (z) } \, \Phi'(z) \right ] = 0 \, ,$$ proving the convexity of $\ell_2 (z)$ and, hence, of the whole cost function.
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abstract: |
In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order $d$ allows for $d$ factorizations of the subdivision operator with respect to the *Gregory operators*: A new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the $d$-th factorization provides a “convergence from contractivity” method for showing $C^d$-convergence of the associated Hermite subdivision scheme. The power of our factorization framework lies in the reduction of computational effort for large $d$: In order to prove $C^d$-convergence, up to now, $d$ factorization steps were needed, while our method requires only one step, independently of $d$. Furthermore, in this paper, we show by an example that the spectral condition is not equivalent to the reproduction of polynomials.
[[**Keywords**]{}: Hermite subdivision schemes; operator factorization; Stirling numbers; Gregory coefficients; polynomial reproduction]{}
[[**MSC**]{}: 65D15; 11B73; 41A15; 65D17]{}
author:
- '[Caroline Moosmüller]{}[^1]'
- '[Svenja Hüning]{}[^2]'
- '[Costanza Conti]{}[^3]'
title: Stirling numbers and Gregory coefficients for the factorization of Hermite subdivision operators
---
Introduction
============
Hermite subdivision schemes are iterative refinement rules, which, applied to discrete vector data, produce a function and its consecutive derivatives in the limit. They find similar applications as classical subdivision schemes [@cavaretta91], but are preferred when the modeling of first derivatives (or even higher derivatives) is of particular interest. This can be the case, for example, for the generation of curves and surfaces [@dyn02; @merrien92; @xue05; @xue06], for the construction of multiwavelets [@cotronei17; @cotronei18], for interpolating and approximating manifold-valued tangent vector data [@moosmueller16; @moosmueller17], and for the analysis of biomedical images [@conti15; @uhlmann14].
The convergence of subdivision schemes as well as the analysis of the regularity of their limit functions are topics of high interest. It is well-known that such analyses are strongly connected to the factorization of the associated subdivision operator [@cohen96; @conti16; @dyn02; @merrien12; @micchelli98].
In this paper we study factorization properties of subdivision operators $S_{{\boldsymbol{A}}}: \ell(\mathbb{Z})^2 \rightarrow \ell(\mathbb{Z})^2$, which correspond to Hermite subdivision schemes producing functions and first derivatives: $$\label{eq:intro_sd}
\left(S_{{\boldsymbol{A}}}{\mathbf{c}}\right)_j=\sum_{k \in {\mathbb{Z}}}{\boldsymbol{A}}_{j-2k}{\mathbf{c}}_j, \quad j \in {\mathbb{Z}}.$$ Here ${\mathbf{c}}$ is a sequence of $2$-dimensional vectors (the input data), and ${\boldsymbol{A}}$ is a finitely-supported sequence of $(2 \times 2)$-matrices, called the *mask* of the operator.
We prove that every Hermite subdivision operator satisfying the *spectral condition of order $d$* (), can be factorized with respect to the operators ${\mathscr{G}}^{[n]}:\ell(\mathbb{Z})^2 \rightarrow \ell(\mathbb{Z})^2$ defined by $$\label{def:gregory_op}
{\mathscr{G}}^{[n]}=\left[\begin{array}{cc}
0 & \Delta^n \\
\Delta & -\sum_{\ell=0}^{n-1}G_{\ell}\Delta^{\ell}
\end{array}\right], \quad n=1,\ldots,d,$$ with $\Delta$ the forward difference operator and with the understanding that $\Delta^0=\operatorname{id}$ and $\Delta^n=\Delta(\Delta^{n-1})$. By $G_n$ we denote the *Gregory coefficients*, which are a well studied sequence in number theory, see e.g. [@candelpergher12; @kowalenko10; @merlini06]. They can be computed from the Stirling numbers of the first kind; see for the first Gregory coefficients $G_n$, $n=0,\dots,6$. We call ${\mathscr{G}}^{[n]}$ the *$n$-th Gregory operator*.
----------------- --- ----- ------- ------ --------- ------- ------------
n 0 1 2 3 4 5 6
\[0.1cm\] $G_n$ 1 1/2 -1/12 1/24 -19/720 3/160 -863/60480
----------------- --- ----- ------- ------ --------- ------- ------------
: First few Gregory coefficients $G_n$.[]{data-label="fig:greg"}
The main results of this paper, proved in , are
\[intro:main\] Let $S_{{\boldsymbol{A}}}$ be a subdivision operator satisfying the spectral condition of order $d\geq 1$. Then for $n=1,\ldots, d$ there exist subdivision operators $S_{{\boldsymbol{B}}^{[n]}}$ such that $$\label{nth_greg}
{\mathscr{G}}^{[n]} S_{{\boldsymbol{A}}}=2^{-n}S_{{\boldsymbol{B}}^{[n]}}{\mathscr{G}}^{[n]}, \quad n=1,\ldots, d.$$
We in addition show that the last factorization gives rise to an easy-to-check condition for the $C^d$-convergence of Hermite subdivision schemes:
\[intro:cor\_main\] With notation as in , if $S_{{\boldsymbol{B}}^{[d]}}$ is contractive, then the Hermite subdivision scheme associated with $S_{{\boldsymbol{A}}}$ is $C^d$-convergent.
Furthermore, in , we show that for primal schemes the spectral condition of order $d$ does not imply that polynomials up to degree $d$ are reproduced, while it is known that the reverse implication holds true [@conti14]. Up to now, these two concepts were conjectured to be equivalent.
Impact of our results
---------------------
Factorization of subdivision operators for proving convergence/regularity of the associated subdivision scheme is a standard method in scalar subdivision [@dyn92], vector subdivision [@charina05; @cohen96; @sauer02] and Hermite subdivision [@conti16; @conti17; @merrien12]. Nevertheless, the results for Hermite subdivision schemes are only concerned with factorizing *once*, that is, with proving the *minimal* regularity of the scheme (for example, in our case the minimal regularity is $1$ since we consider schemes dealing with function values and first derivatives), see e.g. [@merrien12]. Many authors, however, are interested in higher regularity than the minimal one [@conti14; @han05; @jeong17; @moosmueller18]. We show in this paper that for Hermite schemes, the Gregory operators provide the necessary factorization tool to prove regularity higher than $1$.
It is worthwhile noting that *every* Hermite subdivision operator satisfying the spectral condition of order $d$ can be factorized with respect to the Gregory operators. In general, for such an Hermite subdivision operator, there exist infinitely many possibilities to factorize *beyond* the Taylor factorization (i.e. to prove regularity higher than the minimal one). This is due to the theory of factorizing vector schemes [@charina05; @cohen96; @sauer02], which involves choosing an eigenvector of the vector subdivision operator, and completing this vector to a basis of ${\mathbb{R}}^2$ (obviously, there are infinitely many ways to do this). Moreover, the choice of an eigenvector for the $(k+1)$-th factorization depends on the $k$-th factorization. This means that one can only factorize step-by-step, which drastically slows down computations. It also means that different Hermite schemes factorize with respect to very different operators. These facts can be seen from the computations in [@conti14; @jeong17].
We prove that the spectral condition guarantees the existence of *one* factorization that works for *all* Hermite subdivision operators. The key to this factorization is a clever choice of eigenvectors.
We would like to stress the improvement for computations arising from the Gregory factorization. In order to prove that a Hermite scheme is $C^d$-convergent, $d\geq 1$, up to now, $d$ factorization steps were necessary, see again [@conti14; @jeong17]. As shown in , we reduce this procedure to one single factorization: $n=d$ in provides the operator with respect to which one has to factorize.
We mention that for $d=1$ the complete Taylor operator [@merrien12] and ${\mathscr{G}}^{[1]}$ provide the same tool for proving $C^1$-convergence for schemes. In this sense the Gregory operators are direct extensions of the complete Taylor operator of dimension $2$. However, the Taylor operator is more powerful in proving the minimal regularity of a scheme, as it also works for schemes of general dimension $k, k\geq 2$, and for multivariate schemes. We thus consider the Gregory operators as a first step towards an extension of the Taylor operator for proving higher regularity than the minimal one.
Since the Stirling numbers and the Gregory coefficients are closely connected to higher-order finite differences, it is not too surprising that they appear in our construction. Nevertheless, we find it remarkable that the Gregory coefficients appear in such a natural manner and allow for a complete and easy description of the operators .
Organization of the paper
-------------------------
The paper is organized as follows: The preliminary section () fixes the notation and recalls basic facts about subdivision schemes, factorization of subdivision operators, and the convergence of vector and Hermite schemes. introduces Stirling numbers and Gregory coefficients and discusses a recursion involving iterated forward differences. The main results are stated and proved in . Examples of the Gregory factorization and of its use are provided in . In this section we also show that the spectral condition does not imply the reproduction of polynomials. concludes the paper.
Preliminaries {#sec:pre}
=============
Hermite subdivision schemes {#subsec:Hermite}
---------------------------
We denote by $\ell({\mathbb{Z}})^2$ the space of ${\mathbb{R}}^2$-valued sequences ${\mathbf{c}}=\left({\mathbf{c}}_j: j \in {\mathbb{Z}}\right)$, and by $\ell({\mathbb{Z}})^2_{\infty}$ the space of ${\mathbb{R}}^2$-valued sequences with finite infinity-norm: $$\|{\mathbf{c}}\|_{\infty}:=\sup_{j \in {\mathbb{Z}}}|{\mathbf{c}}_j|_{\infty}<\infty,$$ where $|\cdot|_{\infty}$ is the infinity-norm on ${\mathbb{R}}^2$. Similarly, we define the space $\ell({\mathbb{Z}})^{2\times 2}$ of matrix-valued sequences ${\boldsymbol{A}}=\left({\boldsymbol{A}}_j: j \in {\mathbb{Z}}\right)$, and the space $\ell({\mathbb{Z}})^{2\times 2}_{\infty}$ of all such sequences with finite infinity-norm: $$\|{\boldsymbol{A}}\|_{\infty}:=\sup_{j \in {\mathbb{Z}}}|{\boldsymbol{A}}_j|_{\infty}<\infty,$$ where $|\cdot|_{\infty}$ is the operator norm for matrices in ${\mathbb{R}}^{2 \times 2}$ induced by the infinity-norm on ${\mathbb{R}}^2$. We also consider the spaces $\ell({\mathbb{Z}})^{2}_{0}$ and $\ell({\mathbb{Z}})^{2\times 2}_{0}$ which consist of finitely supported vector resp. matrix sequences.
\[def:subd\_op\] A *subdivision operator* with *mask* ${\boldsymbol{A}}\in \ell({\mathbb{Z}})^{2\times 2}_{0}$ is the map $S_{{\boldsymbol{A}}}:\ell({\mathbb{Z}})^{2} \to \ell({\mathbb{Z}})^{2}$ defined by $$\label{eq:sdo}
\left(S_{{\boldsymbol{A}}}{\mathbf{c}}\right)_j=\sum_{k \in {\mathbb{Z}}}{\boldsymbol{A}}_{j-2k}{\mathbf{c}}_j, \quad {\mathbf{c}}\in \ell({\mathbb{Z}})^{2},\, j \in {\mathbb{Z}}.$$
Note that due to the finite support of the mask ${\boldsymbol{A}}$, the sum in is finite. Furthermore, if ${\mathbf{c}}\in \ell({\mathbb{Z}})^{2}_{\infty}$, then $S_{{\boldsymbol{A}}}{\mathbf{c}}\in \ell({\mathbb{Z}})^{2}_{\infty}$. Therefore we can define the norm of a subdivision operator $S_{{\boldsymbol{A}}}$ by $$\|S_{{\boldsymbol{A}}}\|_{\infty}=\sup\{ \|S_{{\boldsymbol{A}}}{\mathbf{c}}\|_{\infty}: \|{\mathbf{c}}\|_{\infty}=1\}.$$
\[def:hermite\] Let $S_{{\boldsymbol{A}}}$ be a subdivision operator . An *Hermite subdivision scheme* is the iterative procedure of constructing vector-valued sequences by $$\label{eq:Hermite_sds}
{\boldsymbol{D}}^{n+1}{\mathbf{c}}^{[n+1]}=S_{{\boldsymbol{A}}}{\boldsymbol{D}}^{n}{\mathbf{c}}^{[n]},\quad n\in {\mathbb{N}},$$ from initial data ${\mathbf{c}}^{[0]} \in \ell({\mathbb{Z}})^{2}$. Here ${\boldsymbol{D}}$ denotes the diagonal matrix ${\boldsymbol{D}}=\operatorname{diag}\left(1, 1/2\right)$.
\[def:convergent\_hermite\] An Hermite subdivision scheme is $C^{d}$-convergent, $d \geq 1$, if for every input data ${\mathbf{c}}^{[0]}\in \ell({\mathbb{Z}})^{2}_{\infty}$ and any compact $K \subset \mathbb{R}$, there exists a function $\varphi \in C^d({\mathbb{R}})$ such that $\Phi=[\varphi,\varphi']^T: {\mathbb{R}}\to {\mathbb{R}}^2$ and the sequence ${\mathbf{c}}^{[n]}$ defined by satisfies $$\label{eq:Hermite_convergence}
\lim_{n \to \infty}\sup_{j\in {\mathbb{Z}}\cap K}|{\mathbf{c}}^{[n]}_j-\Phi\left(2^{-n}j\right)|_{\infty}=0.$$ Furthermore, we request that there exists at least one ${\mathbf{c}}^{[0]} \in \ell({\mathbb{Z}})_{\infty}^{2}$ such that $\varphi \neq 0$.
The regularity of Hermite schemes is studied in many papers, see e.g. [@conti17; @dubuc06; @dubuc05; @dyn95; @dyn99; @merrien12]. Note that these papers are concerned with the *minimal* regularity of an Hermite subdivision scheme (e.g. with regularity $1$). Along the lines of [@conti14; @han05; @jeong17; @moosmueller18], we are interested in the regularity which is higher than one.
For a sequence ${\mathbf{c}}$ we define the forward difference operator by $$\label{def:seq_forward_diff}
\left(\Delta {\mathbf{c}}\right)_j={\mathbf{c}}_{j+1}-{\mathbf{c}}_j, \quad j \in {\mathbb{Z}}.$$ In analogy to , we define the forward difference operator for functions $f$ by $$\begin{aligned}
\label{def:delta}
\left(\Delta f\right)(x)=f(x +1)-f(x), \quad x \in {\mathbb{R}}.\end{aligned}$$ If $f$ is differentiable, we define the differential operator $$\begin{aligned}
\label{def:D}
Df=f',\end{aligned}$$ where we take the derivative component-wise. By sampling $f$ on ${\mathbb{Z}}$, we obtain a vector-valued sequence ${\mathbf{c}}_f=(f(j): j \in {\mathbb{Z}})$. Since in this paper we are only concerned with sampled functions, we denote the sequence ${\mathbf{c}}_f$ again by $f$. Therefore, by $S_{{\boldsymbol{A}}}f=g$ we mean $S_{{\boldsymbol{A}}}{\mathbf{c}}_f={\mathbf{c}}_g$ for two functions $f,g$. Note that this notation is consistent with the forward difference operators for functions and sequences: $$(\Delta {\mathbf{c}}_f)_j=(\Delta f)(j), \quad j \in {\mathbb{Z}}.$$
We denote by $\Pi_k$ the set of polynomials with real coefficients of degree $\leq k$, $k \geq 0$. If $\pi \in \Pi_k$, then we write $$\pi(x)=\sum_{j=0}^k \pi[j]x^j,$$ that is, we denote the $j$-th coefficient of $\pi$ by $\pi[j] \in {\mathbb{R}}$, $j=0,\ldots,k$.
\[def:spectral\] A subdivision operator $S_{{\boldsymbol{A}}}$ satisfies the *spectral condition of order $d$*, $d\geq 1$, if there exist polynomials ${\mathscr{P}}_k \in \Pi_k$, ${\mathscr{P}}_k[k]=1/k!$, such that $$\label{eq:spectral}
S_{{\boldsymbol{A}}}\left[ \begin{array}{c}
{\mathscr{P}}_k\\
D{\mathscr{P}}_k
\end{array}
\right]
= 2^{-k}\left[ \begin{array}{c}
{\mathscr{P}}_k\\
D{\mathscr{P}}_k
\end{array}
\right],$$ $k=0,\ldots, d$. A subdivision operator satisfying the spectral condition of order $d$ is called *Hermite subdivision operator of spectral order $d$*. The polynomials ${\mathscr{P}}_k, k=0,\ldots, d$, are named *spectral polynomials of $S_{{\boldsymbol{A}}}$*.
\[def:reprod\] Let $S_{{\boldsymbol{A}}}$ be a subdivision operator. The Hermite subdivision scheme associated with $S_{{\boldsymbol{A}}}$ is said to *reproduce a function $f\in C^{1}({\mathbb{R}})$* if for initial data ${\mathbf{c}}^{[0]}_j=[f(j),f'(j)]^T$, the iterated sequence ${\mathbf{c}}^{[n]}$ defined by is given by ${\mathbf{c}}^{[n]}_j=[f(2^{-n}j),f'(2^{-n}j)]^T$, $j\in {\mathbb{Z}}, n\geq 1$.
The spectral condition was first introduced by [@dubuc09], see also [@conti14; @merrien12]. In [@dubuc09] it is proved that the spectral condition is equivalent to a special *sum rule* introduced by [@han03b; @han05]. Note that in we use the *primal* parametrization, as opposed to dual or more general parametrizations which can be considered, see e.g. [@conti18; @conti14]. Furthermore, [@conti14] shows that reproduction of $\Pi_d$ implies the spectral condition of order $d$.
The reverse implication was not yet clear, but we here put into evidence that it is actually false. Indeed, the primal Hermite scheme in satisfies the spectral condition of order $d=4$ (for $\theta=1/32$), but polynomials of degree $4$ are not reproduced.
We mention that the spectral condition is a crucial property for the factorizability of an Hermite subdivision operator, but, as proved in [@merrien11; @merrien18], it is not necessary for convergence.
Factorization of subdivision operators
--------------------------------------
In order to discuss factorizations of Hermite subdivision operators and their connection to regularity higher than the minimal, we have to introduce vector subdivision schemes. The following part on vector subdivision schemes presented here is simplified and an adapted version of constructions and results from the general theory of vector subdivision schemes, see e.g. [@charina05; @cohen96; @micchelli98; @sauer02] for details.
Let $S_{{\boldsymbol{B}}}$ be a subdivision operator . A *vector subdivision scheme* is the iterative procedure of constructing vector-valued sequences by $$\label{eq:vector_sd}
{\mathbf{c}}^{[n+1]}=S_{{\boldsymbol{B}}}{\mathbf{c}}^{[n]},\quad n\in {\mathbb{N}},$$ from initial data ${\mathbf{c}}^{[0]}\in \ell({\mathbb{Z}})^2$.
Note that an Hermite subdivision scheme is a *level-dependent* case of vector subdivision, i.e. it can be generated by applying vector subdivision operators that vary with the level $n$: $S_{{\boldsymbol{B}}^{[n]}}={\boldsymbol{D}}^{-(n+1)}S_{{\boldsymbol{A}}}D^{n}$. The crucial difference between Hermite and vector subdivision schemes lies in the definition of convergence:
A vector subdivision scheme is $C^{d}$-convergent, $d \geq 0$, if for every input data ${\mathbf{c}}^{[0]}\in \ell({\mathbb{Z}})^{2}_{\infty}$ and any compact $K \subset \mathbb{R}$, there exists a vector-valued function $\Psi\in C^d({\mathbb{R}},{\mathbb{R}}^2)$ such that the sequence ${\mathbf{c}}^{[n]}$ defined by satisfies $$\label{eq:vector_convergence}
\lim_{n \to \infty}\sup_{j\in {\mathbb{Z}}\cap K}|{\mathbf{c}}^{[n]}_j-\Psi\left(2^{-n}j\right)|_{\infty}=0,$$ and there exists at least one ${\mathbf{c}}^{[0]} \in \ell({\mathbb{Z}})^{2}_{\infty}$ such that $\Psi \neq 0$. $C^0$-convergent vector schemes are simply called “convergent”.
Following [@micchelli98], for a mask ${\boldsymbol{B}}$, we define ${\mathcal{E}}_{{\boldsymbol{B}}}$ by $${\mathcal{E}}_{{\boldsymbol{B}}}=\{v \in {\mathbb{R}}^2: \sum_{j\in {\mathbb{Z}}}{\boldsymbol{B}}_{2j}v=v,\, \sum_{j\in {\mathbb{Z}}}{\boldsymbol{B}}_{2j+1}v=v\}.$$ It is well-known that the convergence of the vector subdivision scheme associated with $S_{{\boldsymbol{B}}}$ implies that there exists $v \neq 0$ such that $v\in{\mathcal{E}}_{{\boldsymbol{B}}}$. The following is clear from the definition of ${\mathcal{E}}_{{\boldsymbol{B}}}$:
Let ${\boldsymbol{B}}$ be a mask. Let $v \in {\mathbb{R}}^2$. Then the following are equivalent:
1. $v \in {\mathcal{E}}_{{\boldsymbol{B}}}$,
2. $S_{{\boldsymbol{B}}}v=v$.
As in , we identify the constant function $v$ with the constant sequence ${\mathbf{c}}_v=\left(v: j\in {\mathbb{Z}}\right)$.
Therefore the space ${\mathcal{E}}_{{\boldsymbol{B}}}$ is the space of all eigenvectors (constant sequences) of $S_{{\boldsymbol{B}}}$ with respect to the eigenvalue $1$.
In this paper we are only concerned with masks ${\boldsymbol{B}}$ with $\dim {\mathcal{E}}_{{\boldsymbol{B}}}=1$. Following [@charina05], we call a matrix $V$ an *${\mathcal{E}}_{{\boldsymbol{B}}}$-generator*, if $V=[v,w]$, where $v\neq 0$ spans ${\mathcal{E}}_{{\boldsymbol{B}}}$, and $v$ and $w$ are linearly independent.
We now introduce a generalization of the forward difference operator $\Delta$ for vector schemes. Let $V$ be an invertible matrix. Define $\Delta_V$ by $$\label{eq:delta_V}
\Delta_V=\left[
\begin{array}{cc}
\Delta & 0 \\
0 & 1
\end{array}
\right]V^{-1}.$$ In [@charina05], the matrix $V$ is assumed to be orthogonal. We choose a slightly more general approach, which, however, does not change the validity of the results below. From [@charina05; @micchelli98; @sauer02] we have the following result concerning the factorization of subdivision operators:
\[thm:vector\] Let $S_{{\boldsymbol{B}}}$ be a subdivision operator and assume that ${\operatorname{dim}}{\mathcal{E}}_{{\boldsymbol{B}}}=1$. For an ${\mathcal{E}}_{{\boldsymbol{B}}}$-generator $V$ there exists a subdivision operator $S_{{\boldsymbol{C}}}$ such that $$\Delta_{V}S_{{\boldsymbol{B}}}=2^{-1}S_{{\boldsymbol{C}}}\Delta_{V}.$$ Furthermore ${\operatorname{dim}}{\mathcal{E}}_{{\boldsymbol{C}}}=1$ or ${\mathcal{E}}_{{\boldsymbol{C}}}=\{0\}$.
From [@charina05 Corollaries 5 and 8] we obtain
\[thm:converge\_vector\] With assumption and notation as in , we have
1. \[it:converge\] If $\|(2^{-1}S_{{\boldsymbol{C}}})^N\|_{\infty}<1$, for some $N\geq 1$, that is, if $2^{-1}S_{{\boldsymbol{C}}}$ is *contractive*, then the vector subdivision scheme associated with $S_{{\boldsymbol{B}}}$ is convergent.
2. \[it:smooth\] If the vector scheme associated with $S_{{\boldsymbol{C}}}$ is $C^{d}$-convergent, then the vector scheme associated with $S_{{\boldsymbol{B}}}$ is $C^{d+1}$-convergent, $d\geq 0$.
Note that [@charina05] shows stronger results than the ones mentioned above. We only need these special cases. Furthermore, in part \[it:smooth\] of the theorem we dropped the assumption that $S_{{\boldsymbol{B}}}$ is convergent. This is possible due to the following reason: If $S_{{\boldsymbol{C}}}$ is convergent, then $2^{-1}S_{{\boldsymbol{C}}}$ is contractive, and thus by part \[it:converge\] of the theorem, $S_{{\boldsymbol{B}}}$ converges.
Note that in order to show $C^{d}$-convergence of a vector subdivision scheme, there are infinitely many ways to factorize $S_{{\boldsymbol{B}}}$, respectively to obtain an operator $S_{{\boldsymbol{C}}}$. Nevertheless, in [@charina05] it is shown that if $2^{-1}S_{{\boldsymbol{C}}}$, coming from a factorization with respect to a matrix $V$, is contractive, then the operator $2^{-1}S_{\mathbf{E}}$ obtained from any other valid factorization is also contractive. Therefore, the choice of $V$ is irrelevant for proving convergence from contractivity.
As in [@charina05], we may iterate , to conclude the following:
\[lem:iteration\_vector\] Let $S_{{\boldsymbol{B}}}$ be a subdivision operator with $\dim{\mathcal{E}}_{{\boldsymbol{B}}}=1$. Let ${\boldsymbol{C}}^{[0]}={\boldsymbol{B}}$ and let $V^{[0]}$ be an ${\mathcal{E}}_{{\boldsymbol{C}}^{[0]}}$-generator. If for $k\geq 0$ and $n=1,\ldots,k+1$ there exist matrices $V^{[n]}$ and masks ${{\boldsymbol{C}}^{[n]}}$ such that $V^{[n]}$ is an ${\mathcal{E}}_{{\boldsymbol{C}}^{[n]}}$-generator and such that $$\label{eq:iteration_vector}
\Delta_{V^{[n]}}\cdots \Delta_{V^{[0]}}S_{{\boldsymbol{B}}}=2^{-(n+1)}S_{{\boldsymbol{C}}^{[n+1]}}\Delta_{V^{[n]}}\cdots \Delta_{V^{[0]}}, \quad n=0,\ldots,k,$$ and $2^{-1}S_{{\boldsymbol{C}}^{[k+1]}}$ is contractive, then the vector scheme associated with $S_{{\boldsymbol{B}}}$ is $C^{k}$-convergent.
Our construction of the $n$-th Gregory operator for Hermite subdivision operators relies heavily on the iteration .
We now continue with Hermite subdivision operators. Denote by $T$ the *Taylor operator* of dimension 2, $$T=\left[
\begin{array}{cr}
\Delta & -1 \\
0 & 1
\end{array}
\right],$$ which was first defined in [@merrien12] for the convergence and smoothness analysis of Hermite schemes. We have the following results from [@merrien12]:
\[thm:Taylor\] If $S_{{\boldsymbol{A}}}$ is an Hermite subdivision operator of spectral order at least $1$, then there exists a subdivision operator $S_{{\boldsymbol{B}}}$ such that $$TS_{{\boldsymbol{A}}}=2^{-1}S_{{\boldsymbol{B}}}T.$$ Also, ${\mathcal{E}}_{{\boldsymbol{B}}}$ is spanned by $[0,1]^T$. In particular, $\dim{\mathcal{E}}_{{\boldsymbol{B}}}=1$.
If ${\mathcal{E}}_{{\boldsymbol{B}}}$ is spanned by $[0,1]^T$, the vector scheme associated with $S_{{\boldsymbol{B}}}$ has limit functions of the form $\Psi=[0,\psi_1]^T$ for all input data (this follows from results in [@micchelli98]; an explicit proof can also be found in [@moosmueller18]). Combining this with results from [@merrien12] we obtain
\[thm:convergent\_Hermite\] Let $S_{{\boldsymbol{A}}}$ be an Hermite subdivision operator of spectral order $d$, $d\geq 1$, and let $S_{{\boldsymbol{B}}}$ be as in . If the *vector subdivision scheme* associated with $S_{{\boldsymbol{B}}}$ is $C^k$-convergent, $k\geq 0$, then the Hermite subdivision scheme associated with $S_{{\boldsymbol{A}}}$ is $C^{k+1}$-convergent.
We mention that this theorem is also stated in [@conti14]. From we see that a tool for checking $C^k$-convergence of a vector subdivision scheme is needed. Combining with , we can state:
\[lem:iteration\_hermite\] Let $d\geq 1$ and let $S_{{\boldsymbol{A}}}$ be a subdivision operator. Suppose that for $n=0,\ldots,d$, there exist matrices $V^{[n]}$ and masks ${{\boldsymbol{C}}^{[n]}}$, such that $V^{[n]}$ is an ${\mathcal{E}}_{{\boldsymbol{C}}^{[n]}}$-generator and such that $$\begin{aligned}
&TS_{{\boldsymbol{A}}}=2^{-1}S_{{\boldsymbol{C}}^{[0]}}T\\
&\Delta_{V^{[n-1]}}\cdots \Delta_{V^{[0]}}TS_{{\boldsymbol{A}}}=2^{-(n+1)}S_{{\boldsymbol{C}}^{[n]}}\Delta_{V^{[n-1]}}\cdots \Delta_{V^{[0]}}T, \quad n=1,\ldots,d.\end{aligned}$$ If $2^{-1}S_{{\boldsymbol{C}}^{[d]}}$ is contractive, then the Hermite subdivision scheme associated with $S_{{\boldsymbol{A}}}$ is $C^{d}$-convergent.
In we prove that the spectral condition of order $d$ stated by equation implies the existence of a factorization as in . The matrices $V^{[n]}$ are given by $$V^{[0]}=\left[
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right], \quad
V^{[n]}=\left[
\begin{array}{cc}
1 & 0 \\
G_n & 1
\end{array}
\right], \quad n \geq 1,$$ where $G_n$ are the Gregory coefficients, see the next section.
A recursion involving iterated forward differences {#sec:forward_differences}
==================================================
Stirling numbers and Gregory coefficients
-----------------------------------------
We do not attempt to give an overview of properties and results concerning the Stirling numbers, as they are fundamental sequences in number theory, but we cite [@graham94] for an introduction. We summarize a few properties relevant for this paper, which are all taken from [@graham94]. The *Stirling numbers of the first kind*, denoted by ${\genfrac{[}{]}{0pt}{}}{n}{m}$, count the numbers of ways to arrange $n$ elements into $m$ cycles. From the initial conditions $${\genfrac{[}{]}{0pt}{}}{0}{0}=1, \quad {\genfrac{[}{]}{0pt}{}}{n}{0}={\genfrac{[}{]}{0pt}{}}{0}{n}=0, \quad n\geq 1,$$ they can be computed via the following recurrence relation: $${\genfrac{[}{]}{0pt}{}}{n+1}{m}=n{\genfrac{[}{]}{0pt}{}}{n}{m}+{\genfrac{[}{]}{0pt}{}}{n}{m-1}, \quad m\geq 1.$$ The *Stirling numbers of the second kind*, denoted by ${\genfrac{\{}{\}}{0pt}{}}{n}{m}$, count the number of ways to split a set of $n$ elements into $m$ non-empty subsets. They can be computed using Binomial coefficients: $$\label{def:stirling2}
{\genfrac{\{}{\}}{0pt}{}}{n}{m}=\frac{1}{m!}\sum_{j=0}^m {m \choose j} (-1)^{m-j}j^n.$$ We further need the following properties: $$\label{prop:stir}
{\genfrac{\{}{\}}{0pt}{}}{n}{n}={\genfrac{\{}{\}}{0pt}{}}{n}{1}=1,\quad {\genfrac{\{}{\}}{0pt}{}}{n}{m}=0 \text{ if } m> n.$$ We introduce the *Gregory coefficients* $$\label{def:greg}
G_n=\frac{1}{n!}\sum_{j=0}^{n}{\genfrac{[}{]}{0pt}{}}{n}{j}\frac{(-1)^{n-j}}{j+1},$$ which are also known as the *Cauchy numbers of the first kind*, the *Bernoulli numbers of the second kind* and the *reciprocal logarithmic numbers*, see e.g. [@blagouchine17; @kowalenko10; @merlini06]. The Gregory coefficients appear in many interesting contexts: As the coefficients in a power series expansion of the reciprocal logarithm [@kowalenko10], in Gregory’s method for numerical integration [@phillips72], in various series representation involving Euler’s constant [@blagouchine16; @candelpergher12], and in a series expansion of the Gompertz constant [@mezo14], to name a few. Our results in are based on the following relation between the Gregory coefficients and the Stirling numbers of the second kind: $$\label{relation_stir_greg}
\sum_{j=0}^n {\genfrac{\{}{\}}{0pt}{}}{n}{j}j!\,G_j=\frac{1}{n+1}.$$ This is proved in e.g. [@merlini06]. Note that [@merlini06] shows for $\mathscr{C}_n=n!\, G_n$, and they call $\mathscr{C}_n$ the Cauchy numbers of the first kind.
Iterated forward differences
----------------------------
In this section we collect properties concerning the operators $\Delta$ and $D$, as well as the iterates $\Delta^{\ell}$, $\ell \geq 1$. They can be derived easily from the respective definitions; for the convenience of the reader, we prove some of them. The following lemma is clear from the definitions and .
The differential operator $D$ and the forward difference operator $\Delta$ defined in and commute: $$\Delta D = D \Delta.$$
\[lem:D\_Delta\_coeff\] For $k\geq 1$, both the differential operator $D$ and the forward difference operator $\Delta$ map $\Pi_k$ to $\Pi_{k-1}$. For $\pi \in \Pi_k$, the coefficients of $D\pi$ resp. $\Delta\pi$ are given by $$\begin{aligned}
&(D\pi)[j]=(j+1)\pi[j+1], \\
&(\Delta \pi)[j]=\sum_{m=j}^{k-1} {m+1 \choose j}\pi[m+1],\end{aligned}$$ $j=0,\ldots,k-1$. For $\pi \in \Pi_0, D\pi=\Delta \pi=0$.
We prove the part involving $\Delta$, the rest is clear. For $\pi \in \Pi_k$, we obtain $$\begin{aligned}
(\Delta \pi)(x)&=\pi(x+1)-{\pi}(x)=\sum_{\ell=0}^{k}{\pi}[\ell]\left((x+1)^{\ell}-x^{\ell}\right)\\
&=\sum_{\ell=0}^{k}{\pi}[\ell]\left(\sum_{m=0}^{\ell}{\ell \choose m}x^m-x^{\ell}\right)\\
&=\sum_{\ell=1}^{k}{\pi}[\ell]\left(\sum_{m=0}^{\ell-1}{\ell \choose m}x^m\right)
=\sum_{\ell=0}^{k-1}{\pi}[\ell+1]\left(\sum_{m=0}^{\ell}{\ell+1 \choose m}x^m\right)\\
&=\sum_{m=0}^{k-1}\sum_{\ell=m}^{k-1}{\pi}[\ell+1]{\ell+1 \choose m}x^m.\end{aligned}$$ This shows that $\Delta \pi \in \Pi_{k-1}$ and verifies the formula for the coefficients as stated in the Lemma.
\[cor:Delta\_minus\_D\] For a polynomial $\pi\in \Pi_k$ with $k\geq 2$, the polynomial $(\Delta-D)\pi$ has degree $k-2$ and its coefficients are given by $$(\Delta-D)\pi[j]=\sum_{m=j+1}^{k-1}{m+1 \choose j}\pi[m+1], \quad j=0,\ldots,k-2.$$ For $\pi \in \Pi_0$ or $\pi\in\Pi_1$, $(\Delta-D)\pi=0$.
\[lem:iterated\_Delta\] For $\pi \in \Pi_k$, $k\geq1$ and $1\leq \ell\leq k$ we have $$\frac{1}{\ell!}\,\Delta^{\ell}\pi(x)=\sum_{j=0}^{k-\ell}\sum_{m=j}^{k}\pi[m]{m \choose j}
{\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}x^{j}.$$ If $k=0$ or $\ell > k$, then $\Delta^{\ell}\pi=0$.
The cases $k=0$ and $\ell > k$ are clear. For $k\geq 1$ and $1\leq \ell\leq k$ we use the following well-known formula (see e.g. [@graham94 p. 188]): $$(\Delta^{\ell}\pi)(x)=\sum_{s=0}^{\ell}{\ell \choose s}(-1)^{\ell-s}{\pi}(x+s).$$ Then by we obtain $$\begin{aligned}
(\Delta^{\ell}\pi)(x)&=\sum_{s=0}^{\ell}{\ell \choose s}(-1)^{\ell-s}\sum_{m=0}^k\pi[m](x+s)^m\\
&=\sum_{s=0}^{\ell}{\ell \choose s}(-1)^{\ell-s}\sum_{m=0}^k\pi[m]\sum_{j=0}^{m}{m \choose j} x^{j}s^{m-j}\\
&=\sum_{j=0}^k\sum_{m=j}^{k}\ell! \,\pi[m]{m \choose j} {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}x^{j}.\end{aligned}$$ Since the degree of $\Delta^{\ell}\pi$ is $k-\ell$, we obtain the result $$(\Delta^{\ell}\pi)(x)=\sum_{j=0}^{k-\ell}\sum_{m=j}^{k}\ell! \,\pi[m]{m \choose j} {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}x^{j}.$$ Note that the vanishing of $(\Delta^{\ell}\pi)[j]$ for $j=k-\ell+1,\ldots,k$ also follows from : For $m=j,\ldots, k$, the value $m-j \in \{0,\ldots,\ell-1\}$, and therefore in these cases ${\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}=0$.
Solving a recursion with iterated forward differences
-----------------------------------------------------
In this section we set the basis for the results in . We solve the recursion of iteratively applying operators of the from to polynomials (more generally, functions).
\[lem:recursion\_pq\] Let $f^{[1]}_k: \mathbb{R} \rightarrow \mathbb{R}$ and $g^{[1]}_k: \mathbb{R} \rightarrow \mathbb{R}$ ($k=0,1,\dots$) be two sequences of real-valued functions and let $(a_{n}, n\geq 1)$, be a sequence of real numbers. We define invertible matrices by $$V^{[n]}=\left[
\begin{array}{cc}
1 & 0 \\
a_{n} & 1
\end{array}
\right], \quad n\geq 1$$ and the sequences of real-valued functions $(f^{[n+1]}_k, k \geq 0)$ and $(g^{[n+1]}_k:k \geq 0)$ by $$\begin{aligned}
\label{al:def_f_g}
\left[ \begin{array}{c}
f_k^{[n+1]}\\
g_k^{[n+1]}
\end{array}
\right]=
\Delta_{V^{[n]}}
\left[ \begin{array}{c}
f_{k+1}^{[n]}\\
g_{k+1}^{[n]}
\end{array}
\right], \quad n \geq 1,\, k\geq 0.\end{aligned}$$ Then for $n \geq 1$ and $k \geq 0$ we have $$\begin{aligned}
\label{al:it_f_g}
f_k^{[n+1]}&=\Delta^{n}f_{k+n}^{[1]},\\ \nonumber
g_k^{[n+1]}&=g_{k+n}^{[1]}-\sum_{\ell=1}^{n}a_{\ell}\Delta^{\ell-1}f_{k+n}^{[1]},\end{aligned}$$ with the understanding that $\Delta^0=\operatorname{id}$.
We prove this claim by induction on $n$. Observe from that $$\Delta_{ V^{[n]}}=\left[
\begin{array}{cc}
\Delta & 0 \\
-a_{n}& 1
\end{array}
\right].$$ Starting with $n=1$, from we obtain $$\begin{aligned}
f_k^{[2]}&=\Delta f_{k+1}^{[1]},\\
g_k^{[2]}&=g_{k+1}^{[1]}-a_{1}f_{k+1}^{[1]},\end{aligned}$$ which is exactly for $n=1$. Assume that the statement is true for $n$, we prove it for $n+1$: $$\begin{aligned}
f_k^{[n+1]}&=\Delta f_{k+1}^{[n]} = \Delta \Delta^{n-1}f_{k+1+n-1}^{[1]}=\Delta^{n}f_{k+n}^{[1]},\\
g_k^{[n+1]}&=g_{k+1}^{[n]}-a_{n}f_{k+1}^{[n]}\\
&=g_{k+1+n-1}^{[1]}-\sum_{\ell=1}^{n-1}a_{\ell}\Delta^{\ell-1}f_{k+1+n-1}^{[1]}
-a_{n}\Delta^{n-1}f_{k+1+n-1}^{[1]}\\
&=g_{k+n}^{[1]}-\sum_{\ell=1}^{n-1}a_{\ell}\Delta^{\ell-1}f_{k+n}^{[1]}
-a_{n}\Delta^{n-1}f_{k+n}^{[1]}\\
&=g_{k+n}^{[1]}-\sum_{\ell=1}^{n}a_{\ell}\Delta^{\ell-1}f_{k+n}^{[1]},\end{aligned}$$ which concludes the induction step.
With a suitable choice of $(f^{[1]}_k, k \geq 0)$ and $(g^{[1]}_k:k \geq 0)$ from we obtain the following
\[cor:pi\_sig\] Let $(h_k:k\geq 0)$ be a sequence of differentiable functions. Setting $f^{[1]}_k=\Delta D h_{k+2}$ and $g^{[1]}_k=\Delta h_{k+2}-Dh_{k+2}$, $k\geq 0$, in , then with $a_{0}=1$ $$\begin{aligned}
\label{def:p}
f_k^{[n]} &=\Delta^{n}Dh_{k+n+1}, \\ \label{def:q}
g_k^{[n]} &=\Delta h_{k+n+1}-\sum_{\ell=0}^{n-1}a_{\ell}
\Delta^{\ell}Dh_{k+n+1},\end{aligned}$$ for $n\geq 1,\, k\geq 0$, and with the understanding that $\Delta^0=\operatorname{id}$.
We now consider the sequence applied to polynomials.
\[prop:p\] Let $(\tau_k: k\geq 0)$ be a sequence of polynomials such that $\tau_k \in \Pi_k,\, k\geq 0$. For $n\geq 1$ and $k\geq 0$ define ${\pi}_k^{[n]}=\Delta^{n}D\tau_{k+n+1}$. Then ${\pi}_k^{[n]}\in \Pi_k$ and its coefficients are given by $$\frac{1}{n!}\, {\pi}_k^{[n]}[j]=\sum_{m=j}^{k+n}(m+1)\tau_{k+n+1}[m+1]{m \choose j}
{\genfrac{\{}{\}}{0pt}{}}{m-j}{n}, \quad j=0,\ldots,k.$$
implies that the degree of $\pi_k^{[n]}$ is ${\operatorname{deg}}(\tau_{k+n+1})-n-1=k$. From and we obtain the coefficients as stated in the Proposition.
Similarly, we now consider the sequence applied to polynomials.
\[prop:q\] Let $(\tau_k: k\geq 0)$ be a sequence of polynomials such that $\tau_k \in \Pi_k,\, k\geq 0$. For $n\geq 1, \, k \geq 0$, define ${\sigma}_k^{[n]}=\Delta \tau_{k+n+1}-\sum_{\ell=0}^{n-1}a_{\ell}
\Delta^{\ell}D\tau_{k+n+1}$, where $(a_n:n\geq 0)$ with $a_0=1$ is a real-valued sequence. Then ${\sigma}_k^{[n]}\in \Pi_{k+n}$ and its coefficients are given by $$\begin{aligned}
{\sigma}_k^{[n]}[j]
=\sum_{m=j}^{k+n}{m+1 \choose j} \left(1
-(m+1-j)\sum_{\ell=0}^{n-1} a_{\ell}\,\ell!\, {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}
\right)\tau_{k+n+1}[m+1],
\end{aligned}$$ $j=0,\ldots,k+n$.
The degree of ${\sigma}_k^{[n]}$ is $k+n$, since by each application of the operators $D$ and $\Delta$ decrease the degree by $1$. Furthermore, from and , for $j=0,\ldots,k+n$ we obtain $$\begin{aligned}
{\sigma}_k^{[n]}[j]=&\sum_{m=j}^{k+n} {m+1 \choose j}\tau_{k+n+1}[m+1]\\
&-\sum_{\ell=0}^{n-1}a_{\ell}\sum_{s=j}^{k+n}(s+1)\tau_{k+n+1}[s+1]{s \choose j}\ell! \, {\genfrac{\{}{\}}{0pt}{}}{s-j}{\ell}\\
=&\sum_{m=j}^{k+n} \left({r+1 \choose j}
-(r+1){r \choose j}\sum_{\ell=0}^{n-1}a_{\ell} \,
\ell! \, {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}\right)\tau_{k+n+1}[m+1]\\
=&\sum_{m=j}^{k+n}{m+1 \choose j} \left(1
-(m+1-j)\sum_{\ell=0}^{n-1}a_{\ell} \,
\ell! \, {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}\right)\pi_{k+n+1}[m+1],
\end{aligned}$$ which proves the claim.
The following observation is the key result that makes our construction work (see the proof of ): If in the elements of the sequence $(a_n:n\geq 0)$ are the Gregory coefficients, then the degree of the polynomial $\sigma^{[n]}_k$ is $k$:
\[prop:degree\_q\] Let $(\tau_k: k\geq 0)$ be a sequence of polynomials such that $\tau_k \in \Pi_k,\, k\geq 0$. For $n\geq 1, \, k \geq 0$, define ${\sigma}_k^{[n]}=\Delta \tau_{k+n+1}-\sum_{\ell=0}^{n-1}G_{\ell}
\Delta^{\ell}D\tau_{k+n+1}$, where $G_n$ are the Gregory coefficients . Then ${\sigma}_k^{[n]}\in \Pi_k$.
Note that ${\sigma}_k^{[n]}$ is the same sequence as in with $a_n=G_n$ and that $G_0=1$. Therefore we know that ${\operatorname{deg}}({\sigma}_k^{[n]})=n+k$. In order to prove the statement of this Proposition, we have to show that $${\sigma}_k^{[n]}[j]=0, \quad j=k+1,\ldots, k+n.$$ This can be deduced from the following observation: For $j\in {\mathbb{N}}$ and $m=j,\ldots,k+n$, we have $m-j=0,\ldots, k+n-j$. Now if $j=k+1,\ldots,k+n$, then $m-j\in \{0,\ldots,n-1\}$. In particular, $m-j \leq n-1$. Therefore, in this case $$\label{eq:greg}
\sum_{\ell=0}^{n-1}G_{\ell}\,
\ell! \, {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}
=\sum_{\ell=0}^{m-j}G_{\ell}\,
\ell! \, {\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}
=\frac{1}{m-j+1},$$ using that ${\genfrac{\{}{\}}{0pt}{}}{m-j}{\ell}=0$ for $\ell \geq m-j$ (see ) and the relation between the Stirling numbers of the second kind and the Gregory coefficients. Now from the form of ${\sigma}_k^{[n]}[j]$ in , we see that implies the vanishing of ${\sigma}_k^{[n]}[j]$ for $j=k+1,\ldots,k+n$.
Statement and proof of the main results {#sec:proof}
=======================================
The main results of this paper are formulated and proved in , and . They show that every Hermite subdivision operator of spectral order $d$ can be factorized as in , and that this factorization is with respect to the Gregory operators . Furthermore, we give an explicit characterization of the eigenspaces in , and an easy-to-check criterion for $C^d$-convergence of an Hermite subdivision scheme of spectral order $d$ ().
\[prop:case0\] Let $S_{{\boldsymbol{A}}}$ be an Hermite subdivision operator of spectral order $d$, $d\geq 1$. Denote by ${\mathscr{P}}_k, k=0,\ldots,d$, its spectral polynomials (). Then there exists a subdivision operator $S_{{\boldsymbol{B}}^{[0]}}$ such that $$\begin{aligned}
\label{al:taylor}
TS_{{\boldsymbol{A}}}=2^{-1}S_{{\boldsymbol{B}}^{[0]}}T, \end{aligned}$$ with ${\mathcal{E}}_{{\boldsymbol{B}}^{[0]}}$ spanned by $[0,1]^T$. Furthermore, for $k=0,\ldots,d-1$, the polynomials $$\begin{aligned}
p_k^{[0]}&:=\Delta {\mathscr{P}}_{k+1}-D{\mathscr{P}}_{k+1}, \\
q_k^{[0]}&:=D{\mathscr{P}}_{k+1},\end{aligned}$$ satisfy $$\label{al:V_factor}
S_{{\boldsymbol{B}}^{[0]}}\left[ \begin{array}{cc}
p_k^{[0]}\\
q_k^{[0]}
\end{array}
\right]
= 2^{-k}
\left[ \begin{array}{cc}
p_k^{[0]}\\
q_k^{[0]}
\end{array}
\right]$$ and $p_0^{[0]}=0,q_0^{[0]}=1$. If $d>1$, then $p_k^{[0]} \in \Pi_{k-1}$, $q_k^{[0]}\in \Pi_k$ for $k=1,\ldots, d-1$.
The existence of $S_{{\boldsymbol{B}}^{[0]}}$ as well as the form of the eigenspace follow from [@merrien12] (which is summarized in our ). We prove the part involving the polynomials $p_k^{[0]},q_k^{[0]}$. By definition $$\begin{aligned}
\left[ \begin{array}{c}
p_k^{[0]}\\
q_k^{[0]}
\end{array}
\right]=
T
\left[ \begin{array}{c}
{\mathscr{P}}_{k+1}\\
D{\mathscr{P}}_{k+1}
\end{array}
\right], \quad k=0,\ldots,d-1.\end{aligned}$$ Therefore $$\begin{aligned}
S_{{\boldsymbol{B}}^{[0]}}\left[ \begin{array}{cc}
p_k^{[0]}\\
q_k^{[0]}
\end{array}
\right]
&= S_{{\boldsymbol{B}}^{[0]}} T \left[ \begin{array}{c}
{\mathscr{P}}_{k+1}\\
D{\mathscr{P}}_{k+1}
\end{array}
\right]
= 2 T S_{{\boldsymbol{A}}}\left[ \begin{array}{c}
{\mathscr{P}}_{k+1}\\
D{\mathscr{P}}_{k+1}
\end{array}
\right]
= 2 \cdot 2^{-k-1}T\left[ \begin{array}{c}
{\mathscr{P}}_{k+1}\\
D{\mathscr{P}}_{k+1}
\end{array}
\right] \\
&= 2^{-k} \left[ \begin{array}{c}
p^{[0]}_k\\
q^{[0]}_k
\end{array}
\right]. \end{aligned}$$ It is easy to see that $q^{[0]}_0=1$. The degree of $q^{[0]}_k$ is $k$ since $D$ decreases the degree of ${\mathscr{P}}_{k+1}$ by $1$. implies that $p_0^{[0]}=0$ and that the degree of $p_k^{[0]}$ is $k-1$, $k=1,\ldots, d-1$, if $d>1$.
\[thm\_main\] Let $d\geq 2$ and let $S_{{\boldsymbol{A}}}$ be an Hermite subdivision operator of spectral order $d$. Denote by ${\mathscr{P}}_k, k=0,\ldots,d$, its spectral polynomials (). Then we have the following:
1. For $n=1,\ldots,d-1$ there exist subdivision operators $S_{{\boldsymbol{B}}^{[n]}}$ such that $$\begin{aligned}
\label{al:taylor2}
\Delta_{V^{[n-1]}} \cdots \Delta_{V^{[0]}} T
S_{{\boldsymbol{A}}}=2^{-(n+1)}S_{{\boldsymbol{B}}^{[n]}}\Delta_{V^{[n-1]}} \cdots
\Delta_{V^{[0]}} T,\end{aligned}$$ where $$V^{[0]}=\left[
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right], \quad
V^{[n]}=\left[
\begin{array}{cc}
1 & 0 \\
G_n & 1
\end{array}
\right], \quad n=1,\ldots,d-1,$$ and $G_{n}$ are the Gregory coefficients of . Furthermore, the eigenspaces ${\mathcal{E}}_{{\boldsymbol{B}}^{[n]}}$ are spanned by $[1,G_n]^T$.
2. For $n=1,\ldots,d-1$ we define polynomials $p_k^{[n]},q_k^{[n]}$ by $$\begin{aligned}
p_k^{[n]} &:=\Delta^{n}D{\mathscr{P}}_{k+n+1}, \quad k=0,\ldots,d-1-n,\\
q_k^{[n]} &:=\Delta {\mathscr{P}}_{k+n+1}-\sum_{\ell=0}^{n-1}G_{\ell}
\Delta^{\ell}D{\mathscr{P}}_{k+n+1}, \quad k=0,\ldots,d-1-n.\end{aligned}$$ They satisfy $$\begin{aligned}
\label{al:poly_reprod}
&S_{{\boldsymbol{B}}^{[n]}}\left[ \begin{array}{cc}
p_k^{[n]}\\
q_k^{[n]}
\end{array}
\right]
= 2^{-k}
\left[ \begin{array}{cc}
p_k^{[n]}\\
q_k^{[n]}
\end{array}
\right], \quad k=0,\ldots, d-1-n
\end{aligned}$$ and $p_k^{[n]},q_k^{[n]} \in \Pi_k$.
Note that $p_k^{[n]},q_k^{[n]}$ are exactly the sequences of and using $\tau_k={\mathscr{P}}_k$ and $a_n=G_n$.
We fix $d\geq 2$ and prove this theorem by recursion on $n$, making use of .
*Case* $n=1$: implies that there exists $S_{{\boldsymbol{B}}^{[0]}}$ such that $TS_{{\boldsymbol{A}}}=2^{-1}S_{{\boldsymbol{B}}^{[0]}}T$ and ${\mathcal{E}}_{{\boldsymbol{B}}^{[0]}}$ is spanned by $[0,1]^T$. Therefore $V^{[0]}$ is an ${\mathcal{E}}_{{\boldsymbol{B}}^{[0]}}$-generator and by there exists $S_{{\boldsymbol{B}}^{[1]}}$ such that $$\Delta_{V^{[0]}} S_{{\boldsymbol{B}}^{[0]}}=2^{-1}S_{{\boldsymbol{B}}^{[1]}}\Delta_{V^{[0]}}.$$ Therefore $$\Delta_{V^{[0]}} T
S_{{\boldsymbol{A}}}=2^{-1}\Delta_{V^{[0]}}S_{{\boldsymbol{B}}^{[0]}}T
=2^{-2}S_{{\boldsymbol{B}}^{[1]}}\Delta_{V^{[0]}}T,$$ for $k=0,\ldots,d-2$, which proves . With $p_{k}^{[0]},q_{k}^{[0]}$ defined in , we have $$\label{eq:def_p1_q1}
p_k^{[1]}=\Delta q_{k+1}^{[0]}, \quad q_k^{[1]}=p_{k+1}^{[0]}, \quad k=0,\ldots,d-2.$$ That is, $$\begin{aligned}
\left[ \begin{array}{c}
p_k^{[1]}\\
q_k^{[1]}
\end{array}
\right]
=
\left[ \begin{array}{cc}
0 & \Delta \\
1 & 0
\end{array}
\right]
\left[ \begin{array}{c}
p_{k+1}^{[0]}\\
q_{k+1}^{[0]}
\end{array}
\right]
=
\Delta_{V^{[0]}}
\left[ \begin{array}{c}
p_{k+1}^{[0]}\\
q_{k+1}^{[0]}
\end{array}
\right]. \end{aligned}$$ Again using , this implies $$\begin{aligned}
\label{al:eigen}
S_{{\boldsymbol{B}}^{[1]}}\left[ \begin{array}{c}
p_k^{[1]}\\
q_k^{[1]}
\end{array}
\right]
& = S_{{\boldsymbol{B}}^{[1]}}\Delta_{V^{[0]}} \left[ \begin{array}{c}
p_{k+1}^{[0]}\\
q_{k+1}^{[0]}
\end{array}
\right]
= 2\Delta_{V^{[0]}} S_{{\boldsymbol{B}}^{[0]}}\left[ \begin{array}{c}
p_{k+1}^{[0]}\\
q_{k+1}^{[0]}
\end{array}
\right]\\ \nonumber
&= 2^{-k}\Delta_{V^{[0]}}\left[ \begin{array}{c}
p^{[0]}_{k+1}\\
q^{[0]}_{k+1}
\end{array}
\right]
= 2^{-k}\left[ \begin{array}{c}
p_k^{[1]}\\
q_k^{[1]}
\end{array}
\right],
\end{aligned}$$ proving .
From we know that $q_{k+1}^{[0]} \in \Pi_{k+1}$ and $p_{k+1}^{[0]}\in \Pi_k$ and thus implies that $p_{k}^{[1]},q_{k}^{[1]} \in \Pi_k$, $k=0,\ldots,d-2$.
In particular $p^{[1]}_0,q^{[1]}_0$ are constants. From the computation we see that $[p^{[1]}_0,q^{[1]}_0]^T$ lies the eigenspace of $S_{{\boldsymbol{B}}^{[1]}}$ with respect to the eigenvalue $1$. Using the explicit form of $p^{[1]}_0,q^{[1]}_0$ from and we compute $$\begin{aligned}
&p_0^{[1]}=p_0^{[1]}[0]=1,\\
&q_0^{[1]}=q_0^{[1]}[0]=1/2=G_1.
\end{aligned}$$ Hence the eigenspace ${\mathcal{E}}_{{\boldsymbol{B}}^{[1]}}\neq \{0\}$ and by it has dimension $1$. Therefore it is spanned by $[p^{[1]}_0,q^{[1]}_0]^T=[1,G_1]^T$. This concludes the proof for $n=1$.
If $d=2$ nothing else needs to be shown, since $n=1$ only. We now prove that if $d>2$ the case $n-1$ implies the case $n$, for $n=2,\ldots,d-1$.
We assume that the statements of the theorem are satisfied for $n-1$ and prove it for $n$.
By assumption there exists a subdivision operator $S_{{\boldsymbol{B}}^{[n-1]}}$ such that $$\begin{aligned}
\Delta_{V^{[n-2]}} \cdots \Delta_{V^{[0]}} T
S_{{\boldsymbol{A}}}=2^{-n}S_{{\boldsymbol{B}}^{[n-1]}}\Delta_{V^{[n-2]}} \cdots
\Delta_{V^{[0]}} T\end{aligned}$$ and the eigenspace ${\mathcal{E}}_{{\boldsymbol{B}}^{[n-1]}}$ is spanned by $[1,G_{n-1}]^T$. Therefore $V^{[n-1]}$ is an ${\mathcal{E}}_{{\boldsymbol{B}}^{[n-1]}}$-generator and by there exists a subdivision operator $S_{{\boldsymbol{B}}^{[n]}}$ such that $$\Delta_{V^{[n-1]}} S_{{\boldsymbol{B}}^{[n-1]}}=2^{-1}S_{{\boldsymbol{B}}^{[n]}}\Delta_{V^{[n-1]}}.$$ This implies $$\begin{aligned}
\Delta_{V^{[n-1]}} \cdots \Delta_{V^{[0]}} T
S_{{\boldsymbol{A}}}
&=2^{-n}\Delta_{V^{[n-1]}}S_{{\boldsymbol{B}}^{[n-1]}}\Delta_{V^{[n-2]}} \cdots
\Delta_{V^{[0]}} T\\
&=2^{-(n+1)}S_{{\boldsymbol{B}}^{[n]}}\Delta_{V^{[n-1]}}\Delta_{V^{[n-2]}} \cdots
\Delta_{V^{[0]}} T,\end{aligned}$$ which proves .
Since $p_k^{[n]},q_k^{[n]}$ are the sequences of and with $\tau_k={\mathscr{P}}_k$ and $a_n=G_n$, we know from that $$\begin{aligned}
\left[ \begin{array}{c}
p_k^{[n]}\\
q_k^{[n]}
\end{array}
\right]=
\Delta_{V^{[n-1]}}
\left[ \begin{array}{c}
p_{k+1}^{[n-1]}\\
q_{k+1}^{[n-1]}
\end{array}
\right], \quad k=0,\ldots, d-1-n.\end{aligned}$$ Therefore $$\begin{aligned}
\label{al:n}
S_{{\boldsymbol{B}}^{[n]}}\left[ \begin{array}{c}
p_k^{[n]}\\
q_k^{[n]}
\end{array}
\right]
& = S_{{\boldsymbol{B}}^{[n]}}\Delta_{V^{[n-1]}} \left[ \begin{array}{c}
p_{k+1}^{[n-1]}\\
q_{k+1}^{[n-1]}
\end{array}
\right]
= 2\Delta_{V^{[n-1]}} S_{{\boldsymbol{B}}^{[n-1]}}\left[ \begin{array}{c}
p_{k+1}^{[n-1]}\\
q_{k+1}^{[n-1]}
\end{array}
\right]\\ \nonumber
&= 2^{-k}\Delta_{V^{[n-1]}}\left[ \begin{array}{c}
p^{[n-1]}_{k+1}\\
q^{[n-1]}_{k+1}
\end{array}
\right]
= 2^{-k}\left[ \begin{array}{c}
p_k^{[n]}\\
q_k^{[n]}
\end{array}
\right],
\end{aligned}$$ which proves .
Since ${\mathscr{P}}_k\in \Pi_k$, from and we can conclude that $p^{[n]}_k,q^{[n]}_k\in \Pi_k$.
In particular, $p^{[n]}_0,q^{[n]}_0$ are constants and from the computation we see that $[p^{[n]}_0,q^{[n]}_0]^T$ lies in the eigenspace of $S_{{\boldsymbol{B}}^{[n]}}$ with respect to the eigenvalue $1$. Using the explicit formula of we get $$\begin{aligned}
p^{[n]}_0
&=p^{[n]}_0[0]
=n!\, \sum_{m=0}^{n}(m+1){\mathscr{P}}_{n+1}[m+1]{m \choose 0}{\genfrac{\{}{\}}{0pt}{}}{m}{n}\\
&=n!\, (n+1){\mathscr{P}}_{n+1}[n+1]{\genfrac{\{}{\}}{0pt}{}}{n}{n}=\frac{(n+1)!}{(n+1)!}\\
&=1,\end{aligned}$$ where we use the properties of the Stirling numbers of the second kind and the fact that ${\mathscr{P}}_{\ell}[\ell]=1 / \ell !$ from .
Continuing with the explicit formula for $q^{[n]}_0$ from we also get $$\begin{aligned}
q^{[n]}_0=&\, q^{[n]}_0[0]\\
=& \sum_{m=0}^{n}{m+1 \choose 0} \left(1
-(m+1)\sum_{\ell=0}^{n-1} G_{\ell}\ell!\, {\genfrac{\{}{\}}{0pt}{}}{m}{\ell}
\right){\mathscr{P}}_{n+1}[m+1]\\
=& \sum_{m=0}^{n} \left(1
-(m+1)\sum_{\ell=0}^{\min\{m,n-1\}} G_{\ell}\ell!\, {\genfrac{\{}{\}}{0pt}{}}{m}{\ell}
\right){\mathscr{P}}_{n+1}[m+1]\\
=& \sum_{m=0}^{n-1}\left(1
-(m+1)\sum_{\ell=0}^{m} G_{\ell}\ell!\, {\genfrac{\{}{\}}{0pt}{}}{m}{\ell}
\right){\mathscr{P}}_{n+1}[m+1]\\
&+
\left(1
-(n+1)\sum_{\ell=0}^{n-1} G_{\ell}\ell!\, {\genfrac{\{}{\}}{0pt}{}}{n}{\ell}
\right){\mathscr{P}}_{n+1}[n+1]\\
=& \sum_{m=0}^{n-1}\left(1
-(m+1)\frac{1}{m+1}\right){\mathscr{P}}_{n+1}[m+1]\\
&+
\left(1
-(n+1)\left(\frac{1}{n+1}-G_n n! {\genfrac{\{}{\}}{0pt}{}}{n}{n}\right)
\right){\mathscr{P}}_{n+1}[n+1]\\
=& \,(n+1)\,G_n\, n! \, {\mathscr{P}}_{n+1}[n+1]\\
=& \,G_n,\end{aligned}$$ where again we use the properties of the Stirling numbers of the second kind , the relation , and the fact that ${\mathscr{P}}_{\ell}[\ell]=1 / \ell !$ from .
The eigenspace ${\mathcal{E}}_{{\boldsymbol{B}}^{[n]}}\neq \{0\}$ and thus, by , it has dimension $1$. It is therefore spanned by $[p_0^{[n]},q_0^{[n]}]^T=[1,G_n]^T$. This concludes the proof.
With notation as in , the operator used for factorizing is the $n$-th Gregory operator , that is $$\Delta_{V^{[n-1]}}\cdots \Delta_{V^{[0]}}T
={\mathscr{G}}^{[n]}=\left[\begin{array}{cc}
0 & \Delta^n \\
\Delta & -\sum_{\ell=0}^{n-1}G_{\ell}\Delta^{\ell}
\end{array}\right], \quad n=1,\ldots,d-1.$$
From and we get one additional factorization:
\[prop:greg\_n\] Let $d\geq 1$ and let $S_{{\boldsymbol{A}}}$ be an Hermite subdivision operator of spectral order $d$. Then for $n=1,\ldots, d$, the operator $S_{{\boldsymbol{A}}}$ factorizes with respect to the $n$-th Gregory operator ${\mathscr{G}}^{[n]}$ , that is, there exist subdivision operators $S_{{\boldsymbol{B}}^{[n]}}$ such that $${\mathscr{G}}^{[n]}S_{{\boldsymbol{A}}}=2^{-(n+1)}S_{{\boldsymbol{B}}^{[n]}}{\mathscr{G}}^{[n]}, \quad n=1,\ldots, d.$$
For $d=1$, from , we get a factorization $$TS_{{\boldsymbol{A}}}=2^{-1}S_{{\boldsymbol{B}}^{[0]}}T,$$ and we know that ${\mathcal{E}}_{{\boldsymbol{B}}^{[0]}}$ is spanned by $[0,1]^T$. Therefore $V^{[0]}$ defined in is an ${\mathcal{E}}_{{\boldsymbol{B}}^{[0]}}$-generator and by there exists $S_{{\boldsymbol{B}}^{[1]}}$ such that $$\Delta_{V^{[0]}}TS_{{\boldsymbol{A}}}=2^{-2}S_{{\boldsymbol{B}}^{[1]}}\Delta_{V^{[0]}}T,$$ and $\Delta_{V^{[0]}}T={\mathscr{G}}^{[1]}$. This concludes the case $d=1$.
Now if $d\geq 2$, from we obtain $S_{{\boldsymbol{B}}^{[n]}}$ such that $${\mathscr{G}}^{[n]}S_{{\boldsymbol{A}}}=2^{-(n+1)}S_{{\boldsymbol{B}}^{[n]}}{\mathscr{G}}^{[n]}, \quad n=1,\ldots,d-1.$$ We know that ${\mathcal{E}}_{{\boldsymbol{B}}^{[d-1]}}$ is spanned by $[1,G_{d-1}]^T$. Therefore, by , we can factorize with respect to $$V^{[d-1]}= \left[ \begin{array}{cc}
1 & 0 \\
G_{d-1} & 1
\end{array}
\right]$$ and obtain that there exists a subdivision operator $S_{{\boldsymbol{B}}^{[d]}}$ such that $${\mathscr{G}}^{[d]}S_{{\boldsymbol{A}}}=2^{-(d+1)}S_{{\boldsymbol{B}}^{[d]}}{\mathscr{G}}^{[d]}.$$ This concludes the proof for $d\geq 2$.
Note that with $S_{{\boldsymbol{C}}^{[n]}}=2^{-1}S_{{\boldsymbol{B}}^{[n]}}$, is exactly the main theorem stated in the introduction ().
The factorization of together with the Taylor factorization of satisfies the assumptions of . Therefore we get a criterion to check the $C^d$-convergence, $d\geq 1$, of an Hermite subdivision scheme of spectral order $d$: If $2^{-1}S_{{\boldsymbol{B}}^{[d]}}$ is contractive, then the Hermite scheme associated with $S_{{\boldsymbol{A}}}$ is $C^d$-convergent. By considering $S_{{\boldsymbol{B}}}=2^{-1}S_{{\boldsymbol{B}}^{[d]}}$, we thus obtain from the introduction:
\[cor:greg\_d\] Let $d\geq1$ and let $S_{{\boldsymbol{A}}}$ be an Hermite subdivision operator of spectral order $d$. Then there exists $S_{{\boldsymbol{B}}}$ such that $${\mathscr{G}}^{[d]}S_{{\boldsymbol{A}}}=2^{-d}S_{{\boldsymbol{B}}}{\mathscr{G}}^{[d]},$$ where ${\mathscr{G}}^{[d]}$ is the $d$-th Gregory operator . If $S_{{\boldsymbol{B}}}$ is contractive, then the Hermite subdivision scheme associated with $S_{{\boldsymbol{A}}}$ is $C^d$-convergent.
Note that since the spectral condition of order $d$ implies the spectral condition of order $\ell$, for every $\ell \leq d$, can be used to prove any regularity $\ell \leq d$ of the Hermite scheme. This is useful for schemes which have lower regularity than polynomial reproduction order, see e.g. some of the examples in [@jeong17].
Examples {#sec:ex}
========
In this section we provide an algorithm for computing the $n$-th Gregory factorization using symbols and apply it to an example of [@jeong17]. We also show that this example is an incident of an Hermite scheme which satisfies the spectral condition but does not reproduce polynomials, proving that these concepts are not equivalent.
\[alg:general\_ex\] We show how the $n$-th Gregory factorization can be computed using *symbols*. The *symbol* of a sequence ${\mathbf{c}}\in \ell({\mathbb{Z}})_0^{2}$ is the Laurent polynomial $${\mathbf{c}}^{\ast}(z)=\sum_{j\in{\mathbb{Z}}}{\mathbf{c}}_j z^j, \quad z\in {\mathbb{C}}\,\backslash \,\{0\}.$$ Similarly, we can define ${\boldsymbol{A}}^{\ast}(z)$ for ${\boldsymbol{A}}\in \ell({\mathbb{Z}})^{2\times 2}_{0}$. It is well-known [@dyn02; @merrien12] that a factorization of the form relates to the following equation in symbols: $${{\mathscr{G}}^{[n]}}^{\ast}(z){\boldsymbol{A}}^{\ast}(z)=2^{-n}{{\boldsymbol{B}}^{[n]}}^{\ast}(z){{\mathscr{G}}^{[n]}}^{\ast}(z^2).$$ With ${\boldsymbol{A}}^{\ast}(z)=\left[{\mathbf{a}}^{\ast}_{jk}(z) \right]_{j,k=1}^2$ and $g^{\ast}_n(z)=-\sum_{\ell=0}^{n-1}G_{\ell}(z^{-1}-1)^{\ell}$ we obtain ${{\boldsymbol{B}}^{[n]}}^{\ast}(z)=\left[{{\mathbf{b}}^{[n]}_{jk}}^{\ast}(z) \right]_{j,k=1}^2$: $$\begin{aligned}
&{{\mathbf{b}}^{[n]}_{11}}^{\ast}(z)=2^{n}\frac{(z^{-2}-1){\mathbf{a}}_{22}^{\ast}(z)-g^{\ast}_n(z^2){\mathbf{a}}_{21}^{\ast}(z)}{(z^{-1}-1)(z^{-1}+1)^{n+1}},\\[0.2cm]
&{{\mathbf{b}}^{[n]}_{12}}^{\ast}(z)=2^{n}\frac{(z^{-1}-1)^{n-1}{\mathbf{a}}_{21}^{\ast}(z)}{z^{-1}+1},\\[0.2cm]
&{{\mathbf{b}}^{[n]}_{21}}^{\ast}(z)=2^{n}\\
&\frac{(z^{-2}-1)((z^{-1}-1){\mathbf{a}}_{12}^{\ast}(z)+g^{\ast}_n(z){\mathbf{a}}^{\ast}_{22}(z))-g^{\ast}_n(z^2)((z^{-1}-1){\mathbf{a}}_{11}^{\ast}(z)+g^{\ast}_n(z){\mathbf{a}}_{21}^{\ast}(z))}{(z^{-2}-1)^{n+1}},\\[0.2cm]
&{{\mathbf{b}}^{[n]}_{22}}^{\ast}(z)=2^{n}\frac{(z^{-1}-1){\mathbf{a}}_{11}^{\ast}(z)+g^{\ast}_n(z){\mathbf{a}}_{21}^{\ast}(z)}{(z^{-2}-1)},\end{aligned}$$ which can be computed, for example, with Mathematica.
\[ex:H1\] We consider the primal Hermite subdivision scheme $H_1$ proposed in [@jeong17]. Its mask is supported in $[-2,2]\cap {\mathbb{Z}}$ with nonzero elements given by $$\begin{aligned}
\left[
\begin{array}{rr} \theta & -\frac{\theta}{2}\\ -\frac{3\omega}{2} & \frac{\omega}{2}\end{array}\right], \,
\left[
\begin{array}{rr} \frac12 & -\frac18\\[0.1cm] \frac34& -\frac{1}{8}\end{array}\right], \,
\left[
\begin{array}{cc} 1-2\theta & 0\\ 0 & \frac{1+4\omega}{2}\end{array}\right], \,
\left[
\begin{array}{rr} \frac12 & \frac18\\[0.1cm] -\frac34& -\frac{1}{8}\end{array}\right], \,
\left[
\begin{array}{rr} \theta & \frac{\theta}{2}\\[0.05cm] \frac{3\omega}{2}& \frac{\omega}{2}\end{array}\right],\end{aligned}$$ with parameters $\theta, \omega \in {\mathbb{R}}$.
In [@jeong17] it is proved that $H_1$ reproduces polynomials up to degree $3$ and thus it satisfies the spectral condition up to order $3$ with spectral polynomials $1,x,\tfrac{1}{2!}x^2,\tfrac{1}{3!}x^3$, see [@conti14]. By , we have Gregory factorizations for $n=1,2,3$.
It is easy to see that the scheme $H_1$ does not satisfy the spectral condition of order $4$ with spectral polynomial $\tfrac{1}{4!}\,x^4$ for all parameters $\theta,\omega$. This implies that it does not reproduce polynomials of degree $4$ for all parameters $\theta, \omega$, see [@conti14]. This can also be proved using the methods of [@conti18]. However, with $\theta=1/32$ it satisfies the spectral condition of order $4$ with $4$-th spectral polynomial given by ${\mathscr{P}}_{4}(x)=\tfrac{1}{4!}\,x^4+\tfrac{1}{360}$. Therefore, $H_1$ with $\theta=1/32$ provides an example of an Hermite scheme which does not reproduce polynomials of degree $4$, but satisfies the spectral condition of order $4$. To the best of our knowledge, this is the first time it is observed that the spectral condition is *not* equivalent to the reproduction of polynomials.
This explains why in [@jeong17] a factorization of $H_1$ up to order $n=4$ is possible, even though the mask only reproduces polynomials up to degree $3$. Of course we also have a $4$-th Gregory factorization for $\theta=1/32$, which we now provide using .
For $\theta=1/32$, the mask ${\boldsymbol{B}}^{[4]}$ of the $4$-th Gregory factorization is supported in $[-4,2]\cap {\mathbb{Z}}$: $$\begin{aligned}
\left[
\begin{array}{cc}
0 & -24\,\omega\\
0 & 0
\end{array}
\right], \,
\left[
\begin{array}{cc}
0 & 96\,\omega+12\\[0.1cm]
0 & \omega
\end{array}
\right], \,
\left[
\begin{array}{cc}
-\omega & -168\,\omega-48\\[0.1cm]
0 & -5\,\omega-\frac{1}{2}
\end{array}
\right],\end{aligned}$$ $$\begin{aligned}
\left[
\begin{array}{cc}
4\,\omega +\frac{1}{2} & 192\,\omega+72\\[0.1cm]
\frac{\omega}{24} & 20\,\omega+3
\end{array}
\right],\,
\left[
\begin{array}{cc}
-6\,\omega-2 & -168\,\omega-48 \\[0.1cm]
-\frac{5\,\omega}{24}-\frac{1}{48} & 4\,\omega-2
\end{array}
\right],\end{aligned}$$ $$\begin{aligned}
\left[
\begin{array}{cc}
4\,\omega +\frac{5}{2} & 96\,\omega+12\\[0.1cm]
\frac{19\,\omega}{24}+\frac{1}{8} & 19\,\omega+3
\end{array}
\right], \quad
\left[
\begin{array}{cc}
-\omega & -24\,\omega\\[0.1cm]
\frac{3\,\omega}{8}-\frac{1}{16} & 9\,\omega +\frac{1}{2}
\end{array}
\right].\end{aligned}$$ We would like to stress that $\theta=1/32$ is the only value for which we obtain a $4$-th Gregory factorization of the scheme $H_1$ (and thus the only value for which the spectral condition of order $4$ is satisfied). With the results of and our Gregory factorzation we can now analyze the smoothness of $H_1$. Numerical computations show that $\Vert(\frac{1}{2}S_{{\boldsymbol{B}}^{[4]}})^6\Vert_\infty<1$ for $\omega \in [-0.10210,-0.09582]$. Thus $H_1$ is $C^4$ for this range of $\omega$ which confirms the result of [@jeong17]. The advantage of our factorization however is that we only need 6 iterations to prove the contractivity of $S_{{\boldsymbol{B}}^{[4]}}$ whereas 24 iterations are needed in [@jeong17]. Therefore, we can enlarge the domain for $\omega$ and still obtain a smoothness result. Computations show that the Hermite scheme $H_1$ is $C^4$ for $\omega \in [-0.12,-0.088]$ since $\Vert(\frac{1}{2}S_{{\boldsymbol{B}}^{[4]}})^{10}\Vert_\infty<1$ for these values of $\omega$.
Conclusion {#sec:conclusion}
==========
In this paper we provide a novel factorization framework for Hermite subdivision operators based on Stirling numbers and Gregory coefficients. We further derive , which allows to easily compute the $n$-th Gregory factorization using symbols. The usefulness of the Gregory factorization is evident from the reduction of computational cost for proving $C^d$-convergence of an Hermite subdivision scheme: Only one factorization needs to be computed, independently of $d$ (). Certainly, the $d$-th Gregory factorization is not the only possible factorization for Hermite schemes of spectral order $d$, but the only one which is explicitly computed for general $d$. Furthermore, in , we provide an instance of an Hermite scheme which satisfies the spectral condition of order $d=4$, but does not reproduce polynomials of degree $4$, showing that the spectral condition is not equivalent to the reproduction of polynomials.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank B. Jeong and J. Yoon for sharing some of the masks from [@jeong17], which we used for validating our method.
S.H. acknowledges the support of the Austrian Science Fund (FWF): W1230.
C.C. acknowledges the support of GNCS-INdAM, Italy.
[^1]: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA. `[email protected]` (corresponding author)
[^2]: Institute of Geometry, TU Graz, Kopernikusgasse 24, 8010 Graz, Austria. `[email protected]`
[^3]: DIEF, Università di Firenze, Viale Morgagni 40/44, 50134 Firenze, Italy. `[email protected]`
|
---
abstract: 'The ability to uniquely identify a quantum state is integral to quantum science, but for non-orthogonal states, quantum mechanics precludes deterministic, error-free discrimination. However, using the non-deterministic protocol of unambiguous state discrimination (USD) enables error-free differentiation of states, at the cost of a lower frequency of success. We discriminate experimentally between non-orthogonal, high-dimensional states encoded in single photons; our results range from dimension $d=2$ to $d=14$. We quantify the performance of our method by comparing the total measured error rate to the theoretical rate predicted by minimum-error state discrimination. For the chosen states, we find a lower error rate by more than one standard deviation for dimensions up to $d=12$. This method will find immediate application in high-dimensional implementations of quantum information protocols, such as quantum cryptography.'
author:
- 'Megan Agnew,$^{1,2}$ Eliot Bolduc,$^{2}$ Kevin J. Resch,$^{1}$ Sonja Franke-Arnold,$^{3}$ Jonathan Leach$^{2}$'
title: 'Discriminating single-photon states unambiguously in high dimensions'
---
Discriminating between different quantum states without error is a fundamental requirement of quantum information science. However, due to the nature of quantum mechanics, only orthogonal states can be exactly discriminated without error 100% of the time. In contrast, the discrimination of non-orthogonal states requires a decrease in either detection accuracy, using minimum-error state discrimination, or detection frequency, using unambiguous state discrimination. Minimum-error state discrimination (MESD) always provides information about the state, though the information may be incorrect [@Helstrom]. Conversely, unambiguous state discrimination (USD) provides either the correct information about a detected state or inconclusive information about the state [@Ivanovic1987; @Dieks1988; @Peres1988; @Chefles1998; @Peres1998; @Sun2001; @Rudolph2003; @Raynal2005; @Raynal2007; @Jafarizadeh2008; @Sugimoto2010; @Waldherr2012; @Zhou2012; @Bergou2013].
High-dimensional quantum states are an important resource for quantum information. In comparison to qubits, the use of qu$d$its, which are states belonging to a $d$-dimensional space, provides access to a larger alphabet and correspondingly higher information rates, and a higher tolerance to noise. The ability to unambiguously discriminate such states is thus of key importance, and successful protocols that accomplish this task will extend the use of these states in quantum information science. Examples of such systems include the time degree of freedom and the spatial light profile, or more specifically the orbital angular momentum degree of freedom, which we use in this work [@Mair2001; @Leach2010; @Agnew2011; @Dada2011; @Agnew2012; @Agnew2013; @Donohue2013]. High-dimensional USD is also potentially relevant for pattern recognition in quantum and classical regimes as images contain typically very large numbers of spatial modes and are non-orthogonal to one another [@Malik2012].
The problem of unambiguous discrimination of qu$d$it states has received a great deal of attention [@Herzog2008; @Pang2009; @Bergou2012; @Chen2012; @Li2012; @Franke-Arnold2012]. USD was first experimentally realised, with a classical light source, to distinguish two non-orthogonal states in the polarisation degree of freedom [@Clarke2001]. A subsequent experiment with a similar source extended this to distinguish three states encoded in three-dimensional photon path information [@Mohseni2004]. USD has also been performed for two mixed polarisation states using a quantum dot single-photon source [@Steudle2011].
In this work, we discriminate unambiguously between non-orthogonal quantum states encoded in single photons, in dimensions ranging from $d=2$ to $d=14$. While USD theoretically promises the unambiguous discrimination of any set of states, real experimental situations always include error sources, and perfect discrimination in an experimental environment is challenging. Even with these unavoidable errors, we show that our scheme successfully discriminates between the chosen states and does so with lower error rates than those predicted by MESD. We note that here we implement USD as a sequential measurement of all required detection states. Using instead simultaneous detection, e.g., based on OAM sorter technology [@Berkhout2010], would allow unambiguous discrimination at the single-photon level.
To perfectly distinguish orthogonal states, one requires projections onto the orthogonal state basis, giving $d$ measurement outcomes in a $d$-dimensional space. To implement the USD protocol, which distinguishes non-orthogonal states, one requires the introduction of an additional measurement outcome – an inconclusive result – into the procedure, providing $d+1$ measurement possibilities. The increased number of measurement outcomes necessitates the introduction of an ancillary dimension or degree of freedom; orbital angular momentum lends itself well to this treatment as it provides an unlimited supply of additional dimensions. The introduction of the inconclusive result enables the remaining measurement outcomes to be orthogonalised [@Neumark1943]. The protocol then provides one of the following: a correct state identification, in which case the state is known with certainty, or an inconclusive result, in which case no information is known about the state.
![[]{data-label="theory"}](Fig1.pdf)
In this work, we choose $d$ states in $d$ dimensions that have an equal overlap with each other; these are referred to as equally probable, linearly independent, symmetrical states and, compared to less symmetric states, have a maximal discrimination probability [@Chefles1998] [^1]. See Fig. \[theory\](a) for an example in three dimensions. Note that all of these states have only real amplitudes. The overlap between any two states is then a function of the parameter $\theta$, given by $$\label{overlap}
{\langle\Psi_i|\Psi_j\rangle}=\frac{d\,{\rm cos}^2\theta-1}{d-1},$$ for $i \neq j$. To ensure positive overlap between the input states, the maximum value of $\theta$ is $\theta_{\rm max}={\rm cos}^{-1}\sqrt{1/d}$ .
In the problem of USD, we must establish a set of measurement states $\{{|D_i\rangle}\}$ to distinguish the set of input states $\{{|\Psi_i\rangle}\}$. To achieve this, for every state ${|\Psi_i\rangle}$ we first identify a preliminary measurement state ${|\Psi^\perp_i\rangle}$; this preliminary state is orthogonal to all other states ${|\Psi_j\rangle}$ (for $j \neq i$) but has a nonzero overlap with ${|\Psi_i\rangle}$. Due to this definition, a detection with ${| \Psi^\perp_i \rangle \langle \Psi^\perp_i |}$ will unambiguously indicate that the photon was in state ${|\Psi_i\rangle}$. These $d$ preliminary measurement states $\{{|\Psi^\perp_i\rangle}\}$, however, do not generally form an orthonormal basis set. This can be achieved by extending the preliminary measurement states to an ancillary dimension, followed by normalisation to obtain $d$ measurement states $\{{|D_i\rangle}\}$. The basis set is completed by including an additional state ${|D_{d+1}\rangle}$ orthogonal to all other measurement states, so that the whole $(d+1)$-dimensional basis of measurement states is $\{{|D_i\rangle}\}$ with ${\langleD_i|D_j\rangle}=\delta_{ij}$.
The probability of obtaining an inconclusive result, $|{\langle\Psi_i|D_{d+1}\rangle}|^2$, and the probability of correctly identifying a state, $|{\langle\Psi_i|D_i\rangle}|^2$, sum to unity as the probability of an error is by definition zero. The probability of an inconclusive result is precisely the overlap between any two input states [@Dieks1988; @Chefles1998]. Thus using Eq. (\[overlap\]), we can write the probabilities of successful identification, erroneous identification, and inconclusive result as
\[p\] $$\begin{aligned}
&p_{\rm suc}=\frac{d}{d-1}{\rm sin}^2\theta\\
&p_{\rm err}=0\\
&p_{\rm inc}=\frac{d\,{\rm cos}^2\theta-1}{d-1}.\end{aligned}$$
Theoretical predictions of these values for states in three dimensions are shown in Fig. \[theory\](b).
We use the process outlined above to find the discrimination states for a range of input states in a range of dimensions, and we use them to implement USD as a sequential measurement on orbital angular momentum states. Our experimental procedure is as follows. We produce entangled photons by spontaneous parametric downconversion (SPDC) [@Walborn2004] in a 3-mm type-I BBO crystal with a phase mismatch factor of approximately $\phi=-1$. We pump the crystal with a 100-mW laser at 405 nm. In each path, we image the plane of the BBO crystal to a different section of a spatial light modulator (SLM), allowing us to manipulate both the phase and the amplitude of each photon’s mode with high fidelity. The simplified experimental setup is shown in Fig. \[setup\].
The photons produced from the BBO crystal are entangled in their orbital angular momentum in the two-photon state ${|\psi\rangle}=\sum_{\ell=-\infty}^{\infty} c_\ell {|\ell\rangle}_A \otimes {|-\ell\rangle}_B$, where $|c_\ell|^2$ is the probability of finding photon $A$ with OAM $\ell \hbar$ and photon $B$ with OAM $-\ell \hbar$ [@Torres2003]. The SLM in our experiment performs two functions in regards to this state: first, it allows us to select a range of OAM values and explore a discrete dimension space, and second, it allows us to equalise the probabilities of detection, a process similar to entanglement concentration [@Bennett1996].
The entanglement of the OAM degree of freedom allows the use of remote state preparation [@Bennett2001; @Liu2007], which enables us to herald the presence of a range of single-photon states ${|\Psi_i\rangle}$. These heralded states are prepared by using one half of the SLM in combination with a single-mode fibre. Consequently, the detection of a single photon in the first arm collapses the photon in the other arm into the desired state. The second path is then used to perform the state discrimination measurements ${|D_j\rangle}$ on the heralded state ${|\Psi_i\rangle}$, and we measure the coincidences between the two paths.
For a given dimension $d$, we measure all $d+1$ measurement outcomes for each input state ${|\Psi_i\rangle}$. We use our measurements to calculate a quantity called the quantum contrast, which is defined by the coincidence rates normalised by the singles $Q_{ij}=C_{ij}/(S_{Ai} S_{Bj} t)$; this accounts for any variations in the quantum efficiency of the detection and generation of particular states. Here $C_{ij}$ is the number of coincidence counts defined by an event in both detectors within a time window of $t=25$ ns. The quantities $S_{Ai}$ and $S_{Bj}$ represent the number of counts in path $A$ (heralding the preparation of ${|\Psi_i\rangle}$) and $B$ (measuring ${|D_j\rangle}$) respectively. We normalise this quantum contrast into probabilities using $P_{ij}=(Q_{ij}-1)/\sum_j(Q_{ij}-1)$. The $-1$ term accounts for the fact that two independent and uncorrelated sources will have a quantum contrast equal to unity. An integration time of 30 s was used for each measurement, and the maximal coincidence count rate was approximately 350 Hz.
![[]{data-label="setup"}](Fig2.pdf)
We have implemented our procedure for unambiguous discrimination of states in high dimensions ranging from $d=2$ to $d=14$ and with varying overlap between the states. In Fig. \[exptdata\], we show the unambiguous discrimination of 6 states in $d=6$ dimensions.
Fig. \[exptdata\](a) shows the results at $\theta=40^\circ$ of measuring all $\{{|\Psi_i\rangle}\}$ states using all $\{{|D_j\rangle}\}$ measurements. The green bars denote successful identifications, the red bars denote erroneous identifications, and the blue bars denote inconclusive results. As the probabilities of successful identification greatly exceed the probabilities of erroneous identification, it follows that each input state ${|\Psi_i\rangle}$ almost always results in either correct detection by ${|D_i\rangle}$ or the inconclusive outcome ${|D_7\rangle}$.
Fig. \[exptdata\](b) shows the results of measuring a specific state, in this case ${|\Psi_2\rangle}$, using all $\{{|D_j\rangle}\}$ measurements, for a range of angles $\theta$. Each angle corresponds to a different overlap between the $\{{|\Psi_i\rangle}\}$ states as in Eq. (\[overlap\]). An angle of $0^\circ$ corresponds to a complete overlap between the states and hence a completely inconclusive result; the probability for correct identification increases with $\theta$, with in principle perfect identification at $\theta\approx66^\circ$. The solid lines indicate theoretical predictions from Eq. (\[p\]); our experimental data is in good agreement with these predictions.
![[]{data-label="exptdata"}](Fig3.pdf)
Whilst USD has the theoretical advantage of never misidentifying a state, in practice this is not possible to achieve. In experimental implementations, errors necessarily occur due to finite detector efficiency and errors caused by transformation optics. To evaluate the performance of our measurements, we compare our experimentally recorded errors to those theoretically predicted for the MESD protocol. A significant advantage is found in the case that the recorded errors for our scheme are smaller than those produced in MESD.
Due to the equal overlap between our input states, the minimum error rate for MESD in $d$ dimensions [@Qiu2008] reduces to $$\label{bound}
p_{\rm err} \geq \frac{1}{2}\left(1-\sqrt{1-|{\langle\Psi_i|\Psi_j\rangle}|^2}\right),$$ where the overlap ${\langle\Psi_i|\Psi_j\rangle}$ is given by Eq. (\[overlap\]). A violation of this inequality indicates that USD provides less ambiguity in state identification than is theoretically possible using MESD.
In Fig. \[errors\], we compare this bound to the mean total error rate observed using our method. To determine our error rate, we first determine the error rate for a single input state ${|\Psi_i\rangle}$; this is the sum of all possible incorrect state identifications. We then average over all input states $\{{|\Psi_i\rangle}\}$ to obtain the mean total error rate.
Fig. \[errors\](a) shows the total error rate as a function of angle for the $d=6$ case. The total error rate for angles up to $\theta=30^\circ$ is at least one standard deviation below the MESD bound, demonstrating that our approach is particularly successful for states with large overlap. The total error rate exceeds the MESD bound at higher angles, where the states have lower overlap and are closer to orthogonal. In this case, the bound converges to 0, matching the theoretical prediction for USD. Since the two schemes converge, it is inevitable that the experimentally measured errors exceed the ideal MESD curve at a sufficiently high angle.
![[]{data-label="errors"}](Fig4.pdf)
Fig. \[errors\](b) shows the total error rate as a function of dimension for a fixed overlap of $1/\sqrt{2}$ between the initial states. We choose a constant overlap so that the MESD bound is equal in all dimensions (in this case, $(1-\sqrt{1/2})/2\approx 0.146$). To achieve the constant overlap, the parameter $\theta$ must change with dimension . The total error rate for dimensions up to $d=12$ is below the MESD bound by at least one standard deviation.
In dimensions $d \geq 13$, the bound for MESD is successfully violated, but by less than one standard deviation. This is due to two main factors. Firstly, for all of these data, the average measured probability of obtaining an error, i.e., measuring a state ${|\Psi_i\rangle}$ with an incorrect detection state ${|D_j\rangle}$ ($i \not\in \{j,d+1\}$), is approximately $1\%$. As the dimension increases, so too does the number of opportunities to misidentify a state. Thus the total error grows accordingly, making it increasingly difficult to obtain a low total error. Secondly, due to the limited spiral bandwidth in the downconverted state, the probability amplitudes of the individual OAM modes decrease as $\ell$ increases. This limits the coincidence rate, and thus increases the uncertainty of the measurements, for high dimensions.
We have demonstrated USD via sequential measurements to distinguish $d$ non-orthogonal single-photon states in $d$-dimensional Hilbert spaces. In a modified set-up, our method could be realised as a true POVM experiment in high dimensions. While experimental constraints prevent completely error-free identification, we have shown that, for a range of high-dimensional states, our method still provides a lower error rate than minimum-error state discrimination. With suitable improvements in SLM resolution, spiral bandwidth production, and detector efficiency, this could be increased to even higher dimensions. This method of state discrimination will allow the use of high-dimensional non-orthogonal states in quantum protocols, enabling secure quantum communication with larger alphabets.
We thank Sarah Croke for valuable discussions regarding this work. MA acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Supplementary Materials
=======================
*Determining $d$ symmetric states in $d$ dimensions*: The $d$ states in $d$ dimensions that we choose to distinguish are maximally separated when projected onto $d-1$ dimensions. We describe first how to construct the $d-1$ projected vectors $\{{|\Psi^\prime_i\rangle}\}$. Without loss of generality, the first of these vectors, ${|\Psi^\prime_1\rangle}$, can be chosen to lie along one axis such that its first component is 1 and its remaining components are 0. We can construct all other vectors from their pairwise overlap, ${\langle\Psi^\prime_i|\Psi^\prime_j\rangle}=-1/(d-1)$ for $i \neq j$, and the normalisation condition, ${\langle\Psi^\prime_i|\Psi^\prime_i\rangle}=1$. The overlap condition requires that the first component of each remaining vector ${|\Psi^\prime_{j\geq 2}\rangle}$ must be $-1/(d-1)$. For the second vector we can determine the second component from the normalisation condition and set all following components equal to zero. The remaining vectors can be iteratively determined in the same way: for the third vector, the second component is determined from the overlap with the second vector, the third from normalisation, and all following components are zero; and similar for all subsequent vectors.
Once we have obtained these states, we transform them into $d$-dimensional states using $$\begin{aligned}
\label{psii}
{|\Psi_i\rangle}={\rm sin}\,\theta\,{|\Psi^\prime_i\rangle} + {\rm cos}\,\theta \,{|d\rangle}.\end{aligned}$$ For example, in dimension $d=3$, the states are the lifted trine states
$$\begin{aligned}
{|\Psi_1\rangle}&=&{\rm sin}\,\theta\,{|\ell_1\rangle}+{\rm cos}\,\theta\,{|\ell_3\rangle}\\
{|\Psi_2\rangle}&=&-\frac{1}{2}{\rm sin}\,\theta\,{|\ell_1\rangle}+\frac{\sqrt{3}}{2}{\rm sin}\,\theta\,{|\ell_2\rangle}+{\rm cos}\,\theta\,{|\ell_3\rangle}\\
{|\Psi_3\rangle}&=&-\frac{1}{2}{\rm sin}\,\theta\,{|\ell_1\rangle}-\frac{\sqrt{3}}{2}{\rm sin}\,\theta\,{|\ell_2\rangle}+{\rm cos}\,\theta\,{|\ell_3\rangle},\end{aligned}$$
where ${|\ell_1\rangle}$, ${|\ell_2\rangle}$, and ${|\ell_3\rangle}$ are the three chosen OAM basis states.
*Determining discrimination states*: We determine the orthogonal states $\{{|\Psi^\perp_i\rangle}\}$, with ${\langle\Psi^\perp_i|\Psi_j\rangle} \propto \delta_{ij}$, by taking each $(d-1)$-sized subset of the $\{{|\Psi_i\rangle}\}$ vectors and applying the Gram-Schmidt algorithm to find a vector orthogonal to this subset.
We then transform the set $\{{|\Psi^\perp_i\rangle}\}$ to an orthonormal basis set $\{{|D_i\rangle}\}$ by extension to an ancillary dimension followed by normalisation. For this, we make use of the fact that due to the inherent symmetry, the inner product of any two of the $\{{|\Psi^\perp_i\rangle}\}$ states, ${\langle\Psi^\perp_i|\Psi^\perp_j\rangle}$, $i \neq j$, is the same. As a result, we obtain $${|D_i\rangle}={|\Psi^\perp_i\rangle}+\sqrt{-{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}}{|d+1\rangle}.$$ Finally, we identify the inconclusive measurement state ${|D_{d+1}\rangle}$ such that ${\langleD_i|D_{d+1}\rangle}=0$, resulting in a complete basis in $d+1$ dimensions, again using the Gram-Schmidt algorithm.
*Transforming ${|\Psi^\perp_i\rangle}$ to ${|D_i\rangle}$*: Here we illustrate the calculation for $d=3$, but it functions similarly in higher dimensions. In order to orthogonalise our three 3-dimensional measurement states
$$\begin{aligned}
{|\Psi^\perp_1\rangle}&=&\sqrt{3}{\rm cos}\,\theta\,{\rm sin}\,\theta\,{|\ell_1\rangle}+\frac{\sqrt{3}}{2}{\rm sin}^2\theta\,{|\ell_3\rangle}\\
{|\Psi^\perp_2\rangle}&=&-\frac{\sqrt{3}}{2}{\rm cos}\,\theta\,{\rm sin}\,\theta\,{|\ell_1\rangle}+\frac{3}{2}{\rm cos}\,\theta\,{\rm sin}\,\theta\,{|\ell_2\rangle}\notag\\&+&\frac{\sqrt{3}}{2}{\rm sin}^2\theta\,{|\ell_3\rangle}\\
{|\Psi^\perp_3\rangle}&=&-\frac{\sqrt{3}}{2}{\rm cos}\,\theta\,{\rm sin}\,\theta\,{|\ell_1\rangle}-\frac{3}{2}{\rm cos}\,\theta\,{\rm sin}\,\theta\,{|\ell_2\rangle}\notag\\&+&\frac{\sqrt{3}}{2}{\rm sin}^2\theta\,{|\ell_3\rangle}\end{aligned}$$
into three 4-dimensional measurement states $\{{|D_1\rangle},{|D_2\rangle},{|D_3\rangle}\}$, we need only to add an arbitrary fourth component to each vector such that $${|D_j\rangle}={|\Psi^\perp_j\rangle}+(a_j+ib_j){|d+1\rangle},$$ where $a_i$ and $b_i$ are real numbers. For orthogonality, these states must satisfy $${\langleD_i|D_j\rangle}=C\delta_{ij},$$ where $C$ is some constant since the vectors are as yet unnormalised.
By examining the inner product ${\langleD_1|D_2\rangle}$, we obtain $${\langleD_1|D_2\rangle}=0={\langle\Psi^\perp_1|\Psi^\perp_2\rangle} + a_1 a_2 + i a_1 b_2 - i b_1 a_2 + b_1 b_2.$$ The real and imaginary parts then independently need to be equal to zero: $$\begin{aligned}
{\langle\Psi^\perp_1|\Psi^\perp_2\rangle} + a_1 a_2 + b_1 b_2 &=& 0\label{d1d2}\\
a_1 b_2 - b_1 a_2 &=& 0.\end{aligned}$$ From the inner products ${\langleD_2|D_3\rangle}$ and ${\langleD_1|D_3\rangle}$, we obtain similar equations. These six equations are solved simultaneously by defining the coefficients $a_i$ and $b_i$ as
$$\begin{aligned}
a_2&=&a_1 \frac{{\langle\Psi^\perp_2|\Psi^\perp_3\rangle}}{{\langle\Psi^\perp_1|\Psi^\perp_3\rangle}}\\
b_2&=&b_1 \frac{{\langle\Psi^\perp_2|\Psi^\perp_3\rangle}}{{\langle\Psi^\perp_1|\Psi^\perp_3\rangle}}\\
a_3&=&a_1 \frac{{\langle\Psi^\perp_2|\Psi^\perp_3\rangle}}{{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}}\\
b_3&=&b_1 \frac{{\langle\Psi^\perp_2|\Psi^\perp_3\rangle}}{{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}}.\end{aligned}$$
By substituting these values back into Eq. (\[d1d2\]), we obtain $$\begin{aligned}
0&=&{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}+\frac{{\langle\Psi^\perp_2|\Psi^\perp_3\rangle}}{{\langle\Psi^\perp_1|\Psi^\perp_3\rangle}}(a_1^2+b_1^2)\\
a_1^2+b_1^2&=&-\frac{{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}{\langle\Psi^\perp_1|\Psi^\perp_3\rangle}}{{\langle\Psi^\perp_2|\Psi^\perp_3\rangle}}.\end{aligned}$$ However, in our particular case, we know that the overlap between each pair of vectors in the set $\{{|\Psi^\perp_i\rangle}\}$ is equal; as a result, this can be reduced to $$a_1^2+b_1^2=-{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}.$$ Recall that we defined $a_i,b_i \in \mathbb{R}$; thus $a_1^2+b_1^2 \geq 0$ and we find that the states $\{{|D_i\rangle}\}$ as defined above can only exist if $${\langle\Psi^\perp_1|\Psi^\perp_2\rangle} \leq 0.$$
In order to form our ${|D_i\rangle}$ states, we choose $b_1=0$ for simplicity and thus $a_1=\sqrt{-{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}}$ and our discrimination states become $${|D_i\rangle}={|\Psi^\perp_i\rangle}+\sqrt{-{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}}{|d+1\rangle}.$$ The normalised form of these states is $${|D_i\rangle}=\frac{{|\Psi^\perp_i\rangle}+\sqrt{-{\langle\Psi^\perp_1|\Psi^\perp_2\rangle}}{|d+1\rangle}}{({\langle\Psi^\perp_i|\Psi^\perp_i\rangle}-{\langle\Psi^\perp_1|\Psi^\perp_2\rangle})^2}.$$ The inconclusive result ${|D_4\rangle}$ can then be found using the Gram-Schmidt algorithm on the first three $\{{|D_i\rangle}\}$ vectors.
For our example of dimension $d=3$, we end up with the following discrimination states:
$$\begin{aligned}
{|D_1\rangle}&=&\sqrt{\frac{2}{6}}{|\ell_1\rangle}+\frac{1}{\sqrt{6}}{\rm tan}\,\theta{|\ell_3\rangle}+\sqrt{\frac{3\,{\rm cos}^2\theta-1}{6}}{\rm sec}\,\theta{|\ell_4\rangle}\\
{|D_2\rangle}&=&-\frac{1}{\sqrt{6}}{|\ell_1\rangle}+\frac{1}{\sqrt{2}}{|\ell_2\rangle}+\frac{1}{\sqrt{6}}{\rm tan}\,\theta\,{|\ell_3\rangle}+\sqrt{\frac{3\,{\rm cos}^2\theta-1}{6}}{\rm sec}\,\theta{|\ell_4\rangle}\\
{|D_3\rangle}&=&-\frac{1}{\sqrt{6}}{|\ell_1\rangle}-\frac{1}{\sqrt{2}}{|\ell_2\rangle}+\frac{1}{\sqrt{6}}{\rm tan}\,\theta\,{|\ell_3\rangle}+\sqrt{\frac{3\,{\rm cos}^2\theta-1}{6}}{\rm sec}\,\theta{|\ell_4\rangle}\\
{|D_4\rangle}&=&-\sqrt{\frac{3\,{\rm cos}^2\theta-1}{2}}{\rm sec}\,\theta{|\ell_3\rangle}+\frac{1}{\sqrt{2}}{\rm tan}\,\theta{|\ell_4\rangle}.\end{aligned}$$
*Angle calculation for fixed overlap*: From the definition of the vector ${|\Psi_i\rangle}$ in Eq. (\[psii\]), we find that the overlap is $${\langle\Psi_i|\Psi_j\rangle}={\rm sin}^2\theta{\langle\Psi^\prime_i|\Psi^\prime_j\rangle}+{\rm cos}^2\theta,$$ where we have used that all vectors ${|\Psi_i\rangle}$ are orthogonal to ${|d\rangle}$. Using furthermore that all vectors have the same overlap ${\langle\Psi^\prime_i|\Psi^\prime_j\rangle}=-1/(d-1)$ we find $$\begin{aligned}
{\langle\Psi_i|\Psi_j\rangle}&=&{\rm sin}^2\theta\left(-\frac{1}{d-1}\right)+{\rm cos}^2\theta \nonumber \\
&=&\frac{d \, \cos^2\theta-1}{d-1}. \label{overlap2}\end{aligned}$$
In order to obtain states with equal overlap but defined in different dimensions, as we have chosen for Fig. 4(b), we can solve the above equation for $\theta$, $$\theta={\rm cos}^{-1}\sqrt{\frac{1}{d}(1+(d-1) {\langle\Psi_i|\Psi_j\rangle})}.$$
*MESD bound in $d$ dimensions*: As shown in Ref. [@Qiu2008], the error obtained using MESD to distinguish $d$ states in $d$ dimensions satisfies the inequality $$p_{\rm err} \geq \frac{1}{2}\left( 1 - \frac{1}{d-1} \sum_{i=1}^d \sum_{j=1}^{i-1} {\rm Tr} \left| \eta_i \rho_i - \eta_j \rho_j \right| \right),\label{MESDbound}$$ where $\eta_i$ is the *a priori* probability of generating the state $\rho_i$ and $|X|=\sqrt{X^\dagger X}$.
This expression can be simplified somewhat in our case. Firstly, our *a priori* probabilities $\eta_i$ are all equal to $1/d$ so that $${\rm Tr} \left| \eta_i \rho_i - \eta_j \rho_j \right|=\frac{1}{d}{\rm Tr} \left| \rho_i - \rho_j \right|.$$ Secondly, we use only pure states, so that [@NielsenChuang] $${\rm Tr} \left| \rho_i - \rho_j \right|=2\sqrt{1-|{\langle\Psi_i|\Psi_j\rangle}|^2}.$$ Then Eq. (\[MESDbound\]) becomes $$p_{\rm err} \geq \frac{1}{2}\left( 1-\frac{1}{d-1}\sum_{i=1}^d \sum_{j=1}^{i-1} \frac{2}{d} \sqrt{1-|{\langle\Psi_i|\Psi_j\rangle}|^2} \right).$$ Since all initial states $\{{|\Psi_i\rangle}\}$ for a particular angle and dimension have a known equal overlap with one another, the term inside the sum is a constant and can be factored out so that $$p_{\rm err} \geq \frac{1}{2}\left( 1-\frac{1}{d-1} \frac{2}{d} \sqrt{1-|{\langle\Psi_i|\Psi_j\rangle}|^2}\sum_{i=1}^d \sum_{j=1}^{i-1}1 \right).$$ By evaluating the sum as $\sum_{i=1}^d \sum_{j=1}^{i-1} 1 = (d^2-d)/2$, we obtain $$p_{\rm err} \geq \frac{1}{2}\left(1-\sqrt{1-|{\langle\Psi_i|\Psi_j\rangle}|^2}\right).$$ As the overlap is defined by Eq. (\[overlap2\]), we finally find the MESD error bound to be $$p_{\rm err} \geq \frac{1}{2}\left[1-\sqrt{1-\left( \frac{d \, \cos^2\theta-1}{d-1} \right)^2}\right].$$ For states that overlap by ${\langle\Psi_i|\Psi_j\rangle}=1/\sqrt{2}$, as in Fig. 4(b), this evaluates to $p_{\rm err} \geq \frac{1}{2}\left( 1-\sqrt{\frac{1}{2}}\right)\approx 0.146.$
*OAM values*: For dimension $d$, we require $d+1$ OAM values: $d$ OAM values to form a basis for our states, and one additional OAM value to facilitate our discrimination measurements. As the probability of production of an OAM value decreases in absolute value, it is advantageous to use OAM values closest to zero to obtain greatest signal. The chosen OAM values for several dimensions are shown in Table \[table\].
Dimension $d$ $\ell$-values for states Ancillary $\ell$-value
--------------- -------------------------- ------------------------
2 0,1 -1
3 -1,0,1 -2
4 -1,0,1,2 -2
5 -2,-1,0,1,2 -3
: OAM values.[]{data-label="table"}
[^1]: See Supplementary Materials for further information.
|
---
abstract: |
We define the flatness and quasi-flatness problems in cosmological models. We seek solutions to both problems in homogeneous and isotropic Brans-Dicke cosmologies with varying speed of light. We formulate this theory and find perturbative, non-perturbative, and asymptotic solutions using both numerical and analytical methods. For a particular range of variations of the speed of light the flatness problem can be solved. Under other conditions there exists a late-time attractor with a constant value of $
\Omega \ $that is smaller than, but of order, unity. Thus these theories may solve the quasi-flatness problem, a considerably more challenging problem than the flatness problem. We also discuss the related $\Lambda $ and quasi-$
\Lambda $ problem in these theories. We conclude with an appraisal of the difficulties these theories may face.
address: |
$^1$Astronomy Centre, University of Sussex, Brighton BN1 9QJ, U.K.\
$^2$ Theoretical Physics, The Blackett Laboratory,\
Imperial College, Prince Consort Road, London, SW7 2BZ, U.K.
author:
- 'John D. Barrow$^1$ and João Magueijo$^2$'
title: 'Solving the Flatness and Quasi-flatness Problems in Brans-Dicke Cosmologies with a Varying Light Speed'
---
The flatness and the quasi-flatness problems
============================================
The observable universe is close to a flat Friedmann model, the so-called Einstein-de Sitter universe, in which the energy density, $\rho ,$ takes the critical value, $\rho _c$, and the homogeneous spatial surfaces are Euclidean. All astronomical evidence shows that we are quite close to this state of flatness, although a value of $\Omega _0$ in the vicinity of $0.2$ is preferred by several observations.
It is therefore disquieting to notice that the flat Friedmann model containing dust or blackbody radiation is unstable as time increases. Small deviations from the exact $\Omega =1$ model grow quickly in time, typically like $a^2$, where $a$ is the expansion factor of the universe. The observed $
\Omega _0\approx 1$ state therefore requires extreme fine tuning of the cosmological initial conditions, assumed to be set at Planck epoch, because $
a$ has increased by a factor of order $10^{32}$ from that epoch to the present. This is the [*flatness problem*]{}. If the universe is slightly open at Planck time, within a few Planck times it would become totally curvature dominated. If it is initially slightly closed, it would quickly collapse to Planck density again. Explaining its current state requires an extraordinarily close proximity to perfect flatness initially, or some sequence of events which can subsequently reverse expectations and render the flat solution asymptotically stable.
Particle physics theories naturally contain self-interacting scalar matter fields which violate the strong energy condition (so the density and pressure, $p$, obey $\rho +3p<0$). These can make the flat solution asymptotically stable with increasing time, allowing the asymptotic state to naturally be close to flatness. Cosmological histories in which a brief period of expansion is dominated by a matter field or other effective stress which violate the strong energy condition, and so exhibit gravitational repulsion, are called “inflationary”.
Solutions to the flatness problem have been proposed in the context of inflationary scenarios [@infl], pre-Big-Bang models [@prebb], and varying speed of light cosmologies [@mof; @mof1; @vsl0; @vsl1]. In all of these theories, $\Omega =1$ becomes an asymptotically stable attractor for the expanding universe. The observed $\Omega _0\approx 1$ state results then from a temporary period of calculable physical processes, rather than from highly tuned initial conditions. In such scenarios the price to be paid is that $\Omega _0$ should be very close to unity, $\left| \Omega _0-1\right| $ $\leq 10^{-5}$, if one is not to invoke an unmotivated fine tuning of the initial conditions again.
If we take the trend of the observational data seriously, then explaining a current value of $\Omega _0$ of, say, $0.2-0.9$ is yet another challenge. We call it the [*quasi-flatness problem*]{}. Solutions to this problem have been proposed in the context of open inflationary models [@open]. In these one has to come to grips with some degree of fine tuning. The Anthropic Principle [@anth0] is usually invoked for this purpose [@anth] but considerable uncertainties exist in the range of predictions that emerge and there does not appear to be scope for a very precise prediction of $\Omega _0$ in, say, the range $0.2.$ Undoubtedly, it would be better if one could find mechanisms which would produce a definite $\Omega _0$ of order one, but different from 1, as an attractor. In a recent letter [@quasi], we displayed one theory in which this possible. Here we present further solutions in support of such models. We explore analytical and numerical solutions to Brans-Dicke (BD) cosmologies with a varying speed of light (VSL). These generalise our earlier investigations of this theory [@vsl0; @vsl1; @vsl2]. We show that if the speed of light evolves as $
c\propto a^n$ with $0<n<-1$ there is a late-time attractor at $\Omega =-n$. Hence, these cosmologies can solve the quasi-flatness problem. This work expands considerably the set of solutions presented in ref. [@quasi].
We note that the existence of the dimensionless fine structure constants allows these varying-$c$ theories to be transformed, by a change of units, into theories with constant $c,$ but with a varying electron change, $e$, or dielectric ’constant’ of the vacuum. This process is described in detail in ref. [@vsl2] where a particular theory is derived from an action principle. A different varying-$e$ theory has been formulated by Bekenstein [@bek] but it is explicitly constructed to produce no changes in cosmological evolution. A study of this theory will be given elsewhere [@btoo].
There also exist analogues of the flatness and quasi-flatness problems with regard to the cosmological constant. The [*lambda problem*]{} is to understand why the cosmological constant term, $\Lambda c^2$ in the Friedmann equation does not overwhelmingly dominate the density and curvature terms today (quantum gravity theories suggest that it ought to be about $10^{120}$ times bigger than observations permit [@hawk; @anth0]). The [*quasi-lambda problem*]{} is to understand how it could be that this contribution to the Friedmann equation could be non-zero and of the same order as the density or curvature terms (as some recent supernovae observations suggest [@super]).
In Section \[eqns\] we write down the evolution equations in these theories. The varying $G$ aspect of the theory will be accommodated in the standard way by means of a Brans-Dicke theory of gravitation. This theory is adapted in a well-defined way to incorporate varying $c.$ We write equations in both the Jordan and Einstein’s frames. In Section \[pert\] we study solutions to the flatness problem in the perturbative regime ($\Omega
\approx 1$), when both $c$ and $G$ may change. In Section \[gcons\] we present non-perturbative solutions when $G$ is constant, and in Section \[gvar\] when $G$ is varying. In Section \[asym\] we derive simple conditions on any power-law variation of $c$ with scale factor if the flatness problem is to be solved. These conclusions are reinforced by some exact solutions in Section \[exact\]. In section \[qflat\] we show that the quasi-flatness problem can be naturally solved in a class of varying-$c$ cosmologies and then we discuss the solution of the quasi-lambda problem by such cosmologies in Section \[qlamb\]. We conclude with a summary of our results, highlighting the cases in which we can claim to have solved the quasi-flatness and quasi-lambda problems.
Cosmological Field Equations {#eqns}
============================
In BD varying speed of light (VSL) theories the Friedmann equations are [@vsl1]: $$\begin{aligned}
3{\frac{\ddot a}a} &=&-{\frac{8\pi }{(3+2\omega )\phi }}[(2+\omega )\rho
+3(1+\omega )p/c^2]-\omega {\left( \frac{\dot \phi }\phi \right) }^2-{\frac{
\ddot \phi }\phi } \label{fr1} \\
{\left( \frac{\dot a}a\right) }^2 &=&{\frac{8\pi \rho }{3\phi }}-{\frac{Kc^2
}{a^2}}-{\frac{\dot \phi a}{\phi a}}+{\frac \omega 6}{\left( \frac{\dot \phi
}\phi \right) }^2 \label{fr2}\end{aligned}$$ where the speed of light, $c,$ is now an arbitrary function of time, $K$ is the curvature constant, and $\omega $ is the constant BD parameter. The wave equation for the BD scalar field $\phi =1/G$ is $$\ddot \phi +3{\frac{\dot a}a}\dot \phi ={\frac{8\pi }{3+2\omega }}(\rho
-3p/c^2) \label{fr3}$$ From these three equations we can obtain the generalised conservation equation. Since after we impose the equation of state, the varying $c$ term only appears in eq. (\[fr2\]), the only new contribution to $2\dot a\ddot
a $ is from the $-2Kc\dot c$ term. Hence, together, these imply the ’non-conservation’ equation: $$\dot \rho +3{\frac{\dot a}a}(\rho +p/c^2)={\frac{3Kc\dot c}{4\pi a^2}}\phi .
\label{noncons}$$ In the radiation-dominated epoch ($p=\rho c^2/3$) the general solution for $
\phi $ is $$\phi =\phi _0+\alpha {\int {\frac{dt}{a^3}}} \label{dotG}$$ If the integration constant $\alpha =0$ the usual solutions for VSL in general relativity with varying $c$ follow illustrating the fact that for a radiation source any solution of general relativity is a particular solution of BD theory with constant $\phi $. In the next sections we explore the solutions to the flatness problem when $\dot \phi \neq 0$. We will also consider the radiation to matter transition in these theories.
Equations (\[fr1\])-(\[dotG\]) apply in the so-called Jordan frame. It will be useful to introduce the Einstein frame, by means of the transformations, $$\begin{aligned}
d{\hat t} &=&{\sqrt{G\phi }}dt \\
{\hat a} &=&{\sqrt{G\phi }}a \\
\sigma &=&{\left( \omega +3/2\right) }^{1/2}\ln ({G\phi )} \\
{\hat \rho } &=&(G\phi )^{-2}\rho \\
{\hat p} &=&(G\phi )^{-2}p, \\
&& \nonumber\end{aligned}$$ which are performed at constant $c$. These may be regarded as merely mathematical transformations of variables.
The Friedmann equations in the new frame are $$\begin{aligned}
3{\frac{{\hat a}^{\prime \prime }}{{\hat a}}} &=&{\frac{-4\pi G}3}[{\hat
\rho }+3{\hat p}/c^2]-{\frac{\sigma ^{\prime }{}^2}3} \\
{\left( \frac{{\hat a}^{\prime }}{{\hat a}}\right) }^2+{\frac{Kc^2}{{\hat a}
^2}} &=&{\frac{8\pi G{\hat \rho }}3}+{\frac{\sigma ^{\prime }{}^2}6}\end{aligned}$$ where $^{\prime }=d/d{\hat t}$. The transformed scalar field equation for $
\sigma $ is $$\sigma ^{\prime \prime }+3{\frac{{\hat a}^{\prime }}{{\hat a}}}\sigma
^{\prime }=-{\frac{8\pi G}{{\sqrt{6+4\omega }}}}({\hat \rho }-3{\hat p}/c^2)$$ These are just the standard Friedmann equations with constant $G$ and a scalar field added to the normal matter. The scalar field behaves like a ’stiff’ perfect fluid with equation of state $${\hat p}_\sigma ={\hat \rho }_\sigma ={\frac{\sigma ^{\prime }{}^2}{16\pi G}.
}$$ In the Einstein frame, if $\dot c=0$, the total stress-energy tensor is conserved, but the scalar field and normal matter exchange energy according to: $${\hat \rho }^{\prime }+3{\frac{{\hat a}^{\prime }}{{\hat a}}}({\hat \rho }+{
\hat p}/c^2)=-({\hat \rho }_\sigma ^{\prime }+3{\frac{{\hat a}^{\prime }}{{
\hat a}}}({\hat \rho }_\sigma +{\hat p}_\sigma /c^2))={\frac{\sigma ^{\prime
}({\hat \rho }-3{\hat p}/c^2)}{{\sqrt{6+4\omega }}}.}$$ If $\dot c\neq 0,$ one has instead $$\begin{aligned}
{\hat \rho }^{\prime }+3{\frac{{\hat a}^{\prime }}{{\hat a}}}({\hat \rho }+{
\hat p}/c^2) &=&{\frac{\sigma ^{\prime }({\hat \rho }-3{\hat p}/c^2)}{{\sqrt{
6+4\omega }}}}+{\frac{3Kc^2}{4\pi G{\hat a}^2}}{\frac{c^{\prime }}c}
\label{conseins} \\
{\hat \rho }_\sigma ^{\prime }+3{\frac{{\hat a}^{\prime }}{{\hat a}}}({\hat
\rho }_\sigma +{\hat p}_\sigma /c^2) &=&-{\frac{\sigma ^{\prime }({\hat \rho
}-3{\hat p}/c^2)}{{\sqrt{6+4\omega }}}}\end{aligned}$$ All equations derived for standard VSL theory, with constant $G$, are valid in the Einstein frame. However, one should always remember to add to normal matter the scalar field energy and pressure (so the total density and pressure are given by ${\hat \rho }_t={\hat \rho }+\rho _\sigma $ and ${\hat
p}_t={\hat p}+{\hat p}_\sigma ,$ respectively).
Perturbative solutions to the flatness problem {#pert}
==============================================
We first study solutions to the flatness problem when there are small deviations from flatness. Let us define the critical density, $\rho _c$ , in a B-D universe by means of the equation: $${\left( \frac{\dot a}a\right) }^2={\frac{8\pi \rho _c}{3\phi }}-{\frac{\dot
\phi a}{\phi a}}+{\frac \omega 6}{\left( \frac{\dot \phi }\phi \right) }^2,$$ that is, Eqn. \[fr2\] with $K=0$. In the Einstein frame the critical density in normal matter is $${\hat \rho }_c={\frac 3{8\pi G}}{\left( {\left( \frac{{\hat a}^{\prime }}{{
\hat a}}\right) }^2-{\frac{{\sigma ^{^{\prime }2}}}6}\right) }={\frac{\rho _c
}{G^2\phi ^2}}$$ In terms of total energy density ${\hat \rho }_t={\hat \rho }+\rho _\sigma $ , the critical energy density in the Einstein frame is: $${\hat \rho }_{ct}={\frac 3{8\pi G}}{\left( \frac{{\hat a}^{\prime }}{{\hat a}
}\right) }^2.$$ Accordingly, we may define a [*relative flatness parameter:*]{} $${\hat \epsilon }_t={\frac{{\hat \rho }_t-{\hat \rho }_{ct}}{{\hat \rho }_{ct}
}}$$ As shown in the previous section, the usual equations for VSL theory should apply to this quantity. Therefore, one has $${\hat \epsilon _t}^{\prime }=(1+{\hat \epsilon }_t){\hat \epsilon }
_t(3\gamma -2){\frac{{\hat a}^{\prime }}{{\hat a}}}+2{\frac{c^{\prime }}c}{
\hat \epsilon }_t$$ with the equation of state given by
$$\gamma -1\equiv p/\rho c^2={\hat p}/{\hat \rho }c^2 \label{gamma}$$
with $\gamma $ constant.
But, since $${\hat \epsilon }_t={\frac \epsilon {1+(2\omega +3){\frac{{\dot \phi }^2}{
8\pi \phi \rho _c}}},}$$ we see that the natural quantity to quantify deviations from flatness in the Jordan frame is not $$\epsilon ={\frac{{\rho }-{\rho }_c}{{\rho }_c}}$$ but an adaptation of it, $$\delta ={\frac{\rho -\rho _c}{\rho _c+{\frac{(2\omega +3){\dot \phi }^2}{
8\pi \phi }}}}<\epsilon , \label{deltaeqn}$$ which satisfies the equation $$\dot \delta =(1+\delta )\delta (3\gamma -2){\left( {\frac{\dot a}a}+{\frac 12
}{\frac{\dot \phi }\phi }\right) }+2{\frac{\dot c}c}\delta . \label{eps}$$ If $\delta \ll 1$ this can be integrated to give $$\delta \propto a^{3\gamma -2}c^2\phi ^{\frac{\ 3\gamma -2}2}$$ Hence, we have the solution $${\frac 1\epsilon }={\frac C{a^{3\gamma -2}c^2\phi ^{\frac{3\gamma -2}2}}}-{
\frac{(2\omega +3){\dot \phi }^2}{8\pi \phi \rho _c}.} \label{perteps}$$ If $G\propto \phi ^{-1}$ varies then the second term on the right always works against solving the flatness problem. Hence solving the flatness problem in BD requires that $\phi $ decreases in the early universe and that the first term on the right of eq. (\[perteps\]) dominates the second.
In a radiation-dominated universe ($\gamma =4/3$): $${\frac 1\epsilon }={\frac C{a^2c^2\phi ^2}}-{\frac{(2\omega +3){\alpha }^2}{
8\pi \phi a^6\rho _c},}$$ where $\alpha $ was defined by eqn.(\[dotG\]). If $\alpha $ is small and $
c=c_0a^n$ one still solves the flatness problem when $n<-1$. Notice that since $\rho _c\propto 1/a^4$ at late times, the second term always eventually becomes negligible.
Non-perturbative solutions with $\phi=const$ {#gcons}
============================================
Matter and radiation-dominated cases
------------------------------------
In order to explore non-perturbative solutions to the flatness problem we solved Eqns. (\[fr1\])-(\[dotG\]) numerically. First we consider the case where $\phi =\phi _0$. Exact solutions exist in this case for matter and radiation-dominated universes. Setting $\phi =1/G$ in Eqn. (\[eps\]) leads to $${\dot \epsilon }=(1+\epsilon )\epsilon (3\gamma -2){\frac{{\dot a}}{{a}}}+2{
\frac{{\dot c}}c}\epsilon$$ which, assuming ${\dot c}/c=n{\dot a}/a,$ becomes $${\dot \epsilon }=\epsilon {\left( (1+\epsilon )(3\gamma -2)+2n\right) }{
\frac{{\dot a}}{{a}}.}$$ The general structure of the attractors can be inferred from this equation. However, it can also be integrated to give $$\epsilon =-{\frac{1+{\frac{2n}{3\gamma -2}}}{1-Ca^{-(2n+3\gamma -2)}},}
\label{soleps}$$ with $n<0$, and where the integration constant $C$ is chosen so as to enforce the specified initial conditions. Our numerical code matched this analytical result to very high accuracy. In Fig. \[fig1\] we plot numerical results for a radiation-dominated universe. We comment on four possible situations, illustrated in Figs. \[fig1\]-\[fig4\].
If $c$ is a constant ($n=0$) then the flat universe ($\epsilon =0$) is unstable (Fig. \[fig1\]). There are two attractors at $\epsilon =-1$ and $
\epsilon =\infty $, showing that the universe tends to become curvature dominated. For slightly closed universes this means evolution towards a Big Crunch singularity. For slightly open universes it means evolution towards a Milne universe ($\epsilon =-1$) which in the case of constant $G$ is simply an expanding empty spacetime with $a\propto t$. If the speed of light were to increase with $a$ ($c\propto a^n$ with $n>0$) the situation is the same, but with a stronger repulsion from $\epsilon =0$.
If $c\propto a^n$ with $n<-(3\gamma -2)/2$, then, as pointed out in [@vsl1], the flatness problem is solved if $\epsilon $ is not too far from zero initially. In fact, $\epsilon =0$ is now an attractor (see Fig. \[fig2\]). The non-perturbative analysis reveals two novelties: there is an unstable node at $\epsilon =-1-2n/(3\gamma -2)$ and $\epsilon =\infty $ is an attractor. This means that if the universe at any given initial time satisfies $\epsilon >-1-2n/(3\gamma -2)$ it will not evolve towards the flat attractor. Instead, it will evolve towards a big crunch with $\epsilon
=\infty $. Therefore, for closed universes, the flatness problem is solved only if they are not too positively curved.
It is curious to note that the situation is different for open universes. No matter how far they are from flatness they will always evolve towards the Einstein-de Sitter model at late times. This was first pointed out in [@vsl0], where it was noted that in VSL theories the Einstein-de Sitter universe is an attractor even if the universe is initially Minkowski (empty) or de Sitter (dominated by a cosmological constant).
The greatest novelty, however, appears for the case in which $-(3\gamma
-2)/2<n<0$. Although $\epsilon =0$ is now unstable, there is a stable attractor at $\epsilon =-1-2n/(3\gamma -2)$ (see Fig. \[fig3\]). The system is now repelled from $\epsilon =-1$ but is attracted to $\epsilon
=\infty $.
Hence we have the following scenario: all closed universes evolve towards a big crunch, no matter how close to flatness they are initially. Thus a selection process excludes even slightly closed universes. [* *]{}In contrast, flat and all open models (no matter how low their density) evolve towards an open model with $\Omega =-2n/(3\gamma -2)$, a value that lies between $0$ and $1$, and is typically of order unity. We therefore have a solution to the quasi-flatness problem, for open, but not for closed universes.
The $n=-(3\gamma -2)/2$ case is a transition case in between the last two situations just described (Fig. \[fig4\]). In this case $\epsilon =0$ is a saddle: stable if approached from below but unstable from above.
The radiation-to-matter transition
----------------------------------
In quasi-flat scenarios the major contribution to the matter density of the universe is the term resulting from violations of energy conservation. This can be seen from $$\rho =\frac B{a^{3\gamma }}+\frac{3Kc_0^2\phi n}{4\pi (3\gamma +2n-2)}
a^{2n-2}$$ derived above. Taking radiation as an example ($\gamma =4/3$), if $n<-1$ we are pushed towards the attractor $\Omega =1$ and the energy-violation term (second term) becomes negligible. In scenarios in which the quasi-flatness problem is solved the converse happens: the second term dominates. Matter is continuously being created and this provides the dominant contribution to cosmic matter at any given time. Matter which was created in the past gets diluted away by expansion very quickly. In effect, we have a state of permanent “reheating”. This scenario has a passing resemblance with the steady state universe, in which the so called $C$ field continuously creates more and more matter.
If we couple a changing $c$ say to the Lagrangian of the standard particle physics model then all particles are produced, and whether they behave like matter or radiation depends on whether $T\ll m$ or $T\gg m$. If $n<0$ the energy density in created matter decreases, and the universe cools down as it expands. Hence there will necessarily be a radiation to matter transition in these scenarios, just like in the standard Big Bang theory with constant $
G$ and $c$.
A complication arises when we attempt to model the evolution through this transition because the structure of the attractors changes. For a radiation-dominated universe one requires $-1<n<0$ for the open attractor $
\Omega =-n$ to be achieved. For a matter-dominated universe the attractors for open universes are given by $\Omega =-2n$ and are achieved when $
-1/2<n<0 $. Hence, if $-1<n<-1/2$ we have attraction towards a (quasi-flat) open universe during the radiation-dominated epoch which is then driven towards flatness in the subsequent matter-dominated epoch. If $-1/2<n<0$ we have attraction to a (quasi-flat) open model in both epochs but the universe will evolve from a trajectory asymptoting to $\Omega =-n$ toward one asymptoting to $\Omega =-2n$ after the matter-radiation equality time. The solution (\[soleps\]) is then no longer valid because $\gamma $ is time dependent.
Solutions with variable $\phi$ {#gvar}
==============================
Exact Radiation solutions
-------------------------
For radiation, $\gamma =4/3$, we can solve the field equations by generalising the method introduced for scalar-tensor theories introduced in ref. [@JDBBD]. Define the conformal time
$$ad\tau =dt \label{time}$$
so $^{\prime }=d/d\tau .$ Eq. (\[fr3\]) integrates to give
$$\phi ^{\prime }=\frac A{a^2} \label{int}$$
with $A$ constant .
Now change the variables of eq.(\[fr2\]) by introducing
$$y=\phi a^2$$ and $d/d\tau .$ It reduces to
$$y^{\prime 2}=A^2(1+\frac{2\omega }3)+\frac{32\pi y\rho a^4}3-4Kc^2y^2
\label{y1}$$
For radiation, eq. (\[den1\]) gives
$$\rho a^4=B+\frac{3Kc_0^2my^{2m+1}}{4\pi (2m+1)} \label{y2}$$
If we substitute (\[y2\]) in (\[y1\]) then we have
$$y^{\prime 2}=A^2(1+\frac{2\omega }3)+\frac{32\pi B\ }3y+4Kc_0^2[\frac{2m}{\
(2m+1)}-1]\ y^{2(m+1)} \label{y5}$$
This can be integrated exactly for appropriate choices of $m$ and its qualitative behaviour is easy to understand. Note that if we can solve (\[y5\]) for $y(\tau )$ $=\phi a^2$, then we know $\phi (\tau ,a)$ and can solve (\[int\]) to get $\phi (\tau )$ hence $a(\tau ),$ and finally $a(t)$ from (\[time\]) since
$$\frac{\phi ^{\prime }}\phi =\frac Ay \label{y6}$$
We note that the curvature term in (\[y5\]) changes sign for a special value of $m,$
$$m=m_{*}=-\frac 12$$ Examining (\[y5\]) we see that the $K=0$ solution appears to be approached if $2(m+1)<1$, ie $m<-1/2.$
When $m=-1$ there is a simple special case which allows an exact solution. Eq. (\[y5\]) is now
$$y^{\prime 2}=\alpha +\beta y+\Gamma \label{y7}$$
where $\alpha ,\beta >0,\Gamma =+4Kc_o^2.$ We integrate (\[y6\]) to get
$$y=\frac \beta 4(\tau +\tau _0)^2-\frac{\alpha +\Gamma }\beta =\phi a^2
\label{y8}$$
Hence,
$$\phi =\phi _0\left[ \frac{\beta (\tau +\tau _0)-2\sqrt{\alpha +\Gamma }}{
\beta (\tau +\tau _0)+2\sqrt{\alpha +\Gamma }}\right] ^{A/\sqrt{\alpha
+\Gamma }}$$ and
$$a^2=\frac y\phi =\frac{\beta ^2(\tau +\tau _0)^2}{4\beta \phi _0}\left[
\frac{\beta (\tau +\tau _0)+2\sqrt{\alpha +\Gamma }}{\beta (\tau +\tau _0)-2
\sqrt{\alpha +\Gamma }}\right] ^{A/\sqrt{\alpha +\Gamma }}$$ We see that if $\alpha +\Gamma >0$ then, as $\tau \rightarrow \infty ,$ we have $a\rightarrow \tau \rightarrow t^{1/2}$ unless the denominator blows up.
If $\alpha +\Gamma <0$ then we get
$$\ln (\phi /\phi _0)=\frac{2A}{\sqrt{-(\alpha +\Gamma )}}\tan ^{-1}\left(
\frac{\beta (\tau +\tau _0)}{\sqrt{-(\alpha +\Gamma )}}\right)$$ and
$$a^2=\frac 1{4\phi _0\beta }\left[ \beta ^2(\tau +\tau _0)+\lambda ^2\right]
\exp \{-\frac{4A}\lambda \tan ^{-1}\left( \frac{\beta (\tau +\tau _0)}
\lambda \right) \}$$ where
$$\lambda ^2\equiv -4(\alpha +\Gamma )$$ Looking back at the defining equation (\[y5\]), we see that for this $m=-1$ case, $\Gamma =4Kc_0^2$ and so
$$\alpha +\Gamma =A^2(1+\frac{2\omega }3)+4Kc_0^2$$ and for large $A$ we have $\alpha +\Gamma >0$ when $K>0$.
Numerical solutions
-------------------
A numerical evolution of the equations with a varying $\phi $ reveals that for reasonable initial values of $\phi $ the structure of attractors is not affected. However, the speed at which attractors are reached may be increased if $\phi $ is allowed to change. As an example we shall consider the case in which one starts from a Milne universe ($\epsilon =-1$). In [@vsl0] it was argued that this is the natural initial condition to consider in the context of VSL cosmologies.
Let us then consider the case $\gamma =4/3$, and integrate Eqns. (\[fr1\] ), (\[fr2\]), and (\[fr3\]). We consider solutions of the form (\[dotG\]) with various integration constants $\alpha $. In Figures \[figbd\] and \[figbd1\] we show the evolution of $\epsilon =\Omega -1$ and $\phi $ from such a state, with $c=c_0a^n$ and $n=-2,-1/2$. In the first case we see that we still have a flat attractor, which is reached much more rapidly if $
\phi $ is increasing or decreasing. In the latter case we have a quasi-flat attractor, with $\epsilon =-1/2$, whether or not $\phi $ is allowed to change. The attractor is achieved faster with a changing $\phi $, especially a decreasing one.
Asymptotic Solutions to the Flatness Problem {#asym}
============================================
Exact solutions are only possible for particular equations of state and $c$-variation laws but it is possible to understand the asymptotic behaviour in general. The VSL theory Brans-Dicke solutions are given for eqns. (\[fr1\])-(\[fr3\]) together with (\[noncons\]). We assume a perfect-fluid equation of state given by (\[gamma\]). In the flat case ($
K=0$) the equations reduce to those of standard BD flat universes with constant $c$. At late times, when $K=0$, the general BD solutions for all $
\gamma $ approach the particular (’matter-dominated’ or ’Machian’) power-law solutions [@nar]
$$\begin{aligned}
G &\propto &\phi ^{-1}\propto t^{-D} \label{s1} \\
a(t) &\propto &t^A \label{s2}\end{aligned}$$
where
$$A=\frac{2+2\omega (2-\gamma )}{
\begin{array}{c}
4+3\omega \gamma (2-\gamma ) \\
\end{array}
} \label{A}$$
$$D=\frac{2(4-3\gamma )}{
\begin{array}{c}
4+3\omega \gamma (2-\gamma ) \\
\end{array}
} \label{D}$$
Hence, we see that $D+3\gamma A=2;$ note that $D=0$ for radiation;.These exact power-law solutions reduce the special $\phi =const,a\propto t^{1/2}$ general relativity solution in the radiation case. We shall therefore look at the stability of these $K=0$ asymptotes when we turn on the $K\neq 0$ terms in eqns. (\[fr1\])-(\[fr3\]) and (\[noncons\]) with $c(t)$ included. If we substitute (\[A\]) and (\[D\]) in (\[noncons\]) with $
c=c_0a^n$ then we have
$$\begin{aligned}
&& \\
(\rho t^{3\gamma A})^{\cdot } &\simeq &\frac{3Kc_0^2An\phi _0}{4\pi }
t^{A(3\gamma +2n-3)+A-1+D}\simeq \frac{3Kc_0^2An\phi _0}{4\pi }t^{1+2A(n-1)}\end{aligned}$$
Integrating, we get
$$\rho t^{3\gamma A}\simeq B+\frac{3Kc_0^2An\phi _0}{8\pi (1+An-A)}
t^{2(1+An-A)} \label{asy}$$
if $A(n-1)\neq -1$. Thus, to solve the flatness problem we will need the $B$ term to dominate the $K$ term on the right-hand side of eq. (\[asy\]) at large $t;$ that is, since $A>0$ for expanding universes, asymptotic approach to flatness requires
$$n<1-\frac 1A.$$ Using (\[A\]), this gives the condition
$$n<\frac{(2-\gamma )(2-3\gamma )\omega -2}{2[1+\omega (2-\gamma )]}.$$
For radiation and dust universes this condition for solving the flatness problem reduces to:
$$\begin{aligned}
rad &:&n<-1 \label{co1} \\
dust &:&n<\frac{-(\omega +2)}{2(\omega +1)}\rightarrow -\frac 12\ as{\rm {\ }
\omega \rightarrow \infty} \label{co2}\end{aligned}$$
where (\[co2\]) approaches the general relativity value for $\omega
\rightarrow \infty ,$ as expected and (\[co1\]) agrees with the radiation case studied above.
Exact solutions to the flatness problem {#exact}
=======================================
Radiation-dominated case
------------------------
The exact solutions found for the VSL theory with $c=c_0a^n$ in [@vsl1] can be generalized to the BD case if one assumes that the variation of the speed of light is governed by a relation of the form $$c=c_0(a\sqrt{G\phi })^n$$ This reduces to the previously studied (non-BD) case when $\phi $ is constant. In the Einstein frame one then has $c=c_0{\hat a}^n$ and in the radiation-dominated epoch the first of equations (\[conseins\]) becomes $${\hat \rho }^{\prime }+3{\frac{{\hat a}^{\prime }}{{\hat a}}}({\hat \rho }+{
\hat p}/c^2)={\frac{3Kc^2}{4\pi G{\hat a}^2}}{\frac{c^{\prime }}c}$$ Hence, for radiation, one has the integral $${\hat \rho }{\hat a}^4=B+{\frac{3Kc_0^2n{\hat a}^{2n+2}}{4\pi G(2n+2)}}$$ with $B$ constant. Since ${\hat \rho }{\hat a}^4=\rho a^4,$ in the Jordan frame, one has $$\rho a^4=B+{\frac{3Kc_0^2na^{2n+2}(G\phi )^{n+1}}{4\pi G(2n+2)}}$$ This solution can be verified directly from eq. (\[noncons\]), although it would have been difficult to guess it without a foray into the Einstein frame.
Conditions for solving the flatness problem can now be derived by inspection of the Friedmann-like equation: $${\left( \frac{\dot a}a\right) }^2={\frac{C_1}{\phi a^4}}+{C_2Ka^{2n-2}\phi ^n
}-{KC_3a^{2n-2}\phi ^n}-{\frac{\dot \phi a}{\phi a}}+{\frac \omega 6}{\left(
\frac{\dot \phi }\phi \right) }^2$$ Since $\phi $ approaches an asymptotic value we still require that $n<-1$ for the term in $C_1$ to dominate the curvature ($K$) term at late times.
If $\phi $ is decreasing we do not need the speed of light to decrease in time so fast in order to solve the flatness problem. Since $${\frac{\dot c}c}=n{{\frac{\dot a}a}+{\frac 12}{\frac{\dot \phi }\phi }}$$ we see that while ${\dot \phi /\phi }$ is non-negligible we have $|\dot
c/c|<|n|\dot a/a$. Weaker gravity in the early universe therefore assists a varying speed of light in solving the flatness problem
Other equations of state
------------------------
Introduce the variable
$$x=\phi a^{3\gamma -2\ } \label{defx}$$
and assume that $c$ varies as
$$c=c_0x^{n/2} \label{cx}$$
Hence, (\[cons2\]) becomes
$$(\rho a^{3\gamma })^{\cdot }=\frac{3Kc_0^2n}{8\pi }x^n\dot x$$ Integrating, we have
$$\rho a^{3\gamma }=B+\frac{3Kc_0^2n\phi ^{n+1}a^{(3\gamma -2)(n+1)}}{8\pi
(n+1)} \label{den1}$$
if $n\neq -1$. This solution can be used to study the evolution for general $
\gamma .$
Solutions to the Quasi-flatness Problem {#qflat}
=======================================
The constant $\phi $ case
-------------------------
We want to discover if it is ever natural to have evolution which asymptotes to a state of expansion with a [*non-critical*]{} density (for example, say, $\Omega _0\simeq 0.2,$ as some observations have implied). For simplicity we consider first the solutions with constant $\phi .$ The conservation equation (\[noncons\]) with constant $\phi $ reduces to
$$\begin{aligned}
\frac{(\rho a^{3\gamma })^{\cdot }}{a^{3\gamma }} &=&\frac{3Kc_0^2n}{4\pi }
a^{2n-3}\dot a\phi \\
&&\ \end{aligned}$$
which integrates to give
$$\rho =\frac B{a^{3\gamma }}+\frac{3Kc_0^2\phi n}{4\pi (3\gamma +2n-2)}
a^{2n-2} \label{exa}$$
with $B$ constant, so substituting in (\[fr2\]) we have
$$\frac{\dot a^2}{a^2}=\frac{8\pi B}{3\phi a^{3\gamma }}-\frac{K(3\gamma
-2)c_0^2\ }{a^{2(1-n)}(3\gamma +2n-2)}.$$ >From this we can easily determine the attractors at large $a$. Specialising to the radiation case, we have
$$\frac{\dot a^2}{a^2}=\frac{8\pi B}{3\phi a^4}-\frac{K\ c_0^2\ }{
a^{2(1-n)}(1+n)}.$$
Now, the density parameter $\Omega $ is defined by: $${\frac \Omega {\Omega -1}}={\frac{\frac{8\pi G\rho }3}{\frac{Kc^2(t)\ }{a^2}}
}$$ For a quasi-flat open universe the ratio between the two terms on the right-hand side is approximately constant. From the solution (\[exa\]) with $\gamma =4/3$ we have $${\frac \Omega {\Omega -1}}={\frac{\frac{8\pi G}3{\left( {\frac B{a^4}}+{
\frac{3Kc_0^2n}{8\pi G(n+1)a^{2(1-n)}}}\right) }}{\frac{Kc_0^2}{a^{2(1-n)}}}}$$ If $n<-1$ the energy production term (second term on the right-hand side of ( \[exa\])) is subdominant at late times, and so this ratio goes to infinite, meaning $\Omega =1$. If $0<n<-1$ then the second term in (\[exa\] ) dominates at late times and therefore the expansion asymptotes to one displaying $${\frac \Omega {\Omega -1}}={\frac n{n+1}}$$ that is $\Omega =-n$. So the key feature is that in these scenarios the curvature terms associated with violations of energy conservation dump energy into the universe at the same rate as the curvature term in Friedmann equation. Therefore the ratio of the two terms in the Friedmann equation stays constant, leading to an open universe with finite $\Omega $ value today.
The varying $\phi $ case
-------------------------
The formulation of the radiation case in terms of the variable $y(\tau
)=\phi a^2$ allows us to extend the analysis to the varying-$\phi $ case. Taking
$$c=c_0y^m$$
we have
$$y^{\prime 2}=A^2(1+\frac{2\omega }3)+\frac{32\pi y\rho a^4}3-4Kc^2y^2$$
and
$$\rho a^4=B+\frac{3Kc_0^2my^{2m+1}}{4\pi (2m+1)}.$$
We introduce the density parameter
$$\Omega =\frac \rho {\rho _c}=\frac{y^{\prime 2}-A^2(1+\frac{2\omega }
3)+4Kc^2y^2}{y^{\prime 2}-A^2(1+\frac{2\omega }3)\ },$$ so that
$$\frac \Omega {\Omega -1}=\frac{8\pi }{3Kc_0^2y^{2m+1}}\left[ B+\frac{
3Kmc_0^2y^{2m+1}}{4\pi (2m+1)}\right] .$$ As $t\rightarrow \infty ,\tau \rightarrow \infty ,y\rightarrow \infty $ we see that $\Omega \rightarrow 1$ if $2m+1<0$ (in Section [*\[gvar\]* ]{}we give a full exact solution for the case $m=-1$ which falls into this class), but if $2m+1>0$ we have approach to a quasi-flat open universe with
$$\Omega \rightarrow -2m$$ Again, we can have a natural quasi-flat asymptote with $0<\Omega <1.$
General asymptotic behaviour
----------------------------
Consider the behaviour of eq. (\[y5\]) at large $\tau $ and $y$ in the case where $2m+1>0;$ that is, where we have the quasi-flat attractor $\Omega
\rightarrow \Omega _\infty =-2m$ as $\tau \rightarrow \infty .$ This assumes the constants in the Friedmann equation allow sufficient expansion to occur (so there is no collapse at a finite early time). Since
$$y^{\prime }\simeq \Gamma y^{m+1}$$ we have
$$y\simeq \left[ \frac{\Gamma \Omega _\infty }2(\tau +\tau _0)\right] ^{\frac
2{\Omega _\infty }}\sim \tau ^{\frac 2{\Omega _\infty }}$$ as $\tau \rightarrow \infty .$ Using (\[y6\]) we get
$$\phi =\phi _0\exp \left[ \frac{\tau ^{1-\frac 2{\Omega _\infty }}}{1-\frac
2{\Omega _\infty }}\right]$$ so, since $\Omega _\infty <1$, we have $\phi \rightarrow \phi _0$ as $\tau
\rightarrow \infty .$ So, we have
$$a(\tau )\sim \tau ^{\frac 1{\Omega _\infty }}\exp \left[ \frac{\tau
^{1-\frac 2{\Omega _\infty }}}{2(1-\frac 2{\Omega _\infty })}\right] \sim
\tau ^{\frac 1{\Omega _\infty }}$$ Thus $t\sim \tau ^{(\Omega _\infty +1)/\Omega _\infty }$ and
$$a\sim t^{\frac 1{(1+\Omega _\infty )}}$$ as $t\rightarrow \infty .$ When $\Omega _\infty =1$ we have the expected $
a\sim t^{\frac 12}$ flat radiation asymptote.
The Lambda and the Quasi-lambda Problems {#qlamb}
========================================
The General Relativity Case
---------------------------
Let us consider the impact of a varying speed of light on the general relativity case. Similar results will occur in the BD case (to which it reduces exactly in the case of radiation with constant $\phi $). If we wish to incorporate a cosmological constant term, $\Lambda $, (which we shall assume to be constant) then we can define a vacuum stress obeying an equation of state
$$p_\Lambda =-\rho _\Lambda c^2, \label{vac}$$
where
$$\rho _\Lambda =\frac{\Lambda c^2}{8\pi G}.$$
Then, since $G$ is constant, and replacing $\rho $ by $\rho +\rho _\Lambda $ in (\[noncons\]), we have the generalisation
$$\dot \rho +3\frac{\dot a}a(\rho +\frac p{c^2})+\dot \rho _\Lambda =\frac{
3Kc\dot c}{4\pi Ga^2}. \label{cons2}$$
We shall assume that the matter obeys an equation of state of the form (\[gamma\]) and the Friedmann equation is
$$\begin{aligned}
\frac{\dot a^2}{a^2} &=&\frac{8\pi G\rho }3-\frac{Kc^2\ }{a^2}+\frac{\Lambda
c^2}3. \label{fr} \\
&&\ \ \nonumber \end{aligned}$$
We also assume that $c=c_0a^n$ again, so eq. (\[cons2\]) integrates immediately to give [@vsl1]
$$\rho =\frac B{a^{3\gamma }}+\frac{3Kc_0^2na^{2(n-1)}}{4\pi G(2n-2+3\gamma )}-
\frac{\Lambda nc_0^2a^{2n}}{4\pi G(2n+3\gamma )},$$ with $B$ a positive integration constant. Substituting in (\[fr\]) we have
$$\frac{\dot a^2}{a^2}=\frac{8\pi GB}{3a^{3\gamma }}+\frac{
Kc_0^2a^{2(n-1)}(2-3\gamma )}{(2n-2+3\gamma )}+\frac{\Lambda \gamma
c_0^2a^{2n}}{(3\gamma +2n)} \label{frnew}$$
Eq. (\[frnew\]) allows us to determine what happens at large $a.$
If $-3\gamma >2n>2n-2$ then we see that the flatness and lambda problems are both solved as before. There are three distinct cases:
### Case 1: $2n>-3\gamma >2n-2$
The $\Lambda $ term dominates the right-hand side of (\[frnew\]), the curvature term becomes negligible, and
$$\frac{\dot a^2}{a^2}\ \rightarrow \frac{\Lambda \gamma c_0^2a^{2n}}{(3\gamma
+2n)}. \label{lamas}$$
So, at large $t,$we have
$$a\sim t^{\frac{-1}n}.$$ Note that for radiation this case requires
$$-1<n_{rad}<-2,$$ while for dust it requires
$$-\frac 12<n_{dust}<-\frac 32.$$ For general fluids it requires
$$-\frac{\ 3\gamma }2<n<\frac{2-3\gamma }2.$$
### Case 2: $2n>2n-2>-3\gamma $
The scale factor approaches (\[lamas\]) but the curvature term dominates the matter density term. Define
$$\Omega =\Omega _m+\Omega _\Lambda =\frac{8\pi G(\rho +\rho _\Lambda )a^2}{
3Kc^2}$$ and then we have that, at large $a$,
$$\frac \Omega {\Omega -1}\rightarrow \frac{8\pi GB}{3Kc_0^2}a^{2-2n-3\gamma
}.$$ Thus $\Omega /(\Omega -1)$ $\rightarrow \infty $ as $a\rightarrow \infty $ for $-3\gamma <-2n+2.$ But when $2n>-3\gamma ,$
$$\frac \Omega {\Omega -1}\rightarrow \frac{\gamma \Lambda a^2}{K(2n+3\gamma )}
\rightarrow \infty$$ in (\[frnew\]), and so $\Omega \rightarrow 1$.
If $\Lambda =0$ we note that
$$\frac \Omega {\Omega -1}\rightarrow \frac{2n}{2n-2+3\gamma }$$ when $-3\gamma <2n-2$ and this is just the solution for the quasi-flatness problem found above for general relativity and Brans-Dicke theory in Section \[qflat\].
For $\Lambda \neq 0,$ when the $\Lambda $ term dominates at large $a$ we see that
$$\frac{\Omega _m}{\Omega _\Lambda }=\frac \rho {\rho _\Lambda }\rightarrow -
\frac{2n}{2n+3\gamma }$$ and we have a ’solution’ to the quasi-lambda problem ([*ie*]{} the problem of why $\Omega _m$ and $\Omega _\Lambda $ are of similar order today). Recall that in this case we have $n<0$, $2n+3\gamma >0$ and so the asymptote is again of the form $a\sim t^{\frac{-1}n}.$
Challenges for quasi-flat and quasi-lambda scenarios
====================================================
Scenarios in which the quasi-flatness problem is solved are considerably more exotic than the VSL solution to the flatness problem. Unlike flat scenarios they have a Planck epoch, something which may be a problem. We discuss this issue in the Appendix.
These scenarios possess several other unusual features. In standard flatness VSL scenarios, the expansion factor in the radiation-dominated phase is still $a\propto t^{1/2}$. Standard nucleosynthesis should still be valid unless there are changes to other aspects of relevant strong and weak interaction physics (which seems likely). However in scenarios which solve quasi-flatness we have $a\propto t^{\frac 1{1-n}}$, which for a $\Omega =0.2$ attractor means $a\propto t^{0.83}$. This could easily conflict with the nucleosynthesis constraints. However it is not enough to state that the expansion factor at nucleosynthesis time is different: the couplings, masses, decay times, etc, going into nucleosynthesis are all different [@bek]. One must rework the whole problem from scratch before ruling out these scenarios on grounds of discordant nucleosynthesis predictions.
Structure formation is another concern. It was shown in [@vsl0] that the comoving density contrast $\Delta $ and gauge-invariant velocity $v$ are subject to the equations: $$\begin{aligned}
\Delta ^{\prime }-{\left( 3(\gamma -1){\frac{a^{\prime }}a}+{\frac{c^{\prime
}}c}\right) }\Delta &=&-\gamma kv-2{\frac{a^{\prime }}a}(\gamma -1)\Pi _T
\label{delcdotm} \\
v^{\prime }+{\left( {\frac{a^{\prime }}a}-2{\frac{c^{\prime }}c}\right) }v
&=&{\left( {\frac{c_s^2k}\gamma }-{\frac 3{2k}}{\frac{a^{\prime }}a}{\left( {
\frac{a^{\prime }}a}+{\frac{c^{\prime }}c}\right) }\right) }\Delta \nonumber
\\
+{\frac{kc^2(\gamma -1)}\gamma }\Gamma - &&\ kc(\gamma -1)\left( \frac
2{3\gamma }+\frac 3{k^2c^2}\left( \frac{a^{\prime }}a\right) ^2\right) \Pi _T
\label{vcdotm}\end{aligned}$$ where $k$ is the comoving wave vector of the fluctuations, and $\Gamma $ is the entropy production rate, $\Pi _T$ the anisotropic stress, and the speed of sound $c_s$ is given by $$c_s^2={\frac{p^{\prime }}{\rho ^{\prime }}}=(\gamma -1)c^2{\left( 1-{\frac
2{3\gamma }\ \frac{c^{\prime }}c}{\frac a{a^{\prime }}}\right) } \label{cs}$$ Note that the thermodynamical speed of sound is given by $c_s^2=(\partial
p/\partial \rho )|_S$. Since in standard Big Bang models evolution is isentropic: $c_s^2=(\partial p/\partial \rho )|_S=\dot p/\dot \rho
=p^{\prime }/\rho ^{\prime }$. When $\dot c\neq 0$ the evolution need not be isentropic. However, we keep the definition $c_s^2=p^{\prime }/\rho ^{\prime
}$ since this is the definition used in perturbative calculations. One must however remember that the speed of sound given in (\[cs\]) is not the usual thermodynamical quantity. With this definition one has $\delta
p/\delta \rho =p^{\prime }/\rho ^{\prime }$ for adiabatic perturbations; that is, the ratio between pressure and density fluctuations mimics the ratio of its background rate of change.
Let us assume a radiation-dominated background. For superhorizon modes ($
ck\eta \gg 1$) there is a power-law solution, $\Delta \propto \eta ^\beta $ with $\beta =2(n+1)$ and $\beta =n-1$. The general solution takes the form: $$\Delta =A\eta ^{2(n+1)}+B\eta ^{n-1}$$ where $A$ and $B$ are constants in time. For a constant $c$ ($n=0$) this reduces to the usual $\Delta \propto \eta ^2$ growing mode, and $\Delta
\propto 1/\eta $ decaying mode. If the flatness problem is to be solved, one must have $n<-1$. If this condition is satisfied there is no growing mode.
This is an expression, in the context of Machian BD scenarios, of the link between solving the flatness problem and suppressing density fluctuations. Flatness is imposed in VSL by violations of energy conservation, acting so as to leave the universe is a state with $\Omega =1$. This process acts locally, so it also suppresses density fluctuations. Alternatively, we can see that the approach to flatness everywhere means that inhomogeneous variations in the spatial curvature must also die away.
Accordingly, we see that in VSL scenarios there is a strong connection between solving the flatness problem, and predicting a perfectly homogeneous universe. Quasi-flat scenarios therefore risk not solving the homogeneity problem. On the other hand there could some mechanism for amplifying thermal fluctuations to become seeds for the large scale structure of the universe [@vsl0]. Indeed the equations above should provide a transfer function converting the thermal white noise spectrum into a tilted or flat spectrum. If this were the case, then these theories would predict a link between the spectral tilt and $\Omega $.
Finally, one may wonder whether such $c(t)$ could ever arise in a dynamical VSL theory. In work in preparation [@BMnew] we address this issue, with the result that indeed if $\psi =log(c/c_0)$ is a scalar field with a Brans Dicke type of dynamics: $${\ddot \psi }+3{\frac{\dot a}a}\dot \psi =4\pi G\omega \rho \label{dyn2}$$ then one has a Machian solution with an exponent $n$ related to the coupling $\omega $ of the theory. Hence there will be a range of coupling values for which the flatness problem is solved, and a range for which the quasi-flatness problem is solved. In the latter case, not only can we predict an open attractor but also the value of $\Omega $ of the universe is related to a coupling constant.
Conclusion
==========
We have performed an extensive analysis solutions to Brans-Dicke theories, with varying $G$, in which the speed of light is also permitted to vary in time. We have found cosmological scenarios with novel features. For power-law variations in the velocity of light with the cosmological scale factor we identified the cases where the flatness problem can be solved. These generalise the conditions found in earlier investigations of this VSL theory. Unlike in inflationary universes which solve the flatness problem, no unusual matter fields are required in the early universe. We have also identified the cases in which the quasi-flatness problem can be solved; that is, where there can be asymptotic approach at late times to an open universe with a density close to that of the critical value. Similarly, we identified those variations of $c$ which provided solutions of the lambda problem and the quasi-lambda problem. The possibility of solving the quasi-flatness and quasi-lambda problems in this way is a genuine novelty of the VSL theory that distinguishes from the standard inflationary universe scenario. We have also discussed some problems with the VSL scenario, highlighting in particular the matter-radiation transition solutions and the role of the Planck epoch in setting initial conditions (see Appendix for further details). We have also examined the behaviour of inhomogeneous perturbations to the homogeneous and isotropic solutions and found the conditions for density perturbations to grow or decay.
Our investigations reinforce the conclusions of our earlier investigations of varying-$c$ theories without varying $G$: the scope for obtaining cosmological models which share a number of appealing properties, which closely mirror those of the observed universe, suggest that cosmologies with varying $c$ should be thoroughly explored. It is a challenge to find observational predictions which would allow future satellite probes of the microwave background radiation structure to distinguish them from inflationary universe models (with constant $c$). We hope that this paper will serve as a further stimulus to undertake those investigations and to search out new ways of testing the constancy of the traditional constants of Nature [@webb].
Acknowledgments {#acknowledgments .unnumbered}
===============
JM acknowledges financial support from the Royal Society and would like to thank A. Albrecht and C. Santos for helpful comments. JDB is supported by a PPARC Senior Fellowship.
Appendix - Planck time in VSL scenarios {#appendix---planck-time-in-vsl-scenarios .unnumbered}
=======================================
The Machian VSL scenario, in which $c=c_0a^n$, introduced by Barrow [@vsl1] has significant advantages to the phase transition scenario, in which the speed of light changes suddenly from $c^{-}$ to $c^{+},$ preferred by Albrecht and Magueijo [@vsl0]. In the phase transition scenario one runs into the problem of having to decide when to lay down “natural initial conditions” (that is $\epsilon $ and $\epsilon _\Lambda $ of order 1). For a phase transition occurring at time $t_c\approx t_P^{+}$, the Planck time $
t_P^{-}$ (built from the constants as they were before the transition) is much smaller than $t_c$. But why should we lay down natural initial conditions just before the phase transition? If the only scales in the problem are the ones set by the constants as they were before the transition, then natural initial conditions should be set at $t_P^{-}$. If we are to lay down natural initial conditions at $t_P^{-}$ then the universe goes off the attractor well before the phase transition. A catastrophic phase precedes the phase transition, in which the universe becomes Milne (curvature dominated) or de Sitter ($\Lambda $ dominated). Albrecht and Magueijo noted that this catastrophic phase is not the end of the universe in VSL scenarios. A varying speed of light would not write off Milne or de Sitter universes, but would still push them towards an Einstein-de Sitter universe. The only universes which would be selected out in this process are the ones with positive curvature, which would end in a Big Crunch, well before the phase transition.
The Machian scenario does not have this problem, if we are content with solving the flatness but not the quasi-flatness problem. If we consider the radiation dominated phase, $c=c_0a^n$ with $n<-1$, and $a\propto t^{1/2}$, then we have $t/t_P\propto t^{n+1}$. Hence there is no Planck time in these scenarios: as $t\rightarrow 0$, one has $t/t_P\rightarrow \infty $. As we go back in time, the universe becomes hotter and hotter, but the Planck temperature also increases, and is never achieved at any time. The idea of setting natural initial conditions at Planck time does not make sense in these scenarios. We have a universe constantly pushed towards an attractor, which is flat, and has zero cosmological constant.
Scenarios in which the quasi flatness problem is solved do not have this desirable feature. We find that $t/t_P\propto t^{\frac{1+n}{1-n}}$. Hence we must have a Planck time in our past in these scenarios.
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|
---
abstract: 'In this article we furnish a new simple proof of a hard identity from the theory of cubature formulas via the method of coefficients.'
address: 'RUSSIA, Krasnoyarsk,'
author:
- 'Georgy P. Egorychev'
date: 'December 3, 2011'
title: Integral representation and computation a multiple sum in the theory of cubature formulas
---
[Introduction]{}
================
Let $\mathbf{\alpha }=(\alpha _{0},\alpha _{1},\ldots ,\alpha _{d}),$ $%
\mathbf{\beta }=(\beta _{0},\beta _{1},\ldots ,\beta _{d})$ be vectors from $E^{d+1}$ with integer non-negative coordinates, and the vector $\mathbf{
\gamma }=(\gamma _{0},\gamma _{1},\ldots ,\gamma _{d})\in E^{d+1}$. Denote ** **$\left\vert \mathbf{\alpha }\right\vert :=\alpha _{0}+\alpha
_{1}+\ldots +\alpha _{d}=2s+1,$ $\mathbf{\alpha }!:=\alpha _{0}!\alpha
_{1}!\ldots \alpha _{d}!,$ $\binom{\mathbf{\alpha }}{\mathbf{\beta }}:=%
\binom{\alpha _{0}}{\beta _{0}}\ldots \binom{\alpha _{d}}{_{d}},$ where $%
\binom{a}{b}:=\frac{\Gamma \left( a+1\right) }{\Gamma \left( b+1\right)
\Gamma \left( a-b+1\right) },$ and $\binom{a}{b}:=0$, if $b\geq a+1.$ Moreover, we write $\mathbf{\alpha }-1/2:=\left( \alpha _{0}-1/2,\alpha
_{1}-1/2,\ldots ,\alpha _{d}-1/2\right) .$
Heo S. and Xu Y.[@Heo99 pp.631-635] with the help of theory of operators and generating functions have proved the following multiple combinatorial identity [@Heo99 the identity (2.9)]:$$2^{2s}\mathbf{\alpha }!\binom{\mathbf{\alpha +\gamma }}{\mathbf{\alpha }}%
=$$ $$\sum_{j=0}^{s}\left( -1\right) ^{j}\binom{d\text{ }+\sum_{i=0}^{d}(\alpha
_{i}+\gamma _{i})}{j}\sum_{\beta _{0}+\beta _{1}+...+\beta
_{d}=s-j}\prod_{i=0}^{d}\binom{\beta _{i}\mathbf{+}\gamma _{i}}{%
\beta _{i}}\left( 2\beta _{i}+\gamma _{i}+1\right) ^{\alpha _{i}}.
\label{K1}$$
At the end of the 1970’s, G.P. Egorychev has developed the method of coefficients, which was successfully applied to many combinatorial sums [@2; @3; @4; @5]. The purpose of this article is finding a new simple proof of identity (\[K1\]) by means of the method of coefficients [@2] and multiple applications of a known theorem on the total sum of residues in the theory of holomorphic functions.
[Proof of the identity ]{}(\[K1\])
==================================
The identity (\[K1\]) can be expressed in the form of$$\sum_{j=0}^{s}\left( -1\right) ^{j}\binom{d+\sum_{i=0}^{d}(\alpha
_{i}+\gamma _{i})}{j}\sum_{\beta _{0}+\beta _{1}+...+\beta
_{d}=s-j}\prod_{i=0}^{d}\binom{\beta _{i}\mathbf{+}\gamma _{i}}{%
\beta _{i}}\frac{\left( 2\beta _{i}+\gamma _{i}+1\right) ^{\alpha _{i}}}{%
\left( \alpha _{i}\right) !}=$$$$=2^{2s}\prod_{i=0}^{d}\binom{\alpha _{i}\mathbf{+}\gamma _{i}}{%
\alpha _{i}}. \label{K2}$$Denote by $T\left( s;\mathbf{\alpha },\mathbf{\beta }\right)$ the left hand side of identity (\[K2\]):$$T\left( s;\mathbf{\alpha },\mathbf{\beta }\right) :=\sum_{j=0}^{s}\left(
-1\right) ^{j}\binom{\sum_{i=0}^{d}(\alpha _{i}+\gamma _{i})\mathbf{+}d}{j}%
\times S_{j}, \label{K3}$$where$$S_{j}:=\sum_{\beta _{0}+\beta _{1}+...+\beta _{d}=s-j}\prod_{i=0}^{d}%
\binom{\beta _{i}\mathbf{+}\mu _{i}}{\beta _{i}}\frac{\left( 2\beta _{i}+\mu
_{i}+1\right) ^{\alpha _{i}}}{\left( \alpha _{i}\right) !}. \label{K4}$$Then by means of the method of coefficients we obtain$$S_{j}=\sum_{\left\vert \mathbf{\beta }\right\vert
=s-j}(\prod_{i=0}^{d}\binom{\beta _{i}\text{ }\mathbf{+}\text{ }%
\gamma _{i}}{\beta _{i}}\frac{\left( 2\beta _{i}+\gamma _{i}+1\right)
^{\alpha _{i}}}{\left( \alpha _{i}\right) !}=$$$$=\sum_{\mathbf{\beta }_{0}=0}^{\infty }\ldots \sum_{\mathbf{\beta }%
_{d}=0}^{\infty }\mbox{\bf res}_{z_{0},\ldots
,z_{d},t}(t^{-s+j-1}\prod_{i=0}^{d}\left( 1-tz_{i}\right) ^{-\gamma
_{i}-1}z_{i}^{-\beta _{i}-1})\times$$ $$\times \mbox{\bf res}_{w_{0},\ldots
,w_{d}}(\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\exp (w_{i}\left(
2\beta _{i}+\gamma _{i}+1\right) )=$$$$=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s+j-1}(\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\exp
(w_{i}\left( \gamma _{i}+1\right) )\times$$ $$\times \prod_{i=0}^{d}\left(
\sum_{\beta _{i}=0}^{\infty }\left( \exp (\beta _{i}\left( 2w_{i}\right)
\right) \mbox{\bf res}_{z_{i}}\left( \left( 1-tz_{i}\right) ^{-\gamma
_{i}-1}z_{i}^{-\beta _{i}-1}\right) \right) \}=$$(the summation by each $\beta _{i}$ and $\mbox{\bf res}_{z_{i}},$ $%
i=0,\ldots ,d$: the substitution rule, the changes $z_{i}=\exp (2w_{i}),$ $i$ $=$ $0,\ldots ,d$)$$=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s+j-1}(\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\exp
(w_{i}\left( \gamma _{i}+1\right) )\times \prod_{i=0}^{d}\left(
1-t\exp (2w_{i})\right) ^{-\gamma _{i}-1}\}=$$$$=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s+j-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left(
\exp (-w_{i})-t\exp (w_{i})\right) ^{-\gamma _{i}-1}\},$$i.e.$$S_{j}=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s+j-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left(
\exp (-w_{i})-t\exp (w_{i})\right) ^{-\gamma _{i}-1}\}. \label{K5}$$According to (\[K3\])**–**(\[K5\])** **we obtain$$T\left( s;\mathbf{\alpha },\mathbf{\beta }\right) =\sum_{j=0}^{s}\mbox{\bf
res}_{w_{0},\ldots
,w_{d},t}\{t^{-s+j-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left(
\exp (-w_{i})-t\exp (w_{i})\right) ^{-\gamma _{i}-1}\}\times$$ $$\times \mbox{\bf res}%
_{x}\{x^{-j-1}\left( 1-x\right) ^{d+\sum_{i=0}^{d}(\alpha _{i}+\gamma
_{i})}\}=$$$$=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left( \exp
(-w_{i})-t\exp (w_{i})\right) ^{-\gamma _{i}-1}\times$$ $$\times(\sum_{j=0}^{\infty }t^{j}%
\mbox{\bf res}_{x}\{x^{-j-1}\left( 1-x\right) ^{d+\sum_{i=0}^{d}(\alpha
_{i}+\gamma _{i})})\}=$$(summation w.r.t. $j$, and $\mbox{\bf res}_{x}$: the substitution rule, the change $x=t$)$$=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left( \exp
(-w_{i})-t\exp (w_{i})\right) ^{-\gamma _{i}-1}\left( 1-t\right)
^{d+\sum_{i=0}^{d}(\alpha _{i}+\gamma _{i})}\}.$$Thus we proved
\[AprL1\]Let parameters $s,\alpha _{0},\alpha _{1},\ldots ,\alpha
_{d},\beta _{0},\beta _{1},\ldots ,\beta _{d}$ be non-negative integers, for which $\alpha _{0}+\ldots +\alpha _{d}=2s+1,$ *and the* vector $(\mu _{0},\mu _{1},\ldots ,\mu _{d})\in \mathbb{R}^{d+1}.$ *Then the following integral formula holds*:$$\sum_{j=0}^{s}\left( -1\right) ^{j}\binom{d+\sum_{i=0}^{d}(\alpha
_{i}+\gamma _{i})}{j}\sum_{\beta _{0}+\beta _{1}\ldots +\beta
_{d}=s-j}\prod_{i=0}^{d}\binom{\beta _{i}\mathbf{+}\gamma _{i}}{%
\beta _{i}}\frac{\left( 2\beta _{i}+\gamma _{i}+1\right) ^{\alpha _{i}}}{%
\left( \alpha _{i}\right) !}=$$$$=\mbox{\bf res}_{w_{0},\ldots
,w_{d},t}\{t^{-s-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left( \exp
(-w_{i})-t\exp (w_{i})\right) ^{-\gamma _{i}-1}\left( 1-t\right)
^{d+\sum_{i=0}^{d}(\alpha _{i}+\gamma _{i})}\}. \label{K6}$$
It is easy to see, that a consecutive calculation of multiple integral in the right hand side of **(**\[K6\]**)** on each variable $t
$ *and* $w_{0},\ldots ,w_{d}$ gives the multiple sum in the left part of (\[K6\]). Now, we provide new proof of identity (\[K2\]**)** by calculation of a multiple residue at zero point in the right part of the formula (\[K6\]) consecutively on each variable $w_{0},\ldots
,w_{d}$ *and* $t$ (see lemmas* *\[AprL1\]–\[AprL3\] and the theorem* *\[AprT1\]).
Let’s introduce some necessary notations. Denote $$f=f\left( w,t\right) :=e^{-w}-te^{w},\text{ }g=g\left( w,t\right)
:=e^{-w}+te^{w}, \label{K7}$$where $\alpha $ is the fixed integer and $\gamma \in \mathbb{R}.$ Obviously$$f^{\prime }:=\frac{df}{dw}=-g,\text{ }g^{\prime }:=\frac{dg}{dw}=-f,\text{ }%
g^{2}-f^{2}=4t,\text{ }(f^{-\gamma })^{\prime }=\gamma f^{-\gamma -1},\text{
}g^{\alpha }=-\alpha g^{\alpha -1}f, \label{K11}$$$$f\left( 0\right) =1-t,\text{ }g\left( 0\right) =1+t. \label{K12}$$
\[AprL2\]*If* $s$* is a non-negative integer and* $\gamma \in \mathbb{R},$ in notation **(**\[K7\]**)** *and* **(**[K11]{}**)** *the following expansion* of the derivative$$(f^{-\gamma -1})_{w}^{\left( \alpha \right) }=\left( \gamma +1\right) \times
\ldots \times \left( \gamma +\alpha \right) f^{-\gamma -\alpha -1}g^{\alpha
}+\sum_{k=1}^{[\alpha /2]}c_{k}\left( \gamma \right) f^{-\gamma +2k-\alpha
-1}g^{\alpha -2k}, \label{K13}$$*with integer coefficients* $c_{1},c_{2},\ldots ,c_{[\alpha /2]}$ *is valid.* According to (\[K12\]) the formula (\[K13\]) generates the following formula$$\mbox{\bf res}_{w}w_{i}^{-\alpha -1}\left( \exp (-w_{i})-t\exp
(w_{i})\right) ^{-\gamma -1}:=[\left( \exp (-w_{i})-t\exp (w_{i})\right)
^{-\gamma -1}]_{w=0}^{\left( \alpha \right) }/\alpha !=$$$$=\binom{\alpha +\gamma }{\alpha }\left( 1-t\right) ^{-\gamma -\alpha
-1}\left( 1+t\right) ^{\alpha }(1+\sum_{k=1}^{[\alpha /2]}h_{k}\left( \alpha
,\gamma \right) \left( 1-t\right) ^{2k}\left( 1+t\right) ^{-2k}),
\label{K14}$$where the rational coefficients $h_{k}\left( \alpha ,\gamma \right)
:=c_{k}\left( \gamma \right) /\alpha !,$ $k=1,\ldots ,[\alpha /2].$
The formula **(**\[K13\]**) ** can be easily proved by induction on parameter $\alpha $. According to **(**\[K11\]**)** we have for initial values $\alpha =1,2,3$:$$(f^{-\gamma -1})^{\prime }=-\left( \gamma +1\right) f^{-\gamma -2}f^{\prime
}=\left( \gamma +1\right) f^{-\gamma -2}g.$$$$(f^{-\gamma -1})^{^{\prime \prime }}=((f^{-\gamma -1})^{\prime })^{\prime
}=(\left( \gamma +1\right) f^{-\gamma -2}g)^{\prime }=\left( \gamma
+1\right) \left( f^{-\gamma -2}\right) ^{\prime }g+\left( \gamma +1\right)
f^{-\gamma -2}f=$$$$=\left( \gamma +1\right) \left( \gamma +2\right) f^{-\gamma -3}g^{2}-\left(
\gamma +1\right) f^{-\gamma -1}.$$$$(f^{-\gamma -1})^{^{\prime \prime \prime }}=((f^{-\gamma -1})^{^{\prime
\prime }})^{\prime }=(\left( \gamma +1\right) \left( \gamma +2\right)
f^{-\gamma -3}g^{2}-\left( \gamma +1\right) f^{-\gamma -1})^{\prime }=$$$$=\left( \gamma +1\right) \left( \gamma +2\right) \left( \gamma +3\right)
f^{-\gamma -4}g^{3}+\left( \gamma +1\right) \left( \gamma +2\right)
f^{-\gamma -3}2gf-\left( \gamma +1\right) ^{2}f^{-\gamma -2}g=$$$$=\left( \gamma +1\right) \left( \gamma +2\right) \left( \gamma +3\right)
f^{-\gamma -4}g^{3}-\left( \gamma +1\right) (3\gamma +2)f^{-\gamma -2}g.$$Further, if formula **(**\[K12\]**)** is valid for current value $\alpha $, with the help of **(**\[K11\]**)** we have$$(f^{-\gamma -1})^{^{\left( \alpha +1\right) }}=((f^{-\gamma -1})^{\left(
\alpha \right) })^{\prime }=(\left( \gamma +1\right) \times \ldots \times
\left( \gamma +\alpha \right) f^{-\gamma -\alpha -1}g^{\alpha
}+\sum_{k=1}^{[\alpha /2]}c_{k}\left( \gamma \right) f^{-\gamma +2k-\alpha
-1}g^{\alpha -2k})^{\prime }=$$$$=\left( \gamma +1\right) \times \ldots \times \left( \gamma +\alpha
+1\right) f^{-\gamma -\alpha -2}g^{\alpha +1}-\left( \gamma +1\right) \times
\ldots \times \left( \gamma +\alpha \right) f^{-\gamma -\alpha }\alpha
g^{\alpha -1}+$$$$+\sum_{k=1}^{[\alpha /2]}c_{k}\left( \gamma \right) \left( \gamma -2k+\alpha
+1\right) f^{-\gamma +2k-\alpha -2}g^{\alpha -2k+1}-\sum_{k=1}^{[\alpha
/2]}c_{k}\left( \gamma \right) f^{-\gamma +2k-\alpha }\left( \alpha
-2k\right) g^{\alpha -2k-1}=$$(replacement in the first sum of an index $k-1$ by $k)$$$=\left( \gamma +1\right) \ldots \left( \gamma +\alpha +1\right) f^{-\gamma
-\alpha -2}g^{\alpha +1}+(c_{1}\left( \gamma \right) \left( \gamma +\alpha
-1\right) -\alpha \left( \gamma +1\right) \ldots \left( \gamma +\alpha
\right) )f^{-\gamma -\alpha +1}g^{\alpha -1}+$$$$+\sum_{k=0}^{[\alpha /2]-1}c_{k+1}\left( \gamma \right) \left( \gamma
-2k+\alpha -1\right) f^{-\gamma +2k-\alpha }g^{\alpha
-2k-1}-\sum_{k=1}^{[\alpha /2]}c_{k}\left( \gamma \right) f^{-\gamma
+2k-\alpha }\left( \alpha -2k\right) g^{\alpha -2k-1}=$$$$=\left( \gamma +1\right) \ldots \left( \gamma +\alpha +1\right) f^{-\gamma
-\alpha -2}g^{\alpha +1}+(c_{1}\left( \gamma \right) \left( \gamma +\alpha
-1\right) -c_{[\alpha /2]}\times$$$$\times\left( \gamma \right) f^{-\gamma +2[\alpha
/2]-\alpha }\left( \alpha -2[\alpha /2]\right) g^{\alpha -2[\alpha /2]-1}+$$$$+\sum_{k=1}^{[\alpha /2]-1}(c_{k+1}\left( \gamma \right) \left( \gamma
-2k+\alpha -1\right) -\left( \alpha -2k\right) c_{k}\left( \gamma \right)
)f^{-\gamma +2k-\alpha }g^{\alpha -2k-1},$$as was to be shown.
\[AprL3\]If $s$ *is a non-negative integer then the* following formulas hold:$$J=\mbox{\bf res}_{t}\left( 1-t\right) ^{-1}\left( 1+t\right)
^{2s+1}t^{-s-1}=2^{2s}, \label{K16}$$$$J_{k}=\mbox{\bf res}_{t}\left( 1-t\right) ^{k-1}\left( 1+t\right)
^{2s-k+1}t^{-s-1}=0,\text{ }\forall k=1,\ldots ,2s. \label{K17}$$
We have$$J=\mbox{\bf res}_{t}\left( 1-t\right) ^{-1}\left( 1+t\right)
^{2s+1}t^{-s-1}:=\mbox{\bf res}_{t=0}\left( 1-t\right) ^{-1}\left(
1+t\right) ^{2s+1}t^{-s-1}=$$(the theorem of the full sum of residues)$$=-\mbox{\bf res}_{t=1}\left( 1-t\right) ^{-1}\left( 1+t\right)
^{2s+1}t^{-s-1}-\mbox{\bf res}_{t=\infty }\left( 1-t\right) ^{-1}\left(
1+t\right) ^{2s+1}t^{-s-1}=$$(directly by definition of a residue at a corresponding point)$$=[\left( 1+t\right) ^{2s+1}t^{-s-1}]_{t=1}-res_{t=0}\left( 1-1/t\right)
^{-1}\left( 1+1/t\right) ^{2s+1}\left( 1/t\right) ^{-s}(-1/t)^{2}=$$$$=2^{2s+1}-\mbox{\bf res}_{t=0}\left( 1-t\right) ^{-1}\left( 1+t\right)
^{2s+1}t^{-s}=2^{2s+1}-J\Leftrightarrow J=2^{2s+1}-J\Rightarrow J=2^{2s}.$$Let $k$ be the any fixed number from set $\{1,\ldots ,2s\}$. Similarly to the previous case, we have$$J_{k}=\mbox{\bf res}_{t}\left( 1-t\right) ^{k-1}\left( 1+t\right)
^{2s-k+1}t^{-s-1}:=\mbox{\bf res}_{t=0}\left( 1-t\right) ^{k-1}\left(
1+t\right) ^{2s-k+1}t^{-s-1}-$$$$-\mbox{\bf res}_{t=\infty }\left( 1-t\right) ^{k-1}\left( 1+t\right)
^{2s+1}t^{-s-1} =$$(as powers of binomials $\left( 1-t\right) ^{k-1}$ and $\left( 1+t\right)
^{2s-k+1}t^{-s-1}$ non-negative at any $k$ from set $\{1,\ldots ,2s\}$)$$= 0-\mbox{\bf res}_{t=0}\left( 1-1/t\right) ^{k-1}\left( 1+1/t\right)
^{5}\left( 1/t\right) ^{-3}(-1/t)^{2}=$$$$=-\mbox{\bf res}_{t=0}\left( 1-t\right) ^{-s-1}\left( 1+t\right)
^{2s-k+1}t^{-s-1}=-J_{k}\Longleftrightarrow J_{k}=-J_{k}\Longleftrightarrow
J_{k}=0.$$
\[AprT1\]The identity (\[K2\]) *is valid*.
According to **(**\[K6\]**)** we have the following integral representation for sum $T\left( s;\mathbf{\alpha },\mathbf{\beta }\right) $ in the left hand side of identity **(**\[K2\]**):**$$T\left( s;\mathbf{\alpha },\mathbf{\beta }\right) =\mbox{\bf res}%
_{w_{0},\ldots ,w_{d},t}\{t^{-s-1}\left( 1-t\right)
^{d+\sum_{i=0}^{d}(\alpha _{i}+\gamma
_{i})}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left( \exp
(-w_{i})-t\exp (w_{i})\right) ^{-\mu _{i}-1}\}=$$$$=\mbox{\bf res}_{t}\{t^{-s-1}\left( 1-t\right) ^{d+\sum_{i=0}^{d}(\alpha
_{i}+\gamma _{i})}(\prod_{i=0}^{d}\mbox{\bf res}_{w_{i}}w_{i}^{-%
\alpha _{i}-1}\left( \exp (-w_{i})-t\exp (w_{i})\right) ^{-\mu _{i}-1})\}$$Calculating in the last expression each of residues w.r.t. variables $w_{0},\ldots
,w_{d}$ by the formula **(**\[K14\]**)** we have$$T\left( s;\mathbf{\alpha },\mathbf{\beta }\right) =\mbox{\bf res}%
_{t}\{t^{-s-1}\left( 1-t\right) ^{d+\sum_{i=0}^{d}(\alpha _{i}+\gamma
_{i})}\times$$$$\times (\prod_{i=0}^{d}\binom{\alpha _{i}+\gamma _{i}}{\alpha _{i}}%
\left( 1-t\right) ^{-\gamma _{i}-\alpha _{i}-1}\left( 1+t\right) ^{\alpha
_{i}}(\sum_{k=1}^{[\alpha _{i}/2]}h_{k}\left( \alpha _{i},\gamma _{i}\right)
\left( 1-t\right) ^{2k}\left( 1+t\right) ^{-2k}))\}=$$(trivial cancelations under the product sign $\prod_{i=0}^{d}\ldots $ according to the assumption $\sum_{i=0}^{d}\alpha _{i}=2s+1)$)$$=\binom{\mathbf{\alpha +\gamma }}{\mathbf{\alpha }}\mbox{\bf res}%
_{t}\{t^{-s-1}\left( 1-t\right) ^{-1}\left( 1+t\right)
^{2s+1}\prod_{i=0}^{d}(1+\sum_{k=1}^{[\alpha _{i}/2]}h_{k}\left(
\alpha _{i},\gamma _{i}\right) \left( 1-t\right) ^{2k}\left( 1+t\right)
^{-2k})\}. \label{K18}$$As $[\alpha _{0}/2]+[\alpha _{1}/2]+\ldots +[\alpha _{d}/2]\leq \lbrack
\sum_{i=0}^{d}\alpha _{i}/2]=s,$ it is easy to see, that after opening the brackets and simplifying the similar terms the product$$\prod_{i=0}^{d}(1+\sum_{k=1}^{[\alpha _{i}/2]}h_{k}\left( \alpha
_{i},\gamma _{i}\right) \left( 1-t\right) ^{2k}\left( 1+t\right) ^{-2k}),$$under the sign $\mbox{\bf res}_{t}$ in (\[K18\])** **is** **is representable in the form of a polynomial$$1+\sum_{k=2}^{2s}\lambda _{k}\left( 1-t\right) ^{k}\left( 1+t\right) ^{-k},$$where coefficients $\lambda _{1},\ldots ,\lambda _{2s-1}$ are some fixed rational numbers. Thus$$T\left( s;\mathbf{\alpha },\mathbf{\beta }\right) =\binom{\mathbf{\alpha
+\gamma }}{\mathbf{\alpha }}\mbox{\bf res}_{t}\{t^{-s-1}\left( 1-t\right)
^{-1}\left( 1+t\right) ^{2s+1}(1+\sum_{k=1}^{2s}\lambda _{k}\left(
1-t\right) ^{k}\left( 1+t\right) ^{-k})\}=$$$$=\binom{\mathbf{\alpha +\gamma }}{\mathbf{\alpha }}\{\mbox{\bf res}%
_{t}\left( 1-t\right) ^{-1}\left( 1+t\right)
^{2s+1}t^{-s-1}+\sum_{k=1}^{2s}\lambda _{k}\mbox{\bf res}_{t}\left(
1-t\right) ^{k-1}\left( 1+t\right) ^{2s-k+1}t^{-s-1}\}=$$(calculation of residues in last expression using formulas **(**[K16]{}**)** and **(**\[K17\]**)**$$=\binom{\mathbf{\alpha +\gamma }}{\mathbf{\alpha }}\{2^{2s}+\sum_{k=1}^{2s}%
\lambda _{k}\times 0\}=\binom{\mathbf{\alpha +\gamma }}{\mathbf{\alpha }}%
2^{2s}.$$
It would be interesting to know what additional information one can obtain from the knowledge of the integral representation of the left hand side of identity (\[K1\]). $$J=\mbox{\bf res}_{w_{0},...,w_{d},t}\{t^{-s-1}\prod_{i=0}^{d}w_{i}^{-\alpha _{i}-1}\left( \exp (-w_{i})-t\exp
(w_{i})\right) ^{-\mu _{i}-1}\left( 1-t\right) ^{d+\sum_{i=0}^{d}(\alpha
_{i}+\gamma _{i})}\}/\mathbf{\alpha }!, \label{K20}$$For example, the integral (\[K20\]) can written in the following form $$J=\mbox{\bf res}_{t}\{(t^{-s-1}\prod_{i=0}^{d}\mbox{\bf res}%
_{w_{i}}w_{i}^{-\alpha _{i}-1}\left( \exp (-\lambda _{i}w_{i})-t\exp
(\lambda _{i}w_{i})\right) ^{-\gamma _{i}-1}\}/\mathbf{\alpha }!.
\label{K21}$$The calculation of integral (\[K21\]) is connected with studying of the hyperbolic $t$-sine* [@FoaHun2005]*$$\sinh _{t}(x):=(\exp (-x)-t\exp (x))/2, \label{K22}$$and the functions $\sinh _{t}^{-\gamma }(x),$ $\gamma \in
\mathbb{N}$, *and*$$J_{a,\gamma }\left( t\right) :=\mbox{\bf res}_{z}(z^{-\alpha -1}\left( \exp
(-z)-t\exp (z)\right) ^{-\gamma -1})/\alpha !=\mbox{\bf res}_{z}(z^{-\alpha
-1}\left( \exp (-z)-t\exp (z)\right) ^{-\gamma -1})/\alpha !. \label{K23}$$In my opinion, the study of these functions is interesting, including their combinatorial interpretation and various corresponding relations.
The author is thankful to E.Zima and I.Kotsireas for useful comments on early drafts of this paper.
[9]{} G.P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Nauka, Novosibirsk, 1977 (in Russian). English transl. Transl. Math. Monographs 59, Amer. Math. Soc., Providence. RI 1984; 2nd ed. in 1989.
G.P. Egorychev, Method of coefficients: an algebraic characterization and recent applications. Labours Waterloo Workshop on Computer Algebra, Waterloo 5-7 May 2008, Springer Verlag, 2009, 1–33.
G.P. Egorychev and E.V. Zima, Integral representation and algorithms for closed form summation. handbook of Algebra, **5** (ed. M. Hazewinkel), Elsevier, 2008, 459–529.
D. Foata and G.-N. Han, The q-tangent and q-secant numbers via basic eulerian polynomials, Proc. Amer. Math. Soc., 138, 2010, 385–393.
S. Heo and Y. Xu, Invariant cubature formulae for spheres and balls by combinatorial methods. SIAM J. Numer. Anal. **38**(2), 2000, 626–638.
V.K. Leont’ev, Selected problems of combinatorial analysis, MSTU, Bauman, 2001 (in Russian).
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abstract: 'We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B$, $A\cap C$, and $B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.'
author:
- 'Eben Blaisdell,$^{1}$ András Gyárfás,$^{2}$ Robert A. Krueger,$^{3}$ Ronen Wdowinski$^{4}$'
title: 'Partitioning the power set of $[n]$ into $C_k$-free parts'
---
Introduction and results
========================
Hypergraph Ramsey problems usually address the existence of large monochromatic structures in colorings of the edges of $K_n^r$, the complete $r$-uniform hypergraph. It is rare that monochromatic structures are sought in colorings of hypergraphs containing [*all subsets*]{} of $[n]$, ${\mathcal{P}}(n)$. An exception is the Finite Unions Theorem of Folkman, Rado, Sanders [@GRS]. A more recent research in this direction is by Axenovich and Gyárfás [@AGY], where Ramsey numbers of Berge-$G$ hypergraphs were studied for several graphs $G$ in colorings of ${\mathcal{P}}(n)$. Ramsey numbers of Berge-$G$ hypergraphs in the [*uniform case*]{} have been investigated also in [@GMOV; @STWZ].
A hypergraph $H=(V,F)$ is called Berge-$G$ if $G=(V,E)$ is a graph and there exists a bijection $g: E(G)\mapsto E(H)$ such that for $e\in E(G)$ we have $e\subseteq g(e)$. Note that for a given graph $G$ there are many Berge-$G$-hypergraphs. Berge-$G$ hypergraphs were defined by Gerbner and Palmer [@GP] to extend the notion of paths and cycles in hypergraphs introduced by Berge in [@B]. In particular, a Berge-$C_3$ hypergraph consists of three subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. When there is no confusion, we will often refer to a Berge-$G$ hypergraph simply as ‘a $G$.’ The graphs $C_k,P_k,S_k$ denote cycle, path, and star with $k$ edges, respectively. It is customary to use the names *triangle* and *claw* for the graphs $C_3$ and $S_3$, respectively.
A hypergraph $H$ with vertex set $[n]$ and whose edges are sets from ${\mathcal{P}}(n)$ is called [*$G$-free*]{}, if it does not contain any subhypergraph isomorphic to a Berge-$G$ hypergraph. The [*intersection graph*]{} of a hypergraph $H$ is a graph $G$ whose vertices represent edges of $H$ and where there is an edge in $G$ if and only if the corresponding edges of $H$ have non-empty intersection. Note that if the intersection graph of $H$ has no subgraph isomorphic to the intersection graph of $G$ (that is, the line graph of $G$), then $H$ is $G$-free. The reverse statement is not true: the intersection graph of the hypergraph $H$ with edges $\{1,2\},\{1,2,3\},\{1,2,4\}$ is a triangle but $H$ is triangle-free.
To define the Ramsey-type problem we address here, let $f(n,G)$ be the smallest number of colors in a coloring of ${\mathcal{P}}(n)$ such that all color classes are $G$-free. In other words, in every coloring of ${\mathcal{P}}(n)$ with $f(n,G)-1$ colors, there is a Berge-$G$ subhypergraph in some color class. We use the terms [*coloring, partitioning*]{} of ${\mathcal{P}}(n)$ in the same sense. Since the presence of empty sets and singleton sets do not influence whether a coloring is $G$-free, we usually construct colorings of ${\mathcal{P}}^*(n)$, what we define to be ${\mathcal{P}}(n)$ with the empty set and the singletons removed. However, the following natural partition of the whole power set of $[n]$ is useful. For every $A\subseteq [n-1]$, the part defined by $A$ is $$\{X_1(A)=A,X_2(A)=[n]\setminus X_1(A), X_3(A)=A\cup \{n\}, X_4(A)=[n]\setminus X_3(A)\}.$$ Since $A$ and $[n-1]\setminus A$ define the same part, we have $2^{n-2}$ parts (each of size four). This partition was used in [@AGY] to show that $f(n,C_3)\le 2^{n-2}$. Observing that $$X_1(A)\cap X_2(A)=X_3(A)\cap X_4(A)=X_1(A)\cap X_4(A)=\emptyset,$$ these parts are $C_3$-free, $C_4$-free and $S_3$-free. Thus we have a natural upper bound for three small graphs:
\[basepart\] $f(n,G)\le 2^{n-2}$ for $G\in \{C_3,C_4,S_3\}$.
How sharp is this upper bound for the three small graphs involved? The easiest lower bound comes for the claw.
\[claw\] $2^{n-2} - n/2 \leq f(n, S_3)$. In general, $\frac{2^{n-1}}{k-1} - O\left(n^{k-2}\right) \le f(n,S_k)$.
Consider a partition $Q$ of ${\mathcal{P}}(n)$ into $S_k$-free parts. Let $H=(V,E)$ be the subhypergraph of ${\mathcal{P}}(n)$ determined by the edges of size at least $k$. Then $Q$ partitions $H$ into $S_k$-free parts $H_i=(V,E_i)$, for $i=1,\dots,t$. Since $k$ edges of size at least $k$ cannot have common intersection by the $S_k$-free property, each hypergraph $H_i$ has maximum degree at most $k-1$. Therefore
$$n2^{n-1} - \left(n + 2\binom{n}{2} + \cdots + (k-1)\binom{n}{k-1}\right) =\sum_{v\in V} d_H(v) = \sum_{i=1}^t \sum_{v\in V} d_{H_i}(v) \le (k-1)nt,$$ implying $t \geq \frac{2^{n-1}}{k-1} - \frac{1}{k-1}\left( 1 + \binom{n-1}{1} + \cdots + \binom{n-1}{k-2} \right) = \frac{2^{n-1}}{k-1} - O\left(n^{k-2}\right)$. For $k=3$, this calculation gives $2^{n-2} - n/2 \leq f(n, S_3)$.
The discrepancy of $-n/2$ between Proposition \[basepart\] and \[claw\] for $f(n,S_3)$ is the consequence of the fact that three edges of ${\mathcal{P}}^*(n)$ intersecting in a vertex $v$ do not define a claw in the special case when the three edges are $\{v,x,y\},\{v,x\},\{v,y\}$. Utilizing this with several different designs, we have small examples in Section \[designs\] showing that the upper bound for $f(n, S_3)$ in Proposition \[basepart\] can sometimes be lowered (in particular, we show that $f(6,S_3)\le 15$ and $f(9,S_3)\le 126$). It is unclear whether one can use this phenomenon to decrease the upper bound for infinitely many $n$.
For the case of the triangle, the upper bound of Proposition \[basepart\] is tight. For odd $n\ge 7$ this was shown with a simple proof in [@AGY]. Somewhat surprisingly, this remains true for the even $n$ case as well (but not for $n=5$).
\[trianglethm\] For $n\ge 3, n\ne 5$, $f(n,C_3)=2^{n-2}$. Additionally, $f(5, C_3) = 7$.
In case of $G=C_4$ we improve the upper bound of Proposition \[basepart\] by a constant factor and slightly improve the lower bound ${2^{n-1}\over 3}(1-o(1))$ from [@AGY].
\[c4theorem\] For even $n$, we have $f(n, C_4) = \frac{2^{n-1}}{3}\left(1 + \Theta\left(\frac{1}{\sqrt{n}}\right)\right)$. Additionally, for all $n \geq 27$, we have $\frac{2^{n-1}}{3} \leq f(n,C_4) \leq {2^{n-1}\over 3} \left(1 + O\left(\frac{1}{\sqrt{n}}\right)\right)$.
While our lower bound for $f(n, C_4)$ for even $n$ is asymptotically larger than our lower bound for odd $n$, we have no reason to believe that the lower bound for odd $n$ cannot be improved. We suspect that a better bound for odd $n$ would follow from a similar proof as that with even $n$, just with more work involved.
For the upper bound on $f(n,C_4)$ we combine designs to include almost all sets in ${\mathcal{P}}(n)$. In fact, we do this to provide an upper bound for $f(n,G)$ *for any connected graph $G$*. The construction is based on [*asymptotically optimal packings*]{}, $D(n,m,r)$, which is a large subset $S\subseteq {[n]\choose m}$ with the property that every $r$-element subset of $[n]$ is contained in [*at most one*]{} member of $S$. The existence of such packings was proved in a breakthrough paper of Rödl [@RO]. For our purposes only a special case is needed, $D(n,m,m-1)$, where constructions were known earlier, for example in [@KUZ].
\[asymcycle\] Let $G$ be a connected graph with $k$ edges, where $k \geq 2$ is fixed. Then $f(n, G) \leq \frac{2^n}{2(k-1)} \left(1 + O\left(\frac{1}{\sqrt{n}}\right)\right)$.
The upper bound of Theorem \[asymcycle\] gives the upper bound in Theorem \[c4theorem\]. In fact, it also matches the corresponding asymptotic lower bound ${2^{n-1}\over |E(G)|-1}(1-o(1))$ in [@AGY] when $G$ is a cycle or path, and the asymptotic lower bound of Proposition \[claw\], implying
$\frac{2^n}{2(k-1)} (1 - o(1))\leq f(n, C_k), f(n, P_k), f(n, S_k) \leq \frac{2^n}{2(k-1)} (1 + o(1))$.
Proof of Theorem \[trianglethm\] {#tr}
================================
It was shown in [@AGY] that $f(n,C_3)=2^{n-2}$ for any odd $n\ge 3, n\ne 5$. The following $C_3$-free partition of ${\mathcal{P}}^*(5)$ shows that $n=5$ is indeed exceptional. (Here and later we represent sets of small numbers without commas and brackets.) $$X_1=\{[5],[4]\}, Y_1=\{124,234,245\}, Y_2=\{123,134,135\}, Z_1=\{12,35,1235,345\},$$ $$\label{f5}
Z_2=\{23,45,2345,145\}, Z_3=\{34,15,1345,125\}, Z_4=\{14,25,1245,235\}.$$
In fact, (\[f5\]) is the only partition of ${\mathcal{P}}^*(5)$ into at most seven $C_3$-free parts (up to permutations), implying $f(5,C_3)=7$. For $n \neq 5$, three sets of size at least $\lfloor n/2 \rfloor +1$ always form a triangle (this is proven for odd $n$ in [@AGY] and generalized for even $n$ in Lemma \[trianglelemma\]). This is indeed not true for $n=5$, as witnessed by the ‘crowns’ $Y_1$ and $Y_2$ in (\[f5\]).
Let ${\mathcal{L}}$ be the set of all subsets of $[n]$ of size at least $\lfloor n/2\rfloor + 1$ (these are the ‘large’ subsets). For even $n$ let ${\mathcal{M}}$ be the set of all subsets of $[n]$ of size $n/2$ (these are the ‘medium’ subsets). Note that $2|{\mathcal{L}}| + |{\mathcal{M}}| = 2^n$.
\[trianglelemma\] For every even $n \geq 6$, we have the following:
1. For any distinct $M_1,M_2,M_3,M_4,M_5 \in {\mathcal{M}}$, some three form a triangle.
2. For any distinct $M_1,M_2,M_3 \in {\mathcal{M}}, L \in {\mathcal{L}}$, some three form a triangle.
3. Any distinct $M \in {\mathcal{M}}, L_1,L_2 \in {\mathcal{L}}$ form a triangle.
4. Any distinct $L_1, L_2, L_3 \in {\mathcal{L}}$ form a triangle.
For odd $n$ the theorem was proved in [@AGY]. By Proposition \[basepart\], we have to prove that $f(n,C_3)\ge 2^{n-2}$ for even $n$. Let $n\geq 6$, and let $Q$ be a partition of ${\mathcal{P}}(n)$ into the minimum number of triangle-free parts. (For $n=4$ a similar lemma and argument works.) Let there be $a$ parts of $Q$ with exactly two sets of ${\mathcal{L}}$, let there be $b$ parts of $Q$ with exactly one set of ${\mathcal{L}}$, and let there be $c$ parts of $Q$ with no sets of ${\mathcal{L}}$. Lemma \[trianglelemma\] implies that these account for all the parts, so that $a + b + c = f(n, C_3)$. Lemma \[trianglelemma\] also implies that $|{\mathcal{M}}| \leq 2b + 4c$, and since $|{\mathcal{L}}| = 2a + b$, we have $$f(n, C_3) = a + b + c \geq \frac{1}{4}(2|{\mathcal{L}}| + |{\mathcal{M}}|) = \frac{1}{4}(2^n)=2^{n-2} .$$
Since a set of ${\mathcal{L}}$ always contains as a subset a set of ${\mathcal{M}}$, it is clear that statement 3 implies statement 4. Thus we only need to prove statements 1, 2, and 3.
Let’s first note some basic intersection properties of sets from ${\mathcal{M}}\cup {\mathcal{L}}$. Let $L_1, L_2 \in {\mathcal{L}}$ and $M_1, M_2, M_3 \in {\mathcal{M}}$ be arbitrary. It is clear that $|L_1 \cap L_2| \geq 2$, $|L_1 \cap M_1| \geq 1$, and either $|M_1 \cap M_2| \geq 1$ or $|M_2 \cap M_3| \geq 1$.
In any of the three cases of the lemma, we first want to find three pairwise intersecting sets. In the first case, WLOG $M_1$ intersects with $M_2$, $M_3$, and $M_4$, and again WLOG $M_2$ intersects with $M_3$. In the second case, $L$ intersects $M_1$, $M_2$, and $M_3$, and WLOG $M_1$ intersects $M_2$. In the third case, every pair of sets intersect.
Let $A, B, C \in {\mathcal{M}}\cup {\mathcal{L}}$ be three distinct pairwise intersecting sets, in any case, and suppose they do not form a triangle. By Hall’s theorem as applied to distinct representatives, there are only a few cases where they may not form a triangle. WLOG, either $|(A \cap B) \cup (A \cap C)| \leq 1$ or $|(A \cap B) \cup (A \cap C) \cup (B \cap C)| \leq 2$. In the first case, it cannot be that $|A \cap B \cap C| = 0$, since the sets are pairwise intersecting, so we must have $|A \cap B \cap C| = 1$ and $|A \cap B \setminus C| = |A \cap C \setminus B| = 0$. In the second case, it likewise cannot be that $|A \cap B \cap C| = 0$. The case where $|A \cap B \cap C| = 1$ falls into the previous case, so this case reduces to $|A \cap B \cap C| = 2$ and $|A \cap B \setminus C| = |B \cap C \setminus A| = |C \cap A \setminus B| = 0$.
Define $\delta_A = |A|-n/2$, and likewise for $B$ and $C$. Furthermore let $\delta = \delta_A + \delta_B + \delta_C$.
**Case 1:** $|A \cap B \cap C| = 2$ and $|A \cap B \setminus C| = |B \cap C \setminus A| = |C \cap A \setminus B| = 0$. Here we count $$n \geq |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \geq \frac{3}{2}n + \delta - 2 - 2 - 2 + 2$$ implying $$n + 2\delta \leq 8 .$$
**Case 1a:** Suppose $n=8$. Then $\delta = 0$ and so $A, B, C \in {\mathcal{M}}$ and WLOG the configuration is isomorphic to $A = 1234$, $B =1256$, and $C = 1278$. A fourth set $D \in {\mathcal{M}}\cup {\mathcal{L}}$ must meet two of $A,B,C$ in a vertex not in $\{1,2\}$, forming a triangle with them.
**Case 1b:** Suppose $n=6$. If $\delta=0$, then WLOG $A = 123$, $B = 124$, and $C = 125$. The only pairs of vertices a fourth set $D \in {\mathcal{M}}\cup {\mathcal{L}}$ may contain without forming a triangle are those pairs containing $6$ and the pair $12$. Thus unless $D = 126$, we have a triangle. If $D = 126$, then we must be in the first case of the lemma, and so we may take a fifth set $E \in {\mathcal{M}}$. Since $\{1,2\} \not\subseteq E$, we have that $E$ must contain two vertices of $\{3,4,5,6\}$, forming a triangle.
If $\delta = 1$, then WLOG we are in the second case of the lemma and $A = 123$, $B = 124$, and $C = 1256$. Since the fourth set $D \in {\mathcal{M}}$ contains some pair of vertices other than $12$ and $56$, we have a triangle.
**Case 2:** $|A \cap B \cap C| = 1$ and $|A \cap B \setminus C| = |A \cap C \setminus B| = 0$. Here we count $$n \geq |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \geq \frac{3}{2}n + \delta - |B \cap C| - 1 ,$$ therefore $$\frac{1}{2} n + \delta - 1 \leq |B \cap C| \leq n - |A| \quad \text{(since $B$ and $C$ are distinct)} ,$$ implying $$2\delta_A + \delta_B + \delta_C \leq 1 .$$ Thus $\delta_A = 0$, and at most one of $\delta_B$ and $\delta_C$ is $1$, meaning that WLOG $A, B \in {\mathcal{M}}$, so we are not in the third case of the lemma. This means that the third case of the lemma was proved in case 1, so we are free to use it to finish the proof here. Let $D \in {\mathcal{M}}$ be a set distinct from $A$, $B$, and $C$.
If $D \subseteq B \cup C$, then $B$, $C$, and $D$ are three sets of size at least $(n-2)/2 + 1$ contained within a set of size $n/2+1 \leq n-2$. We may apply the third case of the lemma (with $n-2$ for $n$) to see that $B, C, D$ form a triangle.
Otherwise, let $x \in A \cap B \cap C$, let $y \in D \cap A \setminus \{x\}$, and let $z \in D \cap (B \cup C) \setminus \{x\}$. Note that $x,y,z$ necessarily exist and are distinct, so either $A, D, B$ or $A, D, C$ form a triangle.
Proof of Theorem \[c4theorem\] {#c4}
==============================
The upper bound of Theorem \[c4theorem\] follows from Theorem \[asymcycle\] (with $k=4$). So we prove the lower bound. As in Section \[tr\], we need a lemma concerning sets of ${\mathcal{M}}\cup {\mathcal{L}}$. We also give the corresponding lemma for odd $n$, which we prove from Lemma \[evenc4lemma\].
\[evenc4lemma\] For every even $n \ge 26$, we have the following:
1. For any distinct $M_1, M_2, M_3, M_4, M_5 \in {\mathcal{M}}$, some four form a $C_4$.
2. For any distinct $M_1, M_2, M_3, M_4 \in {\mathcal{M}}$, $L_1 \in {\mathcal{L}}$, some four form a $C_4$.
3. Any distinct $M_1, M_2 \in {\mathcal{M}}$, $L_1, L_2 \in {\mathcal{L}}$ form a $C_4$.
4. Any distinct $M_1 \in {\mathcal{M}}$, $L_1, L_2, L_3 \in {\mathcal{L}}$ form a $C_4$.
\[oddc4lemma\] For every odd $n \geq 27$, any distinct $L_1, L_2, L_3, L_4 \in {\mathcal{L}}$ form a $C_4$.
Let $n \ge 26$ be even, and let $Q$ be a partition of ${\mathcal{P}}(n)$ into the minimum number of $C_4$-free parts. Say $Q$ has $a$ parts with three sets in ${\mathcal{L}}$, $b$ parts with two sets in ${\mathcal{L}}$, $c$ with one, and $d$ with no sets in ${\mathcal{L}}$. Lemma \[evenc4lemma\] implies that these account for all the parts of $Q$, so $a + b + c + d = f(n, C_4)$. Moreover, Lemma \[evenc4lemma\] implies the relations $|{\mathcal{L}}| = 3a + 2b + c$ and $|{\mathcal{M}}| \le b + 3c + 4d$. Since $|{\mathcal{M}}| = \binom{n}{n/2} = \Theta(2^n/\sqrt{n})$, this gives us (by $b + 3c + 4d \le {3b\over 2}+3c+{9d\over 2})$ that $$f(n, C_4) = a + b + c + d \ge \frac{1}{6}\left(2|{\mathcal{L}}| + \frac{4}{3}|{\mathcal{M}}|\right) = \frac{1}{6}\left(2^n + \frac{1}{3} |{\mathcal{M}}|\right) = \frac{2^{n-1}}{3}\left(1 + \Theta\left(\frac{1}{\sqrt{n}}\right)\right) .$$
For odd $n \geq 27$, again take such a minimal $C_4$-free partition of ${\mathcal{P}}(n)$. Each part has at most three sets in ${\mathcal{L}}$, so $f(n, C_4) \geq \frac{1}{3}|{\mathcal{L}}| = \frac{2^{n-1}}{3}$.
In order to prove Lemma \[evenc4lemma\], we need the following definition:
Assume $n \ge 4$ is even. We say that four distinct sets $A, B, C, D \in {\mathcal{M}}\cup {\mathcal{L}}$ form a *$\Psi$-configuration* if there exists some $x$ such that $A \cap B, A \cap C, A \cap D \subseteq \{x\}$. In such a configuration we call $A$ a *stem*.
Let us elaborate on the structure of a $\Psi$-configuration $A,B,C,D$. Suppose $A$ is a stem and $A \cap (B \cup C \cup D) \subseteq \{x\}$. Since $A,B,C,D$ are distinct sets in ${\mathcal{M}}\cup {\mathcal{L}}$, we have the inequalities $|A| \ge \frac{n}{2}$ and $|B \cup C \cup D| \ge \frac{n}{2} + 1$. But also, $$n+1 \ge |A \cup (B \cup C \cup D)| + |A \cap (B \cup C \cup D)| = |A| + |B \cup C \cup D| \ge \frac{n}{2} + (\frac{n}{2} + 1) = n+1.$$ So in fact, $|A| = \frac{n}{2}$ and $|B \cup C \cup D| = \frac{n}{2} + 1$. That is to say, $A \in {\mathcal{M}}$, and $B,C,D$ are $\frac{n}{2}$- or $(\frac{n}{2}+1)$-subsets of the $(\frac{n}{2} + 1)$-set $([n] \setminus A) \cup \{x\}$. Based on this, it is easy to see that a stem of a $\Psi$-configuration is unique.
Also note that in this $\Psi$-configuration we have $$|B \cap C| = |B| + |C| - |B \cup C| \ge \frac{n}{2} + \frac{n}{2} - (\frac{n}{2} + 1) = \frac{n}{2} - 1,$$ and similarly $|B \cap D|, |C \cap D| \ge \frac{n}{2} - 1$. Thus, the non-stem sets of a $\Psi$-configuration pairwise intersect in at least $\frac{n}{2} - 1$ elements. Finally, observe that a $\Psi$-configuration does not form a $C_4$.
Suppose $n\geq 26$ is even. We first prove the following claim:
**Claim:** Any four distinct $A, B, C, D \in {\mathcal{M}}\cup {\mathcal{L}}$ form either a $C_4$ or a $\Psi$-configuration.
Suppose that $A,B,C,D$ do not form a $\Psi$-configuration. We wish to show that $A,B,C,D$ form a $C_4$
First assume that two of the sets, say $A$ and $C$, are complementary. Since the complement of any set is unique and our sets are in ${\mathcal{M}}\cup {\mathcal{L}}$, the intersections $A \cap B$ and $A \cap D$ are nonempty. Moreover, because $A \cap C = \emptyset$ and $A,B,C,D$ do not form a $\Psi$-configuration, $A \cap B$ and $A \cap D$ cannot be the same singleton set. Thus, there exist distinct representatives $x_1 \in A \cap B$, $x_2 \in A \cap D$. Similarly, there exist distinct representatives $x_3 \in B \cap C$, $x_4 \in C \cap D$. Clearly $x_1$ and $x_2$ are distinct from $x_3$ and $x_4$, since the first two are contained in $A$ while the second two are contained in $C = [n]\setminus A$. Thus $A, B, C, D$ form a $C_4$.
Now assume that $A, B, C, D$ are pairwise intersecting. Consider all perfect matchings $\{X_1, X_2\}, \{X_3, X_4\}$ (that is, partitions into sets of size 2) of $\{A,B,C,D\}$, and let $\{A,C\}, \{B,D\}$ be the one that minimizes $|(X_1 \cap X_2) \cup (X_3 \cap X_4)|$. We will show that if $A,B,C,D$ do not form a $C_4$ in that cyclic order, then there is another cyclic order of $A,B,C,D$ that forms a $C_4$. To do this, we use Hall’s theorem on distinct representatives as we did in Lemma \[trianglelemma\]. WLOG, the following are the only cases in which $A,B,C,D$ may fail to form a $C_4$ in that cyclic order:
**Case 1:** $|(A \cap B) \cup (B \cap C) \cup (C \cap D)| \le 3$. (Note that this case covers when $|(A \cap B) \cup (B \cap C) \cup (C \cap D)| \le 2$ and when $|(A \cap B) \cup (B \cap C) \cup (C \cap D) \cup (D \cap A)| \le 3$.) The intersections in this union must each be a subset of a 3-set $\{x_1,x_2,x_3\}$. By minimality of $|(A \cap C) \cup (B \cap D)|$, $A \cap C$ and $B \cap D$ are subsets of a 3-set $\{y_1,y_2,y_3\}$. It follows that the sets $X \setminus \{x_1,x_2,x_3,y_1,y_2,y_3\}$ for $X \in \{A,B,C\}$ are pairwise disjoint. Counting the number of elements in $[n]$ outside of $\{x_1,x_2,x_3,y_1,y_2,y_3\}$, we get the inequality $$3\left(\frac{n}{2} - 6\right) \le n - 6 ,$$ from which it follows that $n \le 24$. Since we assumed that $n \ge 26$, this is impossible.
**Case 2:** $|(A \cap B) \cup (C \cap D)| \le 1$. Since our sets are pairwise intersecting, $A \cap B = C \cap D = \{x\}$ for some $x$. Then $x \in A \cap C$ and $x \in B \cap D$. Since we assumed that $|(A \cap C) \cup (B \cap D)|$ is minimal, it follows that $(A \cap C) \cup (B \cap D) = \{x\}$. But then $$\begin{aligned}
(A \cup D) \cap (B \cup C) = (A \cap B) \cup (C \cap D) \cup (A \cap C) \cup (B \cap D) = \{x\},\end{aligned}$$ from which we get that $$\begin{aligned}
n + 1 \ge |(A \cup D) \cup (B \cup C)| + |\{x\}| = |A \cup D| + |B \cup C| \ge \left(\frac{n}{2} + 1\right) + \left(\frac{n}{2} + 1\right) = n + 2,\end{aligned}$$ a contradiction.
**Case 3:** $|(A \cap B) \cup (A \cap D)| \le 1$. Similar to case 3, $A \cap B = A \cap D = \{x_1\}$ for some $x_1$. Since $A,B,C,D$ do not form a $\Psi$-configuration, there must be some $x_2 \in A \cap C$ different from $x_1$. Now, $C \cap D$ cannot be $\{x_1\}$ because otherwise $A \cap B = C \cap D = \{x_1\}$, which we showed is an impossible circumstance in Case 2. Moreover, $C \cap D$ cannot contain $x_2$ because otherwise $x_2 \in A \cap D = \{x_1\}$. Thus, there must be some $x_3 \in C \cap D$ different from $x_1$ and $x_2$. Finally, note that $$\begin{aligned}
n &\ge |A \cup B \cup D| \\
&= |A| + |B| + |D| - |A \cap B| - |A \cap D| - |B \cap D| + |A \cap B \cap D| \\
&\ge \frac{n}{2} + \frac{n}{2} + \frac{n}{2} - 1 - 1 - |B \cap D| + 1 \\
&= \frac{3n}{2} - 1 - |B \cap D|,\end{aligned}$$ from which we get that $|B \cap D| \ge \frac{n}{2} - 1 \ge 4$. So there must be some $x_4 \in B \cap D$ different from each of $x_1,x_2,x_3$. It follows that $A,C,D,B$ form a $C_4$ in that cyclic order.
This concludes the proof of the claim.
Now we prove the statements of Lemma \[evenc4lemma\]. Observe that, similar to Lemma \[trianglelemma\], statement 2 follows from statement 1, and statement 4 follows from statement 3. So we prove statements 1 and 3.
Statement 3 follows immediately from the observations about $\Psi$-configurations, specifically that their stem must be an $\frac{n}{2}$-set, and their non-stem sets must be subsets of an $(\frac{n}{2} + 1)$-set, say $X$. There is only one set in ${\mathcal{L}}$ that could be part of such a configuration, namely $X$ itself. Thus, it is impossible for distinct $M_1, M_2 \in {\mathcal{M}}, L_1, L_2 \in {\mathcal{L}}$ to form a $\Psi$-configuration. By the claim, they must form a $C_4$.
For statement 1, first consider the sets $M_1, M_2, M_3, M_4$. If they form a $C_4$, then we are done; otherwise, they must form a $\Psi$-configuration. Say that $M_1$ is the stem. Next consider the sets $M_1, M_2, M_3, M_5$. Again we are done if they form a $C_4$; otherwise, they must form another $\Psi$-configuration. $M_1$ must again be the stem because $M_1 \cap M_2$, $M_1 \cap M_3$ have at most one element, and we have seen that the non-stem sets of $\Psi$-configurations must pairwise intersect in at least $\frac{n}{2} - 1$ elements. So finally consider the sets $M_2, M_3, M_4, M_5$. They are all non-stem sets in our previous two configurations, so $|M_i \cap M_j| \ge \frac{n}{2} - 1$ for all distinct $i,j \in \{2,3,4,5\}$. Thus $M_2,M_3,M_4,M_5$ form a $C_4$.
Note that in case 1 of the proof of the claim, the required lower bound on $n$ of 26 is not tight because the $x_i$’s and $y_i$’s considered in the proof may not all be distinct. This bound can definitely be reduced, but doing so requires extra casework.
Now we prove Lemma \[oddc4lemma\] from Lemma \[evenc4lemma\].
Let $n \geq 27$ be odd, and let $L_1, L_2, L_3, L_4 \in {\mathcal{L}}$, meaning that $|L_i| \geq \frac{n+1}{2}$. We break into two cases.
Suppose there exists $j \in [n]$ such that $j$ is in at most two of the $L_i$. Without loss of generality, $j \not\in L_3, L_4$. This means that $L_3$ and $L_4$ are sets of size at least $\frac{n+1}{2} = \frac{n-1}{2} + 1$ contained in a set of size $n-1$ (namely, $[n] \setminus \{j\}$). Let $M_1 \subseteq L_1 \setminus \{j\}$ and $M_2 \subseteq L_2 \setminus \{j\}$ be distinct sets of size $\frac{n-1}{2}$, which are necessarily contained in the same set of size $n-1$ as before (namely, $[n] \setminus \{j\}$). We may then consider $L_3$ and $L_4$ to be in ${\mathcal{L}}$ and $M_1$ and $M_2$ to be in ${\mathcal{M}}$ in the sense that Lemma \[evenc4lemma\](3) applies in $[n]\setminus \{j\}$: since these sets are distinct, they form a $C_4$.
Otherwise, suppose every $j \in [n]$ is in at least three of the $L_i$. This implies $\sum |L_i| \geq 3n$. No three of the $L_i$ can have size exactly $\frac{n+1}{2}$, since this would imply that the fourth set has size at least $3n - 3\frac{n+1}{2} = \frac{3n}{2} - \frac{3}{2} > n$, an impossibility. Thus at most two of the $L_i$ have size $\frac{n+1}{2}$, meaning that (as in the preceding paragraph) upon the removal of any vertex there are at most two sets of size $\frac{n-1}{2}$. In a similar fashion as the previous paragraph, Lemma \[evenc4lemma\](3) implies that these sets form a $C_4$.
Proof of Theorem \[asymcycle\]
==============================
Here we construct a partition of ${\mathcal{P}}(n)$ where almost all of the sets are in parts of size $2(k-1)$. In fact, these parts of size $2(k-1)$ consist of $k-1$ sets of size less than $n/2$, and $k-1$ sets of size at least $n/2$, in such a way that all of the larger sets are disjoint from the smaller sets. The sets not in parts of size $2(k-1)$ can be placed arbitrarily in parts of size at most $k-1$. This partition is $G$-free since the intersection graph of any partition class has connected components with at most $k-1$ vertices. We assume that $n\ge 2(k-1)$.
Define $A_{m,r} := \{A \in \binom{[n]}{m} : \sum_{a \in A} a \equiv r {\ (\mathrm{mod}\ n)}\}$. Since $\sum_{r=0}^{n-1} |A_{m,r}| = \binom{n}{m}$, there exists some $r_m$ such that $|A_{m,r_m}| \geq \frac{1}{n}\binom{n}{m}$. Fix these $r_m$ for $k-1 \leq m < n/2$. We construct a part in our partition from each $A \in A_{m,r_m}$ for $k-1 \leq m < n/2$.
Let $A \in A_{m,r_m}$ and enumerate $A=\{a_0,\dots,a_{m-1}\}$ and $B=[n]\setminus A=\{b_0,\dots,b_{n-m-1}\}$. For integers $i$ with $0\leq i\leq\lfloor\frac{m}{k-1}\rfloor-1$, construct the part consisting of the sets $$A\setminus\{a_{(k-1)i}\}, A\setminus\{a_{(k-1)i+1}\},\dots, A\setminus\{a_{(k-1)i+(k-2)}\} ,$$ $$B\setminus\{b_{(k-1)i}\}, B\setminus\{b_{(k-1)i+1}\},\dots, B\setminus\{b_{(k-1)i+(k-2)}\} .$$
The sets of the form $A \setminus \{a_j\}$ (in the first line) are all different. Indeed, suppose $A\setminus\{a_j\}=A'\setminus\{a'_{j'}\}$, so necessarily $|A|=|A'|$. Also, $\sum_{a_i\in A\setminus\{a_j\}}a_i\equiv\sum_{a'_i\in A'\setminus\{a'_{j'}\}}a_i' {\ (\mathrm{mod}\ n)}$. This implies $-a_j+\sum_{a_i\in A}a_i\equiv-a'_{j'}+\sum_{a'_i\in A'}a'_i {\ (\mathrm{mod}\ n)}$. By construction, this is equivalent to $r_{|A|}-a_j\equiv r_{|A'|}-a'_{j'} {\ (\mathrm{mod}\ n)}$, and thus $a_j\equiv a'_{j'} {\ (\mathrm{mod}\ n)}$. This means that $a_j=a'_{j'}$, which together with $A\setminus\{a_j\}=A'\setminus\{a'_{j'}\}$ implies that $A=A'$. Thus the two sets were the same. Analogous reasoning concludes that the sets appearing in the second line are also all different. Finally, for any $m$, sets in the first line have size less than ${n\over 2}-1$, while in the second line the sets have size at least ${n\over 2}-1$. Therefore the constructed sets are all different in any part.
For each $A \in A_{m,r_m}$, there are ${\left\lfloor\frac{m}{k-1}\right\rfloor}$ possible values of $i$ in the construction, that is, ${\left\lfloor\frac{m}{k-1}\right\rfloor}$ different parts that $A$ generates. Since $\binom{n}{{\left\lfloorn/2\right\rfloor}} = \Theta(2^n/\sqrt{n})$, this construction creates at least $$\sum_{m=k-1}^{\lceil\frac{n}{2}\rceil-1}\left\lfloor\frac{m}{k-1}\right\rfloor\frac{1}{n}\binom{n}{m}\geq\frac{2^n}{2(k-1)}\left(1 - \frac{c}{\sqrt{n}}\right)$$ parts of size $2(k-1)$, for some constant $c>0$ not depending on $n$ or $k$.
Since this construction yields at least $\frac{2^n}{2(k-1)}\left(1-c/\sqrt{n}\right)$ parts of size $2(k-1)$, there are at least $2^n\left(1-c/\sqrt{n}\right)$ sets placed in parts this way. Thus at most $2^n-2^n(1-c/\sqrt{n})=2^n\left(c/\sqrt{n}\right)$ sets have not been placed into a part. We place these remaining sets arbitrarily into parts of size $k-1$ (with one possible smaller part). Partitioning the rest this way generates at most $\frac{2^n}{k-1}\left(c/\sqrt{n}\right)+1$ additional parts. Thus, in total the partition will have at most $\frac{2^n}{2(k-1)}\left(1-c/\sqrt{n}\right)+\frac{2^n}{k-1}\left(c/\sqrt{n}\right)+1=\frac{2^n}{2(k-1)}\left(1+\Theta\left(1/\sqrt{n}\right)\right)$ parts.
Bounds on $f(6, S_3)$ and $f(9, S_3)$ {#designs}
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$f(6,S_3)\le 15$.
Let $X=\{123,456,12,13,23,45,46,56\}$ be one ($3$-regular but claw-free) partition class. All other classes will be $2$-regular (thus automatically claw-free). Let $Y_1,Y_2,Y_3,Y_4$ contain two pairs of complementary triples, not using the pair $123,456$. Then define $$Z_1=\{14,23456,12356\}, Z_2=\{25,13456,12346\}, Z_3=\{36,12456,12345\}.$$ Let $U_1,U_2,U_3,U_4,U_5$ be defined as the complementary sets of the $1$-factors in a $1$-factorization of $K_6$. Then $W$ is defined by the edges of the $6$-cycle $1,5,3,4,2,6,1$ and $R$ contains $[6]$ together with the one complementary pair of triples not used in $X$ and in $Y_i$. Now we have 15 claw-free partition classes of ${\mathcal{P}}^*(6)$.
$f(9,S_3)\le 126$.
Take a partition $Q$ of $[9]\choose 3$ into $28$ classes, each containing three pairwise disjoint triples - a very special case of Baranyai’s theorem [@BA]. However, we need another property of $Q$: four of these classes $X_1,X_2,X_3,X_4$ must form a Steiner triple system. Then these can be extended by the nine pairs covered by their triples implying that $X_1\cup X_2 \cup X_3\cup X_4$ covers each pair of $[9]$ exactly once. The existence of these $X_i$-s certainly follows from a much stronger result, stating that $[9]\choose 3$ can be partitioned into seven Steiner triple systems (but probably there are easier ways to get them). Then the $X_i$-s provide four claw-free $3$-regular partition classes. Partition classes $Y_i$ can be defined by putting together $12$ pairs of the remaining $24$ classes of $Q$, they form a double cover of $[9]$. Next we can define $28$ partition classes $Z_i$ by the complements of the $28$ classes of $Q$, each of them forms a double cover on $[9]$.
Next we design $9$ double covers of type $(5,5,8)$ and $18$ double covers of type $(4,7,7)$. To prepare, set $A_i=\{i+1,i+2,i+3,i+6\}$, $B_i=\{i+4,i+5,i+7,i+8\}$ with arithmetic mod 9. Then $9$ double covers of $[9]$ are defined as $U_i=\{A_i\cup i,B_i\cup i, A_i\cup B_i\}$. Set $$C_i=[9]\setminus \{i+1,i+2\}, D_i=[9]\setminus \{i+3,i+6\},$$ $$E_i=[9]\setminus \{i+4,i+8\}, F_i=[9]\setminus \{i+5,i+7\}.$$ Then $2\times 9$ double covers of $[9]$ are defined as $W_i=\{A_i,C_i,D_i\}$ and $R_i=\{B_i,E_i,F_i\}$. Note that $U_i,W_i,R_i$ take care of $18$ complementary pairs of sizes $4$ and $5$. The remaining ${9\choose 4}-18$ such pairs can be placed into $54$ partition classes $T_i$ forming double covers on $[9]$. Finally, $[9]$ alone forms a partition class (leaving some hope of improvement).
Altogether we have $4+12+28+9+18+54+1=126$ claw-free partition classes of ${\mathcal{P}}^*(9)$.
Acknowledgement
===============
This paper was written under the auspices of the Budapest Semesters in Mathematics program during the Fall semester of 2018.
[99]{} M. Axenovich, A. Gyárfás, A note on Ramsey numbers for Berge-$G$ hypergraphs, submitted, [*arXiv:1807.10062v2*]{}.
Zs. Baranyai, The edge-colouring of complete hypergraphs, [*Journal of Combinatorial Theory B*]{}, (1979) [**26**]{}, 276–294. C. Berge, Graphs and Hypergraphs, North-Holland, 1973 D. Gerbner, C. Palmer, [*Extremal results for Berge-hypergraphs*]{}, SIAM Journal on Discrete Mathematics, (2015) [**31(4)**]{}, 2314–2327. D. Gerbner, A. Methuku, G. Omidi, M. Vizer, Ramsey problems for Berge hypergraphs, [*arXiv:1808.10434v1*]{}. R.L.Graham, B.L.Rothschild, J.H.Spencer, Ramsey Theory, 1990, 2013, John Wiley, Hoboken, New Jersey. N. N. Kuzrujin, On minimal coverings and maximal packings of $(k-1)$-tuples by $k$-tuples, [*Mat. Zametki*]{} (1977) [**9**]{}, 565–571 (in Russian). V. Rödl, On a packing and covering problem, [*European Journal of Combinatorics*]{} (1985) [**8**]{}, 69–78. N. Salia, C. Tompkins, Z. Wang, O. Zamora, Ramsey numbers of Berge-hypergraphs and related structures, [*arXiv:1808.09863*]{}.
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abstract: 'First principles calculations reveal that adding a metallic overlayer on LaAlO$_3$/SrTiO$_3$(001) eliminates the electric field within the polar LaAlO$_3$ film and thus suppresses the thickness-dependent insulator-to-metal transition observed in uncovered films. Independent of the LaAlO$_3$ thickness both the surface and the interface are metallic, with an enhanced interface carrier density relative to LaAlO$_3$/SrTiO$_3$(001) after the metallization transition. Moreover, a monolayer thick metallic Ti-contact exhibits a finite magnetic moment and for a thin SrTiO$_3$-substrate induces a spin-polarized 2D electron gas at the *n*-type interface due to confinement effects. A diagram of band alignment in $M$/LaAlO$_3$/SrTiO$_3$(001) and Schottky barriers for $M$=Ti, Al, and Pt are provided.'
author:
- 'Victor G.'
- Rémi Arras
- 'Warren E. Pickett'
- Rossitza Pentcheva
title: ' Tuning the two-dimensional electron gas at the LaAlO$_3$/SrTiO$_3$(001) interface by metallic contacts '
---
The (001) interface between the band insulators [LaAlO$_3$]{} (LAO) and [SrTiO$_3$]{} (STO) provides remarkable examples of the novel functionalities that can arise at oxide interfaces, including a quasi two-dimensional electron gas (q2DEG)[@Ohtomo:Hwang2004], superconductivity [@Reyren:Thiel:Mannhart2007], magnetism [@Brinkman:Huijben:etal2007] and even signatures of their coexistence [@Dikin; @Li]. Moreover, a thickness-dependent transition from insulating to conducting behavior was reported in thin [LaAlO$_3$]{} films on [SrTiO$_3$]{}(001) at $\sim4$ monolayers (ML) LAO [@Thiel:Hammerl:Mannhart2006]. This insulator-metal transition (MIT) can be controlled reversibly via an electric field, e.g. by an atomic force microscopy (AFM) tip [@Cen:Thiel:Hammerl:etal2008; @Bi:Levy:2010; @Chen:2010]. Recently, it was shown that an additional STO capping layer can trigger the MIT already at 2 ML of LAO, allowing stabilization of an electron-hole bilayer [@Pentcheva:2010]. Density functional theory calculations (DFT) have demonstrated the emergence of an internal electric field for thin polar LAO overlayers[@Ishibashi:Terakura2008; @Pentcheva:Pickett2009; @Pentcheva:PickettRev2010; @Son:2009]. Screening of the electric field by a strongly compensating lattice polarization in the LAO film[@Pentcheva:Pickett2009; @Willmott:2011] allows several layers of LAO to remain insulating before an electronic reconstruction takes place at around 4-5 monolayers of LAO. Recent AFM experiments provide evidence for such an internal field in terms of a polarity-dependent asymmetry of the signal [@Xie:Hwang:2010], but x–ray photoemission studies[@Segal:Reiner:etal2009; @Sing:Claessen:2009; @Chambers:Ramasse:2010] have not been able to detect shifts or broadening of core-level spectra that would reflect an internal electric field. This discrepancy implies that besides the electronic reconstruction, extrinsic effects play a role, e.g. oxygen defects [@Zhong:2010; @Bristowe:2011], adsorbates such as water or hydrogen [@Son:2010] or cation disorder [@Willmott:2007; @Qiao:2010] (for detailed reviews on the experimental and theoretical work see [@Huijben:2009; @Pentcheva:PickettRev2010; @Chen:Beigi:2010; @Triscone:2011]).
The LAO/STO system is not only of fundamental scientific interest, but is also a promising candidate for the development of electronics and spintronics devices [@Mannhart:2010; @cheng_NatNano2011]. For its incorporation in such devices the influence of metallic leads needs to be considered [@Jany:2010; @Bhalla:2010; @Liu:cm]. Metallic overlayers have been investigated on a variety of perovskite surfaces, such as [SrTiO$_3$]{}(001) [@Asthagiri:Sholl:2002; @mrovec:2009], LaAlO$_3$(001) [@Asthagiri:Sholl:2006; @dong:2006] or BaTiO$_3$(001) [@Sai:Rappe:2005; @Stengel:Vanderbilt:Spaldin:2009]. Furthermore, the magnetoelectric coupling between ferromagnetic Fe and Co films and ferroelectric BaTiO$_3$ and PbTiO$_3$(001)-surfaces [@Duan:Tsymbal:2006; @Fechner:2008] has been explored. However, the impact of a metallic overlayer on a buried oxide interface has not been addressed so far theoretically. Based on DFT calculations we show here that a metallic Ti overlayer not only acts as a Schottky barrier, but has a crucial influence on the electronic properties as it removes the internal electric field in the LAO film on STO(001). Despite the lack of a potential build up, the underlying STO layer is metallic and a significantly enhanced carrier concentration emerges at the $n$-type interface as compared to the system without metallic contact. Although bulk Ti is nonmagnetic, the undercoordinated Ti in the contact layer shows an enhanced tendency towards magnetism with a significant spin polarization and a magnetic moment of 0.60 [$\mu_{\rm B}$]{}. Most interestingly, quantum confinement within the STO-substrate can induce spin-polarized carriers at the interface.
DFT calculations on $n$Ti/$m$LAO/STO(001), where $n$ ($m$) denotes the number of metallic overlayers (monolayers of LAO) were performed using the all-electron full-potential linearized augmented plane wave (FP-LAPW) method in the WIEN2k implementation [@Wien2k] and the generalized gradient approximation (GGA)[@GGA] of the exchange-correlation potential. Including an on-site Coulomb correction within the LDA/GGA+$U$ approach [@Anisimov:Solovyev:etal1993] with $U=5$ eV and $J=1$eV applied on the Ti $3d$-states and $U=7$ eV on the La $4f$ states showed only small differences in electronic behavior. The calculations are carried out using a symmetric slab with LAO and Ti layers on both sides of the STO-substrate and a vacuum region between the periodic images of at least $10$Å. The lateral lattice parameter is set to the GGA equilibrium lattice constant of STO (3.92 Å, slightly larger than the experimental value 3.905 Å) and the atomic positions are fully relaxed within tetragonal symmetry.
![(Color online) Layer resolved density of states (LDOS) of a) 1Ti/4LAO/STO(001) and b) 1Ti/2LAO/STO(001). The potential build up in the uncovered systems (black line) is canceled in the ones with a Ti overlayer (filled area). Additionally, there is a significant spin-polarization both in the surface Ti, as well as the interface TiO$_2$ layers in 1Ti/$N$LAO/STO(001). []{data-label="fig:dos1Ti2-4LAO3STO"}](Fig1)
Fig. \[fig:dos1Ti2-4LAO3STO\]a shows the layer resolved density of states (LDOS) of a 1Ti/4LAO/STO(001) system, where the Ti atoms are adsorbed on top of the oxygen ions in the surface AlO$_2$ layer. A striking feature is that the electric field of the uncovered LAO film (black line), expressed through an upward shift of the O $2p$ bands [@Pentcheva:Pickett2009], is completely eliminated after the adsorption of the Ti overlayer. As a consequence of the vanishing electric field within the LAO layer, no dependence of the electronic properties on the LAO thickness is expected, as opposed to the insulator-metal transition that occurs in the uncovered LAO/STO(001) system. Indeed the LDOS of 1Ti/2LAO/STO(001) (Fig. \[fig:dos1Ti2-4LAO3STO\]b, only 2 LAO layers) confirms a very similar behavior with no shifts of the O $2p$ bands in the LAO part. Despite the lack of a potential build up, there is a considerable occupation of the Ti $3d$ band at the interface and thus both the surface Ti layer and the interface are metallic. The charge transfer from the Ti adlayer allows the Ti+LAO+STO system to equilibrate in charge and potential, with the result being that the Fermi level lies just within the STO conduction band.
The low coordination of the surface Ti atoms enhances their tendency towards magnetism resulting in a magnetic moment of 0.58[$\mu_{\rm B}$]{} in the surface layer. The electron gas at the interface is also spin-polarized with magnetic moments sensitive to the thickness of the LAO-spacer: for $m_{LAO}=4$ the magnetic moments are smaller (0.05/0.11[$\mu_{\rm B}$]{} in the interface (IF)/IF-1 layer) than for $m_{LAO}=2$ (0.10/0.24[$\mu_{\rm B}$]{} in IF/IF-1). Calculations performed within GGA+$U$ for the latter case show a similar behavior but with an even stronger spin-polarization of carriers and magnetic moments of Ti of 0.20/0.30 [$\mu_{\rm B}$]{} in the IF/IF-1 layer, respectively. Thus correlation corrections have only small influence on the overall band alignment, confirming that the observed behavior is not affected by the well known underestimation of band gaps of LDA/GGA.
![(Color online) a) Layer resolved density of states (LDOS) of 1Ti/2LAO/STO(001) within GGA with a 6.5ML thick STO-substrate; b) Side view of the system with the electron density integrated in the interval E$_{\rm F}-0.5$eV to E$_{\rm F}$; c) vertical displacements of cations and anions in 1Ti/2LAO/STO(001); d) majority and minority band structure where the $3d$ bands in the interface layer are emphasized with the $d_{xy}$ orbital being the lowest lying level at $\Gamma$. []{data-label="fig:1Ti2LAO6STO"}](Fig2)
The calculations so far were performed with a rather thin substrate layer of 2.5 ML STO. To examine the dependence on the thickness of the substrate layer, we studied 1Ti/2LAO/STO(001) containing a 6.5 ML thick STO part. As shown in Fig. \[fig:1Ti2LAO6STO\]a, the most prominent difference to the system with a thin STO layer is the suppression of spin-polarization of carriers at the interface, indicating that the spin-polarization is a result of confinement effects in the thin STO layer. Apart from this, a notable band bending occurs in the STO part of the heterostructure. The largest occupation of the Ti $3d$ band arises at the interface, followed by a decreasing occupation in deeper layers. The electron density, integrated over states between E$_{\rm F}-0.5$eV to E$_{\rm F}$ in Fig. \[fig:1Ti2LAO6STO\]b, reveals orbital polarization of the Ti electrons in the conduction band: predominantly $d_{xy}$ character in the interface layer, nearly degenerate $t_{2g}$ occupation in IF-1, and a preferential occupation of $d_{xz}$, $d_{yz}$ levels in the deeper layers. The band structure plotted in Fig. \[fig:1Ti2LAO6STO\]d shows that the conduction band minimum is at the $\Gamma$-point, formed by $d_{xy}$ states of Ti in the interface layer. While the $d_{xy}$ bands have a strong dispersion, the $d_{xz}$, $d_{yz}$ bands lie slightly higher in energy but are much heavier along the $\Gamma-X$ direction. In addition to the orbital polarization of the filled bands, the different band masses indicate a significant disparity in mobilities of electrons in the different $t_{2g}$ orbitals. Similar multiple subband structure has been recently reported for LAO/STO superlattices [@Popovic:Satpathy:Martin:2008; @Janicka:2009], $\delta$-doped LAO in STO [@Ong:2011] as well as doped STO(001)-surfaces [@Santander:11; @Meevasana:2011]. The remaining bands between -2.5eV and E$_{\rm F}$ are associated with the surface Ti-layer.
The altered boundary conditions in 1Ti/LAO/STO(001) affect also the structural relaxations: In the uncapped LAO/STO(001) system a strong lattice polarization emerges primarily in the LAO film (with a buckling of $\sim0.26$Å in the LaO and $\sim0.17$Å in the subsurface AlO$_2$ layers) that opposes the internal electric field [@Pentcheva:Pickett2009]. As seen in the LDOS plots, the formation of a chemical bond between the metallic layer and the LAO film cancels the electric field within LAO, and hence, as shown in Fig. \[fig:1Ti2LAO6STO\]c, cations and anions within the LAO film relax by similar amount with no appreciable buckling of the layers. On the other hand, the excess charge at the LAO/STO interface induces polarization in the STO substrate. The displacement between anions and cations is driven mainly by an outward oxygen shift and is largest at the interface ($0.17$Å in the interface TiO$_2$ layer and $0.16$Å in the next SrO layer) and decays in deeper layers away from the interface. This relaxation pattern resembles the one of $n$-type LAO/STO and LTO/STO superlattices [@Pentcheva:Pickett2008; @Spaldin:2006].
Increasing the thickness of Ti to 2 ML (with Ti in the second layer positioned above Al in the surface AlO$_2$-layer and La in the subsurface LaO-layer) leads to a significant reduction of the spin-polarization of the Ti film: the magnetic moment is 0.25[$\mu_{\rm B}$]{} in the surface and -0.10[$\mu_{\rm B}$]{} in the subsurface layer. This confirms that the high spin-polarization of Ti in 1Ti/$m$LAO/STO(001) is a result of the reduced coordination (for comparison, the magnetic moment of a free standing Ti layer is 0.90[$\mu_{\rm B}$]{}). The weaker binding to the oxide surface in 2Ti/2LAO/STO(001) is expressed in a longer Ti-O bond length of 2.06 Å compared to 2.00 Å in 1Ti/2LAO/STO(001). Despite these differences in the structural and magnetic properties of the metallic overlayer, the occupation of the Ti $3d$ band at the interface is very similar (cf. Fig.\[fig:NoccO1s\]a), but decreases quicker in the deeper layers.
![ a) Layer resolved electron occupation integrated between $E_F-0.65$ eV and $E_F$ for 1ML Ti and thin (green stars)/thick STO substrate (black squares); 2ML Ti (magenta diamonds), as well as 1ML Pt (blue triangles) and 1ML Al (red circles). Occupation within SrTiO$_3$ layers is due to Ti $3d$ electrons, the one in the topmost AlO$_2$ arises from metal induced gap states (MIGS). Note that charge on the AlO$_2$ layer next to the interface is nearly vanishing in all cases, charge on the AO layers is zero (not shown). b) Positions of O$1s$ states with respect to $E_F$.[]{data-label="fig:NoccO1s"}](Fig3)
Besides Ti, we have studied also Pt and Al overlayers and find that the overall behavior – strong reduction/cancellation of the electric field within LAO and metallization of the underlying STO – is robust for all studied metallic contacts. The layer resolved occupation of the Ti $3d$ band in [SrTiO$_3$]{}(001) (Fig.\[fig:NoccO1s\]a) correlates with the positions of O$1s$ core levels with respect to the Fermi level (cf. Fig.\[fig:NoccO1s\]b). The lowest O$1s$ eigenvalue occurs at the interface, with the strongst binding energy for a Ti monolayer with a thin STO substrate (occupation of $0.2e^-$ of the Ti $3d$ orbitals). For the thicker STO substrate, the ordering of $3d$ occupation is Al, Ti, 2Ti with relatively small differences. Finally, the lowest Ti $3d$ band occupation is for a Pt contact with the respective O$1s$ eigenvalue at the interface being highest. These quantitative differences are associated with the different chemical bond between the metal overlayer and the surface AlO$_2$ layer: [[*e.g.*]{}]{} the bonding is strongly ionic in the case of Al with a significant charge transfer from Al to O and is much weaker for a Pt overlayer with its nearly filled $4d$ states. The bond strength variation is also reflected in a sizeable difference in bond distances ($d_{\rm Al-O}=1.97$Å, $d_{\rm Pt-O}=2.31$Å). Furthermore, in the systems with a thick STO-substrate the O1s levels in deeper layers away from the IF shift upwards and converge to a similar value indicating that the inner potential relaxes to the bulk STO value. The total $3d$ band occupation is similar for Ti (thin and thick STO slab) and Al ($\sim0.4e$) and lower for 2Ti ($0.22e$) and Pt ($0.16e$).
![Schematic band diagram of a) LAO/STO(001) at the verge of an electronic reconstruction at the critical LAO thickness; b) LAO/STO(001) covered by a metallic contact layer (M). Note, that the potential build up that leads to an electronic reconstruction in the uncovered LAO/STO(001) system at the critical thickness is strongly reduced/eliminated in M/LAO/STO(001).[]{data-label="fig:banddiagr"}](Fig4)
The distinct mechanisms of formation of a q2DEG in LAO/STO(001) with and without a metallic contact are displayed in the schematic band diagram in Fig. \[fig:banddiagr\]. For LAO/STO(001) a thickness dependent MIT occurs as a result of the potential buildup, where the electronic reconstruction involves the formation of holes at the surface and electrons at the interface. In contrast, for $M$/LAO/STO(001) the potential in LAO is flat regardless of the LAO or STO thickness. Simultaneously, a q2DEG with higher carrier density is formed at the interface. Only in the case of Pt (and possiblly other noble metals), likely due to the weaker bonding and smaller charge transfer to the oxide layer, a small residual slope within LAO is found, consistent with the recently measured potential build up in Pt/LAO/STO(001) [@Bhalla:2010].
The Schottky barrier height (SBH) between the metal and the oxide film is an important quantity which depends critically on the type of metal, the chemical bonding characteristics and the work function (see [[*e.g.*]{}]{} [@mrovec:2009]): the $p$-SBH determined from the LDOS are 3.0 eV (Al), 2.8 eV (Ti) and 2.2 (Pt). The latter two values are slightly higher but show the same trend as recent results for Pt and Al on LAO(001) [@dong:2006]. The corresponding $n$-SBH are 0.7 eV (Al), 0.9 eV (Ti) and 2.0 eV (Pt), determined from LDOS, or 1.6 eV higher in each case if the experimental band gap of LAO is considered. The conduction band offset ($n$-type SBH) is related to the effective work function of the system ($\Phi^{\rm Al}=3.53$ eV, $\Phi^{\rm Ti}=4.05$ eV, and $\Phi^{\rm Pt}=5.60$ eV) and correlates also with the Ti $3d$ band occupation at the LAO/STO interface which is highest for an Al contact, followed by Ti and lowest for Pt.
These results show that metallic contacts ultimately change the electrostatic boundary conditions and represent a further powerful means besides oxygen defects[@Bristowe:2011] or adsorbates[@Son:2010], oxide overlayers [@Pentcheva:2010] or an external electric field [@Cen:Thiel:Hammerl:etal2008; @Bi:Levy:2010; @Chen:2010] to tune the functionality at the LAO/STO(001) interface. Despite analogies to the adsorption of hydrogen on LAO/STO(001)[@Son:2010], there are also subtle differences ([[*e.g.*]{}]{} the strong dependence of the potential slope on coverage in the latter system). These differences emphasize not only the importance of the electrostatic boundary conditions[@Stengel:2011] but also of a precise understanding of structural effects and chemical bonding to LAO/STO(001) in order to achieve better understanding and control device performance.
We acknowledge discussions with J. Mannhart and financial support through the DFG SFB/TR80 (project C3) and grant [*h0721*]{} for computational time at the Leibniz Rechenzentrum. V. G. R. L. acknowledges financial support from CONACYT (Mexico) and DAAD (Germany). W. E. P. was supported by U.S. Department of Energy Grant No. DE-FG02-04ER46111.
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---
abstract: 'We study linear problems $S_d$ defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the eigenvalues $\lambda$ of the operator $W_1=S_1^\dagger S_1$ of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti-) symmetry conditions on the complexity, compared to the classical unrestricted problem. In particular, for symmetric problems with $\lambda_1\leq1$ we give characterizations for polynomial tractability and strong polynomial tractability in terms of $\lambda$ and the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schrödinger equation.'
author:
- 'Markus Weimar[^1]'
bibliography:
- 'Bibliography.bib'
title: 'The Complexity of linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces[^2]'
---
[**Keywords:**]{} Antisymmetry, Hilbert spaces, Tensor Products, Complexity.
Introduction {#sect_Intro}
============
In the theory of linear operators $S_d\colon H_d {\rightarrow}G_d$ defined between Hilbert spaces it is well-known that we often observe the the so-called *curse of dimensionality* if we deal with $d$-fold tensor product problems. That is, the complexity of approximating the operator $S_d$ by algorithms using finitely many pieces of information increases exponentially fast with the dimension $d$.
In the last years there have been various approaches to break this exponential dependence on the dimension, e.g., we can relax the error definitions. Another way to overcome the curse is to introduce weights in order to shrink the space of problem elements $H_d$. In the case of function spaces this approach is motivated by the assumption that we have some additional a priori knowledge about the importance of several (groups of) variables.
In the present paper we describe an essentially new kind of a priori knowledge. We assume the problem elements $f\in H_d$ to be *(anti-) symmetric*. This allows us to vanquish the curse and obtain different types of tractability.
The problem of approximating *wave functions*, e.g., solutions of the *electronic Schrödinger equation*, serves as an important example from computational chemistry and physics. In quantum physics wave functions $\Psi$ describe quantum states of certain $d$-particle systems. Formally, these functions depend on $d$ blocks of variables $y_j$, which represent the spacial coordinates and certain additional intrinsic parameters, e.g., the *spin*, of each particle within the system. Due to the *Pauli principle*, the only wave functions $\Psi$ which are physically admissible are antisymmetric in the sense that $\Psi(y) = (-1)^{{\left| \pi \right|}} \Psi(\pi(y))$ for all $y$ and all permutations $\pi$ on a subset $I\subset\{1,\ldots,d\}$ of particles with the same spin. Here $(-1)^{{\left| \pi \right|}}$ denotes the sign of $\pi$. The above relation means that $\Psi$ only changes its sign if we replace particles by each other which possess the same spin. For further details on this topic we refer to of this paper and the references given there. Inspired by this application we illustrate our results with some simple toy examples at the end of this section.
To this end, let $H_1$ and $G_1$ be infinite dimensional separable Hilbert spaces of univariate functions $f\colon D\subset{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ and consider a compact linear operator $S_1\colon H_1{\rightarrow}G_1$ with singular values $\sigma=(\sigma_j)_{j\in{\mathbb{N}}}$. Further, let $\lambda=(\lambda_j)_{j\in{\mathbb{N}}}=(\sigma_j^2)_{j\in{\mathbb{N}}}$ denote the sequence of the squares of the singular values of $S_1$. Finally, assume $S_d\colon H_d {\rightarrow}G_d$ to be the $d$-fold tensor product problem. We want to approximate $S_d$ by linear algorithms using a finite number of continuous linear functionals.
By $n^{\rm ent}({\varepsilon},d)$ we denote the minimal number of information operations needed to achieve an approximation with worst case error at most ${\varepsilon}>0$ on the unit ball of $H_d$. The integer $n^{\rm ent}({\varepsilon},d)$ is called *information complexity* of the *entire* tensor product problem. Further, consider the subspace of all $f\in H_d$ that are *fully symmetric*, i.e., $$\begin{gathered}
f(x)=f(\pi(x)) \quad \text{for all } x\in D^d \text{ and all permutations } \pi \text{ of } \{1,\ldots,d\}.\end{gathered}$$ The minimal number of linear functionals needed to achieve an ${\varepsilon}$-approximation for this subspace is denoted by $n^{\rm sym}({\varepsilon},d)$. Finally, define the subspace of all functions $f\in H_d$ that are *fully antisymmetric* by the condition $$\begin{gathered}
f(x)=(-1)^{{\left| \pi \right|}} f(\pi(x)) \quad \text{for all } x\in D^d \text{ and all } \pi\end{gathered}$$ and denote the information complexity with respect to this subspace by $n^{\rm asy}({\varepsilon},d)$.
Since $H_d$ is a Hilbert space, the optimal algorithm for the entire tensor product problem is well-known. Moreover, it is known that its worst case error, and therefore also the information complexity, can be expressed in terms of $\lambda$, [i.e. ]{}in terms of the squared singular values of the univariate problem operator $S_1$, see, e.g., Sections 4.2.3 and 5.2 in Novak and Wo[ź]{}niakowski [@NW08]. It turns out that this algorithm, applied to the (anti-) symmetric problem, calculates redundant pieces of information. Hence, it can not be optimal in this setting.
In preparation for our algorithms, is devoted to (anti-) symmetric subspaces in a more general fashion than in this introduction. Moreover, there we study some basic properties. In we conclude formulae of algorithms for linear tensor product problems defined on these subspaces. We show their optimality in a wide class of algorithms and deduce an exact expression for the $n$-th minimal error in terms of the squared singular values of $S_1$. summarizes the main results. Finally, we use this error formula to obtain tractability results in and apply them to wave functions in .
Our results yield that in any case (if we deal with the absolute error criterion) $$\begin{gathered}
n^{\rm asy}({\varepsilon},d) \leq n^{\rm sym}({\varepsilon},d) \leq n^{\rm ent}({\varepsilon},d)
\quad \text{for every } {\varepsilon}>0 \text{ and all } d\in{\mathbb{N}},\end{gathered}$$ where for $d=1$ the terms coincide, since then we do not claim any (anti-) symmetry. To see that additional (anti-) symmetry conditions may reduce the information complexity dramatically consider the simple case of a linear operator $S_1$ with singular values $\sigma$ such that $\lambda_1=\lambda_2=1$ and $\lambda_j=0$ for $j\geq 3$. Then the information complexity of the entire tensor product problem can be shown to be $$\begin{gathered}
n^{\rm ent}({\varepsilon},d)=2^d \quad \text{for all } d\in{\mathbb{N}}\text{ and } {\varepsilon}< 1.\end{gathered}$$ Hence, the problem suffers from the curse of dimensionality and is therefore *intractable*. On the other hand, our results show that in the fully symmetric setting we have *polynomial tractability*, because $$\begin{gathered}
n^{\rm sym}({\varepsilon},d)=d+1 \quad \text{for all } d\in{\mathbb{N}}\text{ and } {\varepsilon}< 1.\end{gathered}$$ It can be proved that in this case the complexity of the fully antisymmetric problem decreases with increasing dimension $d$ and, finally, the problem even gets trivial. In detail, we have $$\begin{gathered}
n^{\rm asy}({\varepsilon},d)={ \mathop{\mathrm{max}}\left\{3-d,0\right\} } \quad \text{for all } d\in{\mathbb{N}}\text{ and } {\varepsilon}< 1,\end{gathered}$$ which yields *strong polynomial tractability*.
Next, let us consider a more challenging problem where $\lambda_1=\lambda_2=\ldots=\lambda_m=1$ and $\lambda_j=0$ for every $j>m\geq2$. For $m=2$ this obviously coincides with the example studied above, but letting $m$ increase may tell us more about the structure of (anti-) symmetric tensor product problems. In this situation it is easy to check that $$\begin{gathered}
n^{\rm ent}({\varepsilon},d) = m^d
\quad \text{and} \quad
n^{\rm asy}({\varepsilon},d) = \begin{cases}
\binom{m}{d}, & d \leq m\\
0, & d > m,
\end{cases}
\quad \text{for every $d\in{\mathbb{N}}$ and all ${\varepsilon}< 1$.}\end{gathered}$$ Since $\binom{m}{d}\geq 2^{d-1}$ for $d\leq{\left\lfloor m/2 \right\rfloor}$, this means that for large $m$ the complexity in the antisymmetric case increases exponentially fast with $d$ up to a certain maximum. Beyond this point it falls back to zero. The information complexity in the symmetric setting is much harder to calculate for this case. However, it can be seen that we have polynomial tractability, but $n^{\rm sym}({\varepsilon},d)$ needs to grow at least linearly with $d$ such that the symmetric problem can not be strongly polynomially tractable, whereas this holds in the antisymmetric setting. The entire problem again suffers from the curse of dimensionality.
The reason why antisymmetric problems are that much easier than their symmetric counterparts is that from the antisymmetry condition it follows that $f(x)=0$ if there exist coordinates $j$ and $l$ such that $x_j=x_l$. Another explanation for the good tractability behavior of antisymmetric tensor product problems might be the *initial error* ${\varepsilon}_d^{\rm init}$. For every choice of $\lambda$ it tends to zero as $d$ grows, what is not necessarily the case for the corresponding entire and the symmetric problem, respectively. In fact, we have $$\begin{gathered}
{\varepsilon}^{\rm init}_{d,\rm ent} = {\varepsilon}^{\rm init}_{d, \rm sym} = \lambda_1^{d/2}, \quad \text{ whereas} \quad {\varepsilon}^{\rm init}_{d, \rm asy} = \prod_{j=1}^d \lambda_j^{1/2}.\end{gathered}$$
For a last illustrative example consider the case $\lambda_1=1$ and $\lambda_{j+1}=j^{-\beta}$ for some $\beta\geq0$ and all $j\in{\mathbb{N}}$. That means that we have the two largest singular values $\sigma_1=\sigma_2$ of $S_1$ equal to one. The remaining series decays like the inverse of some polynomial. If $\beta=0$ the operator $S_1$ is not compact, since $(\lambda_m)_{m\in{\mathbb{N}}}$ does not tend to zero. Hence, all the information complexities are infinite in this case. For $\beta>0$, any $\delta>0$ and some $C>0$ it is $$\begin{gathered}
n^{\rm ent}({\varepsilon},d)\geq 2^{d}, \quad n^{\rm sym}({\varepsilon},d) \geq d+1
\quad \text{and} \quad n^{\rm asy}({\varepsilon},d)\leq C {\varepsilon}^{-(2/\beta+\delta)},
\quad \text{for all } {\varepsilon}<1, d\in{\mathbb{N}}.\end{gathered}$$ Thus, again for the entire problem we have the curse, whereas the antisymmetric problem is strongly polynomially tractable. Once more, the symmetric problem can shown to be polynomially tractable. Note that in this example the antisymmetric case is not trivial, because all $\lambda_j$ are strictly positive. If we replace $j^{-\beta}$ by $\log^{-1}(j+1)$ in this example we obtain (polynomial) intractability even in the antisymmetric setting.
Altogether these examples show that exploiting an a priori knowledge about (anti-) symmetries of the given tensor product problem can help to obtain tractability, but it does not make the problem trivial in general. We conclude the introduction with a partial summary of our main complexity results.
Let $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}$ denote the non-increasing sequence of the squared singular values of $S_1\colon H_1 {\rightarrow}G_1$ and assume $\lambda_2>0$. Then for the information complexity of (anti-) symmetric linear tensor product problems $S_d$ we obtain the following characterizations:
- The *fully symmetric* problem is strongly polynomially tractable [w.r.t. ]{} the normalized error criterion iff $\lambda\in{\ell}_\tau$ for some $\tau>0$ and $\lambda_1>\lambda_2$. Furthermore, in the case $\lambda_1\leq 1$ the problem is strongly polynomially tractable [w.r.t. ]{}the absolute error criterion iff $\lambda\in{\ell}_\tau$ and $\lambda_2<1$.
- The *fully antisymmetric* problem is strongly polynomially tractable [w.r.t. ]{} the absolute error criterion iff $\lambda\in{\ell}_\tau$ for some $\tau>0$.
In contrast, it is known, see Novak and Wo[ź]{}niakowski [@NW08], that
- the *entire* tensor product problem is never (strongly) polynomially tractable [w.r.t. ]{}to normalized error criterion. Moreover, the problem is strongly polynomially tractable [w.r.t. ]{} the absolute error criterion iff $\lambda\in{\ell}_\tau$ for some $\tau>0$ and $\lambda_1<1$.
Spaces with (anti-) symmetry conditions {#sect_antisym}
=======================================
Motivated by the example of wave functions in , we mainly deal with *function spaces* in this section. To this end, we start by defining (anti-) symmetry properties for functions which will lead us to orthogonal projections, mapping the function space onto its subspace of (anti-) symmetric functions. It will turn out that these projections applied to a given basis in the tensor product Hilbert function space lead us to handsome formulae for orthonormal bases of the subspaces. In a final remark we generalize our approach and define (anti-) symmetry conditions for *arbitrary* tensor product Hilbert spaces based on the deduced results for function spaces.
We use a general approach to (anti-) symmetric functions, as it can be found in Section 2.5 of Hamaekers [@H09]. Therefore, for a moment, consider an abstract separable Hilbert space $F$ of real-valued functions defined on a domain $\Omega \subset {\mathbb{R}}^d$. In this part of the paper let $d \geq 2$ be fixed. The inner product on $F$ is denoted by ${\left\langle \cdot, \cdot \right\rangle}_F$. Moreover, let $I=I(d) \subset \{1,\ldots,d\}$ be an arbitrary given non-empty subset of coordinates. Then we define the set $$\begin{gathered}
{\mathcal{S}}_I = \{ \pi \colon \{1,\ldots,d\} {\rightarrow}\{1,\ldots,d\} {\, | \,}\pi \text{ bijective and } \pi \big|_{\{1,\ldots,d \} \setminus I} = {\mathrm{id}}\}\end{gathered}$$ of all permutations on $\{1,\ldots,d\}$ that leave the complement of $I$ fixed. Obviously, the cardinality of this set is given by $\# {\mathcal{S}}_I = (\# I)!$, where $\#$ denotes the number of elements of a set. For a given $\pi \in {\mathcal{S}}_I$ we define the mapping $$\begin{gathered}
\pi' \colon \Omega {\rightarrow}{\mathbb{R}}^d, \quad x=(x_1,\ldots,x_d) \mapsto \pi'(x)=(x_{\pi(1)}, \ldots, x_{\pi(d)}).\end{gathered}$$ To abbreviate the notation we identify $\pi$ and $\pi'$ with each other.
For an appropriate definition of partial (anti-) symmetry of functions $f\in F$ we need the following simple assumptions. For every $\pi \in {\mathcal{S}}_I$ we assume
1. \[A1\] $x\in \Omega$ implies $\pi(x)\in\Omega$,
2. \[A2\] $f\in F$ implies $f(\pi(\cdot)) \in F$ and
3. \[A3\] there exists $c_{\pi} \geq 0$ (independent of $f$) such that ${\left\| f(\pi(\cdot)) {\, | \,}F \right\|} \leq c_{\pi} {\left\| f {\, | \,}F \right\|}$.
Note that these assumptions always hold if $F$ is a $d$-fold tensor product Hilbert space $H_d=H_1 \otimes\ldots\otimes H_1$ equipped with a cross norm, as described in the examples of the previous section.
Now we call a function $f \in F$ *partially symmetric with respect to $I$* (or *$I$-symmetric* for short) if a permutation $\pi\in{\mathcal{S}}_I$ applied to the argument $x$ does not affect the value of $f$. Hence, $$\begin{gathered}
\label{sym}
f(x)=f(\pi(x)) \quad \text{for all} \quad x\in \Omega \quad \text{and every} \quad \pi \in {\mathcal{S}}_I.\end{gathered}$$
Moreover, we call a function $f \in F$ *partially antisymmetric with respect to $I$* (or *$I$-antisymmetric*, respectively) if $f$ changes its sign by exchanging the variables $x_i$ and $x_j$ with each other, where $i,j\in I$. That is, we have $$\begin{gathered}
\label{antisym}
f(x)=(-1)^{{\left| \pi \right|}} f(\pi(x)) \quad \text{for all} \quad x\in \Omega \quad \text{and every} \quad \pi \in {\mathcal{S}}_I,\end{gathered}$$ where ${\left| \pi \right|}$ denotes the *inversion number* of the permutation $\pi$. The term $(-1)^{{\left| \pi \right|}}$ therefore coincides with the *sign*, or *parity of $\pi$* and is equal to the determinant of the associated permutation matrix. In the case $\#I=1$ we do not claim any (anti-) symmetry, since the set ${\mathcal{S}}_I = \{ {\mathrm{id}}\}$ is trivial. For $I=\{1,\ldots, d\}$ functions $f$ which satisfy [(\[sym\])]{} or [(\[antisym\])]{}, respectively, are called *fully (anti-) symmetric*.
Note that, in particular, formula [(\[antisym\])]{} yields that the value $f(x)$ of (partially) antisymmetric functions $f$ equals zero if $x_i=x_j$ with $i \neq j$ and $i,j\in I$. For (partially) symmetric functions such an implication does not hold. Therefore, the (partial) antisymmetry property is a somewhat more restrictive condition than the (partial) symmetry property with respect to the same subset $I$. As we will see in the next sections this will also affect our complexity estimates.
Next, we define the so-called *symmetrizer ${\mathfrak{S}}_I^F$* and *antisymmetrizer ${\mathfrak{A}}_I^F$ on $F$ with respect to the subset $I$* by $$\begin{gathered}
{\mathfrak{S}}_I^F \colon F {\rightarrow}F, \quad f\mapsto {\mathfrak{S}}_I^F (f) = \frac{1}{\#{\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} f(\pi(\cdot))\end{gathered}$$ and $$\begin{gathered}
{\mathfrak{A}}_I^F \colon F {\rightarrow}F, \quad f\mapsto {\mathfrak{A}}_I^F (f) = \frac{1}{\#{\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} f(\pi(\cdot)).\end{gathered}$$ If there is no danger of confusion we use the notation ${\mathfrak{S}}_I$ and ${\mathfrak{A}}_I$ instead of ${\mathfrak{S}}_I^F$ and ${\mathfrak{A}}_I^F$, respectively. The following lemma collects together some basic properties which can be proved easily. For details see the appendix of this paper.
\[projection\] Both the mappings $P_I\in\{{\mathfrak{S}}_I, {\mathfrak{A}}_I\}$ define bounded linear operators on $F$ with $P_I^2=P_I$. Thus, ${\mathfrak{S}}_I$ and ${\mathfrak{A}}_I$ provide orthogonal projections of $F$ onto the closed linear subspaces $$\begin{gathered}
\label{antisymsubspace}
{\mathfrak{S}}_I(F) = \{ f \in F {\, | \,}f \text{ satisfies } {(\ref{sym})} \} \quad \text{and}
\quad {\mathfrak{A}}_I(F) = \{ f \in F {\, | \,}f \text{ satisfies } {(\ref{antisym})} \}
\end{gathered}$$ of all partially (anti-) symmetric functions ([w.r.t. ]{}$I$) in $F$, respectively. Hence, $$\begin{gathered}
\label{orth_decomp}
F = {\mathfrak{S}}_I(F) \oplus ({\mathfrak{S}}_I(F))^\bot = {\mathfrak{A}}_I(F) \oplus ({\mathfrak{A}}_I(F))^\bot.
\end{gathered}$$
Note that the notion of partially (anti-) symmetric functions can be extended to more than one subset $I$. Therefore, consider two non-empty subsets of coordinates $I,J\subset \{1,\ldots,d\}$ with $I\cap J = {\emptyset}$. Then we call a function $f\in F$ *multiple partially (anti-) symmetric with respect to $I$ and $J$* if $f$ satisfies [(\[sym\])]{}, or [(\[antisym\])]{}, respectively, for $I$ and $J$. Since $I$ and $J$ are disjoint we observe that $\pi \circ \sigma = \sigma \circ \pi$ for all $\pi\in {\mathcal{S}}_I$ and $\sigma\in {\mathcal{S}}_J$. Thus, the linear projections $P_I \in \{{\mathfrak{S}}_I, {\mathfrak{A}}_I\}$ and $P_J \in \{{\mathfrak{S}}_J, {\mathfrak{A}}_J\}$ commute on $F$, [i.e. ]{}$P_I \circ P_J = P_J \circ P_I$.
Further extensions to more than two disjoint subsets of coordinates are possible. We will restrict ourselves to the case of at most two (anti-) symmetry conditions, because in particular wave functions can be modeled as functions which are antisymmetric with respect to $I$ and and $J=I^C$, where $I^C$ denotes the complement of $I$ in $\{1, \ldots, d\}$; see, e.g., of this paper.
Up to this point the function space $F$ was an arbitrary separable Hilbert space of real-valued functions. Indeed, for the definition of (anti-) symmetry we did not claim any product structure. On the other hand, it is also motivated by applications to consider tensor product function spaces; see, e.g., Section 3.6 in Yserentant [@Y10]. In detail, it is well-known that so-called spaces of dominated mixed smoothness, e.g. $W_2^{(1,\ldots,1)}({\mathbb{R}}^{3d})$, can be represented as certain tensor products; see Section 1.4.2 in Hansen [@H10].
Nevertheless, if we take into account such a structure, i.e., assume $F=H_d=H_1\otimes \ldots \otimes H_1$ ($d$ times), where $H_1$ is a suitable Hilbert space of functions $f\colon D{\rightarrow}{\mathbb{R}}$, it is known that we can construct an orthonormal basis (ONB) of $F$ out of a given ONB of $H_1$. In fact, if $\{\eta_i {\, | \,}i\in{\mathbb{N}}\}$ is an ONB of the underlying Hilbert function space $H_1$ then the set of all $d$-fold tensor products $\{\eta_{d,j} = \bigotimes_{l=1}^d \eta_{j_l} {\, | \,}j=(j_1,\ldots,j_d)\in{\mathbb{N}}^d\}$, $$\begin{gathered}
\eta_{d,j}(x) = \prod_{l=1}^d \eta_{j_l}(x_l), \quad x=(x_1,\ldots,x_d)\in D^d,\end{gathered}$$ is mutually orthonormal in $H_d$ and forms a basis. To exploit this representation we start with a simple observation.
Let $j\in {\mathbb{N}}^d$ and $x\in D^d$, as well as a non-empty subset $I$ of $\{1,\ldots,d\}$ be arbitrarily fixed. If we define $\sigma = \pi^{-1} \in S_I$ then $$\begin{aligned}
({\mathfrak{A}}_I \eta_{d,j})(x) &=& \frac{1}{\#{\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \eta_{d,j}(\pi(x))
= \frac{1}{\#{\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \prod_{m=1}^d \eta_{j_m}(x_{\pi(m)}) \nonumber\\
&=& \frac{1}{\#{\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \prod_{m=1}^d \eta_{j_{\sigma(m)}}(x_m)
= \frac{1}{\#{\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \sigma^{-1} \right|}} \eta_{d,\sigma(j)}(x) \label{pi_inside}\\
&=& \frac{1}{\#{\mathcal{S}}_I} \sum_{\sigma \in {\mathcal{S}}_I} (-1)^{{\left| \sigma \right|}} \eta_{d,\sigma(j)}(x). \nonumber\end{aligned}$$ For simplicity, once again we identified $\pi(j)=\pi(j_1,\ldots,j_d)$ with $(j_{\pi(1)},\ldots,j_{\pi(d)})$ for multi-indices $j\in {\mathbb{N}}^d$. Obviously, the same calculation can be made for ${\mathfrak{S}}_I$ without the factor $(-1)$. Since $x\in D^d$ was arbitrary we obtain $$\begin{gathered}
\label{antisym_basis}
{\mathfrak{S}}_I \eta_{d,j} = \frac{1}{\#{\mathcal{S}}_I} \sum_{\sigma \in {\mathcal{S}}_I} \eta_{d,\sigma(j)}
\quad \text{and} \quad {\mathfrak{A}}_I \eta_{d,j} = \frac{1}{\#{\mathcal{S}}_I} \sum_{\sigma \in {\mathcal{S}}_I} (-1)^{{\left| \sigma \right|}} \eta_{d,\sigma(j)}
\quad \text{for all} \quad j\in {\mathbb{N}}^d.\end{gathered}$$ Note that in general, [i.e. ]{}for arbitrary $j\in{\mathbb{N}}^d$ and $\sigma\in{\mathcal{S}}_I$, the tensor products $\eta_{d,\sigma(j)}$ and $\eta_{d,j}$ do not coincide, because taking the tensor product is not commutative in general. Therefore, ${\mathfrak{S}}_I$ is not simply the identity on $\{ \eta_{d,j} {\, | \,}j \in {\mathbb{N}}^d \}$. On the other hand, we see that for different $j\in {\mathbb{N}}^d$ many of the functions ${\mathfrak{S}}_I \eta_{d,j}$ coincide. Of course the same holds true for ${\mathfrak{A}}_I \eta_{d,j}$, at least up to a factor of $(-1)$.
We will see in the following that for $P_I\in\{{\mathfrak{S}}_I, {\mathfrak{A}}_I\}$ a linearly independent subset of all projections $\{ P_I\eta_{d,j} {\, | \,}j\in{\mathbb{N}}^d \}$ equipped with suitable normalizing constants can be used as an ONB of the linear subspace $P_I(H_d)$ of $I$-(anti-)symmetric functions in $H_d$. To this end, we need a further definition. For fixed $d\geq 2$ and $I\subset\{1,\ldots,d\}$, let us introduce a function $$\begin{gathered}
M_I =M_{I,d} \colon {\mathbb{N}}^d {\rightarrow}\{0,\ldots,\#I\}^{\#I}\end{gathered}$$ which counts how often different integers occur in a given multi-index $j\in{\mathbb{N}}^d$ among the subset $I$ of coordinates, ordered with respect to their rate. To give an example let $d=7$ and $I=\{1,\ldots,6\}$. Then $M_{I,7}$ applied to $j=(12, 4, 4, 12, 6, 4, 4) \in {\mathbb{N}}^7$ gives the $\#I=6$ dimensional vector $M_{I,7}(j) = (3, 2, 1, 0, 0, 0)$, because $j$ contains the number “$4$” three times among the coordinates $j_1,\ldots,j_6$, “$12$” two times and so on. Since in this example there are only three different numbers involved, the fourth to sixth coordinates of $M_{I,7}(j)$ equal zero. Obviously, $M_{I}$ is invariant under all permutations $\pi\in{\mathcal{S}}_I$ of the argument. Thus, $$\begin{gathered}
M_{I}(j) = M_{I}(\pi(j)) \quad \text{for all} \quad j\in{\mathbb{N}}^d \quad \text{and} \quad \pi \in {\mathcal{S}}_I.\end{gathered}$$ In addition, since $M_{I}(j)$ is again a multi-index, we see that ${\left| M_{I}(j) \right|}=\#I$ and $M_{I}(j)!$ are well-defined for every $j\in{\mathbb{N}}^d$. With this tool we are ready to state the following assertion which can be shown using elementary arguments as well as ; see the appendix.
\[lemma\_basis\] Assume $\{\eta_{d,j} {\, | \,}j\in {\mathbb{N}}^d \}$ to be a given orthonormal tensor product basis of the function space $H_d$ and let ${\emptyset}\neq I=\{i_1,\ldots,i_{\# I}\} \subset\{1,\ldots,d\}$. Moreover, for $P_I\in\{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$ define functions $\xi_j \colon D^d {\rightarrow}{\mathbb{R}}$, $$\begin{gathered}
\xi_j = \sqrt{\frac{\#{\mathcal{S}}_I}{M_{I}(j)!}} \cdot P_I(\eta_{d,j})
\quad \text{ for } \quad j \in {\mathbb{N}}^d.
\end{gathered}$$ Then the set $\{ \xi_k {\, | \,}k \in \nabla_d \}$ builds an orthonormal basis of the partially (anti-) symmetric subspace $P_I(H_d)$, where $\nabla_d$ is given by $$\begin{gathered}
\label{def_nabla}
\nabla_d = \begin{cases}
\{ k \in {\mathbb{N}}^d {\, | \,}k_{i_1} \leq k_{i_2} \leq \ldots \leq k_{i_{\#I}}\},& \text{ if } P_I={\mathfrak{S}}_I,\\
\{ k \in {\mathbb{N}}^d {\, | \,}k_{i_1} < k_{i_2} < \ldots < k_{i_{\#I}}\},& \text{ if } P_I={\mathfrak{A}}_I.
\end{cases}
\end{gathered}$$
Observe that in the antisymmetric case the definition of $\xi_j$ for $j\in\nabla_d$ simplifies, since then $M_I(j)!=1$ for all $j\in\nabla_d$. Moreover, note that in the special case $I=\{1,\ldots,\#I \}$ we have $$\begin{gathered}
P_I(H_d) = P_I \left( \bigotimes_{j\in I} H_1 \right) \otimes \left( \bigotimes_{j\notin I} H_1 \right).\end{gathered}$$ That is, we can consider the subspace of $I$-(anti-)symmetric functions $f\in H_d$ as the tensor product of the set of all fully (anti-) symmetric $\#I$-variate functions with the $(d-\#I)$-fold tensor product of $H_1$. Modifications in connection with multiple partially (anti-) symmetric functions are obvious.
Finally, note that also holds if the index set ${\mathbb{N}}$ of the univariate basis $\{\eta_i {\, | \,}i\in{\mathbb{N}}\}$ is replaced by a more general countable set equipped with a total order. But let us shortly focus on another generalization of the previous results.
Up to now we exclusively dealt with Hilbert *function* spaces. However, the proofs of and yield that there are only a few key arguments in connection with (anti-) symmetry such that we do not need this restriction.
Starting from the very beginning we need to adapt the definition of $I$-(anti-)symmetry due to [(\[sym\])]{} and [(\[antisym\])]{}. Of course it is sufficient to define this property at first only for basis elements. Therefore, if $E_d=\{\eta_{d,k} {\, | \,}k \in {\mathbb{N}}^d \}$ denotes a tensor product ONB of $H_d$ and ${\emptyset}\neq I\subset\{1,\ldots,d\}$ is given then we call an element $\eta_{d,k} = \bigotimes_{l=1}^d \eta_{k_l}$ *partially symmetric with respect to $I$* (or *$I$-symmetric* for short), if $$\begin{gathered}
\eta_{d,k} = \eta_{d,\pi(k)} \quad \text{for all} \quad \pi \in {\mathcal{S}}_I,\end{gathered}$$ where ${\mathcal{S}}_I$ and $\pi(k)=(k_{\pi(1)},\ldots,k_{\pi(d)})$ are defined as above. Analogously we define *$I$-antisymmetry* with an additional factor $(-1)^{{\left| \pi \right|}}$. Moreover, an arbitrary element in $H_d$ is called $I$-(anti-)symmetric if in its basis expansion every element with non-vanishing coefficient possesses this property.
Next, the *antisymmetrizer* ${\mathfrak{A}}_I$ is defined as the uniquely defined continuous extension of the linear mapping $$\begin{gathered}
{\mathfrak{A}}_I \colon E_d {\rightarrow}H_d, \quad {\mathfrak{A}}_I(\eta_{d,k}) = \frac{1}{\# {\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \eta_{d,\pi(k)}\end{gathered}$$ from $E_d$ to $H_d$. Again the *symmetrizer* ${\mathfrak{S}}_I$ is given in a similar way. Hence, in the general setting we define the mappings using formula [(\[antisym\_basis\])]{}, which we derived for the special case. Note that the triangle inequality yields ${\left\| P_I \right\|} \leq 1$, for $P_I\in\{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$.
Once more we denote the sets of all $I$-(anti-)symmetric elements of $H_d$ by $P_I(H_d)$, where $P_I\in \{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$. Observe, that this can be justified since the operators $P_I$ again provide orthogonal projections onto closed linear subspaces as described in .
Finally, also the proof of can be adapted to the general Hilbert space case.
Optimal algorithms {#sect_OptAlgos}
==================
In the present section we conclude optimal algorithms for linear problems defined on (anti-) symmetric subsets of tensor product Hilbert spaces as described in the previous paragraph. Moreover, we deduce formulae for the $n$-th minimal errors of these (anti-) symmetric problems and recover the known assertions for the entire tensor product problem.
Basic definitions and the main result
-------------------------------------
Throughout the whole section we use the following notation. Let $H_1$ be a (infinite dimensional) separable Hilbert space with inner product ${\left\langle \cdot, \cdot \right\rangle}_{H_1}$ and let $G_1$ be some arbitrary Hilbert space. Furthermore, assume $S_1 \colon H_1 {\rightarrow}G_1$ to be a compact linear operator between these spaces and consider its singular value decomposition. That is, define the compact self-adjoint operator $W_1=S_1^\dagger S_1 \colon H_1 {\rightarrow}H_1$ and denote its eigenpairs with respect to a non-increasing ordering of the eigenvalues by $\{(e_i,\lambda_i) {\, | \,}i\in{\mathbb{N}}\}$, i.e. $$\begin{gathered}
\label{univariateeigenpairs}
W_1(e_i) = \lambda_i e_i, \quad
\text{and} \quad {\left\langle e_i, e_j \right\rangle}_{H_1} = \delta_{i,j} \quad
\text{with} \quad \lambda_1 \geq \lambda_2 \geq \ldots \geq 0.\end{gathered}$$ Then $\lambda=(\lambda_i)_{i\in{\mathbb{N}}}$ coincides with the sequence of the squared singular values $\sigma^2=(\sigma_i^2)_{i\in{\mathbb{N}}}$ of $S_1$ and the set $\{ e_i {\, | \,}i \in {\mathbb{N}}\}$ forms an ONB of $H_1$; see, e.g., Section 4.2.3 in Novak and Wo[ź]{}niakowski [@NW08]. In the following we will refer to $S_1$ as the *univariate problem* or *univariate case*.
For $d\geq 2$, let $H_d = H_1 \otimes \ldots \otimes H_1$ be the $d$-fold tensor product space of $H_1$. This means that $H_d$ is the closure of the set of all linear combinations of formal objects $f=\bigotimes_{l=1}^d f_l$ with $f_l \in H_1$, called *simple tensors* or *pure tensors*. Here the closure is taken with respect to the inner product in $H_d$ which is defined such that $$\begin{gathered}
{\left\langle \bigotimes_{l=1}^d f_l, \bigotimes_{l=1}^d g_l \right\rangle}_{H_d} = \prod_{l=1}^d {\left\langle f_l, g_l \right\rangle}_{H_1} \quad \text{for} \quad f_l, g_l \in H_1.\end{gathered}$$ With these definitions $H_d$ is also an infinite dimensional Hilbert space and it is easy to check that $$\begin{gathered}
\label{basis_eta}
E_d=\left\{\eta_{d,j} = \bigotimes_{l=1}^d \eta_{j_l} \in H_d {\, | \,}j = (j_1,\ldots,d) \in {\mathbb{N}}^d\right\}\end{gathered}$$ forms an orthonormal basis in $H_d$ if $\{\eta_i \in H_1 {\, | \,}i\in{\mathbb{N}}\}$ is an arbitrary ONB in the underlying space $H_1$. Similarly, let $G_d = G_1 \otimes \ldots \otimes G_1$, $d$ times, and define $S_d$ as the tensor product operator $$\begin{gathered}
S_d = S_1 \otimes \ldots \otimes S_1 \colon H_d {\rightarrow}G_d.\end{gathered}$$ In detail, we define the bounded linear operator ${ \widetilde{S} }_d \colon E_d {\rightarrow}G_d$ such that for all $j\in{\mathbb{N}}^d$ we have ${ \widetilde{S} }_d(\eta_{d,j}) = { \widetilde{S} }_d (\bigotimes_{l=1}^d \eta_{j_l} ) = \bigotimes_{l=1}^d S_1(\eta_{j_l}) \in G_d$. Then $S_d$ is assumed to be the uniquely defined linear, continuous extension of ${ \widetilde{S} }_d$ from $E_d$ to $H_d$.
We refer to the problem of approximating $S_d \colon H_d {\rightarrow}G_d$ as the *entire $d$-variate problem*. In contrast, we are interested in the restriction $S_d\big|_{P_I(H_d)} \colon P_I(H_d) {\rightarrow}G_d$ of $S_d$ to some [(anti-)]{} symmetric subspace $P_I(H_d)$ with $P_I\in\{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$ as described in the previous section. To abbreviate the notation we denote this restriction again by $S_d$ and refer to it as the *$I$-symmetric problem*.
For the singular value decomposition of the entire problem operator $S_d$ we consider the self-adjoint, compact operator $$\begin{gathered}
W_d = {S_d}^\dagger S_d \colon H_d {\rightarrow}H_d.\end{gathered}$$ Its eigenpairs $\{(e_{d,j},\lambda_{d,j}) {\, | \,}j=(j_1,\ldots,j_d) \in {\mathbb{N}}^d\}$ are given by the set of all $d$-fold (tensor) products of the univariate eigenpairs [(\[univariateeigenpairs\])]{} of $W_1$, i.e., $$\begin{gathered}
\label{Eigenpairs}
e_{d,j}=\bigotimes_{l=1}^d e_{j_l}
\quad \text{and} \quad
\lambda_{d,j}=\prod_{l=1}^d \lambda_{j_l}
\quad \text{for} \quad j=(j_1,\ldots,j_d)\in{\mathbb{N}}^d.\end{gathered}$$ It is well-known how these eigenpairs can be used to construct a linear algorithm $A'_{n,d}$ which is optimal for the entire $d$-variate tensor product problem. In detail, $A'_{n,d}$ minimizes the *worst case error* $$\begin{gathered}
e^{\rm wor}(A_{n,d}; H_d) = \sup_{f\in {\mathcal{B}}(H_d)} {\left\| A_{n,d}(f)-S_d(f) {\, | \,}G_d \right\|}\end{gathered}$$ among all adaptive linear algorithms $A_{n,d}$ using $n$ continuous linear functionals. Here ${\mathcal{B}}(H_d)$ denotes the unit ball of $H_d$. In other words, $A'_{n,d}$ achieves the *$n$-th minimal error* $$\begin{gathered}
e(n,d;H_d) = \inf_{A_{n,d}} e^{\rm wor}(A_{n,d}; H_d).\end{gathered}$$
With this notation our main result reads as follows.
\[theo\_opt\_algo\] Let $\{(e_m, \lambda_m) {\, | \,}m\in {\mathbb{N}}\}$ denote the eigenpairs of $W_1$ given by [(\[univariateeigenpairs\])]{}. Moreover, for $d>1$ let ${\emptyset}\neq I=\{i_1, \ldots, i_{\#I}\} \subset\{1,\ldots,d\}$ and assume $S_d$ to be the linear tensor product problem restricted to the $I$-(anti-)symmetric subspace $P_I(H_d)$, where $P_I\in\{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$, of the $d$-fold tensor product space $H_d$. Finally, let $\nabla_d$ be given by [(\[def\_nabla\])]{} and define $$\begin{gathered}
\label{eigenpairs_sym}
\{(\xi_{\psi(v)}, \lambda_{d,\psi(v)}) {\, | \,}v\in {\mathbb{N}}\}
= \{(\xi_k, \lambda_{d,k}) {\, | \,}k \in \nabla_d\}
\end{gathered}$$ by $\xi_k=\sqrt{\#{\mathcal{S}}_I / M_I(k)!}\cdot P_I(e_{k_1}\otimes \ldots \otimes e_{k_d})$ and $\lambda_{d,k}=\prod_{l=1}^d \lambda_{k_l}$, for $k\in\nabla_d$, where $\psi\colon {\mathbb{N}}{\rightarrow}\nabla_d$ provides a non-increasing rearrangement of $\{ \lambda_{d,k} {\, | \,}k\in \nabla_d\}$.\
Then for every $d>1$ the set [(\[eigenpairs\_sym\])]{} denotes the eigenpairs of $W_d\big|_{P_I(H_d)}={S_d}^\dagger S_d$. Thus, for every $n \in {\mathbb{N}}_0$, the linear algorithm $A_{n,d}^*\colon P_I(H_d) {\rightarrow}P_I(G_d)$, $$\begin{gathered}
\label{opt_algo}
A_{n,d}^*f = \sum_{v=1}^n {\left\langle f, \xi_{\psi(v)} \right\rangle}_{H_d} \cdot S_d \xi_{\psi(v)},
\end{gathered}$$ which uses $n$ linear functionals, is $n$-th optimal for $S_d$ on $P_I(H_d)$ with respect to the worst case setting. Furthermore, it is $$\begin{gathered}
\label{nth_error}
e(n,d; P_I(H_d))
= e^{\rm wor}(A_{n,d}^*;P_I(H_d))
= \sqrt{\lambda_{d, \psi(n+1)}}.
\end{gathered}$$
Let us add some remarks on this theorem. First of all, the sum over an empty index set is to be interpreted as zero such that $A_{0,d}^*f \equiv 0$. Further, note that the worst case error can be attained with the function $\xi_{\psi(n+1)}$. It can be improved neither by non-linear algorithms using continuous information, nor by linear algorithms using adaptive information. Moreover, observe that the classical entire tensor product problem is included as the case $\#I=1$, where we do not claim any (anti-) symmetry. Then $\nabla_d={\mathbb{N}}^d$ and the $\xi_k$’s simply equal the tensor products $e_{d,k}=\otimes_{l=1}^d e_{k_l}$. Hence, $A^*_{n,d}=A'_{n,d}$.
The remainder of this section is devoted to the proof of the main result .
Proof of Theorem 1 {#sect_auxresults}
------------------
We start with an auxiliary result which shows that any optimal algorithm $A^*$ for $S_d$ needs to preserve the (anti-) symmetry properties of its domain of definition, [i.e. ]{}$A^*f \in P_I(G_d)$ for all $f\in P_I(H_d)$. The following proposition generalizes Lemma 10.2 in Zeiser [@Z10] where this assertion was shown for the approximation problem, [i.e. ]{}for $S_d={\mathrm{id}}$. A comprehensive proof can be found in the appendix of this paper.
\[theo\_bestapprox\] Let $d>1$ and assume ${\emptyset}\neq I \subset \{1,\ldots,d\}$. Furthermore, for $X\in\{H,G\}$, let $P_I^X$ denote the (anti-) symmetrizer $P_I\in\{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$ on $X_d$ with respect to $I$ and suppose $A \colon P_I^H(H_d) {\rightarrow}G_d$ to be an arbitrary algorithm for $S_d$. Then, for $g \in H_d$, $$\begin{gathered}
\label{commute}
(S_d \circ P_I^H)(g) = (P_I^G \circ S_d)(g),
\end{gathered}$$ and for all $f \in P_I^H(H_d)$ it holds $$\begin{gathered}
\label{bestapprox}
{\left\| S_d f - Af {\, | \,}G_d \right\|}^2 = {\left\| S_d f - P_I^G (A f) {\, | \,}G_d \right\|}^2 + {\left\| Af - P_I^G (A f) {\, | \,}G_d \right\|}^2.
\end{gathered}$$ Hence, an optimal algorithm $A^*$ for $S_d$ preserves (anti-) symmetry, [i.e. ]{} $$\begin{gathered}
A^*f \in P_I^G(G_d) \quad \text{for all} \quad f \in P_I^H(H_d).
\end{gathered}$$
Beside this qualitative assertion we are interested in explicit error bounds. Therefore, the next proposition shows an upper bound on the worst case error of the algorithm $A_{n,d}^*$ given by [(\[opt\_algo\])]{}.
\[theo\_upperbound\] Under the assumptions of the worst case error of $A_{n,d}^*$ given by [(\[opt\_algo\])]{} is bounded from above by $$\begin{gathered}
e^{\rm wor}(A^*_{n,d}; P_I(H_d)) \leq \sqrt{\lambda_{d,\psi(n+1)}}.
\end{gathered}$$
By we have for all $f\in P_I(H_d)$ the unique representation $$\begin{gathered}
f = \sum_{k \in \nabla_d} {\left\langle f, \xi_k \right\rangle} \cdot \xi_k.\end{gathered}$$ Therefore, the boundedness of $S_d$ together with [(\[eigenpairs\_sym\])]{} implies that $$\begin{gathered}
\label{Sdf}
S_d f = \sum_{k \in \nabla_d} {\left\langle f, \xi_k \right\rangle}_{H_d} \cdot S_d \xi_k = \sum_{v \in {\mathbb{N}}} {\left\langle f, \xi_{\psi(v)} \right\rangle}_{H_d} \cdot S_d \xi_{\psi(v)}
\quad \text{for every} \quad f \in P_I(H_d).\end{gathered}$$ Furthermore, in the case $P_I={\mathfrak{A}}_I$ it is easy to see that we have $$\begin{aligned}
{\left\langle S_d \xi_j, S_d \xi_k \right\rangle}_{G_d}
&=& \frac{\#{\mathcal{S}}_I}{\sqrt{M_I(j)!\cdot M_I(k)!}} \cdot {\left\langle S_d {\mathfrak{A}}_I e_{d,j}, S_d {\mathfrak{A}}_I e_{d,k} \right\rangle}_{G_d} \\
&=& \frac{1}{\# {\mathcal{S}}_I \sqrt{M_I(j)!\cdot M_I(k)!}} \sum_{\pi,\sigma \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}+{\left| \sigma \right|}} {\left\langle S_d e_{d,\pi(j)}, S_d e_{d,\sigma(k)} \right\rangle}_{G_d}\end{aligned}$$ for $i,j\in \nabla_d$, because of the commutativity of $S_d$ and ${\mathfrak{A}}_I$ due to [(\[commute\])]{} in . Obviously, the same calculation can be done in the symmetric case, where $P_I={\mathfrak{S}}_I$. Since $e_{d,\pi(j)}$ and $e_{d,\sigma(k)}$ are orthonormal eigenelements of $W_d={S_d}^\dagger S_d\colon H_d{\rightarrow}H_d$, see [(\[Eigenpairs\])]{}, it is ${\left\langle S_d e_{d,\pi(j)}, S_d e_{d,\sigma(k)} \right\rangle}_{G_d} = \lambda_{d,\pi(j)} {\left\langle e_{d,\pi(j)}, e_{d,\sigma(k)} \right\rangle}_{H_d} = \lambda_{d,\pi(j)} \delta_{\pi(j),\sigma(k)}$. Hence, similar to the proof of the mutual orthonormality of $\{\xi_k {\, | \,}k\in\nabla_d\}$ for we obtain $$\begin{gathered}
\label{Sdxi_orth}
{\left\langle S_d \xi_j, S_d \xi_k \right\rangle}_{G_d} = \lambda_{d,j} \delta_{j,k}
\quad \text{for all} \quad j,k \in \nabla_d.\end{gathered}$$ Therefore, we calculate for $n\in {\mathbb{N}}_0$ and $f\in P_I(H_d)$ $$\begin{gathered}
{\left\| S_d f - A_{d,n}^* f {\, | \,}G_d \right\|}^2 = {\left\| \sum_{v>n} {\left\langle f, \xi_{\psi(v)} \right\rangle}_{H_d} \cdot S_d\xi_{\psi(v)} {\, | \,}G_d \right\|}^2 = \sum_{v>n} {\left\langle f, \xi_{\psi(v)} \right\rangle}_{H_d}^2 \cdot \lambda_{d,\psi(v)}.\end{gathered}$$ On the other hand, for $f\in{\mathcal{B}}(P_I(H_d))$, we have by Parseval’s identity $$\begin{gathered}
1 \geq {\left\| f {\, | \,}P_I(H_d) \right\|}^2 = {\left\| f {\, | \,}H_d \right\|}^2 = {\left\| \sum_{v\in{\mathbb{N}}} {\left\langle f, \xi_{\psi(v)} \right\rangle}_{H_d} \cdot \xi_{\psi(v)} {\, | \,}H_d \right\|}^2 = \sum_{v\in{\mathbb{N}}} {\left\langle f, \xi_{\psi(v)} \right\rangle}_{H_d}^2.\end{gathered}$$ Thus, because of the non-increasing ordering of $( \lambda_{d,\psi(v)} )_{v\in{\mathbb{N}}}$ due to the choice of the rearrangement $\psi$, we can estimate the worst case error $$\begin{gathered}
e^{\rm wor}(A_{n,d}^*; P_I(H_d))^2 = \sup_{f\in {\mathcal{B}}(P_I(H_d))} {\left\| S_d f - A_{n,d}^* f {\, | \,}G_d \right\|}^2 \leq \lambda_{d,\psi(n+1)},\end{gathered}$$ as claimed.
Note that formula [(\[Sdxi\_orth\])]{} in the proof of together with yields that the set [(\[eigenpairs\_sym\])]{} describes the eigenpairs of the self-adjoint operator $$\begin{gathered}
W_d \big|_{P_I(H_d)} = {S_d}^\dagger S_d \colon P_I(H_d) {\rightarrow}P_I(H_d)\end{gathered}$$ as stated in . Therefore, the upper bound given in is sharp and $A_{n,d}^*$ in [(\[opt\_algo\])]{} is $n$-th optimal, due to the general theory; see, e.g., Corollary 4.12 in Novak and Wo[ź]{}niakowski [@NW08]. From the general theory it also follows that adaption does not help to improve this $n$-th minimal error, see [@NW08 Theorem 4.5], and that linear algorithms are best possible; see [@NW08 Theorem 4.8]. Hence, the proof of is complete.
Since it seems to be a little bit unsatisfying to refer to these deep results for the proof of such an easy theorem we refer the reader to the appendix where a nearly self-contained proof of the remaining facts can be found. Moreover, there we describe what we mean by adaption in this context.
Complexity {#sect_complexity}
==========
In this part of the paper we investigate tractability properties of approximating the linear tensor product operator $S_d$ on certain (anti-) symmetric subsets $P_{I}(H_d)=P_{I_d}(H_d)$, where $P \in \{{\mathfrak{S}},{\mathfrak{A}}\}$ and ${\emptyset}\neq I_d \subset\{1,\ldots,d\}$. Therefore, as usual, we express the $n$-th minimal error derived in formula [(\[nth\_error\])]{} in terms of the *information complexity*, [i.e. ]{}the minimal number of information operations needed to achieve an error smaller than a given ${\varepsilon}> 0$, $$\begin{gathered}
n({\varepsilon},d;P_I(H_d)) = { \mathop{\mathrm{min}}\left\{n\in{\mathbb{N}}_0 {\, | \,}e(n,d; P_I(H_d)) \leq {\varepsilon}\right\} }.\end{gathered}$$ To abbreviate the notation we write $n^{\rm ent}({\varepsilon},d)$ if we deal with the entire tensor product problem. Furthermore, as in the introduction, we denote the information complexity of the fully (anti-) symmetric problem by $n^{\rm asy}({\varepsilon},d)$ and $n^{\rm sym}({\varepsilon},d)$, respectively.
Preliminaries {#sect_prelim}
-------------
From we obtain for any ${\varepsilon}>0$ and every $d\in{\mathbb{N}}$ $$\begin{gathered}
n({\varepsilon},d;P_I(H_d))
= { \mathop{\mathrm{min}}\left\{n\in{\mathbb{N}}_0 {\, | \,}\lambda_{d, \psi(n+1)} \leq {\varepsilon}^2\right\} }
= \#\left\{k\in\nabla_d{\, | \,}\prod_{l=1}^d \lambda_{k_l}>{\varepsilon}^2\right\}\end{gathered}$$ by solving [(\[nth\_error\])]{} for $\psi$. Using this expression we can easily conclude the results for the first two problems in the introduction. There we dealt with the case $\lambda_1=\ldots=\lambda_m=1$ and $\lambda_j=0$ for $j>m\geq 2$.
Let us recall some common notions of tractability. If for a given problem the information complexity $n({\varepsilon},d)$ increases exponentially in the dimension $d$ we say the problem suffers from the *curse of dimensionality*. That is, there exist constants $c>0$ and $C>1$ such that for at least one ${\varepsilon}> 0$ we have $$\begin{gathered}
n({\varepsilon},d) \geq c \cdot C^d\end{gathered}$$ for infinitely many $d\in{\mathbb{N}}$. More generally, if the information complexity depends exponentially on $d$ or ${\varepsilon}^{-1}$ we call the problem *intractable*. Since there are many ways to measure the lack of exponential dependence we distinguish between different types of tractability. The most important type is *polynomial tractability*. We say that the problem is polynomially tractable if there exist constants $C,p>0$, as well as $q\geq0$, such that $$\begin{gathered}
n({\varepsilon},d) \leq C \cdot {\varepsilon}^{-p} \cdot d^{q}
\quad \text{for all} \quad d\in{\mathbb{N}}, {\varepsilon}\in(0,1].\end{gathered}$$ If this inequality holds with $q=0$, the problem is called *strongly polynomially tractable*. If polynomial tractability does not hold we say the problem is *polynomially intractable*. For more specific definitions and relations between these and other classes of tractability see, e.g., the monographs of Novak and Wo[ź]{}niakowski [@NW08; @NW10; @NW11].
In the following we distinguish two cases. First we consider the *absolute error criterion*, where we investigate the dependence of $n({\varepsilon},d; P_I(H_d))$ on $1/{\varepsilon}$ and on the dimension $d$ for every ${\varepsilon}\in (0,1]$ and $d\in{\mathbb{N}}$. Note that without loss of generality we can restrict ourselves to ${\varepsilon}\leq { \mathop{\mathrm{min}}\left\{1,{\varepsilon}_d^{\mathrm{init}}\right\} }$ since obviously $n({\varepsilon},d; P_I(H_d))=0$ for all ${\varepsilon}\geq{\varepsilon}_d^{\mathrm{init}}$. Here $$\begin{gathered}
{\varepsilon}_d^{\rm init} = e(0,d; P_I(H_d)) = \sqrt{\lambda_{d,\psi(1)}} = \begin{cases}
\sqrt{\lambda_1^d}, & \text{ if } P={\mathfrak{S}},\\
\sqrt{\lambda_1^{b_d} \cdot \lambda_1 \cdot \ldots \cdot \lambda_{a_d}}, & \text{ if } P={\mathfrak{A}}\end{cases}\end{gathered}$$ describes the *initial error* of the $d$-variate problem on the subspace $P_I(H_d)$ where $\psi\colon {\mathbb{N}}{\rightarrow}\nabla_d$ again is a non-increasing rearrangement of the set of eigenvalues $\{ \lambda_{d,k} {\, | \,}k \in \nabla_d\}$ of $W_d\big|_{P_I(H_d)}={S_d}^\dagger S_d$ and $b_d = d - a_d$ denotes the number of coordinates without (anti-) symmetry conditions in dimension $d$, [i.e. ]{}$a_d = \#I_d$ and $b_d = d - \#I_d$, respectively.
Afterwards, we deal with the *normalized error criterion*, where we especially investigate the dependence of $n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d; P_I(H_d))$ on $1/{\varepsilon}'$ for ${\varepsilon}'\in(0,1)$. That is, we search for the minimal number of information operations needed to improve the initial error by a factor ${\varepsilon}'$ less than one.
To avoid triviality we will assume ${\varepsilon}_d^{\rm init}>0$, for every $d\in{\mathbb{N}}$, in both cases, because otherwise we have strong polynomial tractability by default. From this assumption it follows that $\lambda_1>0$, which simply means that $S_d$ is not the zero operator. Moreover, note that in the case of antisymmetric problems, if the number of antisymmetric coordinates, [i.e. ]{}the set $I=I(d)$, grows with the dimension, the condition ${\varepsilon}_d^{\rm init}>0$ (for every $d\in{\mathbb{N}}$) even implies that $$\begin{gathered}
\lambda_1 \geq \lambda_2 \geq \ldots > 0.\end{gathered}$$ Finally, we always assume $\lambda_2>0$, because otherwise $S_d$ is equivalent to a continuous linear functional which can be solved exactly with one information operation; see Novak and Wo[ź]{}niakowski [@NW08 p.176].\
For the study of tractability for the absolute error criterion we use a slightly modified version of Theorem 5.1, [@NW08]. It deals with the more general situation of arbitrary compact linear operators between Hilbert spaces. In contrast to Novak and Woźniakowski we drop the (hidden) condition ${\varepsilon}_d^{\rm init}=1$ for the initial error in dimension $d$. For the sake of completeness a proof can be found in the appendix. If we denote Riemann’s zeta function by $\zeta$ the assertion reads as follows.
\[prop\_NW\] Consider a family of compact linear operators $\{T_d \colon F_d {\rightarrow}G_d {\, | \,}d\in{\mathbb{N}}\}$ between Hilbert spaces and the absolute error criterion in the worst case setting. Furthermore, for $d\in{\mathbb{N}}$ let $(\lambda_{d,i})_{i\in{\mathbb{N}}}$ denote the non-negative sequence of eigenvalues of ${T_d}^\dagger T_d$ [w.r.t. ]{}a non-increasing ordering.
- If $\{T_d\}$ is polynomially tractable with the constants $C,p>0$ and $q\geq 0$ then for all $\tau > p/2$ we have $$\begin{gathered}
\label{sup_condition}
C_\tau = \sup_{d\in{\mathbb{N}}} \frac{1}{d^{r}} \left( \sum_{i=f(d)}^\infty \lambda_{d,i}^\tau \right)^{1/\tau} < \infty,
\end{gathered}$$ where $r=2q/p$ and $f\colon {\mathbb{N}}{\rightarrow}{\mathbb{N}}$ with $f(d)={\left\lceil (1+C) \, d^q \right\rceil}$.\
In this case $C_\tau \leq C^{2/p} \, \zeta(2\tau/p)^{1/\tau}$.
- If [(\[sup\_condition\])]{} is satisfied for some parameters $r \geq 0$, $\tau >0$ and a function $f\colon{\mathbb{N}}{\rightarrow}{\mathbb{N}}$ such that $f(d)={\left\lceil C\cdot \left({ \mathop{\mathrm{min}}\left\{{\varepsilon}_d^{\rm init},1\right\} }\right)^{-p}\cdot d^{q} \right\rceil}$, where $C>0$ and $p,q \geq 0$, then the problem is polynomially tractable and $n({\varepsilon},d)\leq (C+C_\tau^{\tau}) \, {\varepsilon}^{-{ \mathop{\mathrm{max}}\left\{p,2\tau\right\} }} \, d^{{ \mathop{\mathrm{max}}\left\{q,r\tau\right\} }}$ for every $d\in{\mathbb{N}}$ and any ${\varepsilon}\in (0,1]$.
Let us add some comments on this result. Since, clearly, $$\begin{gathered}
1 \leq \left({ \mathop{\mathrm{min}}\left\{{\varepsilon}_d^{\rm init},1\right\} }\right)^{-p}
\quad \text{for all} \quad p\geq0\end{gathered}$$ provides a characterization for (strong) polynomial tractability, similar to [@NW08 Theorem 5.1]. But, compared with the assertions from the authors of [@NW08], our result yields the essential advantage that the given estimates incorporate the initial error ${\varepsilon}_d^{\rm init}$. Hence, if ${\varepsilon}_d^{\rm init}$ is sufficiently small then we can conclude polynomial tractability while ignoring a larger set of eigenvalues in the summation [(\[sup\_condition\])]{}. Observe that the first statement does not cover any assertion about the initial error, since $f(d)\geq2$. Hence, it might happen that we have (strong) polynomial tractability though the largest eigenvalue $\lambda_{d,1}=({\varepsilon}_d^{\rm init})^2$ tends faster to infinity than any polynomial. To this end, for $d\in{\mathbb{N}}$, consider the sequences $(\lambda_{d,m})_{m\in{\mathbb{N}}}$ given by $$\begin{gathered}
\lambda_{d,1}=e^{2d} \quad \text{and} \quad \lambda_{d,m}=\frac{1}{m}, \quad \text{for} \quad m\geq 2.\end{gathered}$$ Here, obviously, the initial error grows exponentially fast to infinity, but nevertheless the second point of shows that $\{S_d\}$ is strongly polynomially tractable, since [(\[sup\_condition\])]{} holds with $r=p=q=0$, and $C=\tau=2$.
Let us now return to our $I$-(anti-)symmetric tensor product problems $S_d$ as defined in . Therefore, let ${\emptyset}\neq I_d=\{i_1, \ldots, i_{\#I_d}\} \subset\{1,\ldots,d\}$ and $P_{I_d}\in\{{\mathfrak{S}}_{I_d},{\mathfrak{A}}_{I_d}\}$ for every $d>1$. We start by using to conclude a simple necessary condition for (strong) polynomial tractability of $\{S_d\}$ in the worst case setting [w.r.t. ]{}the absolute error criterion. Recall that $\psi\colon {\mathbb{N}}{\rightarrow}\nabla_d$ defines a rearrangement of the parameter set $\nabla_d$ given in [(\[def\_nabla\])]{}. That is, $$\begin{gathered}
\label{reordered_eigenvalues}
\{\lambda_{d,\psi(v)} {\, | \,}v\in{\mathbb{N}}\}=\left\{\lambda_{d,k}=\prod_{l=1}^d \lambda_{k_l} {\, | \,}k \in \nabla_d \right\}\end{gathered}$$ denotes the set of eigenvalues of ${S_d}^\dagger S_d$ with respect to a non-increasing ordering, see .
\[prop\_general\] The fact that $\{S_d\}$ is polynomially tractable with the constants $C,p>0$ and $q\geq 0$ implies that $\lambda=(\lambda_m)_{m\in{\mathbb{N}}} \in {\ell}_\tau$ for all $\tau > p/2$. Moreover, for any such $\tau$ and all $d\in{\mathbb{N}}$ the following estimate holds: $$\begin{gathered}
\frac{1}{\lambda_{d,\psi(1)}^\tau} \sum_{k\in\nabla_d} \lambda_{d,k}^\tau
\leq (1+C)\, d^q
+ C^{2\tau/p} \, \zeta\left(\frac{2\tau}{p}\right) \left( \frac{d^{2q/p}}{\lambda_{d,\psi(1)}} \right)^\tau.
\end{gathered}$$
From we know that for $\tau > p/2$ and $r=2q/p$ it is $$\begin{gathered}
\label{sup_condition3}
\sup_{d\in{\mathbb{N}}} \frac{1}{d^r} \left( \sum_{v=f(d)}^\infty \lambda_{d, \psi(v)}^\tau \right)^{1/\tau} <\infty,\end{gathered}$$ where the function $f\colon {\mathbb{N}}{\rightarrow}{\mathbb{N}}$ is given by $f(d)={\left\lceil (1+C)\,d^q \right\rceil}$.
Note that for the proof of the first assertion we only need to consider the case where all $\lambda_m$ are strictly positive. Then the condition [(\[sup\_condition3\])]{}, in particular, implies that the sum in the brackets converges for every fixed $d\in{\mathbb{N}}$. If we denote the subset of indices $j\in\nabla_d$ of the $f(d)-1$ largest eigenvalues $\lambda_{d,\psi(v)}$ by $L_d$ then there exists a natural number $s=s(d)\geq d$ such that $L_d$ is completely contained in the cube $$\begin{gathered}
\label{cube_Qds}
Q_{d,s} = \{1,\ldots,s\}^d.\end{gathered}$$ Hence, we can crudely estimate the sum from below by $$\begin{gathered}
\sum_{j\in\nabla_d \setminus Q_{d,s}} \lambda_{d,j}^\tau
\leq \sum_{j\in\nabla_d \setminus L_d} \lambda_{d,j}^\tau
= \sum_{v=f(d)}^\infty \lambda_{d, \psi(v)}^\tau < \infty.\end{gathered}$$ Since $R_{d,s}=\{j=(1,2,\ldots,d-1,m)\in{\mathbb{N}}^d {\, | \,}m > s\}$ is a subset of $\nabla_d \setminus Q_{d,s}$, independently of the concrete (anti-) symmetrizer $P_{I_d}$, where $P\in\{{\mathfrak{S}},{\mathfrak{A}}\}$, we obtain $$\begin{gathered}
(\lambda_{1}\cdot\lambda_{2}\cdot\ldots \cdot\lambda_{d-1})^\tau \sum_{m=s+1}^\infty \lambda_{m}^\tau
= \sum_{j\in R_{d,s}} \lambda_{d,j}^\tau \leq \sum_{j\in\nabla_d \setminus Q_{d,s}} \lambda_{d,j}^\tau.\end{gathered}$$ Thus, for each fixed $d\in{\mathbb{N}}$ the tail series $\sum_{m=s(d)+1}^\infty \lambda_m^\tau$ is finite, which is only possible if ${\left\| \lambda {\, | \,}{\ell}_\tau \right\|}<\infty$. Hence, $\lambda \in {\ell}_\tau$ is necessary for (strong) polynomial tractability.
Let us turn to the second assertion. Obviously, [(\[sup\_condition3\])]{} implies the existence of some constant $C_1>0$ such that $$\begin{gathered}
\sum_{v=f(d)}^\infty \lambda_{d,\psi(v)}^\tau \leq C_1 d^{r\tau} \quad \text{for all} \quad d\in{\mathbb{N}}.\end{gathered}$$ Indeed, yields that we can take $C_1 = C^{2\tau/p} \zeta(2\tau/p)$. The rest of the sum can also be bounded easily for any $d\in{\mathbb{N}}$, $$\begin{gathered}
\sum_{v=1}^{f(d)-1} \lambda_{d,\psi(v)}^\tau \leq \lambda_{d,\psi(1)}^\tau (f(d)-1),\end{gathered}$$ due to the ordering provided by $\psi$. Since $\sum_{k\in\nabla_d} \lambda_{d,k}^\tau = \sum_{v=1}^\infty \lambda_{d,\psi(v)}^\tau$, it remains to show that $f(d)-1 \leq (1 + C) d^{q}$ for every $d\in{\mathbb{N}}$ with $\lambda_{d,\psi(1)}>0$, which is also obvious due to the definition of $f$.
Since we know that antisymmetric problems are easier than symmetric problems we have to distinguish these cases in order to conclude sharp conditions for tractability.
Tractability of symmetric problems (absolute error) {#sect_symprob}
---------------------------------------------------
Beside the general assertion $\lambda \in {\ell}_\tau$, we start with necessary conditions for (strong) polynomial tractability in the symmetric setting. By $b_d$ we denote the amount of coordinates without symmetry conditions in dimension $d$.
Let $\{S_d\}$ be the problem considered in and assume $P={\mathfrak{S}}$.
- If $\{S_d\}$ is polynomially tractable and $\lambda_1 \geq 1$ then $b_d \in {\mathcal{O}}(\ln d)$.
- If $\{S_d\}$ is strongly polynomially tractable and $\lambda_1\geq1$ then $b_d \in {\mathcal{O}}(1)$ and $\lambda_2<1/\lambda_1$.
Assume $\lambda_1\geq 1$ and let $\tau$ be given by . Then, independent of the amount of symmetry conditions, we have $\lambda_{d,\psi(1)}=\lambda_1^d \geq 1$ and there exist absolute constants $r\geq 0$ and $C>1$ such that $$\begin{gathered}
\label{stp_estimate}
\frac{1}{\lambda_1^{\tau d}} \sum_{k\in\nabla_d} \lambda_{d,k}^\tau
\leq C \, d^r, \quad d\in{\mathbb{N}},\end{gathered}$$ due to . In the case of strong polynomial tractability we even have $r=0$. For $d\geq 2$ we use the product structure of $\lambda_{d,k}$, $k\in\nabla_d$, and split the sum on the left [w.r.t. ]{}the coordinates with and without symmetry conditions. Hence, we conclude $$\begin{gathered}
\label{splitting}
\sum_{k=(h,j)\in\nabla_d} \lambda_{d,k}^\tau
= \sum_{j\in {\mathbb{N}}^{b_d}} \lambda_{b_d,j}^\tau
\sum_{\substack{h\in{\mathbb{N}}^{a_d},\\h_1\leq\ldots\leq h_{a_d}}} \lambda_{a_d,h}^\tau
= \left( \sum_{m=1}^\infty \lambda_m^\tau \right)^{b_d}
\sum_{\substack{h\in{\mathbb{N}}^{a_d},\\h_1\leq\ldots\leq h_{a_d}}} \lambda_{a_d,h}^\tau,
\qquad d=a_d+b_d\geq 2,\end{gathered}$$ which leads to $$\begin{gathered}
\left( \sum_{m=1}^\infty \left(\frac{\lambda_m}{\lambda_1}\right)^\tau \right)^{b_d}
\sum_{\substack{h\in{\mathbb{N}}^{a_d},\\h_1\leq\ldots\leq h_{a_d}}} \prod_{l=1}^{a_d}
\left( \frac{\lambda_{h_l}}{\lambda_1}\right)^\tau
\leq C \, d^r.\end{gathered}$$ In any case the second sum in the above inequality is bounded from below by $1$. Thus, we conclude that $(1+\lambda_2^\tau/\lambda_1^\tau)^{b_d} \leq \left( \sum_{m=1}^\infty \lambda_m^\tau/\lambda_1^\tau \right)^{b_d}$ needs to be polynomially bounded from above. Since we always assume $\lambda_2>0$ this leads to the claimed bounds on $b_d$ in the case of (strong) polynomial tractability.
It remains to show that $\lambda_2<1/\lambda_1$ is necessary for strong polynomial tractability. To this end, assume for a moment $\lambda_2\geq 1/\lambda_1$. Then it is easy to see that (independent of the number of symmetry conditions) there are at least $1+{\left\lfloor d/2 \right\rfloor}$ different $k\in\nabla_d$ such that $\lambda_{d,k}\geq 1$. Namely, for $l=0,\ldots,{\left\lfloor d/2 \right\rfloor}$ we can take the first $d-l$ coordinates of $k\in\nabla_d$ equal to one. To the remaining coordinates we assign the value two.
In other words, we have $\lambda_{d,\psi(1+{\left\lfloor d/2 \right\rfloor})} \geq 1$. On the other hand, strong polynomial tractability implies $\sum_{v={\left\lceil 1+C \right\rceil}}^\infty \lambda_{d,\psi(v)}^\tau \leq C_1$ for some absolute constants $\tau,C,C_1>0$ and all $d\in{\mathbb{N}}$; see [(\[sup\_condition3\])]{}. Hence, for every $d\geq 2\,{\left\lceil 1+C \right\rceil}$, $$\begin{gathered}
C_1
\geq \sum_{v={\left\lceil 1+C \right\rceil}}^\infty \lambda_{d,\psi(v)}^\tau
\geq \sum_{v={\left\lceil 1+C \right\rceil}}^{1+{\left\lfloor d/2 \right\rfloor}} \lambda_{d,\psi(v)}^\tau
\geq \lambda_{d,\psi(1+{\left\lfloor d/2 \right\rfloor})}^\tau (2+{\left\lfloor d/2 \right\rfloor}-{\left\lceil 1+C \right\rceil})
\geq {\left\lfloor d/2 \right\rfloor}+1-{\left\lceil C \right\rceil},\end{gathered}$$ because of the ordering provided by $\psi$. Obviously, this is a contradiction. Thus, we have $\lambda_2<1/\lambda_1$ and the proof is complete.
Note in passing that the previous argument can also be used to show that (independent of the number of symmetry conditions) the information complexity $n({\varepsilon},d)$ needs to grow at least linearly in $d$ if we assume $\lambda_2 \geq 1/\lambda_1$. In particular, we cannot have strong polynomial tractability if $\lambda_1=\lambda_2=1$.
We continue the analysis of $I$-symmetric problems with respect to the absolute error criterion by proving that the stated necessary conditions are also sufficient for (strong) polynomial tractability. To this end, we need a rather technical preliminary lemma that can be proven by elementary induction arguments. For the convenience of the reader we included also this proof in the appendix.
\[lemma\_symNEW\] Let $(\mu_m)_{m\in{\mathbb{N}}}$ be a non-increasing sequence of non-negative real numbers with $\mu_1>0$. Then, for all $V\in{\mathbb{N}}_0$ and every $d\in{\mathbb{N}}$, it holds $$\begin{gathered}
\label{estimate_VNEW}
\sum_{\substack{k\in{\mathbb{N}}^d,\\1\leq k_1\leq\ldots\leq k_d}} \mu_{d,k}
\leq \mu_1^d \, d^V \left( 1 + V + \sum_{L=1}^d \mu_1^{-L}
\sum_{ \substack{ j^{(L)}\in{\mathbb{N}}^L,\\V+2\leq j_1^{(L)}\leq\ldots\leq j_L^{(L)} } } \mu_{L,j^{(L)}} \right).
\end{gathered}$$
Now the sufficient conditions read as follows. Once again, we denote the number of coordinates without symmetry conditions in dimension $d$ by $b_d$.
\[prop\_suf\_sym\] Let $\{S_d\}$ be the problem considered in , assume $P={\mathfrak{S}}$ and let $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}\in {\ell}_{\tau_0}$ for some $\tau_0\in(0,\infty)$.
- If $\lambda_1<1$ then $\{S_d\}$ is strongly polynomially tractable.
- If $\lambda_1=1>\lambda_2$ and $b_d \in {\mathcal{O}}(1)$ then $\{S_d\}$ is strongly polynomially tractable.
- If $\lambda_1=1$ and $b_d \in {\mathcal{O}}(\ln d)$ then $\{S_d\}$ is polynomially tractable.
*Step 1*. We start the proof by exploiting the property $\lambda\in{\ell}_{\tau_0}$. It is easy to see that the ordering of $(\lambda_m)_{m\in{\mathbb{N}}}$ implies $$\begin{gathered}
m \lambda_m^{\tau_0}
\leq \lambda_1^{\tau_0} + \ldots + \lambda_m^{\tau_0}
< \sum_{i=1}^\infty \lambda_i^{\tau_0} = {\left\| \lambda {\, | \,}{\ell}_{\tau_0} \right\|}^{\tau_0} < \infty\end{gathered}$$ for any $m\in{\mathbb{N}}$. Hence, there exists some $C_{\tau_0}>0$ such that $\lambda_m$ is bounded from above by $C_{\tau_0} \cdot m^{-r}$ for every $r\leq 1/\tau_{0}$. Therefore, there is some index such that for every larger $m\in{\mathbb{N}}$ we have $\lambda_m<1$. We denote the smallest of these indices by $m_0$. Similar to the calculation of Novak and Wo[ź]{}niakowski in [@NW08 p.180] this leads to $$\begin{gathered}
\sum_{m=m_0}^\infty \lambda_m^\tau
\leq (p+1) \lambda_{m_0}^\tau + C_{\tau_0}^\tau \int_{m_0+p}^\infty x^{-\tau r} dx
= (p+1) \lambda_{m_0}^\tau + \frac{C_{\tau_0}^\tau}{\tau r - 1} \cdot \frac{1}{(m_0+p)^{\tau r -1}}\end{gathered}$$ for every $p\in{\mathbb{N}}_0$ and all $\tau$ such that $\tau r > 1$. Thus, in particular, with $r=1/\tau_0$ we conclude $$\begin{gathered}
\sum_{m=m_0}^\infty \lambda_m^\tau
\leq (p+1) \lambda_{m_0}^\tau + \frac{1/\tau}{1/\tau_0 - 1/\tau} \left( \frac{C_{\tau_0}^{1/(1/\tau_0 - 1/\tau)}}{m_0+p} \right)^{\tau (1/\tau_0-1/\tau)} \quad \text{for all} \quad \tau>\tau_0, p\in{\mathbb{N}}_0.\end{gathered}$$ Note that for a given $\delta>0$ there exists some constant $\tau_1\geq \tau_0$ such that for all $\tau > \tau_1$ it is $1/(1/\tau_0 - 1/\tau) \in (\tau_0,\tau_0+\delta)$. Hence, if $p\in{\mathbb{N}}$ is sufficiently large then we obtain for all $\tau > \tau_1$ $$\begin{aligned}
\sum_{m=m_0}^\infty \lambda_m^\tau
&\leq (p+1) \lambda_{m_0}^\tau + \frac{\tau_0+\delta}{\tau} \left( \frac{C_1}{m_0+p} \right)^{\tau (1/\tau_0-1/\tau)} \\
&\leq (p+1) \lambda_{m_0}^\tau + \frac{\tau_0+\delta}{\tau_1} \left( \frac{C_1}{m_0+p} \right)^{\tau/(\tau_0+\delta)},\end{aligned}$$ where we set $C_1 = { \mathop{\mathrm{max}}\left\{1,C_{\tau_0}^{\tau_0+\delta}\right\} }$. Finally, since $\lambda_{m_0}<1$, both the summands tend to zero as $\tau$ approaches infinity. In particular, there need to exist some $\tau > \tau_1\geq \tau_0$ such that $$\begin{gathered}
\sum_{m=m_0}^\infty \lambda_m^\tau \leq \frac{1}{2}.\end{gathered}$$
*Step 2*. All the stated assertions can be seen using the second point of . Indeed, for polynomial tractability, it is sufficient to show that $$\begin{gathered}
\label{sum_pol}
\sum_{k\in\nabla_d} \lambda_{d,k}^\tau \leq C d^{r \tau}\quad \text{for all} \quad d\in{\mathbb{N}}\end{gathered}$$ and some $C,\tau > 0$ as well as some $r\geq 0$. If this even holds for $r=0$ we obtain strong polynomial tractability.
In the case $\lambda_1<1$ we can estimate the sum on the left of [(\[sum\_pol\])]{} from above by $( \sum_{m=1}^\infty \lambda_m^\tau )^d$. Using Step 1 with $m_0=1$ we conclude $\sum_{k\in\nabla_d} \lambda_{d,k}^\tau \leq 2^{-d}$ for some large $\tau > \tau_0$. Hence, the problem is strongly polynomially tractable in this case.
For the proof of the remaining points assume $\lambda_1=1$. In any case $\sum_{k\in\nabla_1} \lambda_{1,k}^\tau \leq \sum_{m=1}^\infty \lambda_m^{\tau_0} = {\left\| \lambda {\, | \,}{\ell}_{\tau_0} \right\|}^{\tau_0} < \infty$ for all $\tau\geq \tau_0$, because of $\lambda \in {\ell}_{\tau_0}$. Therefore, we can assume $d\geq 2$ in the following. Again we split the sum in [(\[sum\_pol\])]{} with respect to the coordinates with and without symmetry conditions, i.e., for $d=a_d+b_d\geq 2$ we have $$\begin{gathered}
\label{splitting}
\sum_{k=(h,j)\in\nabla_d} \lambda_{d,k}^\tau
= \sum_{j\in{\mathbb{N}}^{b_d}} \lambda_{b_d,j}^\tau \sum_{\substack{h\in {\mathbb{N}}^{a_d},\\h_1\leq\ldots\leq h_{a_d}}} \lambda_{a_d, h}^\tau
= \left( 1 + \sum_{m=2}^\infty \lambda_{m}^\tau \right)^{b_d} \sum_{\substack{h\in {\mathbb{N}}^{a_d},\\h_1\leq\ldots\leq h_{a_d}}} \lambda_{a_d, h}^\tau.\end{gathered}$$
If $\lambda_2<1$ and $b_d$ is universally bounded then the first factor can be bounded by a constant and the second factor can be estimated using with $V=0$, $d$ replaced by $a_d$ and $\mu$ replaced by $\lambda^\tau$. It follows that if $\tau$ is large enough we have $$\begin{gathered}
\sum_{\substack{h\in {\mathbb{N}}^{a_d},\\h_1\leq\ldots\leq h_{a_d}}} \lambda_{a_d, h}^\tau
\leq 1 + \sum_{L=1}^{a_d} \sum_{ \substack{ j^{(L)}\in{\mathbb{N}}^L,\\2\leq j_1^{(L)}\leq\ldots\leq j_L^{(L)} } } \lambda_{L,j^{(L)}}^\tau
\leq 1 + \sum_{L=1}^{a_d} \left( \sum_{m=2}^\infty \lambda_{m}^\tau \right)^L
\leq 1 + \sum_{L=1}^\infty 2^{-L} = 2,\end{gathered}$$ where we again used Step 1 and the properties of geometric series. Thus, $\sum_{k\in\nabla_d} \lambda_{d,k}^\tau$ is universally bounded in this case and therefore the problem is strongly polynomially tractable.
To prove the last point we argue in the same manner. Now $b_d \in {\mathcal{O}}(\ln d)$ yields that the first factor in the splitting [(\[splitting\])]{} is polynomially bounded in $d$. For the second factor we again apply , but in this case we set $V=m_0-2$, where $m_0$ denotes the first index $m\in{\mathbb{N}}$ such that $\lambda_{m}<1$. Keep in mind that this index is at least two because of $\lambda_1=1$. On the other hand it needs to be finite, since $\lambda \in {\ell}_{\tau_0}$. Therefore, the second factor in the splitting [(\[splitting\])]{} is also polynomially bounded in $d$ due to the same arguments as above. All in all, this proves [(\[sum\_pol\])]{} and the problem is polynomially tractable in this case.
We summarize the results obtained for $I$-symmetric tensor product problems in the following theorem.
Assume $S_1 \colon H_1 {\rightarrow}G_1$ to be a compact linear operator between two Hilbert spaces and let $\lambda=(\lambda_m)_{ m\in {\mathbb{N}}}$ denote the sequence of non-negative eigenvalues of $W_1=S_1^\dagger S_1$ [w.r.t. ]{}a non-increasing ordering. Moreover, for $d>1$ let ${\emptyset}\neq I_d \subset\{1,\ldots,d\}$. Assume $S_d$ to be the linear tensor product problem restricted to the $I_d$-symmetric subspace ${\mathfrak{S}}_{I_d}(H_d)$ of the $d$-fold tensor product space $H_d$, consider the worst case setting [w.r.t. ]{}the absolute error criterion and let $\lambda_1\leq 1$.\
Then the problem is strongly polynomially tractable if and only if $\lambda \in {\ell}_\tau$ for some $\tau>0$ and
- $\lambda_1<1$, or
- $1=\lambda_1>\lambda_2$ and $(d-\#I_d) \in {\mathcal{O}}(1)$.
Moreover, the problem is polynomially tractable if and only if $\lambda \in {\ell}_\tau$ for some $\tau>0$ and
- $\lambda_1<1$, or
- $\lambda_1=1$ and $(d-\#I_d) \in {\mathcal{O}}(\ln d)$.
Tractability of symmetric problems (normalized error)
-----------------------------------------------------
Here we briefly focus on the normalized error criterion for the $I$-symmetric setting. Since $({\varepsilon}_d^{\rm init})^2=\lambda_{d,\psi(1)}=\lambda_1^d$ for any kind of symmetric problem, this means that we have to investigate the influence of $d$ and $1/{\varepsilon}'$ on $$\begin{aligned}
n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init}, d; {\mathfrak{S}}_{I_d}(H_d))
&= { \mathop{\mathrm{min}}\left\{n \in {\mathbb{N}}{\, | \,}\lambda_{d,\psi(n+1)} \leq ({\varepsilon}')^2 \lambda_{d,\psi(1)}\right\} } \\
&= \# \left\{ k\in\nabla_d {\, | \,}\prod_{l=1}^d \left( \frac{\lambda_{k_l}}{\lambda_1} \right) > ({\varepsilon}')^2 \right\}
\quad \text{for } {\varepsilon}' \in (0,1), d\in{\mathbb{N}}.\end{aligned}$$ Hence, in fact we have to study the information complexity of a scaled tensor product problem $S_d'\colon {\mathfrak{S}}_{I_d}(H_d){\rightarrow}G_d$ with respect to the absolute error criterion. The squared singular values of $S_1'$ equal $\mu = (\mu_m)_{m\in{\mathbb{N}}}$ with $\mu_m = \lambda_m / \lambda_1$. Obviously, we always have $\mu_1=1$. Furthermore, $\mu \in {\ell}_\tau$ if and only if $\lambda \in {\ell}_\tau$. This leads to the following theorem.
Assume $S_1 \colon H_1 {\rightarrow}G_1$ to be a compact linear operator between two Hilbert spaces and let $\lambda=(\lambda_m)_{ m\in {\mathbb{N}}}$ denote the sequence of non-negative eigenvalues of $W_1=S_1^\dagger S_1$ [w.r.t. ]{}a non-increasing ordering. Moreover, for $d>1$ let ${\emptyset}\neq I_d \subset\{1,\ldots,d\}$. Assume $S_d$ to be the linear tensor product problem restricted to the $I_d$-symmetric subspace ${\mathfrak{S}}_{I_d}(H_d)$ of the $d$-fold tensor product space $H_d$ and consider the worst case setting [w.r.t. ]{}the normalized error criterion.\
Then the problem is strongly polynomially tractable if and only if $\lambda \in {\ell}_\tau$ for some $\tau>0$ and $\lambda_1>\lambda_2$ and $(d-\#I_d) \in {\mathcal{O}}(1)$.\
Moreover, $\{S_d\}$ is polynomially tractable if and only if $\lambda \in {\ell}_\tau$ for some $\tau>0$ and $(d-\#I_d) \in {\mathcal{O}}(\ln d)$.
Tractability of antisymmetric problems (absolute error) {#sect_asy_abs}
-------------------------------------------------------
We start this subsection with simple sufficient conditions for strong polynomial tractability.
\[prop\_suf\_asy\] Let $\{S_d\}$ be the problem considered in , assume $P={\mathfrak{A}}$ and let $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}\in {\ell}_{\tau_0}$ for some $\tau_0\in(0,\infty)$.
- If $\lambda_1<1$ then $\{S_d\}$ is strongly polynomially tractable, independent of the number of antisymmetry conditions.
- If $\lambda_1 \geq 1$ and if there exist constants $\tau \geq \tau_0$ and $d_0\in{\mathbb{N}}$ such that for the number of antisymmetric coordinates $a_d$ in dimension $d$ it holds that $$\begin{gathered}
\label{suf_condition}
\frac{\ln{(a_d!)}}{d} \geq \ln({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau) \quad \text{for all} \quad d\geq d_0
\end{gathered}$$ then the problem $\{S_d\}$ is also strongly polynomially tractable.
Like for the symmetric setting, the proof of these sufficient conditions is based on the second point of . We show that under the given assumptions $$\begin{gathered}
\left( \sum_{v=1}^\infty \lambda_{d,\psi(v)}^\tau \right)^{1/\tau} \leq C < \infty
\quad \text{for every} \quad d\in{\mathbb{N}}\end{gathered}$$ and some $\tau \geq \tau_0$. Once again $\psi$ and $\nabla_d$ are given as in [(\[reordered\_eigenvalues\])]{} and [(\[def\_nabla\])]{}, respectively.
Since for $d=1$ there is no antisymmetry condition we have $\psi = {\mathrm{id}}$ and $$\begin{gathered}
\left( \sum_{v=1}^\infty \lambda_{1,\psi(v)}^\tau \right)^{1/\tau} = \left( \sum_{v=1}^\infty \lambda_v^\tau \right)^{1/\tau} = {\left\| \lambda {\, | \,}{\ell}_\tau \right\|} \leq {\left\| \lambda {\, | \,}{\ell}_{\tau_0} \right\|}.\end{gathered}$$ Therefore, due to the hypothesis $\lambda \in {\ell}_{\tau_{0}}$ the term for $d=1$ is finite.
Hence, let $d\geq 2$ be arbitrarily fixed. For $s\in{\mathbb{N}}$ with $s\geq d$ we define the cubes $Q_{d,s}$ of multi-indices similar to [(\[cube\_Qds\])]{}. With this notation we obtain the representation $$\begin{gathered}
\sum_{v=1}^\infty \lambda_{d,\psi(v)}^\tau = \sum_{k \in \nabla_d} \lambda_{d,k}^\tau = \lim_{s {\rightarrow}\infty} \sum_{k \in \nabla_d \cap Q_{d,s}} \lambda_{d,k}^\tau.\end{gathered}$$ Without loss of generality we may reorder the set of coordinates such that $I_d = \{i_1,\ldots,i_{a_d}\}=\{1,\ldots,a_d\}$. That is, we assume partial antisymmetry with respect to the first $a_d$ coordinates. Furthermore, we define $U_{a_d,s}= \{ j \in Q_{a_d,s} {\, | \,}j_1<j_2<\ldots<j_{a_d} \}$ and set $b_d=d-a_d$.
If $b_d > 0$ then the set of multi-indices under consideration splits into two non-trivial parts: $$\begin{gathered}
\nabla_d \cap Q_{d,s} = U_{a_d,s} \times Q_{b_d,s} \quad \text{for all} \quad s \geq d.\end{gathered}$$ Because of the product structure of $\lambda_{d,k}$ ($k\in\nabla_d$) this implies $$\begin{gathered}
\sum_{k=(j,i) \in \nabla_d \cap Q_{d,s}} \lambda_{d,k}^\tau
= \left(\sum_{j \in U_{{a_d},s}} \prod_{l=1}^{a_d} \lambda_{j_l}^\tau \right) \left( \sum_{i \in Q_{b_d,s}} \prod_{l=1}^{b_d} \lambda_{i_l}^\tau\right).\end{gathered}$$ Since the sequence $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}$ is an element of ${\ell}_\tau$ we can easily estimate the second factor for every $s\geq d$ from above by $$\begin{gathered}
\label{est}
\sum_{i \in Q_{b_d,s}} \prod_{l=1}^{b_d} \lambda_{i_l}^\tau
= \prod_{l=1}^{b_d} \sum_{m=1}^s \lambda_{m}^\tau
= \left( \sum_{m=1}^s \lambda_{m}^\tau \right)^{b_d}
\leq \left( \sum_{m=1}^\infty \lambda_{m}^\tau \right)^{1/\tau \cdot b_d \cdot \tau }
= {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^{b_d \cdot \tau}.\end{gathered}$$ To handle the first term we need an additional argument. Note that the structure of $U_{{a_d},s}$ implies $$\begin{gathered}
\sum_{j \in Q_{{a_d},s}} \prod_{l=1}^{a_d} \lambda_{j_l}^\tau
= \sum_{\substack{j \in Q_{{a_d},s}\\\exists k,m: j_k=j_m}} \prod_{l=1}^{a_d} \lambda_{j_l}^\tau + a_d! \sum_{j \in U_{{a_d},s}} \prod_{l=1}^{a_d} \lambda_{j_l}^\tau,\end{gathered}$$ which leads to the upper bound $$\begin{gathered}
\sum_{j \in U_{{a_d},s}} \prod_{l=1}^{a_d} \lambda_{j_l}^\tau
\leq \frac{1}{a_d!} \sum_{j \in Q_{{a_d},s}} \prod_{l=1}^{a_d} \lambda_{j_l}^\tau
\leq \frac{1}{a_d!} {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^{a_d \cdot \tau}, \end{gathered}$$ where we used the same arguments as in [(\[est\])]{}. Once again this upper bound does not depend on $s\geq d$. Hence, due to $d=a_d+b_d$, we conclude $$\begin{gathered}
\sum_{v=1}^\infty \lambda_{d,\psi(v)}^\tau
= \lim_{s {\rightarrow}\infty} \sum_{k \in \nabla_d \cap Q_{d,s}} \lambda_{d,k}^\tau
\leq \frac{1}{a_d!} {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^{\tau d}\end{gathered}$$ for every choice of ${\mathfrak{A}}_{I_d}$. Of course, for every $2\leq d < d_0$ this upper bound is trivially less than an absolute constant. Thus, we can assume $d\geq d_0$. Then, due to the hypothesis of the second point we have $\ln(a_d!) \geq \ln{({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^{\tau d})}$, which implies $$\begin{gathered}
\left( \sum_{v=1}^\infty \lambda_{d,\psi(v)}^\tau \right)^{1/\tau}
\leq \left( \frac{1}{a_d!} {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^{\tau d} \right)^{1/\tau}
\leq 1 \quad \text{for} \quad d\geq d_0.\end{gathered}$$ Hence, [(\[suf\_condition\])]{} is sufficient for strong polynomial tractability, independent of $\lambda_1$. Therefore it suffices to show that $\lambda_1<1$ implies [(\[suf\_condition\])]{} in order to complete the proof. To this end, let $\lambda_1<1$. We know from Step 1 in the proof of that there exists some $\tau \geq \tau_0$ such that $$\begin{gathered}
{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau = \sum_{m=1}^\infty \lambda_m^\tau \leq \frac{1}{2} < 1.\end{gathered}$$ Thus, we see that the right hand side of [(\[suf\_condition\])]{} is negative, whereas the left hand side is non-negative for every choice of $a_d$.
We also briefly comment on this result. First, note that a sequence $\lambda = (\lambda_m)_{m\in{\mathbb{N}}}$ that is not included in any ${\ell}_\tau$-space, $0<\tau<\infty$, has to converge to zero more slowly than the inverse of any polynomial, i.e., $m^{-\alpha}$ for $\alpha > 0$ arbitrarily fixed. Thus, only sequences like $\lambda_m = 1/\ln(m)$ lead to polynomial intractability in the fully antisymmetric setting.
Secondly, observe that [(\[suf\_condition\])]{} is quite a weak assumption. For example if we have $$\begin{gathered}
a_d \geq {\left\lceil \frac{d}{\ln d^\alpha} \right\rceil} \quad \text{with} \quad 0 < \alpha < \frac{1}{\ln ({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau)}\end{gathered}$$ for all sufficiently large $d$ then $$\begin{gathered}
\frac{\ln(a_d!)}{d}
\geq \frac{a_d (\ln(a_d)-1)}{d}
\geq \frac{1}{\alpha} \cdot \frac{\ln \left( \frac{1}{e\alpha} \cdot \frac{d}{\ln d} \right)}{\ln d}
\, \longrightarrow \, \frac{1}{\alpha} > \ln({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau), \quad d{\rightarrow}\infty.\end{gathered}$$ If $\alpha$ equals its upper bound, [i.e. ]{} $\alpha = 1/\ln({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau)$, then the condition [(\[suf\_condition\])]{} does not hold. This also shows that assumptions like $a_d = {\left\lceil d^\beta \right\rceil}$ with $\beta < 1$ are not sufficient to conclude [(\[suf\_condition\])]{}.
Note that allows us to omit the largest $f(d)-1$ eigenvalues $\lambda_{d,\psi(v)}$ where $f(d)$ may grow polynomially in $({\varepsilon}_d^{\rm init})^{-1}$ with $d$. We did not use this fact in the proof of the sufficient conditions.
Let us now turn to the necessary conditions. We will see that we need a condition similar to [(\[suf\_condition\])]{} in order to conclude polynomial tractability if we deal with slowly decreasing eigenvalues $\lambda$.
\[prop\_nec\_asy\] Let $\{S_d\}$ be the problem considered in and assume $P = {\mathfrak{A}}$. Furthermore, let $\{S_d\}$ be polynomially tractable with the constants $C,p>0$ and $q\geq 0$.\
Then, for $d$ tending to infinity, the initial error ${\varepsilon}_d^{\rm init}$ tends to zero faster than the inverse of any polynomial. Furthermore, $\lambda= (\lambda_m)_{m\in{\mathbb{N}}} \in {\ell}_\tau$ for every $\tau > p/2$ and for all $\delta > 0$ there exists some $d_0 \in {\mathbb{N}}$ such that $$\begin{gathered}
\label{bound_a}
\ln\left({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau\right) - \delta
\leq \frac{1}{d} \sum_{k=1}^{a_d} \ln \left( \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_k^\tau} \right)
\quad \text{for all} \quad d\geq d_0.
\end{gathered}$$ Thus, we have $\lambda_1<1$ or $\lim_{d{\rightarrow}\infty} a_d = \infty$.
*Step 1*. For the whole proof assume $\tau > p/2$ to be fixed. Then shows that $\lambda \in {\ell}_\tau$. Moreover, we again use the notation $d=a_d + b_d$, where $a_d = \#I_d$ denotes the number of coordinates with antisymmetry conditions in dimension $d$. Similar to the symmetric case we can split the sum of the eigenvalues such that for all $d\in{\mathbb{N}}$ $$\begin{gathered}
\sum_{k\in\nabla_d} \lambda_{d,k}^\tau = \left( \sum_{m=1}^\infty \lambda_m^\tau \right)^{b_d} \sum_{\substack{j\in{\mathbb{N}}^{a_d},\\1 \leq j_1<\ldots<j_{a_d}}} \lambda_{a_d,j}^\tau \geq {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^{\tau b_d} \cdot \lambda_1^\tau \cdot \ldots \cdot \lambda_{a_d}^\tau.\end{gathered}$$ Hence, implies that $$\begin{gathered}
\label{asy_est}
\left(\frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_1^\tau} \right)^{b_d}
\leq (1 + C)\,d^q
+ C^{2\tau/p} \zeta\left(\frac{2\tau}{p}\right) \left( \frac{d^{2q/p}}{\lambda_{d,\psi(1)}} \right)^\tau.\end{gathered}$$ In what follows we will use this inequality to conclude all the stated assertions.
*Step 2*. Here we prove the limit property for the initial error ${\varepsilon}_d^{\mathrm{init}}=\sqrt{\lambda_{d,\psi(1)}}$, [i.e. ]{}we need to show that for every fixed polynomial ${\mathcal{P}}> 0$ $$\begin{gathered}
\label{zero_limit}
\lambda_{d,\psi(1)} {\mathcal{P}}(d) \longrightarrow 0 \quad \text{if} \quad d {\rightarrow}\infty.\end{gathered}$$ Since $\lambda_{d,\psi(1)} = \lambda_1^{b_d} \cdot \lambda_1 \cdot \ldots \lambda_{a_d} \leq \lambda_1^{b_d} \cdot \lambda_1^{a_d}=\lambda_1^d$ due to the ordering of $\lambda = ( \lambda_m )_{m\in{\mathbb{N}}}$ we can restrict ourselves to the non-trivial case $\lambda_1 \geq 1$ in the following. Now assume that there exists a subsequence $(d_k)_{k\in{\mathbb{N}}}$ of natural numbers as well as some constant $C_0>0$ such that $\lambda_{d_k,\psi(1)} {\mathcal{P}}(d_k)$ is bounded from below by $C_0$ for every $k\in{\mathbb{N}}$. Then for every $d=d_k$ the right hand side of [(\[asy\_est\])]{} is bounded from above by some polynomial ${\mathcal{P}}_1(d_k)>0$. On the other hand, due to the general condition $\lambda_2 > 0$, the term ${\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau / \lambda_1^\tau$ is strictly larger than one. Thus, it follows that there exists some $C_1>0$ such that $$\begin{gathered}
b_{d_k} \leq C_1 \ln(d_k) \quad \text{for every} \quad k\in{\mathbb{N}}.\end{gathered}$$ Therefore, we have in particular $a_{d_k} = d_k - b_{d_k} {\rightarrow}\infty$, for $k{\rightarrow}\infty$. Moreover, the assumed boundedness of $\lambda_{d_k,\psi(1)} {\mathcal{P}}(d_k)$ leads to $$\begin{gathered}
C_0 {\mathcal{P}}(d_k)^{-1}
\leq \lambda_{d_k,\psi(1)}
\leq \lambda_1^{C_1 \ln(d_k)} \cdot \lambda_1 \cdot \ldots \lambda_{a_{d_k}}
= d_k^{C_1 \ln(\lambda_1)} \cdot \lambda_1 \cdot \ldots \lambda_{a_{d_k}}\end{gathered}$$ since $\lambda_1 \geq 1$. As we showed in Step 1 of the proof of the fact $\lambda \in {\ell}_\tau$ yields the existence of some $C_\tau>0$ such that $\lambda_m \leq C_\tau m^{-1/\tau}$ for every $m\in{\mathbb{N}}$. Indeed, this holds for $C_\tau={\left\| \lambda {\, | \,}{\ell}_\tau \right\|}$, which needs to be larger than one because of $\lambda_1 \geq 1$. Hence, $\lambda_1^\tau \cdot \ldots \cdot \lambda_{a_{d_k}}^\tau \leq C_\tau^{\tau a_{d_k}} (a_{d_k}!)^{-1}$, what implies $$\begin{gathered}
\left( \frac{a_{d_k}}{e}\right)^{a_{d_k}} \leq a_{d_k}! \leq (C_\tau^\tau)^{a_{d_k}} {\mathcal{P}}_2(d_k), \quad k\in{\mathbb{N}}\end{gathered}$$ for some other polynomial ${\mathcal{P}}_2 > 0$. Thus, if $k$ is sufficiently large we conclude $$\begin{gathered}
a_{d_k} \leq a_{d_k} \ln \left( \frac{a_{d_k}}{e\,C_\tau^\tau} \right) \leq \ln({\mathcal{P}}_2(d_k)),\end{gathered}$$ since $a_{d_k} {\rightarrow}\infty$ implies $a_{d_k}/(e\, C_\tau^\tau) \geq e$ for $k\geq k_0$. Therefore, the number of antisymmetric coordinates $a_d$ needs to be logarithmically bounded from above for every $d$ out of the sequence $(d_k)_{k\geq k_0}$. Because also $b_{d_k}$ was found to be logarithmically bounded this is a contradiction to the fact $d_k = a_{d_k} + b_{d_k}$. Thus, the hypothesis $\lambda_{d_k,\psi(1)} {\mathcal{P}}(d_k) \geq C_0 > 0$ can not be true for any subsequence $(d_k)_k$. In other words it holds [(\[zero\_limit\])]{}.
*Step 3*. Next we show [(\[bound\_a\])]{}. From the former step we know that there needs to exist some $d^*\in{\mathbb{N}}$ such that $1/\lambda_{d,\psi(1)} \geq 1$ for all $d \geq d^*$. Hence, [(\[asy\_est\])]{} together with $\tau > p/2$ implies $$\begin{gathered}
\left(\frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_1^\tau} \right)^{b_d}
\leq C_2 \left( \frac{d^{2q/p}}{\lambda_{d,\psi(1)}} \right)^\tau
= \frac{C_2 d^{2q\tau/p}}{\lambda_1^{\tau b_d} \cdot \lambda_1^\tau \cdot \ldots \cdot \lambda_{a_d}^\tau}
\quad \text{for} \quad d\geq d^*,\end{gathered}$$ where we set $C_2 = 1 + C + C^{2\tau/p} \zeta(2\tau/p)$. Therefore, we conclude $$\begin{gathered}
\frac{1}{C_2 d^{2q\tau/p}} {\left\| \lambda{\, | \,}{\ell}_\tau \right\|}^{\tau d} \leq \prod_{k=1}^{a_d} \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_k^\tau}\end{gathered}$$ for all $d\geq d^*$, which is equivalent to $$\begin{gathered}
\label{bound_a2}
\ln \left( {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau \right) - \frac{\ln (C_2 \, d^{2q\tau/p})}{d}
\leq \frac{1}{d} \sum_{k=1}^{a_d} \ln\left( \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_k^\tau}\right).\end{gathered}$$ Obviously, for given $\delta > 0$, there is some $d^{**}$ such that $\ln(C_2 \, d^{2q\tau/p})/d<\delta$ for all $d\geq d^{**}$. Hence, we can choose $d_0={ \mathop{\mathrm{max}}\left\{d^*,d^{**}\right\} }$ in order to obtain [(\[bound\_a\])]{}.
*Step 4*. It remains to show that $\lambda_1 \geq 1$ implies that $\lim_{d{\rightarrow}\infty} a_d$ is infinite. To this end, note that every summand in [(\[bound\_a2\])]{} is strictly positive. If we assume for a moment the existence of a subsequence $(d_k)_{k\in{\mathbb{N}}}$ such that $a_{d_k}$ is bounded for every $k\in{\mathbb{N}}$ then the right hand side of [(\[bound\_a2\])]{} is less than some positive constant divided by $d_k$. Hence, it tends to zero if $k$ approaches infinity. On the other hand, for large $d$, the left hand side of [(\[bound\_a2\])]{} is strictly larger than some positive constant, because of $\lambda_1 \geq 1$ and $\lambda_2>0$. This contradiction completes the proof.
As mentioned before there are examples such that the sufficient condition [(\[suf\_condition\])]{} from is also necessary (up to some constant factor) in order to conclude polynomial tractability in the antisymmetric setting. Now we are ready to give such an example.
Consider the situation of for $P={\mathfrak{A}}$ and assume the problem $\{S_d\}$ to be polynomially tractable. In addition, for a fixed $\tau\in(0,\infty)$, let $\lambda=(\lambda_m)_{m\in{\mathbb{N}}} \in {\ell}_\tau$ be given such that $\lambda_1 \geq 1$ and, moreover, assume that there exist $m_0 \in {\mathbb{N}}$ such that $$\begin{gathered}
\label{assumption}
\lambda_m \geq \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}}{m^{\alpha / \tau}}
\quad \text{for all} \quad m>m_0 \quad \text{and some} \quad \alpha>1.\end{gathered}$$ Then we claim that for every $\delta > 0$ there exists $\bar{d}\in{\mathbb{N}}$ such that $$\begin{gathered}
\label{claimed_bound}
\left( \frac{1}{\alpha}-\delta \right) \ln \left( {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau \right) \leq \frac{\ln(a_d!)}{d} \quad \text{for all} \quad d\geq \bar{d}.\end{gathered}$$ Recall that due to , for the amount of antisymmetry $a_d$, it was sufficient to assume $$\begin{gathered}
\ln \left( {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau \right) \leq \frac{\ln(a_d)!}{d}
\quad \text{for every } d \text{ larger than some fixed } d_0\in{\mathbb{N}}\end{gathered}$$ in order to conclude strong polynomial tractability.
Before we prove the claim it might be useful to give a concrete example where [(\[assumption\])]{} holds true. To this end, set $\lambda_m=1/m^2$, $\tau=1$, $\alpha=3$ and $m_0=2$. Then it is easy to check that ${\left\| \lambda {\, | \,}{\ell}_\tau \right\|}=\zeta(2)=\pi^2/6$ and obviously we have $\lambda_1=1$.
To see that the claimed inequality [(\[claimed\_bound\])]{} holds true we can use and, in particular, inequality [(\[bound\_a2\])]{}. Since $\lambda_1 \geq 1$ we know that $\lim_{d} a_d = \infty$, [i.e. ]{}$a_d > m_0$ for every $d$ larger than some $d_1\in{\mathbb{N}}$. Moreover, note that [(\[assumption\])]{} is equivalent to $$\begin{gathered}
\ln \left( \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_m^\tau} \right)
\leq \alpha \ln(m)
\quad \text{for all} \quad m>m_0.\end{gathered}$$ Hence, if $d\geq d_1$ we can estimate the right hand side of [(\[bound\_a2\])]{} from above by $$\begin{gathered}
\frac{1}{d} \sum_{k=1}^{a_d} \ln \left( \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_k^\tau} \right)
\leq \frac{m_0}{d} \cdot \ln \left( \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_{m_0}^\tau} \right) + \frac{\alpha}{d} \sum_{k=m_0+1}^{a_d} \ln(k)
\leq \frac{C_\lambda}{d} + \alpha \frac{\ln (a_d!)}{d}.\end{gathered}$$ Consequently, this leads to $$\begin{gathered}
\frac{1}{\alpha} \ln \left( {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau \right) - \frac{C_\lambda + \ln(C_2 d^{2q\tau/p})}{\alpha \cdot d} \leq \frac{\ln (a_d!)}{d}
\quad \text{for} \quad d\geq { \mathop{\mathrm{max}}\left\{d^*,d_1\right\} }.\end{gathered}$$ Now [(\[claimed\_bound\])]{} follows easily by choosing $\bar{d}\geq { \mathop{\mathrm{max}}\left\{d^*,d_1\right\} }$ large enough such that the negative term on the left is smaller than a given $\delta > 0$ times $\ln({\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau)$.
Although there remains a small gap between the necessary and the sufficient conditions for the absolute error criterion, the most important cases of antisymmetric tensor product problems are covered by our results. We summarize the main facts in the next theorem.
\[thm\_asy\_abs\] Let $S_1 \colon H_1 {\rightarrow}G_1$ be a compact linear operator between two Hilbert spaces and let $\lambda=(\lambda_m)_{ m\in {\mathbb{N}}}$ denote the sequence of non-negative eigenvalues of $W_1=S_1^\dagger S_1$ [w.r.t. ]{}a non-increasing ordering. Moreover, for $d>1$ let ${\emptyset}\neq I_d \subset\{1,\ldots,d\}$. Assume $S_d$ to be the linear tensor product problem restricted to the $I_d$-antisymmetric subspace ${\mathfrak{A}}_{I_d}(H_d)$ of the $d$-fold tensor product space $H_d$ and consider the worst case setting with respect to the absolute error criterion.\
Then for the case $\lambda_1 < 1$ the following statements are equivalent:
- $\{S_d\}$ is strongly polynomially tractable.
- $\{S_d\}$ is polynomially tractable.
- There exists a universal constant $\tau \in (0,\infty)$ such that $\lambda \in {\ell}_\tau$.
Moreover, the same equivalences hold true if $\lambda_1\geq 1$ and $\#I_d$ grows linearly with the dimension $d$.
Finally, before we continue with the normalized error criterion, we want to deduce an exact formula for the complexity in the case of fully antisymmetric functions. Hence, we set $I=I_d=\{1,\ldots,d\}$ for every $d \in {\mathbb{N}}$ and consider $$\begin{gathered}
S_d \colon {\mathfrak{A}}_I(H_d) {\rightarrow}G_d\end{gathered}$$ in the following. In this case the set of parameters $\nabla_d$ is given by $$\begin{gathered}
\nabla_d = \{k=(k_1,\ldots,k_d)\in {\mathbb{N}}^d {\, | \,}k_1 < k_2 < \ldots < k_d\}.\end{gathered}$$ Thus, to obtain a worst case error less or equal than a given ${\varepsilon}>0$ we need at least $$\begin{gathered}
n^{\rm asy}({\varepsilon},d)=n({\varepsilon},d; {\mathfrak{A}}(H_d)) = \# \left\{k\in\nabla_d {\, | \,}\lambda_{d,k} = \prod_{l=1}^d \lambda_{k_l} >{\varepsilon}^2\right\}\end{gathered}$$ linear functionals for the $d$-variate case.
Due to the ordering of $(\lambda_m)_{m\in{\mathbb{N}}}$ the largest eigenvalue $\lambda_{d,k}$ with $k\in\nabla_d$ is given by the square of ${\varepsilon}_d^{\rm init} = \sqrt{\lambda_1 \cdot \lambda_2 \cdot \ldots \cdot \lambda_d}$. In other words, it is $$\begin{gathered}
n^{\rm asy}({\varepsilon},d) = 0 \quad \text{for all} \quad {\varepsilon}\geq {\varepsilon}_d^{\rm init}.\end{gathered}$$ To calculate the cardinality also for ${\varepsilon}< {\varepsilon}_d^{\rm init}$ let us define $$\begin{aligned}
\label{def_i}
i_d(\delta^2) = { \mathop{\mathrm{min}}\left\{i \in {\mathbb{N}}{\, | \,}\alpha_i=\lambda_i \cdot \lambda_{i+1} \cdot \ldots \cdot \lambda_{i+d-1} \leq \delta^2\right\} }, \quad \text{for} \quad \delta>0 \quad \text{and} \quad d\in{\mathbb{N}}.\end{aligned}$$ Using this notation we can formulate the following assertion. Again the proof can be found in the appendix of this paper.
\[theo\_exact\] Let $\{S_d\}$ be the problem considered in and assume $P = {\mathfrak{A}}$ as well as $I=I_d=\{1,\ldots,d\}$.\
Then for every ${\varepsilon}>0$ the information complexity is given by $n^{\rm asy}({\varepsilon},d) = n^{\rm ent}({\varepsilon},1)$, if $d=1$, and $$\begin{aligned}
&n^{\rm asy}({\varepsilon},d) \label{exact}\\
&\quad = \sum_{l_1=2}^{i_d({\varepsilon}^2)} \sum_{l_2=l_1+1}^{i_{d-1}({\varepsilon}^2 / \lambda_{l_1-1})} \ldots \sum_{l_{d-1}=l_{d-2}+1}^{i_{2}({\varepsilon}^2 / [\lambda_{l_1-1}\cdot\ldots\cdot\lambda_{l_{d-2}-1}])}
\left[ n^{\rm ent}\left( {\varepsilon}/\sqrt{\lambda_{l_1-1}\cdot\ldots\cdot\lambda_{l_{d-1}-1}}, 1\right) - l_{d-1} + 1 \right] \nonumber
\end{aligned}$$ if $d\geq 2$. Here the quantities $i_j$, for $j=2,\ldots,d$, are defined as in [(\[def\_i\])]{}.
If we define $$\begin{gathered}
\alpha^{(k)}_m=\prod_{l=0}^{k-1} \lambda_{m+l}, \quad m \in {\mathbb{N}},\end{gathered}$$ for $k\in{\mathbb{N}}$ and a non-increasing sequence $\lambda_1 \geq \lambda_2 \geq \ldots > 0$, then we can interpret the quantities $i_k(\delta^2)$ as information complexities of modified univariate problems $S^{(k)}_1$. In detail, let $S^{(k)}_1 \colon H_1 {\rightarrow}G_1$ define a compact linear operator such that $$\begin{gathered}
W^{(k)}_1 = \left( S^{(k)}_1 \right)^\dagger \left(S^{(k)}_1\right) \colon H_1 {\rightarrow}H_1\end{gathered}$$ possesses the eigenvalues $\{\alpha_m^{(k)} {\, | \,}m\in{\mathbb{N}}\}$. Then $$\begin{gathered}
n^{\rm ent}(\delta,1)
= n^{\rm ent}(\delta,1;S^{(k)}_1 \colon H_1 {\rightarrow}G_1)
= i_k(\delta^2) - 1 \quad \text{for all} \quad \delta > 0.\end{gathered}$$ Further, note that for $k\geq 2$ the quantities $i_k({\varepsilon}^2/[\lambda_{l_1-1}\cdot\ldots\cdot\lambda_{l_{d-k}-1}])$ are non-increasing functions in $l_1,\ldots,l_{d-k}$ and ${\varepsilon}$.
Out of we can conclude bounds on the information complexity. If $d\geq 2$ and ${\varepsilon}< {\varepsilon}_d^{\rm init}$ then the sum in [(\[exact\])]{} contains at least the term with the index $l_1=2, l_2=3,\ldots,l_{d-1}=d$. That is, for any choice of $\lambda$ we get the lower bound $$\begin{gathered}
n^{\rm asy}({\varepsilon},d)
\geq n^{\rm ent} \left( {\varepsilon}/\sqrt{\lambda_{1}\cdot\ldots\cdot\lambda_{d-1}}, 1 \right) - d + 1,\end{gathered}$$ which can be used to show that we cannot expect the same nice conditions for (strong) polynomial tractability as before if we switch from the absolute to the normalized error criterion. We conclude this subsection with a corresponding example.
Assume $\lambda_m = m^{-2\alpha}$ for all $m \in {\mathbb{N}}$ and some $\alpha > 0$. Then we need to estimate $$\begin{aligned}
n^{\rm ent}\left( {\varepsilon}/\sqrt{\lambda_{1}\cdot\ldots\cdot\lambda_{d-1}}, 1 \right)
&= n^{\rm ent}\left( {\varepsilon}\cdot (d-1)!^{\alpha}, 1 \right) \\
&= \# \left\{ m\in{\mathbb{N}}{\, | \,}m^{-2\alpha} > ({\varepsilon}\cdot (d-1)!^{\alpha})^2 \right\} \\
&= \# \left\{ m\in{\mathbb{N}}{\, | \,}m < \frac{1}{{\varepsilon}^{1/\alpha} \cdot (d-1)!} \right\} \geq \frac{1}{{\varepsilon}^{1/\alpha} \cdot (d-1)!} - 1.\end{aligned}$$ Therefore, $$\begin{gathered}
n^{\rm asy}({\varepsilon},d)
\geq \frac{1}{{\varepsilon}^{1/\alpha} \cdot (d-1)!} - d
= d \left( \frac{1}{{\varepsilon}^{1/\alpha} \cdot d!} - 1 \right)
\quad \text{if} \quad d \geq 2 \quad \text{and} \quad {\varepsilon}< {\varepsilon}_d^{\rm init}.\end{gathered}$$ Since in this case the initial error ${\varepsilon}_d^{\rm init}$ for the $d$-variate problem equals $1/d!^\alpha$, we need at least $$\begin{gathered}
n^{\rm asy}({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d) \geq d \left( \frac{1}{({\varepsilon}')^{1/\alpha}} - 1 \right)\end{gathered}$$ linear functionals to improve the initial error by a factor ${\varepsilon}' < 1$. Because this bound grows linearly with the dimension the problem is not strongly polynomially tractable with respect to the *normalized error criterion*. Nevertheless, the sequence $\lambda$ is an element of $l_{1/\alpha}$, say, which implies strong polynomial tractability for the *absolute error criterion* due to .
Tractability of antisymmetric problems (normalized error) {#sect_normkrit}
---------------------------------------------------------
Up to now every complexity assertion in this paper was mainly based on which dealt with the general situation of arbitrary compact linear operators between Hilbert spaces and with the absolute error criterion. While investigating tractability properties of $I$-symmetric problems with respect to the normalized error criterion, we were able to use assertions from the absolute error setting. Since for $I$-antisymmetric problems the structure of the initial error is more complicated, this approach will not work again. Therefore, we start this subsection with a modified version of another known theorem by Novak and Woźniakowski [@NW08 Theorem 5.2].
\[prop\_NW2\] Consider a family of compact linear operators $\{T_d \colon F_d {\rightarrow}G_d {\, | \,}d\in{\mathbb{N}}\}$ between Hilbert spaces and the normalized error criterion in the worst case setting. Furthermore, for $d\in{\mathbb{N}}$ let $(\lambda_{d,i})_{i\in{\mathbb{N}}}$ denote the non-negative sequence of eigenvalues of ${T_d}^\dagger T_d$ [w.r.t. ]{}a non-increasing ordering.
- If $\{T_d\}$ is polynomially tractable with the constants $C,p>0$ and $q\geq0$ then for all $\tau>p/2$ we have $$\begin{gathered}
\label{sup_condition2}
C_\tau = \sup_{d\in{\mathbb{N}}} \frac{1}{d^r} \left( \sum_{i=f(d)}^\infty \left( \frac{\lambda_{d,i}}{\lambda_{d,1}}\right)^\tau \right)^{1/\tau} < \infty,
\end{gathered}$$ where $r=2q/p$ and $f\colon {\mathbb{N}}{\rightarrow}{\mathbb{N}}$ with $f(d) \equiv 1$.\
In this case, $C_\tau^\tau \leq 1+C+C^{2\tau/p}\,\zeta(2\tau/p)$.
- If [(\[sup\_condition2\])]{} is satisfied for some parameters $r \geq 0$, $\tau >0$ and a function $f\colon{\mathbb{N}}{\rightarrow}{\mathbb{N}}$ such that $f(d) = {\left\lceil C\, d^{q} \right\rceil}$, where $C>0$ and $q \geq 0$, then the problem is polynomially tractable and $n({\varepsilon}'\cdot {\varepsilon}_d^{\mathrm{init}},d)\leq (C+C_\tau^\tau)\, ({\varepsilon}')^{-2\tau} d^{{ \mathop{\mathrm{max}}\left\{q,r\tau\right\} }}$.
Note that this shows that (strong) polynomial tractability is characterized by the boundedness of the sum over the normalized eigenvalues, were we are allowed to omit the $C d^q$ largest of them. Of course, our results are equivalent to the assertions given by Novak and Woźniakowski [@NW08], as one can see easily. But now the connection between the different error criterions is more obvious. From this point of view reads more natural than [@NW08 Theorem 5.2]. The key is to apply the same proof technique for both the assertions.
Moreover, observe that also the theorem in [@NW08] for the normalized error criterion includes further assertions concerning, e.g., the exponent of strong polynomial tractability. Again our proof implies the same results.
Similar to the former sections we continue with an application of to our antisymmetric tensor product problems. To this end, assume $S_1 \colon H_1 {\rightarrow}G_1$ to be a compact linear operator between two Hilbert spaces and let $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}$ denote the sequence of non-negative eigenvalues of $W_1=S_1^\dagger S_1$ [w.r.t. ]{}a non-increasing ordering. Moreover, for $d > 1$ let ${\emptyset}\neq I_d = \{1,\ldots,d\}$. Assume $S_d$ to be the linear tensor product problem restricted to the $I_d$-antisymmetric subspace ${\mathfrak{A}}_{I_d}(H_d)$ of the $d$-fold tensor product space $H_d$ and consider the worst case setting [w.r.t. ]{}the normalized error criterion. Finally, let $b_d$ denote the number of coordinates without antisymmetry conditions in dimension $d$, [i.e. ]{}$b_d=d-a_d$, where $a_d=\#I_{d}$ for $d\in{\mathbb{N}}$.
Under these assumptions the fact that $\{S_d\}$ is polynomially tractable with the constants $C,p>0$ and $q\geq 0$ implies that $\lambda \in {\ell}_\tau$ for all $\tau>p/2$.\
Moreover, for $d$ tending to infinity, ${\varepsilon}_d^{\rm init}$ tends to zero faster than the inverse of any polynomial and $b_d\in{\mathcal{O}}(\ln d)$. Thus, $\lim_{d{\rightarrow}\infty} a_d/d= 1$.\
In addition, if $\{S_d\}$ is strongly polynomially tractable then $b_d \in {\mathcal{O}}(1)$.
From it follows that there is some $C_1>0$ such that for every $d\in{\mathbb{N}}$ $$\begin{gathered}
\frac{1}{\lambda_{d,\psi(1)}^\tau} \sum_{k\in\nabla_d} \lambda_{d,\psi(v)}^\tau
= \sum_{v=1}^\infty \left( \frac{\lambda_{d,\psi(v)}}{\lambda_{d,\psi(1)}} \right)^\tau
\leq C_1 d^{2\tau q/p},\end{gathered}$$ if $\tau > p/2$. Once more the rearrangement function $\psi$ and the index set $\nabla_d$ are given as in [(\[reordered\_eigenvalues\])]{}. Indeed, the proof of yields that it is sufficient to take $C_1 = 1+C+C^{2\tau/p} \zeta(2\tau/p)$. In particular, for $d=1$ it is $\nabla_1={\mathbb{N}}$ and $\lambda_{1,k}=\lambda_k$, for $k\in{\mathbb{N}}$, such that we have $\psi={\mathrm{id}}$ because of the ordering of $\lambda$. Hence, we conclude $$\begin{gathered}
{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau = \sum_{k=1}^\infty \lambda_k^\tau \leq C_1 \lambda_1^\tau < \infty.\end{gathered}$$ In other words, $\lambda \in {\ell}_\tau$. Moreover, like with the arguments of Step 1 in the proof of , it follows $$\begin{gathered}
\label{est_norm}
\left( \frac{{\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau}{\lambda_1^\tau} \right)^{b_d}
\leq C_1 d^{2\tau q/p}, \quad d\in{\mathbb{N}},\end{gathered}$$ since $\lambda_{d,\psi(1)}=\lambda_1^{b_d}\cdot \lambda_1 \cdot \ldots \cdot \lambda_{a_d}$ and ${\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau > \lambda_1^\tau$ due to the general assertion $\lambda_2>0$. Thus, polynomial tractability of $\{S_d\}$ implies $b_d \leq C_2 \ln(d)$ for some $C_2>0$, [i.e. ]{}$b_d \in {\mathcal{O}}(\ln d)$. Therefore, obviously, we have $$\begin{gathered}
1\geq \frac{a_d}{d}
= 1- \frac{b_d}{\ln d} \cdot \frac{\ln d}{d}
\geq 1-C_2 \cdot \frac{\ln d}{d} \longrightarrow 1,
\quad d{\rightarrow}\infty.\end{gathered}$$ The proof that strong polynomial tractability leads to $b_d \in {\mathcal{O}}(1)$ can be obtained using [(\[est\_norm\])]{} with the same arguments as before and $q=0$. Finally, we need to show the assertion concerning ${\varepsilon}_d^{\rm init}$. To this end, we refer to Step 2 in the proof of .
Application: wave functions {#sect_wave}
===========================
During the last decades there has been considerable interest in finding approximations of *wave functions*, e.g., solutions of the electronic Schrödinger equation. Due to the so-called *Pauli principle* of quantum physics only functions with certain (anti-) symmetry properties are of physical interest.
In this last section of the present paper we briefly introduce wave functions and show how our results allow to handle the approximation problem for such classes of functions. For a more detailed view, see, e.g, Hamaekers [@H09], Yserentant [@Y10], or Zeiser [@Z10]. Furthermore, for a comprehensive introduction to the topic, as well as a historical survey, we refer the reader to Hunziker and Sigal [@HS00] and Reed and Simon [@RS78].
In particular, the notion of multiple partial antisymmetry with respect to two sets of coordinates is useful for describing wave functions $\Psi$. In computational chemistry such functions occur as models which describe quantum states of certain physical $d$-particle systems. Formally, these functions depend on $d$ blocks of variables $y_i=(x^{(i)},s^{(i)})$, for $i=1,\ldots,d$, which represent the spacial coordinates $x^{(i)}=(x_1^{(i)},x_2^{(i)},x_3^{(i)})\in{\mathbb{R}}^3$ and certain additional intrinsic parameters $s^{(i)} \in C$ of each particle $y$ within the system. Hence, rearranging the arguments such that $x=(x^{(1)},\ldots,x^{(d)})$ and $s=(s^{(1)},\ldots,s^{(d)})$ yields that $$\begin{gathered}
\Psi \colon ({\mathbb{R}}^{3})^{d} \times C^d {\rightarrow}{\mathbb{R}}, \quad (x,s) \mapsto \Psi(x,s).\end{gathered}$$ In the case of systems of electrons one of the most important parameters is called *spin* and it can take only two values, i.e., $s^{(i)}\in C=\{-\frac{1}{2}, + \frac{1}{2}\}$. Due to the Pauli principle the only wavefunctions $\Psi$ that are physically admissible are those which are antisymmetric in the sense that for $I\subset\{1,\ldots,d\}$ and $I^C=\{1,\ldots,d\}\setminus I$ $$\begin{gathered}
\Psi(\pi(x),\pi(s))) = (-1)^{{\left| \pi \right|}} \Psi(x,s) \quad \text{for all} \quad \pi \in {\mathcal{S}}_I \cup {\mathcal{S}}_{I^C}.\end{gathered}$$ Thus, $\Psi$ changes its sign if we replace any particles $y_i$ and $y_j$ by each other which posses the same spin, [i.e. ]{}$s^{(i)}=s^{(j)}$. So, the set of particles, and therefore also the set of spacial coordinates, naturally split into two groups $I_+$ and $I_-$. In detail, for wave functions of $d$ particles $y_i$ we can (without loss of generality) assume that the first $\#I_+$ indices $i$ belong to the group of positive spin, whereas the rest of them possess negative spin, [i.e. ]{}$I_+=\{1,\ldots,\#I_+\}$ and $I_-=\{\#I_+ + 1,\ldots, d\}$.
In physics it is well-known that some problems, e.g., the electronic Schrödinger equation, which involve (general) wave functions can be reduced to a bunch of similar problems, where each of them only acts on functions $\Psi_s$ out of a certain Hilbert space $F_d=F_d(s)$. That is, $$\begin{gathered}
\Psi_s =\Psi(\cdot,s) \in F_d=\{f \colon ({\mathbb{R}}^{3})^{d} {\rightarrow}{\mathbb{R}}\}\end{gathered}$$ with a given fixed spin configuration $s\in C^d$. Of course, every possible spin configuration $s$ corresponds to exactly one choice $I_+\subset\{1,\ldots,d\}$ of indices. Moreover, it is known that $F_d$ is a Hilbert space which possesses a tensor product structure. Therefore, we can model wave functions as elements of certain classes of smoothness, e.g., $F_d \subset H_d=W_2^{(1,\ldots,1)}({\mathbb{R}}^{3d})$, as Yserentant [@Y10] recently did, and incorporate spin properties by using the projections of the type ${\mathfrak{A}}= {\mathfrak{A}}_{I_+} \circ {\mathfrak{A}}_{I_-}$, as defined in .
In particular, yields $$\begin{gathered}
F_d = {\mathfrak{A}}(H_d) = {\mathfrak{A}}_{I_+}(H_{\#I_+}) \otimes {\mathfrak{A}}_{I_-}(H_{\#I_-})\end{gathered}$$ and the system of all $$\begin{gathered}
{ \widetilde{\xi_k} } = \sqrt{\#S_{I_+} \cdot \#S_{I_-}} \cdot {\mathfrak{A}}(\eta_k), \quad k \in { \widetilde{\nabla} }_d,\end{gathered}$$ with $$\begin{gathered}
{ \widetilde{\nabla} }_d = \{ k=(i,j) \in {\mathbb{N}}^{\#I_++\#I_-} {\, | \,}i_1 < i_2 < \ldots < i_{\#I_+} \text{ and } j_{1} < \ldots < j_{\#I_-} \}\end{gathered}$$ builds an orthonormal basis of $F_d={\mathfrak{A}}(H_d)$, where the set $\{\eta_k {\, | \,}k=(k_1,\ldots,k_d)\in {\mathbb{N}}^d \}$ is once again assumed to be an orthonormal tensor product basis of $H_d=H_1\otimes \ldots \otimes H_1$ constructed with the help of $\{\eta_i {\, | \,}i\in{\mathbb{N}}\}$, an arbitrary orthonormal basis of $H_1$.
Note that in the former sections the underlying Hilbert space $H_1$ always consists of univariate functions. In contrast wave functions of one particle depend on three variables, but we want to stress the point that this is just a formal issue. However, this approach radically decreases the degrees of freedom and improves the solvability of certain problems $S_d$ like the approximation problem, [i.e. ]{}$S_1={\mathrm{id}}\colon H_1 {\rightarrow}G_1$, considered in connection with the electronic Schrödinger equation.
then provides an algorithm which is optimal for the $G_d$-approximation of $d$-particle wave functions in $F_d$ with respect to all linear algorithms that use at most $n$ continuous linear functionals. Moreover, the error can be calculated exactly in terms of the squared singular values $\lambda = (\lambda_m)_{m\in{\mathbb{N}}}$ of $S_1$.
Furthermore, it is possible to prove a modification of for problems dealing with wave functions. In fact, for the mentioned approximation problem polynomial tractability as well as strong polynomial tractability are equivalent to the fact that the sequence $\lambda$ of the squared singular values of the univariate problem belong to some ${\ell}_\tau$-space if we consider the absolute error criterion. The reason is that all the assertions in can be easily extended to the multiple partially antisymmetric case. In detail, if we denote the number of antisymmetric coordinates $x^{(i)}$ within each antisymmetry group $I_d^{m}\subset\{1,\ldots,d\}$ by $a_{d,m}$, $m=1,\ldots,M$, then the constraint $a_d + b_d = d$ extends to $a_{d,1}+\ldots+a_{d,M}+b_d=d$. Here $b_d$ again denotes the number of coordinates without any antisymmetry condition. In conclusion, the sufficient condition [(\[suf\_condition\])]{} in transfers to $$\begin{gathered}
\frac{1}{d} \sum_{m=1}^M \ln(a_{d,m}!) \geq {\left\| \lambda {\, | \,}{\ell}_\tau \right\|}^\tau,
\quad \text{for all} \quad d\geq d_0,\end{gathered}$$ which is always satisfied in the case of wave functions, since then $M=2$ and at least the cardinality $a_{d,m}$ of one of the groups of the same spin needs to grow linearly with the dimension $d$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks E. Novak and H. Wo[ź]{}niakowski for their valuable comments on this paper.
Appendix {#appendix .unnumbered}
========
Proof of in {#proof-of-in .unnumbered}
------------
We show in the case of function spaces.
Obviously, $P \in \{{\mathfrak{S}}_I, {\mathfrak{A}}_I\}$ is well-defined due to the assumptions \[A1\] and \[A2\]. The linearity directly follows from the definition and, using \[A3\], the operator norm is bounded by ${ \mathop{\mathrm{max}}\left\{c_\pi {\, | \,}\pi \in {\mathcal{S}}_I\right\} }$.
To show that the operators are idempotent, [i.e. ]{}$P^2=P$, we first prove that ${\mathfrak{A}}_I (f)$ satisfies [(\[antisym\])]{} for every $f \in F$. Therefore, we use the representation $$\begin{gathered}
({\mathfrak{A}}_I(f))(\pi(x))
= \sum_{\sigma \in {\mathcal{S}}_I} (-1)^{{\left| \sigma \right|}} f(\sigma(\pi (x)))
= \sum_{\lambda \in {\mathcal{S}}_I} (-1)^{{\left| \lambda \right|} + {\left| \pi \right|}} f(\lambda(x))
= (-1)^{{\left| \pi \right|}} ({\mathfrak{A}}_I(f))(x)\end{gathered}$$ for every fixed $\pi\in {\mathcal{S}}_I$. Here we imposed $\lambda = \sigma \circ \pi \in {\mathcal{S}}_I$ and used $$\begin{gathered}
{\left| \lambda \circ \pi^{-1} \right|} = {\left| \lambda \right|} + {\left| \pi^{-1} \right|} = {\left| \lambda \right|} + {\left| \pi \right|}.\end{gathered}$$ Hence, we have ${\mathfrak{A}}_I(F) \subset \{ f \in F {\, | \,}f \text{ satisfies } {(\ref{antisym})} \}$. In a second step, it is easy to check that for every function $g\in F$ which satisfies [(\[antisym\])]{} it is ${\mathfrak{A}}_I(g)=g$. Thus, $\{ f \in F {\, | \,}f \text{ satisfies } {(\ref{antisym})} \} \subset {\mathfrak{A}}_I(F)$ and ${\mathfrak{A}}_I$ is a projector onto ${\mathfrak{A}}_I(F)$.
Since the same arguments also apply for the symmetrizer ${\mathfrak{S}}_I$ this shows [(\[antisymsubspace\])]{}, as well as $P^2=P$ for $P\in\{{\mathfrak{S}}_I, {\mathfrak{A}}_I\}$. Because of the boundedness of the operators the subsets ${\mathfrak{S}}_I(F)$ and ${\mathfrak{A}}_I(F)$ are closed linear subspaces of $F$ and we obtain the orthogonal decompositions $$\begin{gathered}
F = {\mathfrak{S}}_I(F) \oplus ({\mathfrak{S}}_I(F))^\bot = {\mathfrak{A}}_I(F) \oplus ({\mathfrak{A}}_I(F))^\bot,\end{gathered}$$ where the $\bot$ denotes the orthogonal complement with respect to ${\left\langle \cdot, \cdot \right\rangle}_F$, [i.e. ]{}the image of the projectors $({\mathrm{id}}- {\mathfrak{S}}_I)$ and $({\mathrm{id}}-{\mathfrak{A}}_I)$, respectively.
The proof of in the case of arbitrary tensor product Hilbert spaces works exactly in the same way.
Proof of in {#proof-of-in-1 .unnumbered}
------------
We prove in the case of function spaces. For the case of arbitrary tensor product Hilbert spaces only slight modifications are needed. Indeed, the only difference is the conclusion of formula [(\[formula\_coeff\])]{} in Step 2. In the general setting this simply follows from our definitions.
*Step 1*. We start by proving orthonormality. Therefore, let us recall [(\[antisym\_basis\])]{}. To abbreviate the notation further, we suppress the index $H_d$ at the inner products ${\left\langle \cdot, \cdot \right\rangle}_{H_d}$ in this proof. For $P_I={\mathfrak{A}}_I$ and $j,k \in \nabla_d$ easy calculus yields $$\begin{aligned}
{\left\langle \xi_j, \xi_k \right\rangle}
&= \frac{\# {\mathcal{S}}_I}{\sqrt{M_{I}(j)! \cdot M_{I}(k)!}} {\left\langle {\mathfrak{A}}_I (\eta_{d,j}), {\mathfrak{A}}_I (\eta_{d,k}) \right\rangle} \\
&= \frac{1}{\# {\mathcal{S}}_I \sqrt{M_{I}(j)! \cdot M_{I}(k)!}} \sum_{\pi,\sigma \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}+{\left| \sigma \right|}} {\left\langle \eta_{d,\pi(j)}, \eta_{d,\sigma(k)} \right\rangle}.\end{aligned}$$ Of course, up to the factor controlling the sign, the same is true for the case $P_I={\mathfrak{S}}_I$. Assume now there exists $m\in \{1,\ldots,d\}$ such that $j_m \neq k_m$. Then the ordering of $j,k\in\nabla_d$ implies that $\pi(j)\neq\sigma(k)$ for all $\sigma,\pi \in {\mathcal{S}}_I$, since $\pi$ and $\sigma$ leave the coordinates $m\in I^C$ fixed. Hence, we conclude that we have $\pi(j) = \sigma(k)$ only if $j=k$.
At this point we have to distinguish the antisymmetric and the symmetric case. For $P={\mathfrak{A}}_I$ the only way to conclude $\pi(j) = \sigma(k)$ is to claim $j=k$ and $\pi=\sigma$. Furthermore, we see that in the antisymmetric case we have $M_{I}(j)!=1$ for all $j\in\nabla_d$, since all coordinates $j_l$, where $l\in I$, differ. Therefore, in this case the last inner product coincides with $\delta_{j,k} \cdot \delta_{\pi,\sigma}$ because of the mutual orthonormality of $\{\eta_{d,j} {\, | \,}j\in {\mathbb{N}}^d \}$. Hence, $$\begin{gathered}
{\left\langle \xi_j, \xi_k \right\rangle} = \frac{1}{\# {\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{2{\left| \pi \right|}} \delta_{j,k} = \delta_{j,k} \quad \text{for all} \quad j,k\in\nabla_d\end{gathered}$$ as claimed.
So, let us consider the case $P_I={\mathfrak{S}}_I$ and $j=k\in \nabla_d$, because we already saw that otherwise ${\left\langle \xi_j, \xi_k \right\rangle}$ equals zero. Then for fixed $\sigma \in {\mathcal{S}}_I$ there are $M_{I}(j)!$ different permutations $\pi \in {\mathcal{S}}_I$ such that $\pi(j) = \sigma(j)$. This leads to $$\begin{gathered}
{\left\langle \xi_j, \xi_j \right\rangle} = \frac{1}{\# {\mathcal{S}}_I \cdot M_{I}(j)!} \sum_{\sigma \in {\mathcal{S}}_I} M_{I}(j)! = 1\end{gathered}$$ and completes the proof of orthonormality.
*Step 2*. It remains to show that the span of $\{\xi_k {\, | \,}k\in \nabla_d\}$ is dense in $P_I(H_d)$ for $P\in \{{\mathfrak{S}}_I,{\mathfrak{A}}_I\}$. To this end, note that every multi-index $j\in{\mathbb{N}}^d$ can be represented by a uniquely defined multi-index $k\in \nabla_d$ and exactly $M_I(k)!$ different permutations $\pi\in{\mathcal{S}}_I$ such that $j=\pi(k)$.
Now assume $f \in {\mathfrak{A}}_I(H_d)$, [i.e. ]{}$f\in H_d$ satisfies [(\[antisym\])]{}. Then the expansion of $f(\pi(x))$ with respect to the basis functions $\{\eta_{d,j} {\, | \,}j\in{\mathbb{N}}^d\}$ in $H_d$ yields for $\pi \in {\mathcal{S}}_I$ $$\begin{gathered}
(-1)^{{\left| \pi \right|}} \sum_{j\in {\mathbb{N}}^d} {\left\langle f, \eta_{d,j} \right\rangle} \cdot \eta_{d,j}(x)
= \sum_{k\in {\mathbb{N}}^d} {\left\langle f, \eta_{d,k} \right\rangle} \cdot \eta_{d,k}(\pi(x)) \quad \text{for every}
\quad x\in D^d.\end{gathered}$$ Similar to the arguments used in [(\[pi\_inside\])]{} we have $\eta_{d,k}(\pi(x)) = \eta_{d,\sigma(k)}(x)$ with $\sigma = \pi^{-1}$. Therefore, we conclude for $x\in D^d$ $$\begin{gathered}
\sum_{j\in {\mathbb{N}}^d} \left( (-1)^{{\left| \pi \right|}} \cdot {\left\langle f, \eta_{d,j} \right\rangle} \right) \cdot \eta_{d,j}(x)
= \sum_{k\in {\mathbb{N}}^d} {\left\langle f, \eta_{d,\pi(\sigma(k))} \right\rangle} \cdot \eta_{d,\sigma(k)}(x)
= \sum_{j\in {\mathbb{N}}^d} {\left\langle f, \eta_{d,\pi(j)} \right\rangle} \cdot \eta_{d,j}(x).\end{gathered}$$ Because the expansion is uniquely defined we get $$\begin{gathered}
\label{formula_coeff}
(-1)^{{\left| \pi \right|}} \cdot {\left\langle f, \eta_{d,j} \right\rangle}
= {\left\langle f, \eta_{d,\pi(j)} \right\rangle} \quad \text{for all} \quad j\in{\mathbb{N}}^d \quad \text{and} \quad \pi \in {\mathcal{S}}_I.\end{gathered}$$ Using the observations from the beginning of this step we can decompose the basis expansion of $f\in {\mathfrak{A}}_I(H_d)\subset H_d$ and use the derived formula [(\[formula\_coeff\])]{} to get $$\begin{gathered}
f = \sum_{j\in{\mathbb{N}}^d} {\left\langle f, \eta_{d,j} \right\rangle} \eta_{d,j}
= \sum_{k\in \nabla_d} \sum_{\sigma \in {\mathcal{S}}_I}\frac{ {\left\langle f, \eta_{d,\sigma(k)} \right\rangle} \eta_{d,\sigma(k)} }{M_I(k)!}
= \sum_{k\in \nabla_d} \frac{1}{M_I(k)!} \sum_{\sigma \in {\mathcal{S}}_I} (-1)^{{\left| \sigma \right|}} {\left\langle f, \eta_{d,k} \right\rangle} \eta_{d,\sigma(k)}.\end{gathered}$$ Now [(\[antisym\_basis\])]{} yields that $$\begin{gathered}
f = \sum_{k\in \nabla_d} \sqrt{\frac{\# {\mathcal{S}}_I}{M_I(k)!}} \cdot {\left\langle f, \eta_{d,k} \right\rangle} \cdot \sqrt{\frac{\# {\mathcal{S}}_I}{M_I(k)!}} \cdot {\mathfrak{A}}_I(\eta_{d,k}).\end{gathered}$$ Furthermore, summing up [(\[formula\_coeff\])]{} with respect to $\pi$ leads to $$\begin{gathered}
{\left\langle f, \eta_{d,k} \right\rangle}
= \frac{1}{\# {\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} {\left\langle f, \eta_{d,\pi(k)} \right\rangle}
= {\left\langle f, \frac{1}{\# {\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \eta_{d,\pi(k)} \right\rangle}
= {\left\langle f, {\mathfrak{A}}_I (\eta_{d,k}) \right\rangle}, \end{gathered}$$ for $k\in \nabla_d$, such that finally $f\in {\mathfrak{A}}_I(H_d)$ possesses the following representation $$\begin{gathered}
f = \sum_{k \in \nabla_d} {\left\langle f, \xi_k \right\rangle} \cdot \xi_k,\end{gathered}$$ since $\xi_k=\sqrt{\# {\mathcal{S}}_I / M_I(k)!} \cdot {\mathfrak{A}}_I(\eta_{d,k})$ per definition. This proves the assertion for the case $P_I={\mathfrak{A}}_I$. The remaining case $P_I={\mathfrak{S}}_I$ can be treated in the same way.
Proof of in {#proof-of-in-2 .unnumbered}
------------
The proof is organized as follows. First we show that the problem operator $S_d$ and the (anti-) symmetrizer $P_I$ commute on $H_d$, [i.e. ]{}it holds [(\[commute\])]{}. In a second step we conclude [(\[bestapprox\])]{} out of this. The (anti-) symmetry of $A^*f$ for an optimal algorithm $A^*$ then follows immediately.
*Step 1*. Assume $E_d=\{\eta_{d,j} {\, | \,}j\in{\mathbb{N}}^d\}$ to be an arbitrary tensor product ONB of $H_d$, as defined in [(\[basis\_eta\])]{}. Then, for fixed $j\in{\mathbb{N}}^d$, formula [(\[antisym\_basis\])]{} and the structure of the linear tensor product operator $S_d=S_1 \otimes \ldots \otimes S_1$ yields in the case $P_I={\mathfrak{A}}_I$ $$\begin{aligned}
S_d({\mathfrak{A}}_I^H (\eta_{d,j}))
&=& S_d\left(\frac{1}{\# {\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \bigotimes_{l=1}^d \eta_{j_{\pi(l)}}\right)\\
&=& \frac{1}{\# {\mathcal{S}}_I} \sum_{\pi \in {\mathcal{S}}_I} (-1)^{{\left| \pi \right|}} \bigotimes_{l=1}^d S_1(\eta_{j_{\pi(l)}})
= {\mathfrak{A}}_I^G (S_d (\eta_{d,j})).\end{aligned}$$ Obviously, the same is true for $P_I={\mathfrak{S}}_I$. Hence, it holds [(\[commute\])]{} at least on the set of basis elements $E_d$ of $H_d$. Because of the representation $$\begin{gathered}
g = \sum_{j\in{\mathbb{N}}^d} {\left\langle g, \eta_{d,j} \right\rangle}_{H_d} \cdot \eta_{d,j}, \quad \quad g\in H_d,\end{gathered}$$ as well as the linearity and boundedness of the operators $P_I^H, P_I^G$ and $S_d$ we can extend the relation [(\[commute\])]{} from $E_d$ to the whole space $H_d$.
*Step 2*. Now let $f\in P_I^H(H_d)$ and let $Af$ denote an arbitrary approximation of $S_d f$. Then $S_d f = S_d(P_I^H f) = P_I^G(S_d f)$, due to Step 1. Using the fact that $P_I^G$ provides an orthogonal projection onto $P_I^G(G_d)$, see [(\[orth\_decomp\])]{}, we obtain [(\[bestapprox\])]{}, $$\begin{aligned}
{\left\| S_d f - Af {\, | \,}G_d \right\|}^2
&=& {\left\| P_I^G (S_d f) - [P_I^G (Af) + ({\mathrm{id}}^G - P_I^G)(Af)] {\, | \,}G_d \right\|}^2 \\
&=& {\left\| P_I^G (S_d f - Af) {\, | \,}G_d \right\|}^2 + {\left\| ({\mathrm{id}}^G - P_I^G)(Af) {\, | \,}G_d \right\|}^2 \\
&=& {\left\| S_d f - P_I^G(Af) {\, | \,}G_d \right\|}^2 + {\left\| Af - P_I^G(Af) {\, | \,}G_d \right\|}^2,\end{aligned}$$ as claimed.
A self-contained proof of in {#a-self-contained-proof-of-in .unnumbered}
-----------------------------
In order to deduce a lower bound on the $n$-th minimal error of approximating $S_d$ on (anti-) symmetric subspaces $P_I(H_d)$, where $P\in\{{\mathfrak{A}},{\mathfrak{S}}\}$, let us define the classes of algorithms under consideration.
An algorithm $A_{n,d}$ for $S_d\colon F_d=P_I(H_d){\rightarrow}G_d$ which uses $n$ pieces of information is modeled as a mapping $\phi \colon {\mathbb{R}}^n {\rightarrow}G_d$ and a function $N \colon F_d {\rightarrow}{\mathbb{R}}^n$ such that $A_{n,d} = \phi \circ N$. In detail, the information map $N$ is given by $$\begin{gathered}
\label{non-adapt}
N(f)=\left( L_1(f), L_2(f), \ldots, L_n(f) \right), \qquad f\in F_d,\end{gathered}$$ where $L_j \in \Lambda$. Here we distinguish certain classes of information operations $\Lambda$. In one case we assume that we can compute continuous linear functionals. Then $\Lambda=\Lambda^{\rm all}$ coincides with $F_d^*$, the dual space of $F_d$. If $L_v$ depends continuously on $f$ but is not necessarily linear the class is denoted by $\Lambda^{\rm cont}$. Note that in both the cases also $N$ is continuous and we obviously have $\Lambda^{\rm all} \subset \Lambda^{\rm cont}$.
Furthermore, we distinguish between *adaptive* and *non-adaptive* algorithms. The latter case is described above in formula [(\[non-adapt\])]{}, where $L_v$ does not depend on the previously computed values $L_1(f),\ldots,L_{v-1}(f)$. In contrast, we also discuss algorithms of the form $A_{n,d}=\phi \circ N$ with $$\begin{gathered}
N(f)= \left( L_1(f), L_2(f;y_1), \ldots, L_n(f;y_1,\ldots,y_{n-1})\right), \qquad f\in F_d,\end{gathered}$$ where $y_1=L_1(f)$ and $y_v=L_v(f;y_1,\ldots,y_{v-1})$ for $v=2,3,\ldots,n$. If $N$ is adaptive we restrict ourselves to the case where $L_v$ depends linearly on $f$, [i.e. ]{}$L_v(\,\cdot\,; y_1,\ldots,y_{v-1}) \in \Lambda^{\rm all}$.
In all cases of information maps, the mapping $\phi$ can be chosen arbitrarily and is not necessarily linear or continuous. The smallest class of algorithms under consideration is the class of linear, non-adaptive algorithms of the form $$\begin{gathered}
A_{n,d}f=\sum_{v=1}^n L_v(f) \cdot g_v,\end{gathered}$$ with some $g_v \in G_d$ and $L_v \in \Lambda^{\rm all}$. We denote the class of all such algorithms by ${\mathcal{A}}_{n}^{\rm lin}$. On the other hand, the most general classes consist of algorithms $A_{n,d}=\phi \circ N$, where $\phi$ is arbitrary and $N$ either uses non-adaptive continuous or adaptive linear information. We denote the respective classes by ${\mathcal{A}}_n^{\rm cont}$ and ${\mathcal{A}}_n^{\rm adapt}$.
For the proof that the upper bound given in is sharp we use a generalization of Lemma 1 in W. [@W11].
\[needed\_lemma\] Suppose $S$ to be a homogeneous operator between linear normed spaces $X$ and $Y$, [i.e. ]{}$S(\alpha x)=\alpha S(x)$ for all $x\in X$ and $\alpha\in{\mathbb{R}}$. Furthermore, assume that $V \subset X$ is a linear subspace with dimension $m$ and there exists a constant $a \geq 0$ such that $$\begin{gathered}
a \cdot {\left\| f {\, | \,}X \right\|} \leq {\left\| S(f) {\, | \,}Y \right\|} \quad \text{for all} \quad f\in V.
\end{gathered}$$ Then for every $n<m$ and every algorithm $A_n \in {\mathcal{A}}_n^{\rm cont} \cup {\mathcal{A}}_n^{\rm adapt}$ $$\begin{gathered}
e^{\rm wor}(A_n; S, X) = \sup_{f\in {\mathcal{B}}(X)} {\left\| S(f) - A_n(f) {\, | \,}Y \right\|} \geq a.
\end{gathered}$$
It is well-known that for $A_n=\phi \circ N$ with $n<m$ there exists $f^* \in V$ such that $N(f^*)=N(-f^*)$ and ${\left\| f{\, | \,}X \right\|}=1$. Thus, $A_n(f^*)=A_n(-f^*)$. For a more detailed view, see, W. [@W11 Lemma 1] and the references in there. Using the triangle inequality for $Y$ we obtain $$\begin{aligned}
e^{\rm wor}(A_n; X)
&\geq& { \mathop{\mathrm{max}}\left\{{\left\| S(\pm f^*)-A_n(\pm f^*) {\, | \,}Y \right\|}\right\} } = { \mathop{\mathrm{max}}\left\{{\left\| S(f^*) \pm A_n(f^*) {\, | \,}Y \right\|}\right\} } \\
&\geq& \frac{1}{2} ({\left\| S(f^*) + A_n(f^*) {\, | \,}Y \right\|} + {\left\| S(f^*) - A_n(f^*) {\, | \,}Y \right\|}) \\
&\geq& \frac{1}{2} {\left\| 2 \cdot S(f^*) {\, | \,}Y \right\|} \geq a {\left\| f^* {\, | \,}X \right\|} = a\end{aligned}$$ and the proof is complete.
Now let $X=P_I(H_d)$ and $Y=G_d$. Furthermore, for a given $n\in{\mathbb{N}}_0$, define $a=\sqrt{\lambda_{d,\psi(n+1)}}$ and consider $V = {\mathop{\mathrm{span}}\left\{\xi_{\psi(1)},\ldots, \xi_{\psi(n+1)}\right\}} \subset P_I(H_d)$. Then, obviously, $\dim V = n+1=m>n$ . With the representation [(\[Sdf\])]{} and formula [(\[Sdxi\_orth\])]{} from the proof of we conclude $$\begin{gathered}
{\left\| S_d f {\, | \,}G_d \right\|}^2
= \sum_{v=1}^{n+1} {\left\langle f, \xi_{\psi(v)} \right\rangle}^2 \cdot \lambda_{d,\psi(v)}
\geq \lambda_{d,\psi(n+1)} \sum_{v=1}^{n+1} {\left\langle f, \xi_{\psi(v)} \right\rangle}^2
= a^2 {\left\| f {\, | \,}H_d \right\|}^2, \quad f\in V,\end{gathered}$$ where we used the monotonicity of $\{ \lambda_{d,\psi(v)} \}_{v\in{\mathbb{N}}}$ and Parseval’s identity. This leads to the desired lower bound result:
Under the assumptions of the $n$-th minimal error with respect to the class ${\mathcal{A}}_n^{\rm cont} \cup {\mathcal{A}}_n^{\rm adapt}$ is bounded from below by $$\begin{gathered}
e(n,d; P_I(H_d))
= \inf_{A_{n,d}} e^{\rm wor}(A_{n,d}; P_I(H_d))
\geq \sqrt{\lambda_{d,\psi(n+1)}}
\quad \text{for all} \quad d\in{\mathbb{N}}, n\in{\mathbb{N}}_0.
\end{gathered}$$
Hence, this together with shows that $A^*_{n,d}$ given in [(\[opt\_algo\])]{} is $n$-th optimal with respect to the class ${\mathcal{A}}_n^{\rm cont} \cup {\mathcal{A}}_n^{\rm adapt}$ as claimed in .
Proof of in {#proof-of-in-3 .unnumbered}
------------
If the problem is polynomially tractable then there exist constants $C,p>0$ and $q\geq0$ such that for all $d\in {\mathbb{N}}$ and ${\varepsilon}\in ( 0, 1]$ $$\begin{gathered}
n({\varepsilon},d) = n({\varepsilon},d; F_d) \leq C \cdot {\varepsilon}^{-p} \cdot d^q.\end{gathered}$$ Here, ${\varepsilon}_d^{\rm init} = e(0,d)=\sqrt{\lambda_{d,1}}>0$ denotes the initial error of $T_d$. Since $e(n,d)=\sqrt{\lambda_{d,n+1}}$ it is $n({\varepsilon},d)= \# \{i\in{\mathbb{N}}{\, | \,}\lambda_{d,i} > {\varepsilon}^2\}$ and therefore $\lambda_{d, n({\varepsilon},d)+1} \leq {\varepsilon}^2$. The non-increasing ordering of $(\lambda_{d,i})_{i\in{\mathbb{N}}}$ implies $$\begin{gathered}
\lambda_{d, {\left\lfloor C{\varepsilon}^{-p}d^q \right\rfloor}+1} \leq {\varepsilon}^2.\end{gathered}$$ If we set $i={\left\lfloor C\cdot {\varepsilon}^{-p} \cdot d^q \right\rfloor}+1$ and vary ${\varepsilon}\in (0,1]$ then $i$ takes the values ${\left\lfloor C \cdot d^q \right\rfloor}+1$, ${\left\lfloor C\cdot d^q \right\rfloor}+2$, and so forth. On the other hand, we have $i \leq C{\varepsilon}^{-p}d^q+1$, which is equivalent to ${\varepsilon}^2 \leq (Cd^q/(i-1))^{2/p}$ if $i\geq 2$. Thus, $$\begin{gathered}
\lambda_{d,i} \leq \lambda_{d, n({\varepsilon},d)+1} \leq {\varepsilon}^2 \leq \left( \frac{Cd^q}{i-1} \right)^{2/p}
\quad \text{for all} \quad i \geq { \mathop{\mathrm{max}}\left\{2,{\left\lfloor C\cdot d^q \right\rfloor}+1\right\} }.\end{gathered}$$ Choosing $\tau \geq 0$ and $f(d) = {\left\lceil (1+C)\cdot d^q \right\rceil} \geq { \mathop{\mathrm{max}}\left\{2,{\left\lfloor C\cdot d^q \right\rfloor}+1\right\} }$ we conclude $$\begin{gathered}
\sum_{i = f(d)}^\infty \lambda_{d,i}^\tau
\leq \sum_{i = f(d)}^\infty \left( \frac{Cd^q}{i-1} \right)^{2\tau/p}
= (Cd^q)^{2\tau/p} \sum_{i = f(d)-1}^\infty \frac{1}{i^{2\tau/p}}
\leq (Cd^q)^{2\tau/p} \cdot \zeta\left(\frac{2\tau}{p}\right),\end{gathered}$$ where $\zeta$ denotes the Riemann zeta function. In other words, if $\tau > p/2>0$ then $$\begin{gathered}
\frac{1}{d^{2q/p}} \left( \sum_{i = f(d)}^\infty \lambda_{d,i}^\tau \right)^{1/\tau}
\leq C^{2/p} \cdot \zeta\left(\frac{2\tau}{p}\right)^{1/\tau}<\infty
\quad \text{for every} \quad d\in{\mathbb{N}}.\end{gathered}$$ Setting $r = 2q/p$ proves the assertion, as well as the claimed bound on $C_\tau$.
Conversely, assume now that [(\[sup\_condition\])]{} holds with $$\begin{gathered}
f(d)
={\left\lceil C\cdot \left({ \mathop{\mathrm{min}}\left\{{\varepsilon}_d^{\rm init},1\right\} }\right)^{-p}\cdot d^{q} \right\rceil}
\quad \text{where} \quad C>0 \quad \text{and} \quad p,q \geq 0.\end{gathered}$$ That is, for some $r\geq 0$ and $\tau>0$ we have $$\begin{gathered}
0 < C_2 = \sup_{d\in{\mathbb{N}}} \frac{1}{d^r} \left( \sum_{i=f(d)}^\infty \lambda_{d,i}^\tau \right)^{1/\tau} < \infty.\end{gathered}$$ For $n \geq f(d)$, the ordering of $(\lambda_{d,i})_{i\in{\mathbb{N}}}$ implies $\sum_{i=f(d)}^n \lambda_{d,i}^\tau \geq \lambda_{d,n}^\tau \cdot (n - f(d)+1)$. Hence, $$\begin{gathered}
\lambda_{d,n} \cdot (n - f(d)+1)^{1/\tau} \leq \left( \sum_{i=f(d)}^n \lambda_{d,i}^\tau \right)^{1/\tau} \leq \left( \sum_{i=f(d)}^\infty \lambda_{d,i}^\tau \right)^{1/\tau} \leq C_2 \, d^r,\end{gathered}$$ or, respectively, $\lambda_{d,n+1} \leq C_2 \, d^r \cdot ((n+1) - f(d)+1)^{-1/\tau}$, for all $n\geq f(d)-1$. Note that for ${\varepsilon}\in (0, { \mathop{\mathrm{min}}\left\{{\varepsilon}_d^{\rm init},1\right\} }]$ we have $C_2 \, d^r \cdot ((n+1) - f(d)+1)^{-1/\tau} \leq {\varepsilon}^2$ if and only if $$\begin{gathered}
n \geq n^* = {\left\lceil \left( \frac{C_2 \, d^r}{{\varepsilon}^2} \right)^\tau \right\rceil}+ f(d)-2.\end{gathered}$$ In particular, it is $\lambda_{d,n+1} \leq {\varepsilon}^2$ at least for $n\geq { \mathop{\mathrm{max}}\left\{n^*,f(d)-1\right\} }$. Therefore, for every $d\in{\mathbb{N}}$ and for all ${\varepsilon}\in (0, { \mathop{\mathrm{min}}\left\{{\varepsilon}_d^{\rm init},1\right\} }]$ it is $$\begin{aligned}
n({\varepsilon},d;F_d)
&\leq { \mathop{\mathrm{max}}\left\{n^*,f(d)-1\right\} }
\leq f(d)-1 + \left( \frac{C_2 \, d^r}{{\varepsilon}^2} \right)^\tau \\
&\leq C \cdot \left({ \mathop{\mathrm{min}}\left\{{\varepsilon}_d^{\rm init},1\right\} }\right)^{-p}\cdot d^{q} + C_2^\tau\, {\varepsilon}^{-2\tau} \, d^{r\tau}\\
&\leq (C+C_2^\tau) \cdot {\varepsilon}^{-{ \mathop{\mathrm{max}}\left\{p,2\tau\right\} }} \cdot d^{{ \mathop{\mathrm{max}}\left\{q, r\tau\right\} }}.\end{aligned}$$ Hence, the problem is polynomially tractable.
An explicit proof of in {#an-explicit-proof-of-in .unnumbered}
------------------------
*Step 1*. By induction on $s$ we first show for every fixed $m\in{\mathbb{N}}$ $$\begin{gathered}
\label{case_mNEW}
\sum_{\substack{k\in{\mathbb{N}}^s,\\m\leq k_1\leq\ldots\leq k_s}} \mu_{s,k}
= \mu_m^s + \sum_{l=1}^s \mu_m^{s-l}\sum_{ \substack{ j^{(l)}\in{\mathbb{N}}^l,\\m+1\leq j_1^{(l)}\leq\ldots\leq j_l^{(l)} } } \mu_{l,j^{(l)}}
\quad \text{for all} \quad s\in{\mathbb{N}}.\end{gathered}$$ Easy calculus shows that this holds at least for the initial step $s=1$. Therefore, assume [(\[case\_mNEW\])]{} to be true for some $s\in{\mathbb{N}}$. Then $$\begin{aligned}
\sum_{\substack{k\in{\mathbb{N}}^{s+1},\\m\leq k_1\leq\ldots\leq k_{s+1}}} \mu_{s+1,k}
&= \sum_{k_1=m}^\infty \mu_{k_1} \sum_{\substack{h\in{\mathbb{N}}^{s},\\k_1\leq h_1\leq\ldots\leq h_{s}}} \mu_{s,h}
= \mu_m \sum_{\substack{h\in{\mathbb{N}}^{s},\\m\leq h_1\leq\ldots\leq h_{s}}} \mu_{s,h} + \sum_{k_1=m+1}^\infty \mu_{k_1} \sum_{\substack{h\in{\mathbb{N}}^{s},\\k_1\leq h_1\leq\ldots\leq h_{s}}} \mu_{s,h} \\
&= \mu_m \sum_{\substack{h\in{\mathbb{N}}^{s},\\m\leq h_1\leq\ldots\leq h_{s}}} \mu_{s,h} + \sum_{\substack{k\in{\mathbb{N}}^{s+1},\\m+1\leq k_1\leq\ldots\leq k_{s+1}}} \mu_{s+1,k}\end{aligned}$$ Now, by inserting the induction hypothesis for the first sum and renaming $k$ to $j^{(s+1)}$ in the remaining sum, we conclude $$\begin{gathered}
\sum_{\substack{k\in{\mathbb{N}}^{s+1},\\m\leq k_1\leq\ldots\leq k_{s+1}}} \mu_{s+1,k}
= \mu_m^{s+1} + \sum_{l=1}^s \mu_m^{s+1-l}\sum_{ \substack{ j^{(l)}\in{\mathbb{N}}^l,\\m+1\leq j_1^{(l)}\leq\ldots\leq j_l^{(l)} } } \mu_{l,j^{(l)}} + \sum_{\substack{j^{(s+1)}\in{\mathbb{N}}^{s+1},\\m+1\leq j^{(s+1)}_1\leq\ldots\leq j^{(s+1)}_{s+1}}} \mu_{s+1,j^{(s+1)}}.\end{gathered}$$ Hence, [(\[case\_mNEW\])]{} also holds for $s+1$ and the induction is complete.
*Step 2*. Here we prove [(\[estimate\_VNEW\])]{} via another induction on $V\in{\mathbb{N}}_0$. Therefore, let $d\in{\mathbb{N}}$ be fixed arbitrarily. The initial step, $V=0$, corresponds to [(\[case\_mNEW\])]{} for $s=d$ and $m=1$. Thus, assume [(\[estimate\_VNEW\])]{} to be true for some fixed $V\in{\mathbb{N}}_0$. Then it is $$\begin{aligned}
\sum_{\substack{k\in{\mathbb{N}}^d,\\1\leq k_1\leq\ldots\leq k_d}} \mu_{d,k}
&\leq \mu_1^d \, d^V \left( 1 + V + \sum_{L=1}^d \mu_1^{-L} \sum_{ \substack{ j^{(L)}\in{\mathbb{N}}^L,\\V+2\leq j_1^{(L)}\leq\ldots\leq j_L^{(L)} } } \mu_{L,j^{(L)}} \right) \\
&= \mu_1^d \, d^V \left( 1 + V + \sum_{L=1}^d \mu_1^{-L} \left( \mu_{V+2}^L + \sum_{l=1}^L \mu_{V+2}^{L-l} \sum_{ \substack{ j^{(l)}\in{\mathbb{N}}^l,\\(V+2)+1\leq j_1^{(l)}\leq\ldots\leq j_l^{(l)} } } \mu_{l,j^{(l)}} \right) \right),\end{aligned}$$ using [(\[case\_mNEW\])]{} for $s=L$ and $m=V+2$. Now we estimate $1+V$ by $d(1+V)$, take advantage of the non-increasing ordering of $(\mu_m)_{m\in{\mathbb{N}}}$ and extend the inner sum from $L$ to $d$ in order to obtain $$\begin{aligned}
\sum_{\substack{k\in{\mathbb{N}}^d,\\1\leq k_1\leq\ldots\leq k_d}} \mu_{d,k}
\leq \mu_1^{d} \, d^{V+1} \left( 1 + (V + 1) + \sum_{l=1}^d \mu_1^{-l} \sum_{ \substack{ j^{(l)}\in{\mathbb{N}}^l,\\(V+1)+2\leq j_1^{(l)}\leq\ldots\leq j_l^{(l)} } } \mu_{l,j^{(l)}} \right).\end{aligned}$$ Since this estimate corresponds to [(\[estimate\_VNEW\])]{} for $V+1$ the claim is proven.
Proof of in {#proof-of-in-4 .unnumbered}
------------
Note that due to $\lim_{m{\rightarrow}\infty} \lambda_m=0$ the quantity $i_d$ is well defined, because $(\alpha_i)_{i\in{\mathbb{N}}}$ is a non-increasing sequence which tends to zero for $i$ tending to infinity, and $i_d(\delta^2)>1$ for $\delta < {\varepsilon}_d^{\rm init}$. Furthermore, we have $$\begin{gathered}
i_1(\delta^2) = \#\{m\in{\mathbb{N}}{\, | \,}\lambda_m > \delta^2\} + 1 = n^{\rm ent}(\delta,1) + 1 = n^{\rm asy}(\delta,1) +1\end{gathered}$$ and if $d\geq 2$ we can rewrite $i_d$ to obtain $$\begin{gathered}
i_d(\delta^2) = { \mathop{\mathrm{min}}\left\{i \in {\mathbb{N}}{\, | \,}\lambda_{i+1} \cdot \ldots \cdot \lambda_{i+d-1} \leq \frac{1}{\lambda_i} \delta^2\right\} }.\end{gathered}$$ Hence, for every $k=(k_1,\ldots,k_{d-1}) \in \nabla_{d-1}$ with $k_1 > i_d(\delta^2)$ it is $$\begin{gathered}
\lambda_{d-1,k}=\lambda_{k_1}\cdot\ldots\cdot\lambda_{k_{d-1}} \leq \lambda_{i_d(\delta^2)+1}\cdot\ldots\cdot\lambda_{i_d(\delta^2)+d-1} \leq \frac{1}{\lambda_{i_d(\delta^2)}} \delta^2\end{gathered}$$ or, equivalently, $$\begin{gathered}
\left\{k \in \nabla_{d-1} {\, | \,}i < k_1 \text{ and } \lambda_{d-1,k} > \frac{1}{\lambda_i} \delta^2 \right\} = {\emptyset}\quad \text{for all} \quad i \geq i_d(\delta^2).\end{gathered}$$ This leads to the disjoint decomposition of $$\begin{aligned}
\{j \in \nabla_d {\, | \,}\lambda_{d,j} > \delta^2\}
&=& \left\{ j=(i,k) \in {\mathbb{N}}\times\nabla_{d-1} {\, | \,}i < k_1 \text{ and } \lambda_{d-1,k} > \frac{1}{\lambda_i} \delta^2 \right\} \\
&=& \bigcup_{i=1}^{i_d(\delta^2)-1} \left\{ (i,k) {\, | \,}k\in\nabla_{d-1} \text{ such that } i<k_1 \text{ and } \lambda_{d-1,k} > \frac{1}{\lambda_{i}} \delta^2 \right\}.\end{aligned}$$ Therefore, the information complexity of the $d$-variate problem is given by $$\begin{aligned}
n^{\rm asy}({\varepsilon},d)
&=& \# \{ j \in \nabla_d {\, | \,}\lambda_{d,j}>{\varepsilon}^2\}
= \sum_{i=1}^{i_d({\varepsilon}^2)-1} \# \left\{ k \in \nabla_{d-1} {\, | \,}i<k_1 \text{ and } \lambda_{d-1,k} > \frac{1}{\lambda_i} {\varepsilon}^2 \right\} \\
&=& \sum_{l_1=2}^{i_d({\varepsilon}^2)} \# \left\{ k \in \nabla_{d-1} {\, | \,}l_1 \leq k_1 \text{ and } \lambda_{d-1,k} > \frac{1}{\lambda_{l_1-1}} {\varepsilon}^2 \right\}.\end{aligned}$$ Obviously, for fixed $l_1 \in \{2,\ldots,i_d({\varepsilon}^2)\}$, we can repeat this procedure and obtain $$\begin{aligned}
&\# \{ j \in \nabla_{d-1} {\, | \,}l_1 \leq j_1 \text{ and } \lambda_{d-1,j}>\delta^2\} \\
&\qquad\qquad = \sum_{l_2=l_1+1}^{i_{d-1}(\delta^2)} \# \left\{ k \in \nabla_{d-2} {\, | \,}l_2 \leq k_1 \text{ and } \lambda_{d-2,k} > \frac{1}{\lambda_{l_2-1}} \delta^2 \right\},\end{aligned}$$ if $d>2$ and $\delta^2 = {\varepsilon}^2 / \lambda_{l_1-1}$. Note that ${\varepsilon}< {\varepsilon}_d^{\rm init}$ implies $i_{d-1}(\delta^2)\geq l_1+1$ such that $\{l_1+1,\ldots,i_{d-1}(\delta)^2\} \neq {\emptyset}$. Iterating the argument we get $$\begin{aligned}
&n^{\rm asy}({\varepsilon},d) \\
&\qquad = \sum_{l_1=2}^{i_d({\varepsilon}^2)} \sum_{l_2=l_1+1}^{i_{d-1}({\varepsilon}^2 / \lambda_{l_1-1})} \ldots \sum_{l_{d-1}=l_{d-2}+1}^{i_{2}({\varepsilon}^2 / [\lambda_{l_1-1}\cdot\ldots\cdot\lambda_{l_{d-2}-1}])}
\# \left\{ k \in \nabla_{1} {\, | \,}l_{d-1} \leq k_1 \text{ and } \lambda_{1,k} > \frac{1}{\lambda_{l_{d-1}-1}}\delta^2 \right\}\end{aligned}$$ with $\delta^2 = {\varepsilon}^2 / [\lambda_{l_1-1} \cdot\ldots\cdot\lambda_{l_{d-2}-1}]$. It remains to calculate the cardinality of the last set. Of course, we have $$\begin{aligned}
&\left\{ k \in \nabla_{1} {\, | \,}l_{d-1} \leq k_1 \text{ and } \lambda_{1,k} > \frac{1}{\lambda_{l_{d-1}-1}} \delta^2 \right\} \\
&\qquad\qquad= \left\{ k \in {\mathbb{N}}{\, | \,}l_{d-1} \leq k \text{ and } \lambda_{k} > \frac{1}{\lambda_{l_{d-1}-1}} \delta^2 \right\}\\
&\qquad\qquad= \left\{ k \in {\mathbb{N}}{\, | \,}\lambda_{k} > \frac{1}{\lambda_{l_{d-1}-1}} \delta^2 \right\} \setminus \left\{ k \in \{1,\ldots,l_{d-1}-1\} {\, | \,}\lambda_{k} > \frac{1}{\lambda_{l_{d-1}-1}} \delta^2 \right\}.\end{aligned}$$ The first of these sets in the last line contains exactly $n^{\rm ent}(\delta/\sqrt{\lambda_{l_{d-1}-1}}, 1)$ elements. On the other hand, if $k\leq l_{d-1} \leq i_2(\delta^2)$ then $$\begin{gathered}
\lambda_k\lambda_{l_{d-1}-1}\geq \lambda_{i_2(\delta^2)} \lambda_{i_2(\delta^2)-1} > \delta^2,\end{gathered}$$ where the last inequality holds due to the definition of $i_2(\delta^2)$. Therefore, the last set coincides with $\{1,\ldots,l_{d-1}-1\}$ and its cardinality is equal to $l_{d-1}-1$. Furthermore, note that the estimate also shows that $n^{\rm ent}(\delta/\sqrt{\lambda_{l_{d-1}-1}}, 1)$ is at least equal to $l_{d-1}$. Thus, $$\begin{gathered}
\# \left\{ k \in \nabla_{1} {\, | \,}l_{d-1} \leq k_1 \text{ and } \lambda_{1,k} > \frac{1}{\lambda_{l_{d-1}-1}} \delta^2 \right\}
= n^{\rm ent}\left( \delta/\sqrt{\lambda_{l_{d-1}-1}}, 1\right) - l_{d-1} + 1 \geq 1\end{gathered}$$ and the proof is complete.
Proof of in {#proof-of-in-5 .unnumbered}
------------
One possibility to prove the second point of is to apply to a scaled problem $\{{ \widetilde{T} }_d\}$ such that ${ \widetilde{W_d} }={ { \widetilde{T} }_d }^{\dagger} { \widetilde{T} }_d$ possesses the eigenvalues ${ \widetilde{\lambda} }_{d,i} = \lambda_{d,i}/\lambda_{d,1}$ for $i\in{\mathbb{N}}$. Then the initial error of ${ \widetilde{T} }_d$ equals $1$ such that $f$ in does not depend on $p$. That is, we can choose $f(d)={\left\lceil C\, d^{q} \right\rceil}+1$ for some $q\geq 0$ in both the assertions. In order to see why we even can take $f(d) \equiv 1$ in the first point and for the sake of completeness we also add a direct proof for this proposition.
If $\{T_d\}$ is polynomially tractable with respect to the normalized error criterion then there exist constants $C,p>0$ and $q\geq0$ such that for all $d\in {\mathbb{N}}$ and ${\varepsilon}' \in ( 0, 1 ]$ $$\begin{gathered}
n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d) =
n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d; F_d)
\leq C \cdot \left({\varepsilon}'\right)^{-p} \cdot d^q.\end{gathered}$$ As before the quantity ${\varepsilon}_d^{\rm init} = \sqrt{\lambda_{d,1}}>0$ denotes the initial error of $T_d$ and ${\varepsilon}'$ is the (multiplicative) improvement of it. Since $e(n,d)=\sqrt{\lambda_{d,n+1}}$ it is $n({\varepsilon},d)= \# \{i\in{\mathbb{N}}{\, | \,}\lambda_{d,i} > {\varepsilon}^2\}$ where ${\varepsilon}= {\varepsilon}' \cdot {\varepsilon}_d^{\rm init}$. Therefore, $\lambda_{d, n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d)+1} \leq \left({\varepsilon}'\right)^2 \cdot \lambda_{d,1} $. Hence, the non-increasing ordering of $(\lambda_{d,i})_{i\in{\mathbb{N}}}$ implies in this setting $$\begin{gathered}
\lambda_{d, {\left\lfloor C \left({\varepsilon}'\right)^{-p} d^{q} \right\rfloor}+1} \leq \left({\varepsilon}'\right)^2 \cdot \lambda_{d,1}.\end{gathered}$$ If we set $i={\left\lfloor C \left({\varepsilon}'\right)^{-p} d^{q} \right\rfloor}+1$ and vary ${\varepsilon}' \in (0,1]$ then $i$ takes the values ${\left\lfloor C d^q \right\rfloor}+1$, ${\left\lfloor C d^q \right\rfloor}+2$ and so on. Again we have $1 \leq i \leq C \left({\varepsilon}'\right)^{-p}d^q+1$ on the other hand, which is equivalent to $\left({\varepsilon}'\right)^2 \leq (Cd^q/(i-1))^{2/p}$ if $i\geq2$. Thus, $$\begin{gathered}
\lambda_{d,i}
\leq \lambda_{d, n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d)+1}
\leq \left({\varepsilon}'\right)^2 \cdot \lambda_{d,1}
\leq \left( \frac{Cd^q}{i-1} \right)^{2/p} \cdot \lambda_{d,1} \quad \text{for all} \quad i \geq { \mathop{\mathrm{max}}\left\{2, {\left\lfloor C d^q \right\rfloor}+1\right\} }.\end{gathered}$$ Choosing $\tau \geq 0$ and $f^{*}(d)= {\left\lceil (1+C)\, d^q \right\rceil} \geq { \mathop{\mathrm{max}}\left\{2, {\left\lfloor C d^q \right\rfloor}+1\right\} }$ we conclude here $$\begin{gathered}
\sum_{i = f^{*}(d)}^\infty \left(\frac{\lambda_{d,i}}{\lambda_{d,1}}\right)^\tau
\leq \sum_{i = f^{*}(d)}^\infty \left( \frac{Cd^q}{i-1} \right)^{2\tau/p}
\leq (Cd^q)^{2\tau/p} \cdot \zeta\left(\frac{2\tau}{p}\right),\end{gathered}$$ where $\zeta$ again is the Riemann zeta function. On the other hand, it is obvious that $$\begin{gathered}
\sum_{i = 1}^{f^{*}(d)-1} \left(\frac{\lambda_{d,i}}{\lambda_{d,1}}\right)^\tau \leq f^*(d)-1 \leq (1+C) \, d^{q\cdot 2\tau/p},\end{gathered}$$ because $\lambda_{d,i}\leq \lambda_{d,1}$ for all $i\in{\mathbb{N}}$. Therefore, if $\tau > p/2$, $$\begin{gathered}
\frac{1}{d^{2\tau q/p}} \sum_{i = 1}^\infty \left(\frac{\lambda_{d,i}}{\lambda_{d,1}}\right)^\tau
\leq 1+C + C^{2\tau/p} \cdot \zeta\left(\frac{2\tau}{p}\right)<\infty\end{gathered}$$ for all $d\in{\mathbb{N}}$. This proves the assertion setting $r \geq 2q/p$.
The proof of the second point again works like for . Assume that [(\[sup\_condition2\])]{} holds with $f(d) = {\left\lceil C \, d^{q} \right\rceil}$, where $C>0$ and $q \geq 0$. That is, for some $r\geq 0$ and $\tau>0$ we have $$\begin{gathered}
C_\tau = \sup_{d\in{\mathbb{N}}} \frac{1}{d^r} \left( \sum_{i=f(d)}^\infty \left(\frac{\lambda_{d,i}}{\lambda_{d,1}}\right)^\tau \right)^{1/\tau} < \infty.\end{gathered}$$ Since $(\lambda_{d,i})_{i\in{\mathbb{N}}}$ is assumed to be non-increasing the same also holds for the rescaled sequence $(\lambda_{d,i} / \lambda_{d,1})_{i\in{\mathbb{N}}}$ such that $\sum_{i=f(d)}^n (\lambda_{d,i}/\lambda_{d,1})^\tau \geq (\lambda_{d,n}/\lambda_{d,1})^\tau \cdot (n - f(d)+1)$ for $n \geq f(d)$. Hence, $$\begin{gathered}
\frac{\lambda_{d,n}}{\lambda_{d,1}} \cdot (n - f(d)+1)^{1/\tau}
\leq \left( \sum_{i=f(d)}^\infty \left(\frac{\lambda_{d,i}}{\lambda_{d,1}}\right)^\tau \right)^{1/\tau}
\leq C_\tau \, d^r,\end{gathered}$$ or, respectively, $\lambda_{d,n+1} \leq C_\tau \, d^r \cdot ((n+1) - f(d)+1)^{-1/\tau} \cdot \lambda_{d,1}$ for all $n\geq f(d)-1$. As before we have $C_\tau \, d^r \cdot ((n+1) - f(d)+1)^{-1/\tau} \leq \left({\varepsilon}'\right)^2$, for ${\varepsilon}' \in (0, 1]$, if and only if $$\begin{gathered}
n \geq n^{*} = {\left\lceil \left( \frac{C_\tau \, d^r}{\left({\varepsilon}'\right)^2} \right)^\tau \right\rceil}+ f(d)-2.\end{gathered}$$ In particular, $\lambda_{d,n+1} \leq \left({\varepsilon}'\right)^2\cdot \lambda_{d,1}$ at least for $n\geq { \mathop{\mathrm{max}}\left\{n^{*},f(d)-1\right\} }$. Therefore, we conclude in this setting for all ${\varepsilon}' \in (0, 1]$ and every $d\in{\mathbb{N}}$ $$\begin{aligned}
n({\varepsilon}' \cdot {\varepsilon}_d^{\rm init},d; F_d)
&\leq { \mathop{\mathrm{max}}\left\{n^{*},f(d)-1\right\} }
\leq f(d)-1 + \left( \frac{C_\tau \, d^r}{\left( {\varepsilon}'\right)^2} \right)^\tau
\leq C\, d^q + C_\tau^\tau \left( {\varepsilon}' \right)^{-2\tau} d^{r\tau} \\
& \leq (C+C_\tau^\tau) \cdot \left( {\varepsilon}'\right)^{-2\tau} \cdot d^{{ \mathop{\mathrm{max}}\left\{q, r\tau\right\} }}.\end{aligned}$$ Hence, the problem is polynomially tractable. Furthermore, strong polynomial tractability holds if $r=q=0$.
[^1]: Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany. Email: [email protected]. Web: http://users.minet.uni-jena.de/\~weimar.
[^2]: This is an extended version of a same-named paper by the author which was published in the Journal of Approximation Theory [@W12]. Here all the proofs, as well as some additional assertions, are explicitly included.
|
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abstract: |
Hot interstellar gas in elliptical galaxies has two sources: mass lost from evolving stars and a much older component that accompanied galaxy formation or arrived subsequently by secondary cosmic infall toward the galaxy group containing the elliptical. We present here an approximate but comprehensive study of the dynamical evolution of the hot gas in massive elliptical galaxies born into a simple flat universe. Baryonic and dark matter are both conserved. We use NGC 4472 as a prototypical massive elliptical having a well-observed hot interstellar medium. We allow for star formation in a simple single burst using a Salpeter IMF but treat the gas dynamics in detail. The galaxy has a de Vaucouleurs stellar core and a Navarro-Frenk-White dark halo surrounded by inflowing cosmic matter.
Using rather standard assumptions and parameters, we are able to successfully reproduce the gas density and temperature distributions – $n(r)$ and $T(r)$ – in the hot interstellar gas determined by recent X-ray observations. Our model is sensitive to the baryon fraction of the universe, the Type II supernova energy released per unit stellar mass, and the time of galaxy formation. But there is some degeneracy; as each of these parameters is varied, the effect on model fits to $n(r)$ and $T(r)$ is similar. Nevertheless, secondary inflow of cosmic gas is essential for successful fits to $n(r)$ and $T(r)$.
Some gas is expelled from the stellar galactic core at early times when the Type II supernova energy was released. As a result, the present day baryonic fraction has a deep minimum in the outer galactic halo. Interstellar gas that cooled since the time of maximum star formation cannot have all collected at the galactic center but must be spatially dispersed; otherwise both gas temperatures and stellar dispersions in the galactic center would be larger than those observed.
Finally, when relatively gas-rich, X-ray luminous models are spatially truncated at early times, simulating tidal events that may have occurred during galaxy group dynamics, the current locus of truncated models lies just along the $L_x$, X-ray size correlation among well-observed ellipticals. This is another striking confirmation of our model of elliptical evolution.
author:
- 'Fabrizio Brighenti$^{2,3}$ and William G. Mathews$^2$'
title: 'EVOLUTION OF HOT GAS AND DARK HALOS IN GROUP-DOMINANT ELLIPTICAL GALAXIES: INFLUENCE OF COSMIC INFLOW$^1$'
---
\#1\#2\#3\#4[ ]{} \#1\#2
.2in
INTRODUCTION
============
In this paper we present a new, more comprehensive model for the evolution of hot gas in massive ellipticals and dominant ellipticals in galaxy groups. For the first time we combine gas produced by stellar mass loss with additional gas that flows into the galaxy regarded as a perturbation in the Hubble flow. The radial distribution of gas density and temperature in bright ellipticals determined by recent X-ray observations can be understood only by combining these two sources of gas, stellar and cosmic.
Traditionally, it has been assumed that most or all of the hot gas observed in ellipticals arises as a result of mass loss from the aging stellar population, but recent observations with the ROSAT satellite indicate a more complicated origin for the gas. Davis & White (1996) showed that the temperature of the hot gas is generally $\sim 1.5$ times hotter than the equivalent temperature of the stars $T_*$. This implies that the gas is in virial equilibrium in a deeper, and therefore larger, potential than the stars. Even more relevant, X-ray images from ROSAT have revealed very extended regions of X-ray emission surrounding many massive ellipticals, extending far beyond the optical image. Mathews and Brighenti (1998) have shown that the emission from these extended regions can dominate the overall X-ray luminosity. This realization resolves a long-standing puzzle for elliptical galaxies: the enormous variation in X-ray luminosity $L_x$ in ellipticals having similar optical luminosity. However, the total mass of hot gas in the outer regions of many of these galaxies is too large to be understood in terms of stellar mass loss alone. In Brighenti & Mathews (1998) we showed that agreement with the observed hot gas density and temperature profiles is greatly improved by assuming that a large mass of “circumgalactic” gas existed around the galaxy at early times. Much of the gas that filled the galaxy at the time of galaxy formation is still present today. This raises the possibility that the hot gas in elliptical galaxies can reveal, like growth rings in a tree, the sequence of events that occurred long ago during the epoch of galaxy formation.
In this paper we extend this idea to consider the continuous inflow of gas into the galactic halo over the Hubble time. This type of flow is a natural consequence of the large perturbation in the Hubble flow that formed the galaxy and the small group in which it may reside. Our approach in constructing these models is to adopt the simplest or most plausible procedure. We assume for example that the evolution of the dark matter can be described by the normal self-similar flow in a perturbed flat universe, but as the central stationary dark halo grows in size, it adapts to a shape dictated by more detailed N-body calculations. The baryonic gaseous component flows in with the dark matter but eventually shocks and compresses to the virial temperature $\sim 10^7$ K of the dark halo. From the very beginning, as X-ray energy is radiated away, cold gas condenses at the center of the flow. When a sufficient mass of baryons has accumulated, we form the stellar distribution of the elliptical in a single event. In this way we circumvent the complex dynamical and merging processes that occurred during the formation of the big elliptical. At the same instant when the galaxy forms the remaining gas near the galaxy is heated by Type II supernovae and the galactic stars continue to evolve as a single burst thereafter. In spite of the simplicity of this model, the subsequent evolution of the gas, both that expelled from the stars and gas continuously flowing in toward the perturbation, is treated in a self-consistent, reasonably realistic manner. This is confirmed by the agreement of our results with current X-ray observations. It is gratifying that we can match so well the observed mass and entropy distribution in large ellipticals with our models and still conform to global cosmological and astrophysical constraints set by the conservation of baryonic mass, dark mass and supernova energy.
In this study we are concerned with establishing the global properties of hot gas in ellipticals as a consequence of gas dynamical evolution from the earliest times. Therefore, in order to avoid unnecessary detail, we shall not discuss here the detailed metal abundance in the gas nor shall we consider the important effects of galactic rotation.
Although our results are generally applicable to all massive ellipticals and the group environment they inhabit, as a rigorous test of the general success of our models it is useful to use parameters specific to a particular well-observed galaxy; for this purpose we have chosen NGC 4472, the central dominant elliptical in a subcluster in the Virgo cluster.
PROPERTIES OF NGC 4472, A LUMINOUS ELLIPTICAL
=============================================
The X-ray surface brightness variation in massive ellipticals can be inverted to determine the spatial variation of electron density with physical radius. The radial plasma temperature profile can be similarly determined from variations in the apparent X-ray spectrum viewed in projection. Since the hot interstellar gas is in near hydrostatic equilibrium, the total gravitating mass $M(r)$ can be found from the temperature and density profiles. At present the best galaxies for this purpose are those relatively nearby and luminous ellipticals that have been imaged by both [*Einstein*]{} HRI and [*ROSAT*]{} HRI & PSPC: NGC 4472, 4649, and 4636 (Brighenti & Mathews 1997b). Of these we chose NGC 4472 for comparison with our gas dynamical models since the X-ray data for NGC 4472 exists over the largest range, $\sim 0.016$ to $\sim 16$ effective radii where $r_e = 8.57$ kpc at a distance of 17 Mpc. Brighenti & Mathews (1997b) showed that the hot gas in NGC 4472 is at rest in the [*stellar*]{} potential in the range $0.1 - 1$ $r_e$ where the total mass (assuming hydrostatic equilibrium) is identical to the stellar mass predicted from the dynamical mass to light ratio based on a two-integral stellar distribution function (van der Marel 1991). This excellent agreement indicates that gas pressure alone is sufficient to support the hot gas; no additional pressure (magnetic, cosmic rays, turbulence, etc.) is needed nor indicated in $0.1 - 1$ $r_e$. Unfortunately, beyond a radius of $\sim$3.5 $r_e$ the X-ray image of NGC 4472 becomes asymmetric probably as a result of an environmental ram pressure interaction or an interaction with the dwarf irregular galaxy UGC 7636 (Irwin & Sarazin 1996; Irwin, Frayer, & Sarazin 1997). However, the azimuthally averaged X-ray surface brightness used by these authors to determine the average density profile is globally representative of luminous ellipticals. NGC 4472 is a prototypical bright elliptical that exhibits all of the characteristics of these massive galaxies.
The upper panel of Figure 1 shows the gas density (Trinchieri, Fabbiano & Canizares 1986; Irwin & Sarazin 1996) and temperature (Irwin and Sarazin 1996) profiles in NGC 4472 together with analytic fits as described by Brighenti & Mathews (1997b). The corresponding total mass $M(r)$ is shown in the lower panel of Figure 1 together with best fitting stellar and dark halo mass distributions. The stellar mass is described with a de Vaucouleurs profile with effective radius $r_e = 8.57$ kpc. The stellar luminosity is $L_B = 7.89 \times 10^{10}$ $L_{B \odot}$ corresponding to a total stellar mass $M_* = 7.26 \times 10^{11}$ $M_{\odot}$ using a mass to light ratio of $M_{*}/L_B = 9.20$ (van der Marel 1991). The dark halo in NGC 4472 dominates the total mass for $r \gta r_e$. As shown in Figure 1 a dark halo described by the Navarro, Frenk & White (1996) density profile (NFW halo) using a total halo mass $M_h = 4 \times 10^{13}$ $M_{\odot}$ fits the dark matter distribution quite well at large radii. \[A reservation about the applicability of the NFW profile to NGC 4472 expressed by Brighenti & Mathews (1997b) is now seen to be in error.\] The density distribution in an NFW halo is described by $$\rho_h = {200 \over 3}~
{c^3 \over \ln (1 + c) - c/(1+c)}~
{\rho_H \over (r/r_s) [1 + (r/r_s)]^2 }$$ where $\rho_H$ is the current mean density of the universe, $r_s = r_{gn}/c$ and $r_{gn}$ is the current virial radius of the halo where the density is $\sim 200 \rho_H$. The concentration parameter, $c = 10.72 (M_h/3.3 \times 10^{13} M_{\odot})^{-0.1233}$, is based on Figure 8 of NFW. All galactic parameters are scaled to our adopted distance to NGC 4472, $D = 17$ Mpc.
The agreement of the mass distribution of NGC 4472 with an NFW halo plus a de Vaucouleurs core is good but not perfect, particularly at the transition region near $\sim 10$ kpc. The shape of the inner part of galaxy-sized dark halos is controversial, e.g. Kravtsov et al. (1998) predict a flatter density profile within the NFW characteristic radius $r_s$; such a halo would improve the fit to the total mass in Figure 1. However, any mass profile determined by N-body calculations in the absence of baryons will be centrally steepened and redistributed somewhat by the dissipative inward migration of the baryon component. Nevertheless, in the calculations described here we shall adopt the combined de Vaucouleurs and NFW mass distribution for NGC 4472. The relatively small discrepancy apparent in Figure 1 has essentially no significant influence on our gas dynamical models.
We have taken published profiles of the electron density $n(r)$ directly from the literature although Trinchieri, Fabbiano & Canizares (1986) and Irwin & Sarazin (1996) have made different assumptions about $T(r)$ and the metallicity variation $z(r)$ in the gas. Since the indicated gas density depends on $T(r)$ and $z(r)$ as they influence the cooling rate, $n(r) \propto \Lambda(T,z)^{-1/2}$, the “observed” values of the gas density in Figure 1 could change somewhat when more definitive information on $T(r)$ and $z(r)$ becomes available.
EVOLUTION OF THE DARK HALO BY SECONDARY INFALL
==============================================
For simplicity in estimating the growth of the dark halo in NGC 4472, we assume a critical $\Omega = 1$ universe and a current age $t_n = 13$ Gyrs corresponding to $H = 50$ km s$^{-1}$ Mpc$^{-1}$. The self-similar flow of dark matter toward an overdensity perturbation in a critical universe (Bertschinger 1985) consists of two regions: a smooth outer flow in which the Hubble flow reverses at a turnaround radius $r_{ta}(t)$ and an inner flow consisting of an ensemble of caustics or density cusps produced by orbital wrapping of the collisionless dark fluid as it oscillates about the center of the perturbation. As infalling dark matter accumulates in the dark halo, its mass is added to the outer part, leaving the central mass distribution essentially undisturbed. Although newly arrived shells of dark matter pass through the central regions of the previously existing dark halo, they move very rapidly there and do not contribute appreciably to the time-averaged density except near the outermost caustic. Averaging over the caustic peaks, the total dark mass density in Bertschinger’s similarity solution varies as $\rho_d \propto r^{-9/4}$, but this is much too centrally peaked when compared to typical N-body simulations (Navarro, Frenk & White 1996).
We resolve this difficulty by assuming that an NFW halo forms as a stationary core at the center of an outer smooth flow described by Bertschinger’s similarity solution. The outer flow must be attached to the stationary NFW halo so as to conserve total dark mass at all times. In terms of the similarity variable $$\lambda = { r \over r_{ta}(t) }$$ the outer solution can be fit quite precisely with $${\cal M}(\lambda) = \lambda^3 ~ { a + b (\lambda / \lambda_1)^2
\over 1 + (\lambda / \lambda_1)^2 }$$ where $a = 3.4970$, $b = 5.8981$ and $\lambda_1 = 1.1296$. This analytic fit to the smooth outer flow onto a point mass is exact at $\lambda = 0$, 1 and $\infty$.
The current virial radius $r_{gn}$ within the self-similar dark halo corresponds to an overdensity of $\sim 200$, i.e. $$r_{gn} = \left( {3 M_n \over 4 \pi 200 \rho_{Hn}}
\right)^{1/3}$$ where $\rho_{Hn} = 1/6 \pi G t_n^2$ is the current total cosmic density and $M_n = 4 \times 10^{13}$ $M_{\odot}$ is taken to be the current virial mass. In the self-similar solution the approximately stationary halo confronts the secondary inflow at the radius of the outermost caustic $r_{cn} = 1.528 r_{gn}$ at time $t_n$. The current turnaround radius $r_{tan} = r_{ta}(t_n)$ can be found by insisting that the outer self-similar solution meet the NFW halo at radius $r_{cn}$ where the interior masses of the two mass distributions must agree: $${4 \over 3} \pi r_{tan}^3 \rho_{Hn} {\cal M}(r_{cn}/r_{tan})
= M_n { f(r_{cn}/r_s) \over f(c) }$$ where $$f(x) = \ln (1 + x) - {x \over 1 + x}$$ describes the NFW mass profile. With our chosen parameters, the current turnaround radius is $r_{tan} = 1.096 \times 10^{25}$ cm. By similarity arguments, the turnaround radius at earlier times is $r_{ta}(t) = r_{tan} (t/t_n)^{8/9}$.
Finally, the radius $r_c$ at which the NFW halo meets the secondary inflow is found by solving $${4 \over 3} \pi [r_{ta}(t)]^3 \rho_{H}(t)
{\cal M}[r_{c}(t)/r_{ta}(t)]
= M_n { f[r_{c}(t)/r_s] \over f(c) }$$ for $r_c(t)$, keeping $r_s$ constant. Although ${\cal M}(\lambda)$ is unvarying with time in the similarity solution, we have relaxed this condition in the expression above to achieve a fit to the NFW halo. The total mass distribution in the stationary halo and the external flow are illustrated at several times in Figure 2.
BARYONS AND SINGLE BURST STAR FORMATION
=======================================
Within the context of the $\Omega = 1$ cosmology the baryonic contribution to the total density $\Omega_b$ is constrained by primordial nucleosynthesis to a narrow range, $$0.04 \lta \Omega_b (H/50)^2 \lta 0.06$$ (Walker et al. 1991). For a given value of $\Omega_b$ the mass distribution in Figure 2, multiplied by $(1 - \Omega_b)$, represents the varying potential of dark matter. Provided the temperature of the baryonic gas is much less than the virial temperature of the dark halo, $T \lta 10^6$ K, cosmic gas will flow inward toward the galaxy precisely following the dark matter until the gas encounters a shock front. Within this accretion shock the mass distribution of the baryonic component is determined by gas dynamics and the assumed model for star formation.
Interior to the outward facing shock the cosmic gas is heated to $T \sim 10^7$ K, similar to the virial temperature of the dark halo. After a few Gyrs a large mass of gas will have collected and radiatively cooled within the dark matter halo; it is natural to assume that this gas forms the old stellar system observed today. In our very simple model for star formation we assume that all the stars in NGC 4472 form as a single burst at time $t_*$. One possible choice of $t_*$ would be to assume that stars form from the cooled gas as as soon as a mass of cold gas equals the current stellar mass of NGC 4472. However, dynamic considerations suggest that the de Vaucouleurs stellar profile results from the merging of stellar systems that must have formed prior to the merging events, so stars are likely to have predated NGC 4472. Furthermore, gas cools more efficiently and can form stars somewhat sooner in an earlier generation of small galaxies in which the dark matter and cooling gas are more concentrated than in NGC 4472. For these reasons we regard the time of star formation $t_*$ as a variable parameter in our models; if the total mass of radiatively cooled gas is less than the mass $M_*$ of NGC 4472 at time $t_*$, some of the hot gas is used to make up the deficiency since it could have cooled by this time in smaller pre-galactic stellar systems. At the time of star formation the gas density is reduced by a factor $[M_g(r_{sh}) - M_*]/M_g(r_{sh})$ where $M_{g}(r_{sh})$ is the total (all of the cold and perhaps some hot) gas mass within the shock radius $r_{sh}$ just before $t_*$. Experience has shown that this artificial mass rearrangement at $t_*$ results in a relatively short-lived gas dynamical transient.
Soon after time $t_*$ massive stars ($m > 8$ $M_{\odot}$) begin to produce SNII explosions which heat the remaining gas considerably. We deposit this SNII energy instantaneously at time $t_*$ by increasing the specific thermal energy density $\varepsilon (r)$ by an appropriate amount everywhere within the shock radius at $t_*$. The possibility that some SNII events predated the physical formation of NGC 4472 will not be considered here. The combined energy of the SNII explosions is sufficiently large to drive an outflow from the galaxy and the associated galaxy group, creating a second outward moving shock that moves out beyond the accretion shock described above.
As soon as stars form they begin to lose mass by normal stellar evolution. Most of this stellar mass loss occurs at early times shortly after $t_*$. Unlike more realistic, rotating galaxies in which radiatively cooled gas forms into disks of size $r_{disk} \gta r_e$ (Brighenti & Mathews 1996; 1997a), in the purely spherical models discussed here cold gas can only form at the very center. From $t_*$ until the present time enough cold gas can form at the center of the galaxy to influence the gravitational potential there, causing the gas temperature within $r_e$ to rise considerably above values observed. We avoid this possibility here by assuming that the total mass of gas that cools after time $t_*$ also forms into stars distributed at all radii within the stellar galaxy. The mass of stars that is assumed to form in this diffuse manner after time $t_*$ is much less than the current stellar mass of NGC 4472 and can be ignored in evaluating the global galactic potential. Therefore to implement this assumption, we simply ignore the self gravity of the cooled gas at the galactic center. We also ignore the relatively small SNII contribution resulting from this type of additional star formation.
Although our model for star formation is obviously [*ad hoc*]{} and oversimplified, we believe that the subsequent dynamics of the hot gas, which is our main interest here, is modeled reasonably well.
GAS DYNAMICS
============
The standard gas dynamics equations that describe the evolution of hot interstellar gas in ellipticals are the usual conservation equations with additional source and sink terms: $${ \partial \rho \over \partial t}
+ {1 \over r^2} { \partial \over \partial r}
\left( r^2 \rho u \right) = \alpha \rho_*,$$ $$\rho \left( { \partial u \over \partial t}
+ u { \partial u \over \partial r} \right)
= - { \partial P \over \partial r}
- \rho {G M_{tot}(r) \over r^2} - \alpha \rho_* u,$$ and $$\rho {d \varepsilon \over dt } =
{P \over \rho} {d \rho \over d t}
- { \rho^2 \Lambda \over m_p^2}
+ \alpha \rho_*
\left[ \varepsilon_o - \varepsilon - {P \over \rho}
+ {u^2 \over 2} \right]$$ where $\varepsilon = 3 k T / 2 \mu m_p$ is the specific thermal energy; $\mu = 0.62$ is the molecular weight and $m_p$ the proton mass. $M_{tot}(r)$ is the total mass of stars, dark matter and hot gas within radius $r$.
All three equations involve source terms describing the rate $\alpha \rho_*$ (gm s$^{-1}$ cm$^{-3}$) that gas is ejected from evolving stars and supernovae, i.e. $\alpha = \alpha_* + \alpha_{sn}$. To estimate $\alpha_*(t)$ we assume a single burst of star formation with a power law initial mass function (IMF) between lower and upper mass limits $m_{\ell}$ and $m_u$. Using the procedure described by Mathews (1989) we find $$\alpha_*(t) = \alpha_*(t_n - t_*)
[t/(t_n - t_*)]^{-s(t)}$$ where $t_n = 13$ Gyrs is the current time. For a Salpeter IMF of slope $x = 1.35$ with $m_{\ell} = 0.08$ and $m_u = 100$ $M_{\odot}$ we find $\alpha_*(t_n - t_*) = 5.6 \times 10^{-20}$ s$^{-1}$ and $s(t)$ varies slowly with time ($s =$ 0.83, 0.97, 1.11, 1.34 at 0.001, 0.1, 1, 11 Gyrs respectively).
In the thermal energy equation $\Lambda(T,z)$ is the optically thin radiative cooling coefficient which is a function of temperature and metallicity (Sutherland & Dopita 1993). Thermal energy lost from the gas by radiation is largely restored by the gravitational compression of the gas as it flows deeper into the galaxy. Radiative cooling eventually dominates in the densest parts of the flow. In the computed models we allow the gas to cool to $10$ K. In addition we assume that the interstellar gas is heated by the dissipation of the orbital energy of mass-losing stars and by supernovae. The mean gas injection energy is $\varepsilon_o = 3 k T_o /2 \mu m_p$ where $T_o = (\alpha_* T_* + \alpha_{sn} T_{sn})/\alpha$. The stellar temperature $T_*(r)$ is found by solving the Jeans equation for a de Vaucouleurs profile with additional dark matter beyond $r_e$. Type Ia supernova heating is assumed to be distributed smoothly in the gas; we ignore the buoyancy of gas locally heated by supernovae that may transport energy and metals to larger galactic radii. The heating by supernovae is described by multiplying the specific mass loss rate of supernovae $\alpha_{sn}$ by the characteristic temperature of the ejected mass $m_{sn}$, $T_{sn} = m_p E_{sn} / 3 k m_{sn}$, i.e. $$\alpha_{sn} T_{sn} =
2.13 \times 10^{-8}~ {\rm SNu}(t)~ (E_{sn}/10^{51} {\rm ergs})~$$ $$h^{-1.7}~ (L_B/L_{B \odot})^{-0.35} ~~~ {\rm K}~{\rm s}^{-1}$$ where $h \equiv H/100 = 0.5$ is the reduced Hubble constant and we adopt $E_{sn} = 10^{51}$ ergs as the typical energy released in both Type Ia and Type II supernovae. The Type Ia supernova rate is represented in SNu units, the number of supernovae in 100 years expected from stars of total luminosity $10^{10} L_{B \odot}$. The dependence $\alpha_{sn} \propto L_B^{-0.35}$ derives from the mass to light ratio for ellipticals given by van der Marel (1991), $M_{*t}/L_B = 2.98 \times 10^{-3} L_B^{0.35} h^{1.7}$ in solar units.
The equations are solved in one dimensional spherical symmetry using an appropriately modified version of the Eulerian code ZEUS. Spatial zones increase logarithmically and extend far out beyond the turnaround radius where the cosmic gas is supersonically receding from the galaxy. Therefore we can assume “outflow” boundary conditions at the outermost spherical zone, confident that no disturbance at the boundary can propagate back into the galactic flow. For our purposes here it is not necessary to consider the detailed thermal physics of the intergalactic gas; we simply assume that the temperature of this gas is isothermal at $T = 10$ K until it passes through the accretion shock. Although the intergalactic temperature is expected to be greater than this (due to photoelectric heating by quasar radiation for example), the value of the intergalactic gas temperature makes no difference in our calculations as long as it is small compared to the virial temperature of the dark halo of NGC 4472, i.e. $\lta 10^6$ K.
SUPERNOVA HEATING
=================
Supernovae in ellipticals today are infrequent and are all of Type Ia. Cappellaro et al. (1997) find a current rate of $0.066 \pm 0.027 (H/50)^2$ in SNu units. The frequency of SNIa in the past is currently unknown although it seems reasonable to assume that it has been decreasing over cosmic time, $${\rm SNu}(t) = {\rm SNu}(t_n) (t / t_n)^{-p}.$$ Ciotti et al. (1991) recognized that the relative rates of stellar mass ejection ($\sim t^{-s}$) and Type Ia supernova ($\sim t^{-p}$) determine the gas dynamical history of hot interstellar gas in ellipticals. For example if $p > s$ then galactic winds may occur at early times but if $p < s$ winds may occur at late times. However, in the presence of large amounts of additional cosmic gas flowing into the galaxy, galactic winds driven by Type Ia supernovae will tend to be suppressed. We assume here that $p = 1$ and SNu$(t_n)$ = 0.03 so Type Ia supernovae are unable to generate winds by themselves.
However, the enormous energy released by supernovae of Type II produced by massive stars at very early times can drive an outflow from the galaxy even in the presence of secondary infall. With a single burst, power law IMF described by $$\phi(m) dm = \phi_o m^{-(1+x)} dm~~~~~~m_{\ell} < m < m_u$$ the number of SNII per unit stellar mass formed is $$\eta_{II} = { N_{II} \over M_{*t}} = {x - 1 \over x}
{ m_8^{-x} - m_u^{-x} \over m_{\ell}^{1-x} - m_u^{1-x}}$$ where $M_{*t}$ is the total initial mass of stars. All stars greater than eight solar masses ($m > m_8$) are assumed to become SNII. For the Salpeter IMF discussed above ($x = 1.35$, $m_{\ell} = 0.08$, $m_u = 100$) we find $\eta_{II} = 6.81 \times 10^{-3}$ SNII per $M_{\odot}$. The specific supernova number $\eta_{II}$ is sensitive both to the IMF slope and the mass limits; for example $\eta_{II} = 11.6 \times 10^{-3}$ for $x,m_{\ell},m_u = 1.35,
0.3, 100$ and $\eta_{II} = 16.1 \times 10^{-3}$ for $x,m_{\ell},m_u = 1., 0.08, 100$. We conclude that $\eta_{II}$ is uncertain by a factor of at least 2 or 3. If $\eta_{II} = 6.81 \times 10^{-3}$, the total amount of SNII energy produced during the single burst formation of NGC 4472 ($M_{*t} = 7.26 \times 10^{11}$ $M_{\odot}$) is $4.9 \times 10^{60}$ ergs assuming $E_{sn} = 10^{51}$ ergs per supernova. An additional uncertainty is the efficiency $\epsilon_{sn}$ of communicating this SNII energy to the hot interstellar gas; if massive stars form near cold, dusty gas clouds a considerably fraction of SNII energy could be radiated away by dust and dense gas. In galaxy formation simulations it is assumed that $\epsilon_{sn} \sim 0.1 - 0.2$ of the supernova energy is returned to the interstellar gas (e.g. Kauffmann, White & Guiderdoni 1993), but in general $\epsilon_{sn}$ must be regarded as an adjustable parameter.
THE STANDARD MODEL
==================
Our objective is to solve the gas dynamical equations from the earliest times to the present ($t_n = 13$ Gyrs) and identify model parameters that result in gas density and temperature profiles similar to those observed today. While the number of uncertain parameters is formidable – $\Omega$, $\Omega_b$, $t_*$, $t_n$, IMF, $\Lambda(T,z)$, SNII energy, $\epsilon_{sn}$, etc. – there are also robust constraints set by cosmological density variations and the global conservation of dark mass, baryonic mass and energy.
We begin by describing the results of our “standard” model. The standard model is not chosen for its perfect agreement with $n(r)$ and $T(r)$ observed in NGC 4472, although that agreement is quite satisfactory, but because the parameters and procedures used in the standard model are the most plausible and least controversial. In our standard model we assume $\Omega_b = 0.05$ and form the galaxy at time $t_* = 2$ Gyrs. A Salpeter IMF is used with $m_{\ell} = 0.08$ and $m_u = 100$ for which $\alpha_*(t_s) = 4.7 \times 10^{-20}
(t/t_s)^{-1.26}$ s$^{-1}$ is an approximate, time-averaged value of the stellar mass loss rate and $t_s = t_n - t_*$ is the current age of the stars. The IMF also determines the specific energy generated by SNII explosions, $\eta_{std} = 6.81 \times 10^{-3}$ SNII per $M_{\odot}$, and we assume that this energy is fully converted to the hot gas, $\epsilon_{sn} = 1$. Clearly the important parameter for any model is $\eta_{II} \epsilon_{sn}$. The current rate of Type Ia supernovae is $SNu(t_n) = 0.03$ and it has decreased with time as $t^{-p}$ with $p = 1$. In the standard model the radiative cooling was varied with radius to simulate an increasing gas metallicity closer to the center of the galaxy, $\Lambda(T,r) = \Lambda(T)f(r)$ where $f(r) = \max[0.3,~2 - 1.7(r/r_z)]$ with $r_z = 40$ kpc. Such a variation is consistent with the expected radial variation of gaseous metallicity and the metal-dependent cooling coefficients described by Sutherland & Dopita (1993). The function $f(r)$ is designed to match a radial abundance variation similar to that observed in the hot interstellar gas in NGC 4472 (Forman et al. 1993; Matsushita 1997).
In Figure 3 we compare the computed gas density $n(r,t_n)$ and temperature $T(r,t_n)$ for the standard model with observations of NGC 4472. The fit to the observed temperature is excellent and the the density agreement is acceptable, within a factor of 2 of the observed values over a range of 100 in radius. Recall that the “observed” density variation is also somewhat uncertain as discussed earlier. Nevertheless, it appears that the gas density observed beyond 10 kpc is higher than that of the standard model while the reverse is true within 10 kpc. We have found that the first of these problems can be corrected while the second is more difficult.
Figure 4 shows in more detail the spatial variation of gas density, temperature, pressure and velocity over the full range of our calculations at three times including $t_n$. It is remarkable how little the central flow profiles have changed since $t = 4$ Gyrs. The fraction of the gas density within 100 kpc due to stellar mass loss decreases slowly with $\alpha_*(t)$. While new gas continues to flow into the galaxy at later times, this gas accumulates largely in the outer regions which grow in size while the solution within $\sim100$ kpc remains in a quasi-steady state. Notice that there are two shocks. At time $t_n$ the main accretion shock is at $\log r_{kpc} = 3$ and it is preceded by a second, smaller shock at $\log r_{kpc} = 3.4$. Beyond this outer shock, at the turnaround radius, $\log r_{tan} = 3.6$, the perturbed Hubble flow velocity vanishes. The influence of the outer shock can be clearly seen in all flow variables. This outer shock begins at time $t_*$ when the SNII energy is released; as it moves outward it creates a transient baryonic outflow that pushes some gas far out into the galactic halo and beyond. At later times radiative losses just behind the outer shock are significant. The small increase in gas density before entering this outer shock is an expected feature of the perturbed Hubble flow. This calculation was performed with an intergalactic gas temperature of $10$ K, but the flow pattern within the outer shock was unchanged when the calculation was repeated with intergalactic temperature increased to $10^4$ K.
The radial variation of the local baryonic fraction at time $t_n$ is shown in Figure 5 for the standard solution; the hot gaseous contribution to the baryon fraction is shown by the dashed line. Interior to $\sim 100$ kpc most of the baryons are stellar, but the gaseous contribution becomes significant at 1 Mpc and dominant beyond 10 Mpc. The most interesting feature in this plot is the large baryonic minimum centered at $\sim 500$ kpc – this feature is a relic of the gaseous evacuation created by the release of SNII energy. Baryons formerly in this region were pushed outward, forming a region of baryonic overdensity ($\Omega_b > 0.05$) just beyond a radius of 2.5 Mpc. Clearly such a large radius is irrelevant for NGC 4472 since it would enclose the entire Virgo cluster. However, the baryonic depletion is real and the rapid spatial variation of $\Omega_b/\Omega$ in the outer galactic halo must be considered in interpreting observed values of the baryon fraction. For example, [*ROSAT*]{} observations in NGC 4472 extend only to radius $\sim 150$ kpc where the local baryon fraction is seen to be very steeply declining in Figure 5. Clearly, measurements of the baryon fraction at such galactic radii could easily be greater [*or less*]{} than the true cosmic value ($\Omega_b/\Omega = 0.05$ for our standard model).
We show in Figure 6 the time variation of the mass of hot gas, cooled gas and stars in the standard model. At time $t_* = 2$ Gyrs all of the cold gas and some of the hot gas is consumed to form the single burst of stars in NGC 4472. Thereafter the stellar mass is assumed to be constant, $M_*(t_*) = M_*(t_n)$; we have ignored the small ($\sim 20$ percent) decrease in $M_*$ expected from stellar mass loss since time $t_n$. Although hot gas ($T > 10^6$ K) continues to increase within the main accretion shock, the amount of hot gas within 150 kpc is constant for $t \gta 3$ Gyrs, suggesting a quasi-steady state in this part of the flow. Cold gas continues to form after time $t_*$ reaching a final mass $M_{cg}(t_n) = 4.8 \times 10^{10}$ $M_{\odot}$ at the end of the calculation. While $M_{cg}(t_n)$ is comparable to the total mass of hot gas within 150 kpc, it is only $\sim 7$ percent of the current stellar mass. Unlike our earlier models in which we assumed that all of the hot interstellar gas is generated by stellar mass loss, the additional supply of cosmic gas at later times in the standard solution also results in more cold gas overall.
In computing the gravitational potential we have ignored the mass of cold gas, assuming that some or all of it contributed to later generations of stars, formed throughout a significant volume of the galaxy. However, it is interesting to relax this assumption and include the gravity of cold gas which, in our idealized non-rotating models, forms a concentrated point mass at the very center of the galaxy. This variant of the standard model is shown in Figure 7. Compared to Figure 3, the density profile in Figure 7 is slightly improved within a few kpc, but gas within about 5 kpc of the galactic core in Figure 7 becomes heated as it compresses in the point mass potential established by the cold gas. This type of central heating, with temperatures extending to $2 - 3 \times 10^7$ K, has not been observed. We have examined the convective stability of this core of hot gas and find that it is stable apart from a small rather insignificant region $\lta 500$ pc from the very center. However, the massive, compact cloud of cold gas that creates the central potential would also increase the central stellar velocity dispersion in a manner similar to that of a central massive black hole. But the total mass of cooled gas in this model, $M_{cg}(t_n) = 4.8 \times 10^{10}$ $M_{\odot}$, is about 15 times more massive than the largest known central black hole in elliptical galaxies, $3 \times 10^9$ $M_{\odot}$ for M87 (Kormendy & Richstone 1995). [*We conclude that all the interstellar gas that cools since $t_*$ cannot reside within 2 or 3 kpc of the galactic center since its gravity would unrealistically heat both the stars and hot gas.*]{} We therefore propose that this cooled gas goes into a stellar subsystem or disk of dimension $\gta 3$ kpc where its gravitational influence is greatly lessened. This assumption is consistent with our neglect of the gravity of the cold gas and, since $M_{cg}(t_n) \ll M_*$, the additional contribution of the cold gas mass to stars is relatively insignificant. The stellar mass within 1 kpc $M_*(1kpc) = 4.1 \times 10^{10}$ $M_{\odot}$ is comparable to $M_{cg}(t_n)$, but the stellar mass within 3 kpc is considerably larger, $13 \times 10^{10}$ $M_{\odot}$. Only about half of the cold gas mass was available to form additional stars several Gyrs after $t_*$, more recently cooled gas must be accommodated in another way if there is to be no optical evidence for recent star formation. Disks are attractive, but we recognize that the formation of disks in rotating ellipticals (Brighenti & Mathews 1996; 1997a) is accompanied by centrally flattened X-ray images that have not yet been observed. Options for the final disposition of cold gas will be clarified in the future when we include galactic rotation in models such as those described here.
To illustrate the critical importance of additional circumgalactic gas in determining the global properties of hot gas in ellipticals, we show in Figure 8 the density and temperature variations at time $t_n$ for the standard model when the dark halo within 150 kpc is non-varying and the only source of interstellar gas is stellar mass loss. The dramatic inadequacy of this model is seen in the density plot where the mass of hot gas is far too low in the outer galaxy. But the gas temperature is also too low. The central gas temperature in this model is consistent with observed temperatures only at the very center where the observed temperature is nearly equal to the stellar virial temperature $\langle T_* \rangle
\approx 10^7$ K.
It is important to emphasize that the low temperature in this model cannot be brought into agreement with the observations by increasing the current and past rate of Type Ia (or Type II) supernovae. There are two reasons for this. First, increasing the SNIa rate also increases the interstellar iron abundance far above observed values (e.g. Loewenstein & Mathews 1991). Second, Brighenti & Mathews (1998) found that the gas temperature profile at $t_n$ is surprisingly insensitive to increases in the SNIa rate until that rate becomes large enough to drive a galactic wind. When a wind develops the gas temperature rises far too much and the gas density falls very far below observed values at every galactic radius. We could find no value of SNu$(t)$ that corrected the obvious temperature discrepancies like that shown in Figure 8. Davis and White (1996) found that gas temperatures in all bright ellipticals exceed the stellar virial temperature $T_*$ by factors of $\sim$1.5, as we find in Figure 3. We conclude that the long term accumulation of cosmic gas in the dark halo – that naturally results in both densities and temperatures similar to those observed – is essential to understand the nature of hot gas in ellipticals today.
VARIATIONS ON THE STANDARD MODEL
================================
We now briefly discuss additional models in which some of the parameters are varied from those used in the standard model; unless otherwise indicated the parameters are the same as those used in the standard model.
Variation of $\Omega_b$
-----------------------
Within the restrictions imposed by primordial nucleosynthesis, $\Omega_b (H/50)^2 = 0.05 \pm 0.01$ is tightly constrained. Nevertheless, we find that our models are sensitive to these small changes in $\Omega_b$. For example Figure 9 shows the density and temperature profiles at $t_n = 13$ Gyrs for $\Omega_b = 0.04$ and 0.06. The overall agreement with the observed density and temperature profiles is noticeably degraded compared to the $\Omega_b = 0.05$ standard model described earlier. Evidently the amount of cosmically inflowing gas that enters the galaxy has a profound influence on its X-ray appearance.
The inflow of gas at the outer edge of the galaxy, some of which may have previously flowed out of the galaxy during the release of SNII energy, also strongly influences the metallicity of hot gas deep inside the galaxy. Interstellar gas within the stellar galaxy is a blend of gas that has come from stellar mass loss and from cosmic inflow; its net metallicity is also a blend. To illustrate the contribution from these two sources of gas and its dependence on $\Omega_b$, we show in Figure 10 the fraction of gas at time $t_n$ that has come from stellar mass loss within the galaxy since time $t_*$ when the SNII energy was released. In the $\Omega_b = 0.04$ solution almost all of the gas within the galaxy must have about the same average metallicity as the stars. In fact some of the gas lost from the stars in this model since $t_*$ has moved out beyond the stellar system which extends only to 100 kpc. By contrast, only 0.7 and 0.3 of the gas at $r_e = 8.57$ kpc has come from post-$t_*$ stellar mass loss for $\Omega_b = 0.05$ or 0.06 respectively. The blending of two sources of interstellar gas, having different metallicities, complicates further the current controversy regarding the apparent discrepancy of the gaseous iron abundance with that in the stars, $z_{Fe,gas} < z_{Fe,*}$ (Arimoto et al. 1997; Renzini 1998). Our $\Omega_b = 0.04$ model may be inconsistent with this inequality.
Variation of SNII Energy and $t_*$
----------------------------------
The discrepancies seen in Figure 9 must not be taken too seriously since some of the misalignment with observations can be removed by adjusting the total SNII energy ($\eta_{II}$) released and the epoch $t_*$ of galaxy formation. For example, when $\Omega_b = 0.06$ the excess gas density in Figure 9 can be expelled from the galaxy by increasing the total SNII energy released at time $t_*$ by $2$. Alternatively, if the SNII energy is left unchanged, the $\Omega_b = 0.06$ model can be improved by forming the stars at an earlier time, $t_* \sim 1.5$ Gyr, since there is just enough baryonic gas available at that early time to construct the stellar galaxy of mass $M_*$. Conversely, the $\Omega_b = 0.04$ solution in Figure 9 can be improved by lowering the SNII energy by $2$ and by assuming $t_* \approx 3$ Gyrs. Several models adjusted in this manner are shown in Figure 11. As $t_*$ is increased the mass of gas within the accretion shock is greater so the post-SNII temperature is lower; this follows since we suppose that the SNII energy is deposited within the shock radius at $t_*$. In all our models we have assumed complete efficiency in converting SNII energy to the hot gas, $\epsilon_{sn} = 1$, so we have probably somewhat overestimated the SNII heating. Each model depends on the product $\epsilon_{sn} \eta_{II}$ so if $\epsilon_{sn} < 1$ then $\eta_{II}$ would need to be increased by $1/\epsilon_{sn}$ for identical results.
In principle the degeneracy in the choice of the parameters $\Omega_b$, $\eta_{II}$ and $t_*$ can be partially removed by considering in detail the enrichment by elements produced in SNII events. We shall not attempt this here, but simply note that to compensate a relatively small, $\pm 0.01$ change in $\Omega_b$ requires a factor of $\sim 2$ change in the SNII energy or in the mass available to be heated. It is therefore unlikely that successful models are possible if $\Omega_b$ is much greater or less than the bounds set by nucleosynthesis.
Finally, we have computed models in which SNII heating is ignored altogether. In such models the gas density at $t_n$ greatly exceeds (by factors of 2 - 5) the observed gas density at all galactic radii. The amount of cold gas at time $t_n$ is also unacceptably large, $M_{cg}(t_n) \gta 10^{11}$ $M_{\odot}$. We conclude that SNII heating plays a critical role in establishing the amount of hot gas observed today in massive elliptical galaxies.
Variation of Cooling or Heating
-------------------------------
In our standard model we allow for a modest negative gradient in the gas metallicity by increasing $\Lambda(T,z)$ toward the center of the galaxy. When compared to models with spatially constant $\Lambda$, the $d \Lambda/ d r < 0$ assumption slightly improves the fit to the observed gas density for $r \lta 10$ kpc. There is an inconsistency with this approach since the observed densities have been determined assuming uniform metallicity. However, the sensitivity of our models to reasonable variations of $\Lambda$ is not large.
We have also explored the hypothesis advanced by Tucker & David (1997), Binney & Tabor (1995), and Ciotti & Ostriker (1997) that an additional heating process is present in the hot gas that reduces the total amount of cold gas that forms over a Hubble time. However, we have not succeeded in finding a heating source term for the thermal energy equation that significantly improved our models. If a small amount of heating due to some spatially distributed source is present, it is either rather ineffective or, when increased, generates outward propagating waves or shocks that result in peculiar density and temperature profiles and unsatisfactory X-ray surface brightness distributions. The limiting version of such models are those in which there is no radiative cooling at all, as if heating and cooling were in perfect balance, as some have suggested. When the standard model is recalculated with $\Lambda = 0$, for example, we find that the gas density in $r \lta 40$ kpc rises as $n \propto r^{-2.5}$, completely diverging from the observations; the gas temperature is too low by $\sim 0.5$ in this same region. In general therefore, we find that additional sources of distributed heating can introduce undesirable side effects that tend to degrade the agreement with observations.
Two kinds of “dropout”
----------------------
The deviation between our theoretical models and the observed gas density is almost always in the sense that the model profile is too centrally peaked. This type of deviation has been noticed for many years and has led to the hypothesis of distributed mass “dropout” in which hot gas is assumed to disappear from the flow according to some radial dropout function that can be varied to obtain agreement with observations (e.g. Fabian & Nulsen 1977; Sarazin & Ashe 1989). Originally it was hoped that the mass dropout could be understood as some type of thermal instability resulting from an inhomogeneous flow. Unfortunately, detailed hydrodynamical calculations have shown that thermal instabilities are disrupted by a variety of gas dynamical instabilities before an appreciable amount of cold gas forms; see Brighenti & Mathews (1998) for a more detailed discussion of dropout and additional references.
Nevertheless, in view of the popularity of mass dropout models we illustrate in Figure 12 the density and temperature structure at time $t_n$ in the standard model when an additional sink term $-q(\rho/t_{do})$ is included on the right side of the continuity equation. The model in Figure 12 is based on $q = 1$ and the dropout time is given by the local radiative cooling time $t_{do} = 5 m_p k T /2 \rho \mu \Lambda$ (Sarazin & Ashe 1989). The dense, cooling regions that are dropping out contribute substantially to the total emission, reducing the apparent gas temperature $T_{obs}$ below that of the the undisturbed ambient gas $T$; both temperatures are shown in Figure 12.
However, almost the same density profile can be found if the stellar mass ejection rate $\alpha_*$ is uniformly lowered. For comparison, a model based on lowering $\alpha_*$ by 0.5 at all times is shown with dotted lines in Figure 12.
A decrease in $\alpha_*(t)$ can be understood from two astrophysical arguments. First it is possible to reduce $\alpha_*$ by altering the (power law) IMF parameters $x$, $m_{\ell}$ and $m_u$. Unfortunately the stellar mass to light ratio also changes. For example, for values of $x$, $m_{\ell}$ and $m_u$ that reduce $\alpha_*(t_n)$ by 2, the mass to light ratio $M/L_V$ must increase by $\sim 2$ for any choice of the three adjustable IMF parameters. Such a large stellar $M/L_V$ is inconsistent with the value determined by van der Marel (1991) and with the excellent fit to the X-ray data for NGC 4472 using his $M/L_V$ (Brighenti & Mathews 1997b). Second, $\alpha_*$ can be “virtually” lowered if some of the gas ejected from evolving stars (having a normal IMF) never enters the hot gas phase. This idea was discussed by Mathews (1990) but discarded because of the many hydrodynamic instabilities that can disrupt stellar ejecta, increase its surface area and promote dissipative and conductive heating by the hot gas environment. However, if these difficulties can be avoided, the results in Figure 12 suggests that observations of NGC 4472 may be consistent with this second type of “dropout”.
TIDAL TRUNCATION AND THE RANGE OF X-RAY PROPERTIES AMONG ELLIPTICALS
====================================================================
For many years X-ray astronomers have noted that the X-ray luminosities of ellipticals $L_x$ span a huge range for galaxies of similar optical luminosity $L_B$ (Eskridge, Fabbiano & Kim 1995). In spite of much effort, no intrinsic property of elliptical galaxies had been found that correlated with this scatter (e.g. White & Sarazin 1991). However, in a recent paper (Mathews & Brighenti 1998) we showed that $L_x/L_B$ correlates significantly with the physical size of the X-ray source, $L_x/L_B \propto (r_{ex}/r_e)^{0.60 \pm 0.30}$, where $r_{ex}$ is the radius that contains half of the X-ray luminosity in projection. The discovery of this correlation was not accidental, but was inspired by the hypothesis that halo gas and dark matter could be tidally traded among massive early type galaxies in small groups. Such a tidal exchange of halo material would result in large, tidally dominant and centrally located ellipticals having huge, extensive halos and secondary donor (giant) ellipticals with more modest, tidally truncated dark matter and hot gas halos. Because of the short dynamical time scales in galaxy groups, many authors have noted that groups are ideal for both the production of ellipticals by mergers and for promoting tidal warfare among the component galaxies (Merritt 1985; Bode et al. 1994; Garciagomez et al. 1996; Athanassoula et al. 1997; Dubinski 1997). It was this hypothesis that led us to seek the correlation between $L_x/L_B$ and $r_{ex}/r_e$.
We can now test this hypothesis by truncating the dark matter and hot gas halos in the models described in this paper. Although we cannot describe fully realistic tidal effects with our spherically symmetric models, tidal truncations can be approximately simulated simply by removing all dark matter and hot gas beyond a truncation radius $r_{tr}$ at some time $t_{tr}$ in the past. For this purpose we begin not with a model that describes NGC 4472 at time $t_n$, but with an elliptical with somewhat higher $L_x$ and containing more hot gas. This can be accomplished simply by lowering the SNII energy below that of our standard model. Immediately after the truncation, hot gas just within $r_{tr}$ flows outward and a rarefaction wave passes into the interstellar gas at the sound speed. After a sound crossing time from $r_{tr}$ to the center of galaxy, $\sim 1$ Gyr, a new cooling flow equilibrium is established.
In Figure 13 we show as dashed lines the final density and temperature profiles at time $t_n$ for the standard model but modified with $\eta_{II} = 0.3 \eta_{std}$. As expected, this model is overdense relative to NGC 4472 at large radii. Also shown with solid lines are the current profiles of a model that was truncated at $t_{tr} = 9$ Gyrs at radius $r_{tr} = 400$ kpc but is otherwise identical to the first model. The outer density profile of NGC 4472 is nicely fit by the truncated model and agreement with observed gas temperatures is satisfactory. [*Since the size of the observed X-ray image of NGC 4472, $r_{ex} = 2.87 r_e$, is average for X-ray bright ellipticals of similar $L_B$, it is very likely that NGC 4472 suffered such a truncation in its past.*]{} This truncation could have been accomplished by nearby group galaxies which may not be as massive as 4472. We find that the final $n(r)$ and $T(r)$ profiles are rather insensitive to the exact time when the truncation occurred.
Finally, in Figure 14 we show the trajectory in the ($L_x/L_B,~r_{ex}/r_e$)-plane caused by truncations of this $\eta_{II} = 0.3 \eta_{std}$ model at five different radii $r_{tr}$, all at time $t_{tr} = 9$ Gyrs. Data for the observed galaxies is provided in Table 1 of Mathews & Brighenti (1998). The large $\times$ at the right in Figure 14 shows the final location of the untruncated, gas-rich galaxy shown in Figure 13. The position of each small $\times$ on the left shows the final locus of the truncated models. Note that models truncated at $r_{tr} = 300$ and (particularly) 400 kpc lie rather close to the observed locus of NGC 4472, as one would expect from Figure 13. [*As models of gas-rich ellipticals are truncated at different radii, they move precisely along the correlation among observed ellipticals.*]{} This most encouraging result strongly supports our hypothesis that the spread in the ($L_x/L_B,~r_{ex}/r_e$)-plane is generated by tidal exchanges of halo material in small groups.
FURTHER REMARKS
===============
We have not considered here the detailed metal enrichment of the hot gas expected from Type Ia and II supernovae; this will be the subject of a future study. Clearly, accurate observations and theoretical predictions of the radial variation of the abundance of iron, silicon and other elements in the hot gas will provide important additional constraints on the formation and evolution of elliptical galaxies. In the absence of this detailed information, it is interesting to compare the total amount of iron produced by SNII in NGC 4472 with the iron present in the stellar system today. According to Trager (1997) the ratio of iron to $\alpha$ elements is less than solar in most bright ellipticals, suggesting that the mean stellar iron abundance in NGC 4472 is about $\langle z_{Fe,*} \rangle \sim 0.7$. The total mass of stellar iron in NGC 4472 is then $M_{Fe} \approx \langle z_{Fe,*} \rangle M_*/1.4 =
6 \times 10^8$ $M_{\odot}$ where 1.4 is the ratio of total to hydrogen mass. If $\sim 0.14$ $M_{\odot}$ of iron is produced in each SNII event (Gibson, Loewenstein, & Mushotzky 1998), then Type II supernovae should produce a total iron mass of $\sim 7 \times 10^8$ $M_{\odot}$, comparable but somewhat larger than the mass of iron observed in stars today. Evidently the iron-producing capability of the Salpeter IMF is sufficient to account for all the iron in stars. In making this estimate we have assumed that the iron enrichment due to Type Ia supernovae goes primarily into the hot gas phase rather than into stars; the total amount of iron in the hot gas within $150$ kpc is rather small, $\sim 3 \times 10^7$ $M_{\odot}$.
In general the density profiles in our most successful models resemble power laws throughout most of the observed galaxy while the observed $n(r)$ suggests a double power law with a flatter slope within $\sim 16$ kpc. As a result our models predict a steeper gas gradient within the central $\sim 16$ kpc than is observed. It is this small deviation that the “dropout” hypothesis was designed to correct. While future observations will clarify this curious systematic deviation, it is possible that new physical effects become important in $r \lta 10$ kpc. We have already noted that a small, but realistic galactic rotation could flatten the azimuthally-averaged gas density profile in this region. Another possibility within $\sim$1 kpc is the growth of magnetic pressure toward the galactic center. Brighenti & Mathews (1997b) have shown that the total mass predicted from hydrostatic models is systematically lower than the known stellar mass within the central kiloparsec in each of the three bright ellipticals studied with [*Einstein*]{} HRI. This deviation can be understood if the gas pressure is assisted by an additional component of non-thermal pressure. The required magnetic field is large – we estimated $B \sim 10^{-4}$ gauss for NGC 4636 – but not unexpected. Mathews & Brighenti (1997) showed that fields of this magnitude are a natural outcome of the ejection of fields by mass-losing stars. Fields ejected in stellar envelopes are amplified by turbulent dynamo action in the interstellar medium and by the natural intensification that accompanies the inward flow of interstellar gas in which the magnetic pressure increases more rapidly than the gas pressure. The importance of magnetic or other non-thermal pressure components in the interstellar cores of ellipticals will be clarified when AXAF observations become available.
CONCLUSIONS
===========
We have described here the overall evolution of massive elliptical galaxies with particular attention to the hot interstellar gas. Although the stellar systems in giant ellipticals are thought to be very old, the mixing of the stellar orbits masks many details of the dynamical and enrichment processes that occurred during the time of galaxy formation. The metallicity gradient in the stars provides some clue to these early events but its interpretation is far from obvious.
In contrast, the hot interstellar gas in ellipticals today does retain a large amount of information about star formation and metal enrichment events that occurred during the earliest phases of galaxy evolution. Interstellar gas in ellipticals can help us solve one of the thorniest problems in modern studies of galaxy formation: the importance of “feedback” SNII energy on the gas. Among the major feedback uncertainties are: the total amount of supernova energy involved, its efficiency in heating the gas, and the time that SNII enrichment occurred relative to the dynamical formation of the galaxies. Regarding this last point, it is currently unclear if some fraction of the metals in the hot gas were made and ejected from an earlier generation of low mass galaxies before the formation of galaxy groups, or if all the metals were made during the formation of group galaxies that merged soon thereafter to create the ellipticals observed today.
We believe that studies of the creation and evolution of hot gas in large ellipticals provide a unique insight into the feedback problem and may help us unravel some of these complex events that occurred so long ago. Many lines of astronomical evidence indicate that the formation sites of most massive ellipticals are small groups of galaxies that are very old. Dynamical studies of the evolution of galaxies in elliptical-forming groups indicate that ellipticals formed early in the dynamical evolution of these groups when mergers and tidal interactions were most likely; if so, both the elliptical galaxy and the hot interstellar gas surrounding it are very old. Remarkably, the hot gas we see today still contains a record of these past events. Since SNII enrichment is accompanied by an increase in entropy due to heating of the gas, enrichment and gas dynamics are closely linked.
In this study we have described the first comprehensive theoretical description of the evolution of hot gas in massive ellipticals that is fully consistent with important cosmological constraints. The goal of our calculations is not merely to explain the total mass and energy in the interstellar medium, but to account for the detailed radial dependence of these quantities throughout the observable range. In spite of the simplicity of our galaxy-forming model and our choice of a currently unfashionable flat cosmology, we have been remarkably successful in explaining X-ray observations of bright ellipticals in a cosmological context that conserves baryonic and dark mass, allows for all energy sources involved, and includes the detailed interaction of gas with mass-losing stars.
We now summarize some of the main conclusions of the calculations we have discussed here:
\(1) Hot gas density and temperature profiles in massive ellipticals cannot be understood solely with gas ejected from the old stellar population that accounts for the optical light observed today.
\(2) Low interstellar gas temperatures in models without cosmic inflow cannot be heated to the observed values by current and past Type Ia supernovae; if this is attempted, strong galactic winds develop that are incompatible with observation.
\(3) Excellent agreement with observed hot gas mass and thermal energy distributions – $n(r)$ and $T(r)$ – can be achieved by the addition of cosmic gas from secondary infall. This gas shocks and compresses to the virial temperature of the galactic halo, accounting for the hot gas temperatures observed in all massive ellipticals. As this gas flows deeper into the galaxy, it mixes with somewhat cooler gas expelled from the stars, resulting in positive gas temperature gradients ($dT/dr > 0$) in $r \lta 3 r_e$ as observed in most or all bright ellipticals (see Fig. 1 of Brighenti & Mathews 1997b).
\(4) Within the framework of a flat cosmology, we can account for observed gas density and temperature profiles only for a rather narrow range in baryon mass, $\Omega_b \approx
0.05 \pm 0.01$, the same range that is consistent with primordial nucleosynthesis.
\(5) The fraction of gas near the optical effective radius that derives from stellar mass loss is strongly and inversely dependent on $\Omega_b$ when other parameters are held constant.
\(6) However, our successful fits to observed $n(r)$ and $T(r)$ are degenerate in some of the parameters. For example, increasing $\Omega_b$ is approximately equivalent to (i) decreasing the SNII energy ($\eta_{II}$) or its efficiency in heating the gas ($\epsilon_{sn}$) or (ii) increasing the time $t_*$ that the bulk of galactic stars formed so that more gas (within the accretion shock) needs to be heated by SNII. Both (i) and (ii) reduce the energy per gram delivered to the gas by SNII explosions. This degeneracy is due in part to the simple, single burst assumption we have made in forming stars and our assumption that SNII energy is deposited within the shock radius at time $t_*$. More complex models that consider the details of metal enrichment and other options for the spatial distribution of SNII energy may remove some or all of this degeneracy.
\(7) The current fraction of mass in baryons is expected to have a large minimum surrounding large ellipticals and their associated groups. This minimum results from an outward displacement of baryons that occurred when the energy of Type II supernovae was released. As a result, observational measurements of $\Omega_b/\Omega$ within large galactic radii can either overestimate or underestimate the true cosmic value of this ratio.
\(8) Since the epoch of star formation the amount of gas that has cooled is comparable to the total mass of hot gas observed today. This gas cannot all cool within the central few kiloparsecs since its mass would unrealistically heat both the gas and stars there. However, even a tiny galactic rotation is sufficient to avoid this central concentration of cooled gas.
\(9) Finally, when a relatively gas-rich version of one of our models is tidally truncated at some time in the past, the X-ray luminosity $L_x$ and the effective radius for X-ray emission $r_{ex}$ both decrease. The vector due to truncation in the ($L_x/L_B,~r_{ex}/r_e$)-plane lies almost exactly along the correlation discovered by Mathews & Brighenti (1998). This give additional support to the notion, described by Mathews & Brighenti (1998), that the enormous spread in $L_x$ for fixed galactic $L_B$ is due in some large measure to the tidal exchange of halo gas and dark matter among galaxies within groups of galaxies.
We thank our Santa Cruz colleagues for enlightening remarks and helpful criticism. Thanks also to Caryl Gronwall for determining some stellar mass to light ratios for us. Our work on the evolution of hot gas in ellipticals is supported by NASA grant NAG 5-3060 for which we are very grateful. In addition FB is supported in part by Grant ARS-96-70 from the Agenzia Spaziale Italiana.
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abstract: 'In the cosmological blast-wave model for gamma ray bursts (GRBs), high energy ($\gtrsim 10$ GeV) $\gamma$ rays are produced either through Compton scattering of soft photons by ultrarelativistic electrons, or as a consequence of the acceleration of protons to ultrahigh energies. We describe the spectral and temporal characteristics of high energy $\gamma$ rays produced by both mechanisms, and discuss how these processes can be distinguished through observations with low-threshold Čerenkov telescopes or GLAST. We propose that Compton scattering of starlight photons by blast wave electrons can produce delayed flares of GeV – TeV radiation.'
address:
- |
Space Physics and Astronomy Department, Rice University, MS 108\
6100 S. Main Street, Houston, TX 77005-1892
- |
E. O. Hulburt Center for Space Research, Code 7653\
Naval Research Laboratory, Washington, DC 20375-5352
author:
- Markus Böttcher
- '& Charles D. Dermer'
title: |
Prompt and Delayed High-Energy Emission\
from Cosmological Gamma-Ray Bursts
---
Introduction
============
The blast-wave model has met with considerable success in explaining the X-ray, optical and radio afterglows of GRBs [@mes97]. The evolving blast wave accelerates particles to ultrarelativistic energies and has been suggested as the source of ultrahigh-energy cosmic rays (UHECRs) [@wax95; @vie95]. GeV – TeV photon production, due primarily to proton synchrotron radiation from UHECR acceleration in GRBs during the afterglow phases, has recently been predicted [@vie97; @boe98]. A large Compton luminosity when relativistic electrons scatter soft photons is expected as well. One source of seed photons is the electron synchrotron radiation, and the synchrotron self Compton (SSC) process has been calculated by [@chi98]. External soft photon sources could also be important in the Compton scattering process. Models for the GRB origin such as the hypernova/collapsar [@pac98; @woo93] scenario suggest that the sources of GRBs are associated with star-forming regions and might thus be embedded in a starlight radiation field. Here we examine the Compton scattering of starlight by relativistic electrons in the blast wave.
Compton Emission
================
We compare high energy $\gamma$-ray emissions due to SSC and SSL (Scattered StarLight) processes, which can be estimated by comparing the comoving blast-wave frame energy densities $u'_{SL}$ and $u'_{sy}$ of starlight and synchrotron photons, respectively. Knowledge of the peak energy of the seed photon spectrum, which determines whether Compton scattering is suppressed due to Klein-Nishina effects, is important for this comparison. The relativistic electrons are assumed to be distributed according to a power-law with index $\gtrsim 3$ and a low-energy cutoff $\gamma_{\rm e, min} =
\xi_e \, ( m_p / m_e) \, \Gamma$ [@mes94], where $\xi_e = 0.1 \, \xi_{e, -1}$ is an electron equipartition factor, and $\Gamma = 300 \, \Gamma_{300}$ is the bulk Lorentz factor of the blast wave. The magnetic field is $H = \sqrt{8 \pi \, r \, m_p c^2 \, n_0 \, \xi_H}
\Gamma$, where $r = 10 \, r_1$ is the shock compression ratio, $n_0$ is the number density of external matter, assumed uniform, and $\xi_H = 10^{-6} \, \xi_{H, -6}$ is a magnetic-field equipartition factor. A value of $\xi_H \sim 10^{-6}$ is required to produce synchrotron spectra resembling observed spectra in the prompt phase of GRBs [@chi98].
Synchrotron and SSC Processes
-----------------------------
The peak photon energy of the synchrotron spectrum is $\epsilon'_{sy} \simeq 10^{-4} \Gamma_{300}^3 \, n_0^{1/2}
\, q_{-4}$, where $q = \sqrt{r_1 \, \xi_H} \, \xi_e^2 = 10^{-4}
\, q_{-4}$ is a combined equipartition factor. The energy density of the synchrotron radiation field may be estimated using $u'_{sy} \approx \tau_T \, \gamma_{\rm e, min}^2 \,
u'_{H}$, where $\tau_T$ is the radial Thomson depth of the blast wave. Specifying the width of the shell, $\Delta x'
\approx x' / \Gamma$, at the deceleration radius, we find $$u'_{sy}\;
[{\rm erg
\> cm}^{-3}] \simeq 7.3 \cdot 10^{-6} \, r_1^2\, n_0^{5/3} \xi_{H, -6}
\, E_{52}^{1/3} \, \xi_{e, -1}^2 \; \Gamma_{300}^{10/3}\; ,$$ where $E_0 = 10^{52} \, E_{52}$ erg is the kinetic energy of the blast wave. SSC scattering can occur efficiently in the Thomson regime if $\xi_{e, -1} \, \Gamma_{300}^4 \, n_0^{1/2} \, q_{-4} \lesssim
1$. In this case, the SSC spectrum in the observer’s frame peaks at $$\epsilon_{\rm SSC} = h\nu_{\rm SSC}/m_ec^2 \simeq 10^8 \, \Gamma_{300}^6 \,
\xi_{e, -1}^2
\, n_0^{1/2} \, q_{-4} \, / \, (1 + z),$$ which cannot exceed $\epsilon_{\rm IC, max} \simeq 1.6 \cdot 10^7 \,
\xi_{e, -1} \, \Gamma_{300}^2/(1+z)$.
Scattered Starlight
-------------------
We assume that the blast wave approaches a star of luminosity $L_{\ast} = \l \, L_{\odot}$ at a distance $d = d_{15}$ cm from a given point within the blast wave. Then the energy density of the stellar radiation field in the blast-wave frame is $$u'_{SL} [{\rm erg \> cm}^{-3}] \simeq {1.3\cdot 10^{-3} \,
\l\Gamma_{300}^2 \over \, d_{15}^{2} }$$ [@ds94]. By comparing eqs. (1) and (3), we see that $u'_{SL}$ dominates $u'_{sy}$ when $d_{15} \lesssim d_{SSL}
\equiv 13 \, \l^{1/2} \, \Gamma_{300}^{-2/3} \, r_1^{-1} \,
n_0^{-5/6} \, \xi_{H, -6}^{-1/2} \, \xi_{e, -1}^{-1} \,
E_{52}^{-1/6}$.
If the starlight spectrum is approximated by a thermal blackbody of temperature $T_0$\[eV\], then this spectrum peaks in the comoving blast-wave frame at dimensionless photon energy $\epsilon'_{SL} \cong 1.6 \cdot 10^{-3} \,
\Gamma_{300} \, T_0$. Compton scattering can occur in the Thomson regime if $\Gamma_{300}^2 \, \xi_{e, -1} \, T_0 \lesssim 10^{-2}$, indicating that the SSL process becomes efficient only if $\Gamma \lesssim 30 \, (\xi_{e, -1} \, T_0)^{-1/2}$. This can be realized in dirty fireballs [@dcb99] or in the later afterglow phase of blast-wave deceleration. When the above condition is satisfied, the SSL spectrum in the observer’s frame peaks at $$\epsilon_{SSL} \cong 1.5 \cdot 10^5 \, \Gamma_{30}^4
\, T_0 \, \xi_{e, -1}^2 \, / \, (1 + z),$$ where $\Gamma_{30} = \Gamma/30$. The SSL component can dominate once scattering occurs primarily in the Thomson regime, at which point it is likely to be the dominant electron radiation mechanism in the 10 – 100 GeV regime.
The minimum duration of flares from the SSL process is $\Delta
t_{\rm min}\approx d/(\Gamma^2 c) \approx 40 d_{15}/\Gamma_{30}^2$ s. The fraction of energy in GeV – TeV photons produced by the SSL mechanism rather than through other processes is roughly given by $[d_{SSL}/(x_{bw}/\Gamma)]^2$, which represents the fraction of the observable blast wave area where $u'_{SL}$ dominates $u'_{sy}$. The term $x_{bw}$ is the distance of the blast wave from the explosion site. Čerenkov telescopes with threshold energies below $\sim$ 100 GeV and the planned GLAST satellite might be able to detect the predicted SSL radiation from nearby GRBs if they are indeed associated with star-forming regions.
Comparison with Gamma Rays from Hadronic Processes
==================================================
The high energy $\gamma$-ray signatures of UHECR acceleration in GRB blast waves have been investigated in detail in [@boe98]. If protons are accelerated efficiently in GRB blast waves, the spectrum in the $\sim 10$ – $100$ GeV range is expected to be dominated by a hard power-law with $\nu F_{\nu} \propto \nu^{+0.5}$ due to proton synchrotron radiation. Since protons are expected to cool inefficiently, while electrons suffer strong radiative cooling, the temporal decay of the proton synchrotron radiation is slower than for the synchrotron component component. Let $g$ be the index parametrizing the deceleration and thereby the radiative regime of the blast wave, then both components decay as $F_{\nu} (t) \propto t^{-\chi}$, but with different temporal indices. For synchrotron radiation from cooling electrons, $\chi_{\rm sy} = (4 g - 2)/(1 + 2 g)$, while for uncooled syncrotron radiation from UHECR, $\chi_{\rm p,sy}
= (4 g - 3) / (1 + 2 g)$. The temporal decay of the SSC radiation is more complicated, because scattering in the Klein-Nishina is important. Comparison of decay observations over a large range of photon energies with model calculations are necessary to accurately discriminate between the various processes. The SSL radiation can be distinguished from the SSC and the hadronic $\gamma$-ray emission by its rapid variability and by the flares it produces in the GeV range. Observations of flares of GeV - TeV radiation would support the hypothesis that GRBs occur within stellar associations and star-forming regions.
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|
---
abstract: 'We study Rokhlin dimension of ${{\mathbb Z}}^m$-actions on simple separable stably finite nuclear $C^*$-algebras. We prove that under suitable assumptions, a strongly outer ${{\mathbb Z}}^m$-action has finite Rokhlin dimension. This extends the known result for automorphisms. As an application, we show that for a large class of $C^*$-algebras, the ${{\mathbb Z}}^m$-Bernoulli action has finite Rokhlin dimension.'
author:
- 'Hung-Chang Liao'
bibliography:
- 'Biblio-Database.bib'
title: 'Rokhlin dimension of ${{\mathbb Z}}^m$-actions on simple $C^*$-algebras'
---
Introduction
============
Extending topological covering dimension to the context of nuclear $C^*$-algebras, Winter and Zacharias introduced *nuclear dimension* in [@WZ09]. Recent advances in the study of nuclear $C^*$-algebras have revealed that:
- Finiteness of nuclear dimension implies other structural properties, including Jiang-Su stability and strict comparison ([@Win12; @Ror04]). In fact, a conjecture of Toms and Winter predicts that these are all equivalent for unital separable simple nuclear $C^*$-algebras.
- Unital separable simple $C^*$-algebras of finite nuclear dimension have been completely classified by $K$-theoretic invariants, provided they satisfy a technical condition known as the UCT (Universal Coefficient Theorem) ([@Kir; @Phi00; @TWW15; @EGLN15]).
Given the success above, it is important to understand how nuclear dimension behaves under various constructions, such as crossed products. Motivated by the Kakutani-Rokhlin lemma in ergodic theory ([@Kak43; @Roh48]), its non-commutative analogues for finite von Neumann algebras ([@Con75; @Ocn85]) and later developments in the $C^*$-realm, Hirshberg, Winter, and Zacharias introduced *Rokhlin dimension* in [@HWZ15] for actions of finite groups and the set of integers ${{\mathbb Z}}$ on unital $C^*$-algebras. One of the main results in [@HWZ15] is that finiteness of nuclear dimension passes to crossed products when the action has finite Rokhlin dimension. In addition, they showed that:
- Actions with finite Rokhlin dimension are ubiquitous; more precisely, given a unital Jiang-Su stable $C^*$-algebra $A$, the automorphisms with finite Rokhlin dimension form a $G_\delta$-subset of the automorphism group $\operatorname{Aut}(A)$ ([@HWZ15 Theorem 3.4]).
- There are plenty of natural examples coming from actions on commutative $C^*$-algebras, i.e., topological dynamics. In fact, given any minimal homeomorphism of a compact metrizable space with finite covering dimension, the induced $C^*$-system has finite Rokhlin dimension ([@HWZ15 Theorem 6.1]).
The notion of Rokhlin dimension has been generalized to actions of ${{\mathbb Z}}^m$ by Szabó [@Sza15], and more generally to actions of residually finite groups by Szabó, Wu, and Zacharias [@SWZ15]. Similar results regarding nuclear dimension, genericity, and topological dynamics were also obtained.
The present paper focuses on $C^*$-dynamical systems of a purely non-commutative nature. We consider ${{\mathbb Z}}^m$-actions on simple separable stably finite nuclear (hence tracial) $C^*$-algebras. Our first main result is a Rokhlin type theorem: under suitable assumptions, a certain strong form of outerness, which is a necessary condition for the action to have finite Rokhlin dimension (see Proposition \[prop:main-converse\]), turns out to be sufficient. The precise statement is given below (see Section 2 and Section 5 for relevant definitions) Recall that *Property (SI)*, as introduced by Sato in [@Sat10], is a comparison-by-traces type property for central sequence algebras (see Definition \[defn:properptySI\] for the precise definition). This property is possessed by a large class of simple nuclear $C^*$-algebras, in particular the ones of finite nuclear dimension (cf. [@MS12a; @Win12; @Ror04]).
\[thm:1-1\] (Theorem \[thm:main\]) Let $A$ be a unital simple separable nuclear $C^*$-algebra with nonempty trace space $T(A)$, and let $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose
1. $A$ has property (SI),
2. $\alpha$ is strongly outer,
3. $T(A)$ is a Bauer simplex with finite dimensional extreme boundary,
4. $\tau\circ \alpha = \tau$ for every $\tau\in T(A)$.
Then ${\mathrm{dim}_{\mathrm{Rok}}}(\alpha) \leq 4^m-1$.
This type of results dates back to Connes’ non-commutative Rokhlin lemma [@Con76], where he showed that outer ${{\mathbb Z}}$-actions on finite von Neumann algebras have the ($W^*$)-Rokhlin property. Ocneanu in [@Ocn85] obtained a similar result for actions of discrete amenable groups. On the $C^*$-side, Hermann and Ocneanu introduced a Rokhlin property for automorphisms of $C^*$-algebras [@HO84]. The definition was later refined by Kishimoto [@Kis95], who proved that for certain AF and A${\mathbb{T}}$ algebras, strong outerness implies the Rokhlin property [@Kis96; @Kis98]. Nakamura studied ${{\mathbb Z}}^2$-actions on UHF algebras and obtained a similar Rokhlin type result [@Nak99]. Various Rokhlin type definitions and theorems also appeared in the work of Osaka and Phillips [@OP06a; @OP06b], Sato [@Sat10], and Matui and Sato [@MS12b; @MS14]. Recently, we proved a Rokhlin type theorem similar to Theorem \[thm:1-1\] for automorphisms [@Lia16], which was built upon a remarkable breakthrough made by Matui and Sato (unpublished; see [@Lia16 Section 6]).
In Section 2 we will prove the converse of Theorem \[thm:1-1\] (see Proposition \[prop:main-converse\]). Combining this and Remark \[rem:trace\], we obtain the following corollary:
Let $A$ be a unital simple separable nuclear $C^*$-algebra with nonempty trace space $T(A)$, and let $\alpha: {{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose
1. $A$ has property (SI),
2. $T(A)$ is a Bauer simplex with finite dimensional extreme boundary, and
3. $\tau\circ \alpha = \tau$ for every $\tau\in T(A)$.
Then the following are equivalent.
1. $\dim_{Rok}(\alpha) \leq 4^m-1$.
2. For every $v\in {{\mathbb Z}}^m$, the restriction map $r:T( A \rtimes_{\alpha^v}{{\mathbb Z}}^m )\to T(A)^{\alpha^v}$ is injective.
3. The action $\alpha$ is strongly outer.
While in general it is not easy to verify that a given action has finite Rokhlin dimension, strong outerness can often be checked explicitly. As an example, we study Bernoulli shifts in Section 6 and make the following observation.
(Proposition \[prop:bernoulli-shift\]) Let $A$ be a unital simple $C^*$-algebra with a unique trace. Suppose $A$ is not isomorphic to the algebra of complex numbers ${{\mathbb C}}$. Then the Bernoulli action $\sigma:{{\mathbb Z}}^m\to \operatorname{Aut}(\bigotimes_{{{\mathbb Z}}^m} A)$ is strongly outer. In particular, $\sigma$ has finite Rokhlin dimension.
In addition to this proposition, we observe that the Rokhlin dimension of a Bernoulli shift can be bounded by the Rokhlin dimension of a subshift (Proposition \[prop:subalgebra-trick\]). Combining these two results, we obtain a large class of examples of ${{\mathbb Z}}^m$-actions with finite Rokhlin dimension.
Let us describe how the paper is organized. In Section 2 we review the definition of Rokhlin dimension for ${{\mathbb Z}}^m$-actions. We show that as in the case of automorphisms, strong outerness is a necessary condition for an ${{\mathbb Z}}^m$-action to have finite Rokhlin dimension. We also review the notion of continuous $W^*$-bundles and discuss group actions on them. In Section 3 we prove a Rokhlin type theorem for ${{\mathbb Z}}^m$-actions on continuous $W^*$-bundles. The ideas and techniques are almost identical to the case of automorphisms, hence we will only outline the important steps and we refer the reader to [@Lia16] for full details. Section 4 is devoted to the technical notion we call the “approximate Rokhlin property”, which already appeared in our previous paper. Here it serves the same purpose: a bridge connecting the Rokhlin property at the level of $W^*$-bundles and Rokhlin dimension at the level of $C^*$-algebras. Section 5 contains a simple but crucial observation (Proposition \[prop:Zm-Rokhlin\]), which allows us to obtain finite Rokhlin dimension for ${{\mathbb Z}}^m$-actions from “equivariant Rokhlin towers for automorphisms” This section also contains a technical result which establishes the existence of these Rokhlin towers. Combining the observations and results from previous sections, we establish Theorem \[thm:1-1\]. In the final section, Section 6, we study Bernoulli shifts on infinite tensor products. Combining the Rokhlin type theorem from Section 5 and an observation involving subshifts, we show that for a large class of $C^*$-algebras, the ${{\mathbb Z}}^m$-Bernoulli shift has finite Rokhlin dimension.\
**Acknowledgment:** Much of this work was done under the Pritchard Dissertation Fellowship granted by the Department of Mathematics at Pennsylvania State University. The author wishes to thank the Department of Mathematics for its support. The author also wishes to thank his dissertation advisor, Nate Brown, for many valuable discussions, including the suggestion of Proposition \[prop:subalgebra-trick\].
Preliminaries
=============
Let us first fix some notations. Let $A$ be a $C^*$-algebra, $\Gamma$ be a discrete group, and $\alpha:\Gamma\to \operatorname{Aut}(A)$ be a group action. We write $\alpha^g$ for the automorphism $\alpha(g)\in \operatorname{Aut}(A)$. Suppose the trace space $T(A)$ of $A$ is nonempty, then the set of $\alpha^g$-invariant traces will be denoted $T(A)^{\alpha^g}$. Given any $\tau\in T(A)$, we write $(\pi_\tau, {{\mathcal H}}_\tau)$ for the GNS representation associated to $\tau$.
([@MS12b Definition 2.7]) Let $A$ be a unital $C^*$-algebra with nonempty trace space $T(A)$, and let $\Gamma$ be a discrete group. An action $\alpha:\Gamma\to \operatorname{Aut}(A)$ is called *strongly outer* if for every $g\in \Gamma\setminus\{e\}$ and every $\tau\in T(A)^{\alpha^g}$, the weak extension $\tilde{\alpha}^g\in \operatorname{Aut}(\pi_\tau(A)'')$ is an outer automorphism.
From now on we focus on the case $\Gamma = {{\mathbb Z}}^m$. We use $\xi_1,...,\xi_m$ for the standard generators of ${{\mathbb Z}}^m$ (i.e., $\xi_i$ is the unit vector with 1 on the i-th place and 0’s elsewhere). The automorphisms $\alpha^{\xi_i}$ will be denoted $\alpha^i$ for short. For each $n\in {{\mathbb N}}$, we write $B_n^m$ for the “box” $\{0,1,...,n-1\}^m$ in ${{\mathbb Z}}^m$. When there is no possible confusion, we drop the dimension $m$ and simply write $B_n$.
Below we recall the definition of Rokhlin dimension for ${{\mathbb Z}}^m$-actions. Roughly speaking, an action has finite Rokhlin dimension if we can find families of positive contractions in the algebra which are almost central, almost cyclically permuted by the action, almost mutually orthogonal, and almost sum to 1.
\[defn:Zm-Rok\] ([@Sza15 Definition 1.6]) Let $A$ be a unital $C^*$-algebra, and let $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. We say that the action $\alpha$ has Rokhlin dimension at most $d$, and write ${\mathrm{dim}_{\mathrm{Rok}}}(\alpha) \leq d$, if for any finite subset $F\subseteq A$, ${\varepsilon}> 0$, and $n\in {{\mathbb N}}$, there exist positive contractions $$(f_v^{(\ell)})_{v\in B_n}^{\ell=0,1,...,d}$$ in $A$ satisfying the following properties:
1. $\| \alpha^w(f_v^{(\ell)}) - f^{(\ell)}_{v+w}\| < {\varepsilon}\mod n\;\;\;\;\;\; (0\leq \ell\leq d,\; v\in B_n,\; w\in {{\mathbb Z}}^m)$.
2. $\| f^{(\ell)}_v f^{(\ell)}_{v'} \| < {\varepsilon}\;\;\;\;\;\; (0\leq \ell \leq d, \; v,v'\in B_n,\; v\neq v')$.
3. $\| [f_v^{(\ell)}, a ] \| < {\varepsilon}\;\;\;\;\;\; (0\leq \ell\leq d,\; v\in B_n,\; a\in F)$.
4. $\left\| \sum_{\ell=0}^d\sum_{v\in B_n}f_v^{(\ell)} - 1_A \right\| < {\varepsilon}$.
As in the case of ${{\mathbb Z}}$-actions, strong outerness is a necessary condition for an action to have finite Rokhlin dimension (when the $C^*$-algebra has traces).
\[prop:main-converse\] Let $A$ be a unital $C^*$-algebra with nonempty trace space $T(A)$, and let $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. If ${\mathrm{dim}_{\mathrm{Rok}}}(\alpha) = d < \infty$, then $\alpha$ is strongly outer.
Since the argument is almost identical to the one for implication (1) $\implies$ (3) of [@Lia16 Corollary 1.2], we only sketch the proof here. Given $v\in {{\mathbb Z}}^m$, let $u$ be the implementing unitary in the crossed product $A\rtimes_{\alpha^v}{{\mathbb Z}}$. Given any tracial state $\varphi$ on $A\rtimes_{\alpha^v}{{\mathbb Z}}$, one can show that $\varphi(au^n) = 0$ for any $a\in A$ and $n\in {{\mathbb N}}$ by approximating $1_A$ by the sum of the Rokhlin elements. Then the proof of [@Kis96 Lemma 4.4] shows that a weak extension of $\alpha^v$ cannot be inner.
\[rem:trace\] Note that the argument above also shows that for each $v\in {{\mathbb Z}}^m$, the restriction map $r:T(A\rtimes_{\alpha^v}{{\mathbb Z}}^m)\to T(A)^{\alpha^v}$ is injective (hence bijective).
We briefly review some basic definitions of continuous $W^*$-bundles, which was introduced by Ozawa in [@Oza13] (see also [@BBSTWW15]). A *continuous $W^*$-bundle* consists of the following data:
- a $C^*$-algebra ${{\mathcal M}}$,
- a compact metrizable space $K$ such that $C(K)$ sits inside the center of ${{\mathcal M}}$, and
- a faithful conditional expectation $E:{{\mathcal M}}\to C(K)$ satisfying the tracial condition $E(ab) = E(ba)$ for all $a,b\in {{\mathcal M}}$,
such that the norm-closed unit ball of ${{\mathcal M}}$ is complete in the *uniform 2-norm* $\|\cdot\|_{2,u}$, defined by $$\|x\|_{2,u} := \| E(x^*x) \|^{1/2}.$$ For each $\lambda\in K$, the GNS representation $\pi_\lambda({{\mathcal M}})$ coming from the tracial state $\tau_\lambda := ev_\lambda\circ E$ is called a *fiber* of the bundle $({{\mathcal M}}, K, E)$.
Given a unital simple separable $C^*$-algebra with nonempty trace space $T(A)$, we define the *uniform 2-norm* $\|\cdot\|_{2,u}$ on $A$ by $$\|a\|_{2,u} := \sup_{\tau\in T(A)}\tau(a^*a)^{1/2}.$$ Let $\overline{A}^u$ be the completion of $A$ with respect to $\|\cdot\|_{2,u}$. Ozawa proved in [@Oza13] that when $T(A)$ is a *Bauer simplex*, i.e., when the extremem boundary $\partial_e T(A)$ is compact, the set of continuous functions on $\partial_e T(A)$ can be embedded into the center of $\overline{A}^u$. Moreover, there is a faithful tracial conditional expectation $$E:\overline{A}^u \to C(\partial_e T(A))$$ satisfying $E(a)(\tau) = \tau(a)$ for each $a\in A$ and $\tau\in \partial_e T(A)$. Therefore $(\overline{A}^u, \partial_e T(A), E)$ has the structure of a continuous $W^*$-bundle. In this case each fiber $\pi_\tau(\overline{A}^u)$ is isomorphic to the weak closure $\pi_\tau(A)''$.
Given a free ultrafilter $\omega$ on ${{\mathbb N}}$, one can form the *ultrapower* $({{\mathcal M}}^\omega, E^\omega, K^\omega)$ and the *central sequence $W^*$-bundle* $({{\mathcal M}}^\omega\cap {{\mathcal M}}', E^\omega, K^\omega)$ of a continuous $W^*$-bundle $({{\mathcal M}}, K, E)$ in a way similar to the case of tracial von Neumann algebras. The uniform 2-norm on ${{\mathcal M}}^\omega$ will be denoted $\|\cdot\|_{2,u}^{(\omega)}$. For details, we refer the reader to [@BBSTWW15] and [@Lia16].
Now we discuss group actions on continuous $W^*$-bundles. Let $({{\mathcal M}}, K, E)$ be a continuous $W^*$-bundle and let $\Gamma$ be a discrete group. A *$\Gamma$-action* on $({{\mathcal M}}, K, E)$ consists of a pair $(\beta, \sigma)$, where $$\beta:\Gamma\to \operatorname{Aut}({{\mathcal M}})\;\;\;\;\;\; \text{ and }\;\;\;\;\;\; \sigma:\Gamma \to \text{Homeo}(K)$$ are group homomorphisms, such that the diagram
\(1) [${{\mathcal M}}$]{}; (2) \[right of=1\] [${{\mathcal M}}$]{}; (6) \[below of=1\] [$C(K)$]{}; (7) \[below of=2\] [$C(K)$]{}; (1) to node [$\beta^g$]{} (2); (6) to node [$\bar{\sigma}^g$]{} (7); (2) to node [$E$]{} (7); (1) to node [$E$]{} (6);
.
commutes for every $g\in \Gamma$. Given a group action $(\beta, \sigma):\Gamma\to \operatorname{Aut}({{\mathcal M}}, K, E)$, for each $g\in \Gamma$ and $\lambda\in K$, there is a well-defined \*-isomorphism $$\begin{aligned}
\beta_\lambda^g: \pi_\lambda({{\mathcal M}}) &\to \pi_{(\sigma_g)^{-1}(\lambda)}({{\mathcal M}}),\\
\pi_\lambda(x)&\mapsto \pi_{(\sigma_g)^{-1}(\lambda)}(\beta^g(x)).\end{aligned}$$ When $\sigma$ is the trivial action on $K$, this gives rise to a group action $$\beta_\lambda:\Gamma\to \operatorname{Aut}(\pi_\lambda({{\mathcal M}})).$$
Let $\omega$ be a free ultrafilter on ${{\mathbb N}}$. A group action $(\beta, \sigma)$ can be promoted to an action on the ultrapower or the central sequence $W^*$-bundle. More precisely, for each $g\in \Gamma$ we have automorphisms $$\begin{aligned}
\beta_\omega^g:{{\mathcal M}}^\omega&\to {{\mathcal M}}^\omega,\\
(x_n)_n&\mapsto (\beta^g(x_n))_n,\end{aligned}$$ and $$\begin{aligned}
\bar{\sigma}_\omega^g:C(K^\omega)&\to C(K^\omega),\\
(f_n)_n&\mapsto (f_n\circ \sigma_g)_n.\end{aligned}$$ Therefore we obtain actions $$\beta_\omega:\Gamma\to \operatorname{Aut}({{\mathcal M}}^{\omega})\;\;\;\;\;\; \text{ and }\;\;\;\;\;\; \bar{\sigma}_\omega:\Gamma \to \operatorname{Aut}(C(K^\omega)).$$ One easily checks that the diagram
\(1) [${{\mathcal M}}^\omega$]{}; (2) \[right of=1, node distance=2.5cm\] [${{\mathcal M}}^\omega$]{}; (6) \[below of=1\] [$C(K^\omega)$]{}; (7) \[below of=2\] [$C(K^\omega)$]{}; (1) to node [$\beta_\omega^g$]{} (2); (6) to node [$\bar{\sigma}_\omega^g$]{} (7); (2) to node [$E^\omega$]{} (7); (1) to node [$E^\omega$]{} (6);
.
commutes for every $g\in \Gamma$. We write $({{\mathcal M}}^\omega\cap {{\mathcal M}}')^{\beta_\omega}$ for the fixed point subalgebra of the central sequence $W^*$-bundle ${{\mathcal M}}^\omega\cap {{\mathcal M}}'$.
$W^*$-Rokhlin property
======================
The goal of this section is to prove a Rokhlin type theorem for ${{\mathbb Z}}^m$-actions on continuous $W^*$-bundles. Since the argument is almost the same as [@Lia16 Theorem 4.1], we will be terse for most of the time. Our first objective is to embed arbitrarily large matrices into the fixed point algebra of the central sequence $W^*$-bundle, as in the following proposition and corollary. Recall that a completely positive (c.p.) map $\varphi:A\to B$ between $C^*$-algebras is called *order zero* if $\varphi(a)\varphi(b) = 0$ for all positive elements $a,b\in A$ satisfying $ab=0$.
\[prop:fixedpointsubalgebra\] (cf. [@Lia16 Proposition 3.4]) Let $({{\mathcal M}}, K, E)$ be a separable continuous $W^*$-bundle and $(\beta,\sigma):{{\mathbb Z}}^m\to \operatorname{Aut}({{\mathcal M}}, K, E)$ be a group action. Suppose
1. the action is trivial on $K$, i.e., $\sigma = {\mathrm{id}}_K$;
2. $\dim(K) = d < \infty$;
3. each fiber $\pi_\lambda({{\mathcal M}})$ is isomorphic to the hyperfinite II$_1$ factor ${{\mathcal R}}$;
4. the fiber action $\beta:{{\mathbb Z}}^m\to \operatorname{Aut}(\pi_\lambda({{\mathcal M}}))$ is pointwise outer for each $\lambda\in {{\mathbb K}}$.
Then for each $p\in {{\mathbb N}}$, there exist completely positive contractive (c.p.c.) order zero maps $\psi_1,...,\psi_{d+1}:M_p({{\mathbb C}})\to ({{\mathcal M}}^\omega\cap {{\mathcal M}}')^{\beta_\omega}$ with commuting ranges such that $\psi_1(1)+\cdots + \psi_{m+1}(1) = 1_{{\mathcal M}}$.
(cf. [@Lia16 Proposition 3.3]) Let $({{\mathcal M}}, K, E)$ be a separable continuous $W^*$-bundle and $(\beta,\sigma):{{\mathbb Z}}^m\to \operatorname{Aut}({{\mathcal M}}, K, E)$ be a group action. Suppose
1. the action is trivial on $K$, i.e., $\sigma = {\mathrm{id}}_K$;
2. $\dim(K) = d < \infty$;
3. each fiber $\pi_\lambda({{\mathcal M}})$ is isomorphic to the hyperfinite II$_1$ factor ${{\mathcal R}}$;
4. the fiber action $\beta:{{\mathbb Z}}^m\to \operatorname{Aut}(\pi_\lambda({{\mathcal M}}))$ is pointwise outer for each $\lambda\in {{\mathbb K}}$.
Then for every $p\in {{\mathbb N}}$, there is a unital \*-homomorphism $\sigma:M_p({{\mathbb C}})\to ({{\mathcal M}}^\omega\cap {{\mathcal M}}')^{\beta_\omega}$.
This follows immediately from Proposition \[prop:fixedpointsubalgebra\] and [@KR14 Lemma 7.6].
To prove Proposition \[prop:fixedpointsubalgebra\], we make use of the finite dimensionality of $K$ and build a collection of c.p.c. maps from a matrix algebra $M_p({{\mathbb C}})$ into ${{\mathcal M}}$ which are almost central, almost invariant under the action $\beta$, and have almost commuting ranges. The following lemma is simply a ${{\mathbb Z}}^m$-version of [@Lia16 Lemma 3.8] (with identical proof).
\[lem:fixpointcpc\] Under the same assumptions as Proposition \[prop:fixedpointsubalgebra\], for every ${\varepsilon}> 0, p\in {{\mathbb N}}$, and norm bounded $\|\cdot\|_{2,u}$-compact subset $\Omega\subseteq {{\mathcal M}}$, there exist c.p.c. maps $\psi^{(1)},...,\psi^{(d+1)}:M_p({{\mathbb C}})\to {{\mathcal M}}$ (remember that $d = \dim(K)$) such that
1. $\|[\psi^{(\ell)}(e),x]\|_{2,u} < {\varepsilon}$ $(1\leq \ell\leq d+1,\; e\in M_p({{\mathbb C}})_1,\; x\in \Omega)$;
2. $\| [\psi^{(\ell)}(e), \psi^{(k)}(f) ] \|_{2,u} < {\varepsilon}$ $(1\leq \ell\neq k \leq d+1,\;e, f\in M_p({{\mathbb C}})_1)$;
3. $\| \psi^{(\ell)}(1)\psi^{(\ell)}(e^*e) - \psi^{(\ell)}(e)^*\psi^{(\ell)}(e) \|_{2,u} < {\varepsilon}$ $(1\leq \ell\leq d+1,\; e\in M_p({{\mathbb C}})_1)$;
4. $\sum_{\ell=1}^{d+1}\psi^{(\ell)}(1) = 1_{{\mathcal M}}$;
5. $\| \beta^i(\psi^{(\ell)}(e))-\psi^{(\ell)}(e)\|_{2,u} < {\varepsilon}$ $(1\leq i\leq m,\; 1\leq \ell\leq d+1,\; e\in M_p({{\mathbb C}})_1)$.
Now the proof of Proposition \[prop:fixedpointsubalgebra\] can be completed by mapping sequences of c.p.c. maps as found in Lemma \[lem:fixpointcpc\] into the ultrapower:
(of Proposition \[prop:fixedpointsubalgebra\]) Let $\{x_1,x_2,...\}$ be a countable $\|\cdot\|_{2,u}$-dense subset of ${{\mathcal M}}$. By Lemma \[lem:fixpointcpc\], for each $n\in {{\mathbb N}}$ there exist c.p.c. maps $\psi_n^{(1)}, \psi_n^{(2)},...,\psi_n^{(d+1)}:M_p({{\mathbb C}})\to {{\mathcal M}}$ such that (ii) to (v) in Lemma \[lem:fixpointcpc\] hold with ${\varepsilon}= \frac{1}{n}$ and
1. $\| [\psi_n^{(\ell)}(e), x_j] \|_{2,u} < \frac{1}{n}\;\;\;\;\;\; (1\leq \ell\leq d+1,\; e\in M_p({{\mathbb C}})_1,\; 1\leq j\leq n)$.
For each $1\leq \ell\leq d+1$ define c.p.c. maps $\psi_\ell:M_p({{\mathbb C}})\to {{\mathcal M}}^{\omega}$ by $$\psi_\ell(e) = \pi_\omega(\psi_1^{(\ell)}(e), \psi_2^{(\ell)}(e), \psi_3^{(\ell)}(e),... ),$$ where $\pi_\omega$ is the quotient map from $\ell^\infty({{\mathbb N}},{{\mathcal M}})$ onto ${{\mathcal M}}^\omega$. Then by construction
- the image of $\psi_\ell$ belongs to $({{\mathcal M}}^\omega\cap {{\mathcal M}}')^{\beta_\omega}$;
- $\psi_1, \psi_2,...,\psi_{d+1}$ have commuting ranges;
- $\psi_1(1)+\cdots + \psi_{d+1}(1) = 1_{{\mathcal M}}$;
- $\psi_\ell(1)\psi_{\ell}(e^*e) = \psi_\ell(e)^*\psi_\ell(e)$ $(1\leq \ell\leq d+1,\; e\in M_p({{\mathbb C}}))$.
It remains to show that each $\psi_\ell$ is order zero, but this follows from the fourth bullet (see [@Lia16 Lemma 3.9]).
Now we define the $W^*$-Rokhlin property and state the Rokhlin type theorem.
\[defn:W-Rok\] Let $({{\mathcal M}}, K, E)$ be a separable continuous $W^*$-bundle and $(\beta,\sigma):{{\mathbb Z}}^m\to \operatorname{Aut}({{\mathcal M}}, K, E)$ be a group action. We say $(\beta, \sigma)$ has the *$W^*$-Rokhlin property* if for every $n\in {{\mathbb N}}$ there exist projections $\{p_v\}_{v\in B_n}$ in ${{\mathcal M}}^\omega\cap {{\mathcal M}}'$ such that
1. $(\beta_\omega)^u(p_v) = p_{v+u} \mod n \;\;\;\;\;\; (v\in B_n,\; u\in {{\mathbb Z}}^m)$.
2. $\sum_{v\in B_n} p_v = 1_{{\mathcal M}}$.
3. $\| 1_{{\mathcal M}}-p_0 \|_{2,u}^{(\omega)} < 1$.
\[thm:W-Rokhlin\] Let $({{\mathcal M}}, K, E)$ be a separable continuous $W^*$-bundle and $(\beta,\sigma):{{\mathbb Z}}^m\to \operatorname{Aut}({{\mathcal M}}, K, E)$ be a group action. Suppose
1. the action is trivial on $K$, i.e., $\sigma = {\mathrm{id}}_K$;
2. $\dim(K) = d < \infty$;
3. each fiber $\pi_\lambda({{\mathcal M}})$ is isomorphic to the hyperfinite II$_1$ factor ${{\mathcal R}}$;
4. the fiber action $\beta:{{\mathbb Z}}^m\to \operatorname{Aut}(\pi_\lambda({{\mathcal M}}))$ is pointwise outer for each $\lambda\in K$.
Then $(\beta, \sigma)$ has the $W^*$-Rokhlin property.
The idea of proving Theorem \[thm:W-Rokhlin\] is the same as [@Lia16 Theorem 4.1]. We first obtain a partition of unity which are almost mutually orthogonal in terms of the uniform 2-norm, and then glue the Rokhlin tower in each fiber (the fiberwise result was due to Ocneanu [@Ocn81]) to a global section. The following lemma provides the partition of unity we need.
Let $({{\mathcal M}}, K, E)$ be a separable continuous $W^*$-bundle and $(\beta, \sigma):{{\mathbb Z}}^m\to \operatorname{Aut}({{\mathcal M}},K,E)$ be a group action. Suppose
1. the action is trivial on $K$, i.e., $\sigma = {\mathrm{id}}_K$;
2. for each $p\in {{\mathbb N}}$ there exists a unital \*-homomorphism $\sigma:M_p({{\mathbb C}})\to ({{\mathcal M}}^\omega \cap {{\mathcal M}}')^{\beta_\omega}$.
Then given a norm bounded $\|\cdot\|_{2,u}$-compact subset $\Omega$ of ${{\mathcal M}}$, ${\varepsilon}> 0$, and a partition of unity $\{g_j\}_{j=1}^N$ subordinate to some open cover of $K$, there exist positive contractions $a_1,a_2,...,a_N$ in ${{\mathcal M}}$ such that
1. $\| [a_j,x] \|_{2,u} < {\varepsilon}\;\;\;\;\;\; (1\leq j\leq N,\; x\in \Omega)$;
2. $\| (\beta_\omega)^i(a_j) - a_j \|_{2,u} < {\varepsilon}\;\;\;\;\;\; (1\leq i\leq m,\; 1\leq j\leq N)$.
3. $\| a_ia_j\|_{2,u} < {\varepsilon}\;\;\;\;\;\; (1\leq i\neq j\leq N)$;
4. $\| [a_j^{1/2}, x] \|_{2,u} < {\varepsilon}\;\;\;\;\;\;\; (1\leq j\leq N,\; x\in \Omega)$;
5. $\| a_i^{1/2}a_j^{1/2} \|_{2,u} < {\varepsilon}\;\;\;\;\;\;\; (1\leq i\neq j \leq N)$;
6. $\left| \tau_\lambda(a_jx) - g_j(\lambda)\tau_\lambda(x) \right| < {\varepsilon}\;\;\;\;\;\; (1\leq j\leq N,\; x\in \Omega,\; \lambda\in K)$;
7. $\sum_{j=1}^N a_j = 1_{{\mathcal M}}$.
The proof is identical to [@Lia16 Lemma 4.4].
(of Theorem \[thm:W-Rokhlin\]) The only difference with the proof of [@Lia16 Theorem 4.1] is that in order to obtain a Rokhlin tower in each fiber, we invoke Ocneanu’s Rokhlin type theorem [@Ocn81 Theorem 2] instead of Connes’ Rokhlin lemma for automorphisms.
Approximate Rokhlin Property
============================
In this section we extend the notion of “invariant $\sigma$-ideal” to actions of an arbitrary discrete amenable group. The main result here is Lemma \[lem:alpha-sigma-ideal\], which generalizes a similar result for automorphisms and shows that the trace kernel ideal $J_A$ is an invariant $\sigma$-ideal under an action of any discrete amenable group.
The notion of $\sigma$-ideals was first introduced by Kirchberg and Rørdam in [@KR14]. It was designed to capture the behavior of a quasicentral approximate unit. When $B$ is a $C^*$-algebra, $\Gamma$ is a discrete countable amenable group, and $\alpha:\Gamma\to \operatorname{Aut}(B)$ is a group action, we define a notion which we shall call “$\alpha$-$\sigma$-ideal” (cf. [@Lia16 Definition 5.5]), which essentially packages an almost invariant quasicentral approximate unit.
Let $B$ be a $C^*$-algebra, $\Gamma$ a discrete countable amenable group, and $\alpha:\Gamma\to \operatorname{Aut}(B)$ a group action. Let $J$ be a closed ideal in $B$ such that $\alpha^g(J) = J$ for every $g\in \Gamma$. We say $J$ is an *$\alpha$-$\sigma$-ideal* if for every separable $C^*$-subalgebra $C$ of $B$ satisfying $\alpha^g(C)=C$ for every $g\in \Gamma$, there exists a positive contraction $u\in C'\cap J$ such that
1. $\alpha^g(u) = u$ for every $g\in \Gamma$.
2. $uc = c$ for every $c\in C\cap J$.
Let $A$ be a unital separable $C^*$-algebra and let $A_\omega$ be the ultrapower of $A$ (here we fix once and for all a free ultrafilter $\omega$ on ${{\mathbb N}}$). Suppose that $A$ has nonemepty trace space $T(A)$. Given an element $a$ in $A_\omega$ represented by a sequence $(a_1,a_2,...)$, recall from [@KR14] that the seminorm $\|\cdot\|_{2,\omega}$ on $A_\omega$ is defined by $$\|a\|_{2,\omega} := \lim_{n\to \omega} \sup_{\tau\in T(A)} \tau(a_n^*a_n)^{1/2}.$$ The *trace kernel ideal* $J_A$ is defined to be $$J_A := \{ e\in A_\omega : \|e\|_{2,\omega} = 0 \}.$$
\[lem:alpha-sigma-ideal\] Let $A$ be a unital separable $C^*$-algebra with $T(A)\neq \emptyset$, $\Gamma$ a discrete countable amenable group, and $\alpha:\Gamma\to \operatorname{Aut}(A)$ a group action. Then the trace-kernel ideal $J_A$ is an $\alpha_\omega$-$\sigma$-ideal in $A_\omega$.
First note that $(\alpha_\omega)^g(J_A) = J_A$ for every $g\in \Gamma$ since $(\alpha_\omega)^g$ preserves the uniform 2-norm. By [@KR14 Proposition 4.6] $J_A$ is a $\sigma$-ideal in $A_\omega$, so there exists a positive contraction $e\in C'\cap J_A$ such that $ec = c$ for every $c\in C\cap J_A$. Let $F_1\subseteq F_2\subseteq \cdots \subseteq \Gamma$ be an increasing sequence of finite subsets whose union is $\Gamma$. By amenability for each $n\in {{\mathbb N}}$ we can find a $(F_n, 1/n)$-invariant (finite) subset $K_n$ of $\Gamma$, in the sense that $$\max_{s\in F_n} \frac{ \left|sK_n\triangle K_n\right|}{|K_n|} < \frac{1}{n}.$$ For each $n\in {{\mathbb N}}$, define $$e^{(n)} := \frac{1}{|K_n|}\sum_{g\in K_n} (\alpha_\omega)^g(e).$$ One checks that $[e^{(n)}, c] = 0$ for every $c\in C$ and $e^{(n)}c = c$ for every $c\in C\cap J$. Moreover, for each $s\in F_n$ we have $$\begin{aligned}
\| (\alpha_\omega)^s(e^{(n)}) - e^{(n)}\| &= \frac{1}{|K_n|}\left\| \sum_{g\in K_n}(\alpha_\omega)^{sg}(e) - \sum_{g\in K_n}(\alpha_\omega)^g(e) \right\| \\
&\leq \frac{1}{|K_n|}|sK_n\triangle K_n| < \frac{1}{n}.\end{aligned}$$
To finish the proof, we invoke Kirchberg’s ${\varepsilon}$-test [@KR14 Lemma 3.1]. Let $d = (d_1,d_2,...)$ be a strictly positive contraction in the separable $C^*$-algebra $C\cap J_A$, and $\{g_k\}_{k=1}^\infty$ be a list of elements in $\Gamma$. Take a dense sequence $\{c^{(k)}\}_{k=1}^\infty$ in the unit ball of $C$, and represent each $c^{(k)}$ by $(c_1^{(k)},c_2^{(k)},...)$. Let each $X_n$ be the set of positive contractions in $A$, and define functions $f_n^{(k)}:X_n\to [0,\infty)$ by $$\begin{aligned}
f_n^{(1)}(x) &= \| (1-x)d_n\|,\\
f_n^{(2)}(x) &= \|x\|_2,\\
f_n^{(2k+1)} &= \|\alpha^{g_k}(x)-x\|\;\;\;\;\;\; (k\in {{\mathbb N}}),\\
f_n^{(2k+2)} &= \|c_n^{(k)}x-xc_n^{(k)}\|\;\;\;\;\;\;\;(k\in {{\mathbb N}}).\end{aligned}$$ Given $m\in {{\mathbb N}}$ and ${\varepsilon}>0$, there exists $\ell\in {{\mathbb N}}$ sufficiently large (along the filter $\omega$) so that $$e^{(\ell)} = (e_1^{(\ell)},e_2^{(\ell)},...)$$ in $A_\omega$ satisfies $$f_\omega^{(k)}(e_1^{(\ell)},e_2^{(\ell)},...) < {\varepsilon}\;\;\;\;\;\; (1\leq k\leq m).$$ Now the proof is finished by applying the ${\varepsilon}$-test.
As in [@Lia16], we define a technical property called the *approximate Rokhlin property*, which would serve as a bridge connecting the $W^*$-Rokhlin property and the Rokhlin dimension.
Let $A$ be a unital $C^*$-algebra and $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. We say $\alpha$ has the *approximate Rokhlin property* if for every $p\in {{\mathbb N}}$ there exist positive contractions $\{f_{v}\}_{v\in B_p}$ in $F(A)$ such that
- $(\alpha_\omega)^u(f_v) = f_{v+u} \mod p\;\;\;\;\;\; (v\in B_p,\;u\in {{\mathbb Z}}^m)$.
- $f_vf_u = 0\;\;\;\;\;\; (v,u\in B_p,\; v\neq u)$.
- $1_A - \sum_{v\in B_p}f_v$ belongs to $J_A\cap F(A)$.
- $\sup_{k\in {{\mathbb N}}} \| 1_A - f_0^k \|_{2,\omega} < 1$.
(here 0 is the origin in ${{\mathbb Z}}^m$).
With the aid of Lemma \[lem:alpha-sigma-ideal\], it is not hard to pass from the $W^*$-Rokhlin property to the approximate Rokhlin property.
\[lem:W-Rok-to-Approx-Rok\] (cf. [@Lia16 Propoition 5.8]) Let $A$ be a unital $C^*$-algebra and $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose the induced action $(\tilde{\alpha}, \bar{\alpha}):{{\mathbb Z}}^m\to \operatorname{Aut}(\overline{A}^u,\partial_e T(A),E)$ has the $W^*$-Rokhlin property. Then $\alpha$ has approximate Rokhlin property.
To simplify the notion, we write $({{\mathcal M}},K, E)$ for the triple $(\overline{A}^u, \partial_e T(A), E)$. Let $\{p_v\}_{v\in B_p}$ be projections in ${{\mathcal M}}^\omega\cap {{\mathcal M}}'$ satisfying the $W^*$-Rokhlin property. Recall that the canonical map $F(A)\to {{\mathcal M}}^\omega\cap {{\mathcal M}}'$ is surjective ([@KR14]). Lifting the projections $\{p_v\}$ along this surjection, we obtain positive contractions $\{f_v'\}$ in $F(A)$ such that
- $(\alpha_\omega)^u(f_v') - f'_{v+u}\in J_A\cap F(A) \mod p\;\;\;\;\;\;\; (v\in B_p,\; u\in {{\mathbb Z}}^m)$,
- $f'_vf'_u\in J_A\cap F(A)\;\;\;\;\;\; (u,v\in B_p,\; u\neq v)$,
- $1_A - \sum_v f'_v\in J_A\cap F(A)$.
Put $C := C^*( A, \{ (\alpha_\omega)^u(f_v'): u\in {{\mathbb Z}}^m,\; v\in B_p \} )$. Then $C$ is a separable $C^*$-subalgebra of $A_\omega$ satisfying $(\alpha_\omega)^u(C) =C$ for every $u\in {{\mathbb Z}}^m$. Since $J_A$ is an $\alpha_\omega$-$\sigma$-ideal, there exists a positive contraction $u\in C'\cap J_A$ such that $(\alpha_\omega)^w(u) = u$ for all $w\in {{\mathbb Z}}^m$ and $uc = c$ for all $c\in C\cap J_A$. Define $$f_v := (1-u)f_v'(1-u)\;\;\;\;\;\; (v\in B_{p}).$$ One checks that the family $\{f_v\}_{v\in B_p}$ has the desired properties.
A Rokhlin type theorem
======================
In this section we establish our Rokhlin type theorem for ${{\mathbb Z}}^m$-actions on $C^*$-algebras (Theorem \[thm:main\]). Let $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. The key observation, motivated by [@Nak99], is that suppose for every canonical generator $\alpha^i$, we can find Rokhlin towers for $\alpha^i$ which are almost fixed by all the other generators $\alpha^1,...,\alpha^{i-1},\alpha^{i+1},...,\alpha^m$, then we can simply “multiply” the towers together and get Rokhlin towers for $\alpha$. Here is the precise statement:
\[prop:Zm-Rokhlin\] Let $A$ be a unital $C^*$-algebra and $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose there exists $d\in {{\mathbb N}}$ such that for each $i\in \{1,...,m\}$ the following holds: for every ${\varepsilon}> 0$, $p\in {{\mathbb N}}$, finite subset $F\subseteq A$, there exist positive contractions $$\{g_{i,k}^{(\ell)}: 0\leq k\leq p-1, \;0\leq \ell\leq d\}$$ in $A$ such that
- $\|\alpha^i(g_{i,k}^{(\ell)}) - g_{i,k+1}^{(\ell)} \| < {\varepsilon}\mod p\;\;\;\;\;\; (0\leq k\leq p-1,\; 0\leq \ell\leq d)$,
- $\| \alpha^j(g_{i,k}^{(\ell)}) - g_{i,k}^{(\ell)} \| < {\varepsilon}\;\;\;\;\;\; (1\leq j\leq m, \; j\neq i,\; 0\leq k\leq p-1,\; 0\leq \ell\leq d)$,
- $\| [ g_{i,k}^{(\ell)}, x] \| < {\varepsilon}\;\;\;\;\;\; (0\leq k\leq p-1,\; 0\leq \ell\leq d,\; x\in F)$,
- $\| g_{i,k}^{(\ell)} g_{i,n}^{(\ell)} \| < {\varepsilon}\;\;\;\;\;\; (0\leq k\neq n\leq p-1,\; 0\leq \ell\leq d)$,
- $\left\| \sum_{k,\ell}g_{i,k}^{(\ell)} - 1_A \right\| < {\varepsilon}$.
Then ${\mathrm{dim}_{\mathrm{Rok}}}(\alpha) \leq (d+1)^m-1$.
Fix ${\varepsilon}> 0$, a finite subset $F\subseteq A$, and $p\in {{\mathbb N}}$. Let $$\{g_{1,k}^{(\ell)}: 0\leq k\leq p-1,\; 0\leq \ell\leq d \}$$ be positive contractions in $A$ satisfying (1) to (5) with respect to ${\varepsilon}' > 0$ and $F_1 := F$, where ${\varepsilon}'$ is a small number to be determined later.
Define, for $i = 1,2,...,m-1$, inductively $F_{i+1} := F_i \cup \{g_{i,k}^{(\ell)} \}$ and let $\{ g_{i+1,k}^{(\ell)}:0\leq k\leq p-1,\; 0\leq \ell \leq d \}$ be positive contractions satisfying conditions (1) to (5) with respect to ${\varepsilon}'$ and $F_i$. For $v = (v_1,...,v_m)\in B_p$ and $(\ell_1,...,\ell_m)\in \{0,...,d \}^m$, define (positive contractions) $$f_v^{(\ell_1,...,\ell_m)} := \left( g_{m,v_m}^{(\ell_m)}\right)^{\frac{1}{2}}\cdots \left( g_{2,v_2}^{(\ell_2)}\right)^{\frac{1}{2}} \left( g_{1,v_1}^{(\ell_1)} \right) \left( g_{2,v_2}^{(\ell_2)}\right)^{\frac{1}{2}}\cdots \left( g_{m,v_m}^{(\ell_m)}\right)^{\frac{1}{2}}.$$ One checks that these elements form $(d+1)^m$ Rokhlin towers for $\alpha$ with respect to ${\varepsilon}$ and $F$, provided that ${\varepsilon}'$ is sufficiently small.
For the rest of the section we will focus on establishing the conditions described in Proposition \[prop:Zm-Rokhlin\]. One technical notion we need is the so-called *Property (SI)*; it was first introduced by Sato in [@Sat10], and later reformulated by Kirchberg and Rørdam in [@KR14].
([@KR14 Definition 2.6]) \[defn:properptySI\] Let $A$ be a unital separable $C^*$-algebra with nonempty trace space. We say $A$ has *property (SI)* if for all positive contractions $e,f\in F(A)$ with $e\in J_A$ and $\sup_{m\in {{\mathbb N}}}\|1_A-f^m\|_{2,\omega} < 1$, there exists an element $s\in F(A)$ with $fs = s$ and $s^*s = e$.
Let $A$ be a unital separable $C^*$-algebra with $T(A)\neq \emptyset$, and $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose $\alpha$ has the approximate Rokhlin property and $A$ has property (SI). Then for every $i\in \{1,...,m\}$ and $p\in {{\mathbb N}}$ there exist $g_0,...,g_{p-1},v$ in $F(A)$ such that
1. $g_0,...,g_{p-1}$ are positive contractions,
2. $g_kg_n = 0\;\;\;\;\;\; (0\leq k\neq n \leq p-1)$,
3. $(\alpha_\omega)^i (g_k) = g_{k+1} \mod p \;\;\;\;\;\; (0\leq k\leq p-1)$,
4. $g_0v = v$,
5. $\sum_{k=0}^{p-1}g_k +v^*v = 1_A$,
6. $(\alpha_\omega)^{pi}(v) = v$,
7. $(\alpha_\omega)^j(g_k) = g_k\;\;\;\;\;\; (1\leq j\neq i\leq m)$,
8. $(\alpha_\omega)^j(v) = v\;\;\;\;\;\; (1\leq j\neq i\leq m)$.
The main idea is almost contained in [@MS14 Proposition 4.5]. However, since the assumptions are slightly different, we include full details for the reader’s convenience. Fix $i\in \{1,...,m\}$ and $p\in {{\mathbb N}}$. First of all, since we are working with ultraproducts, it suffices to find elements satisfying (i) to (viii) up to an arbitrarily small ${\varepsilon}> 0$. Let $\ell$ be a positive integer such that $\frac{2}{\sqrt{\ell p}} < {\varepsilon}$. By the approximate Rokhlin property, there exist $\{f_v\}_{ v\in B_{\ell p} }$ satisfying
1. $(\alpha_\omega)^u(f_v) = f_{v+u} \mod (\ell p)\;\;\;\;\;\;\; (v\in B_{\ell p},\; u\in {{\mathbb Z}}^m)$,
2. $f_vf_u = 0 \;\;\;\;\;\; (v,u\in B_{\ell p},\; v\neq u)$,
3. $e : = 1_A - \sum_{v\in B_{\ell p}}f_v \in J_A\cap F(A)$,
4. $\sup_{n}\| 1_A - f_0^n \|_{2,\omega} < 1$.
Since $A$ has property (SI), there is $w$ in $F(A)$ such that $$w^*w = e,\;\;\;\;\;\; \text{and}\;\;\;\;\;\;\; f_0w = w.$$ In what follows, the coordinates of a point $v\in {{\mathbb Z}}^m$ will be denoted $v_1,v_2,...,v_m$. Define, for $0\leq k\leq \ell p -1$, $$g_k':= \sum_{\{v\in B_{\ell p}: v_i = k \} } f_v$$ and $$s := \frac{1}{ \sqrt{ (\ell p)^{m-1}} } \sum_{\{u\in B_{\ell p}: u_i = 0\} } (\alpha_\omega)^u(w).$$ Then clearly we have
- $(\alpha_\omega)^j(g_k') = g_k'$ ($0\leq k\leq \ell p - 1,\; 1\leq j\neq i\leq m$);
- $(\alpha_\omega)^i(g_k') = g_{k+1}'; \mod (\ell p)$ ($0\leq k\leq \ell p -1$);
- $\sum_{k=0}^{\ell p -1}g_k' + e = 1_A$.
We make several more computations. First of all, $$\begin{aligned}
s^*s &= \frac{1}{ (\ell p)^{m-1} } \sum_{\{u\in B_{\ell p}:u_i = 0\}} \sum_{\{u'\in B_{\ell p}:u_i' = 0\}} \alpha_\omega^{u}(w^*)\alpha_\omega^{u'}(w) \\
&= \frac{1}{ (\ell p)^{m-1} } \sum_{\{u\in B_{\ell p}:u_i = 0\}} \sum_{\{u'\in B_{\ell p}:u_i' = 0\}} \alpha_\omega^{u}(w^*f_0)\alpha_\omega^{u'}(f_0w) \\
&= \frac{1}{ (\ell p)^{m-1} } \sum_{\{u\in B_{\ell p}:u_i = 0\}} \sum_{\{u'\in B_{\ell p}:u_i' = 0\}} \alpha_\omega^{u}(w^*) f_u f_{u'} \alpha_\omega^{u'}(w) \\
&= \frac{1}{ (\ell p)^{m-1} } \sum_{\{u\in B_{\ell p}:u_i = 0\}} \alpha_\omega^{u}(w^*) f_u f_{u} \alpha_\omega^{u}(w) \\
&= \frac{1}{ (\ell p)^{m-1} } \sum_{\{u\in B_{\ell p}:u_i = 0\}} \alpha_\omega^{u}(w^*w) \\
&= e.\end{aligned}$$ Second, $$\begin{aligned}
g_0's &= \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \sum_{ \{v\in B_{\ell p}:v_i = 0\} }\sum_{\{ u\in B_{\ell p}: u_i = 0\}} f_v(\alpha_\omega)^u(w) \\
&= \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \sum_{ \{v\in B_{\ell p}:v_i = 0\} }\sum_{\{ u\in B_{\ell p}: u_i = 0\}} f_vf_u(\alpha_\omega)^u(w) \\
&= \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \sum_{\{ u\in B_{\ell p}: u_i = 0\}} f_uf_u(\alpha_\omega)^u(w) \\
&= \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \sum_{\{ u\in B_{\ell p}: u_i = 0\}} (\alpha_\omega)^u(w) \\
&= s.\end{aligned}$$ Finally, when $j\neq i$ we have, $$\begin{aligned}
\| (\alpha_\omega)^j(s) - s\| &= \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \left\| \sum_{\{u\in B_{\ell p}:u_i=0\}} (\alpha_\omega)^j(\alpha_\omega)^u(w) - \sum_{\{u\in B_{\ell p}:u_i = 0\}} (\alpha_\omega)^u(w) \right\| \\
&= \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \left\| \sum_{ \{ u\in B_{\ell p}: u_i = 0, u_j = \ell p \} } (\alpha_\omega)^u(w) - \sum_{\{ u\in B_{\ell p}: u_i=0, u_j = 0 \}} (\alpha_\omega)^u(w) \right\| \\
&\leq \frac{1}{ \sqrt{ (\ell p)^{m-1} } } \left( \left\| \sum_{ \{ u\in B_{\ell p}: u_i = 0, u_j = \ell p \} } (\alpha_\omega)^u(w) \right\| + \left\| \sum_{\{ u\in B_{\ell p}: u_i=0, u_j = 0 \}} (\alpha_\omega)^u(w) \right\|\right).\end{aligned}$$ Since the terms in the norms are orthogonal, the $C^*$-identity implies that $$\begin{aligned}
\left\| \sum_{ \{ u\in B_{\ell p}: u_i = 0, u_j = \ell p \} } (\alpha_\omega)^u(w) \right\| &\leq \left\| \sum_{ \{ u\in B_{\ell p}: u_i=0, u_j = \ell p \} } (\alpha_\omega)^u(w^*w) \right\|^{1/2} \leq \sqrt{ (\ell p)^{m-2} } \cdot \|w^*w\|^{1/2} \\
&\leq \sqrt{ (\ell p)^{m-2} }\end{aligned}$$ and a similar bound holds for the second norm. Therefore, $$\begin{aligned}
\| (\alpha_\omega)^j(s) -s \| \leq 2\cdot \sqrt{ \frac{(\ell p)^{m-2}}{(\ell p)^{m-1}} } = \frac{2}{ \sqrt{\ell p} } < {\varepsilon}.\end{aligned}$$ Since we work in the ultrapower, we may actually assume that $$(\alpha_\omega)^j(s) = s\;\;\;\;\;\;\; (j\neq i).$$ Now define, for $0\leq k\leq m-1$, $$g_k := \sum_{r=0}^{\ell -1 } g'_{k+rp}$$ and $$v := \frac{1}{\sqrt{\ell}} \sum_{r=0}^{\ell -1} (\alpha_\omega)^{(rp)i}(s).$$ One checks that $g_0, g_1,...,g_{m-1}, v$ satisfy the desired properties.
Let $A$ be a unital separable $C^*$-algebra with nonempty trace space $T(A)$, and $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose $\alpha$ has the approximate Rokhlin property and $A$ has property (SI). Then for each $i\in \{1,...,m\}$ and any $p\in {{\mathbb N}}$ there exist positive contractions $$a_0,...,a_{p-1}, b_0,...,b_{p-1},c_0,...,c_p$$ in F(A) such that
1. $(\alpha_\omega)^i (a_k) = a_{k+1}$ and $(\alpha_\omega)^i(b_k) = b_{k+1} \mod p\;\;\;\;\;\; (0\leq k\leq p-1)$.
2. $(\alpha_\omega)^i(c_k) = c_{k+1} \mod (p+1)\;\;\;\;\;\; (0\leq k\leq p)$.
3. $(\alpha_\omega)^{j}(a_k) = a_k$ and $(\alpha_\omega)^{j}(b_k) = b_k\;\;\;\;\;\; (1\leq j\neq i\leq m,\; 0\leq k\leq p-1)$
4. $(\alpha_\omega)^{j}(c_k) = c_k\;\;\;\;\;\; (1\leq j\neq i\leq m,\; 0\leq k\leq p)$
5. $a_k a_n = b_kb_n = 0\;\;\;\;\;\; (0\leq k\neq n\leq p-1)$.
6. $c_kc_n = 0\;\;\;\;\;\; (0\leq k\neq n\leq p)$.
7. $b_k c_n = 0\;\;\;\;\;\; (0\leq k\leq p-1,\; 0\leq n\leq p)$.
8. $\sum_{k}a_k + \sum_{k} b_k + \sum_{k} c_k = 1_A$.
Copy the proof of [@Lia16 Theorem 6.4] verbatim, replacing [@Lia16 Proposition 6.2] by the previous lemma. Note that since all “matrix units” $x_i$’s are fixed by $(\alpha_\omega)^j$, the entire image of $\psi:M_{\ell p + 1}({{\mathbb C}})\to F(A)$ is fixed by $(\alpha_\omega)^j$ (here $j\neq i$). Therefore (iii) and (iv) are satisfied.
\[thm:main\] Let $A$ be a unital simple separable nuclear $C^*$-algebra with nonempty trace space $T(A)$, and let $\alpha:{{\mathbb Z}}^m\to \operatorname{Aut}(A)$ be a group action. Suppose
1. $A$ has property (SI),
2. $\alpha$ is strongly outer,
3. $T(A)$ is a Bauer simplex with finite dimensional extreme boundary,
4. $\tau\circ \alpha = \tau$ for every $\tau\in T(A)$.
Then ${\mathrm{dim}_{\mathrm{Rok}}}(\alpha) \leq 4^m-1$.
Write $({{\mathcal M}}, K,E)$ for the bundle $(\overline{A}^u, \partial_e T(A), E)$. Under the current assumptions, the induced group action $(\tilde{\alpha}, \bar{\alpha}):{{\mathbb Z}}^m \to \operatorname{Aut}({{\mathcal M}}, K, E)$ satisfies conditions (1) to (4) in Theorem \[thm:W-Rokhlin\]. Therefore the action $(\tilde{\alpha}, \bar{\alpha})$ has the $W^*$-Rokhlin property. By Lemma \[lem:W-Rok-to-Approx-Rok\], $\alpha$ has the approximate Rokhlin property. The previous lemma, together with [@HWZ15 Proposition 2.8] (which converts towers of different heights to towers of the same height), shows that the assumptions of Proposition \[prop:Zm-Rokhlin\] are satisfied with $d=3$. By the same proposition we have ${\mathrm{dim}_{\mathrm{Rok}}}{(\alpha)} \leq 4^m -1$.
Under the same assumptions as Theorem \[thm:main\], the crossed product $A\rtimes_\alpha {{\mathbb Z}}^m$ is a unital simple separable $C^*$-algebra of finite nuclear dimension.
From Theorem \[thm:main\] and [@Sza15 Proposition 2.4], we see that the crossed product $A\rtimes_\alpha {{\mathbb Z}}^m$ has finite nuclear dimension. Simplicity follows from [@Kis81 Theorem 3.1].
Example: Bernoulli Actions
==========================
In this final section we study the Bernoulli actions. We will show that for a large class of $C^*$-algebras, the ${{\mathbb Z}}^m$-Bernoulli shift has finite Rokhlin dimension. Throughout the section all tensor products are understood as minimal tensor products.
We recall the definition of Bernoulli actions. Let $A$ be a unital $C^*$-algebra and $\Gamma$ be a discrete group. Define $\bigotimes_{\Gamma} A$ to be the inductive limit of the system $\{ \bigotimes_F A : F\subseteq \Gamma \text{ finite subset} \}$, where the connecting maps are the natural inclusions. The (left) translation action of $\Gamma$ on itself gives rise to an action $\sigma:\Gamma\to \operatorname{Aut}(\bigotimes_\Gamma A)$, called the *Bernoulli action* (or *Bernoulli shift*).
The following proposition is a slight variation of [@Sat10 Proposition 4.4].
\[prop:bernoulli-shift\] Let $A$ be a unital simple $C^*$-algebra with a unique trace. Suppose $A$ is not isomorphic to the algebra of complex numbers ${{\mathbb C}}$. Then the Bernoulli action $\sigma:{{\mathbb Z}}^m\to \operatorname{Aut}(\bigotimes_{{{\mathbb Z}}^m} A)$ is strongly outer.
For convenience we write $C := \bigotimes_{n\in {{\mathbb Z}}} A$. Let $\tau$ be the unique trace on $C$ and let $\pi_\tau$ be the GNS representation of $C$ associated to $\tau$. Let $\tilde{\sigma}\in \operatorname{Aut}(\pi_\tau(C)'')$ be the weak extension of the action $\sigma$. We show that for every $v\in {{\mathbb Z}}^m\setminus\{0\}$, the automorphism $\tilde{\sigma}^v$ is not inner in $\pi_\tau(C)''$.
Assume on the contrary that $\tilde{\sigma}^v = \operatorname{Ad}W_0$ for some unitary $W_0$ in $\pi_\tau(C)''$. Note that $\tilde{\sigma}^{nv}(W_0) = W_0$ for all $n\in {{\mathbb N}}$. Fix ${\varepsilon}> 0$ and let $x$ be a contraction in some finite tensor product $\bigotimes_F A \subseteq C$. There exist a finite subset $F'\subseteq {{\mathbb Z}}^m$ and an element $w_0\in \bigotimes_{F'} A\subseteq C$ such that $$\| W_0 - \pi_\tau(w_0) \|_{2,\tau} < {\varepsilon}.$$ Then $$\| W_0 - \pi_\tau( \sigma^{nv}(w_0) ) \|_{2,\tau} < {\varepsilon}$$ for any $n\in {{\mathbb N}}$. If $n$ is sufficiently large, then $\sigma^{nv}(w_0)$ commutes with $x$. Therefore $W_0$ commutes with $\pi_\tau(x)$ up to $2{\varepsilon}$ in the 2-norm. Since ${\varepsilon}$ is arbitrary and $\tau$ is faithful (simplicity passes to inductive limits), we see that $W_0$ commutes with $\pi_\tau(x)$ exactly, and hence belongs to the center of $\pi_\tau(C)''$. Since $C$ has a unique trace, $\pi_\tau(C)''$ is a factor (the center is trivial). This shows that $\operatorname{Ad}W_0$ is the identity map, a contradiction.
\[cor:bernoulli-shift\] Let $A$ be either the matrix algebra $M_n({{\mathbb C}})$ ($n\geq 2$) or the Jiang-Su algebra $\mathcal{Z}$ (see [@JS99]), and let $\sigma:{{\mathbb Z}}^m\to \operatorname{Aut}(\bigotimes_{{{\mathbb Z}}^m} A)$ be the Bernoulli action. Then $\sigma$ has finite Rokhlin dimension.
In either case, $A$ is a unital simple separable nuclear $C^*$-algebra with a unique trace. Therefore the previous proposition shows that the Bernoulli action $\sigma$ is strongly outer. It is well-known that the infinite tensor product of a nuclear $C^*$-algebra remains to be nuclear. To apply Theorem \[thm:main\], it remains to check that $\bigotimes_{{{\mathbb Z}}^m} A$ has property (SI). In the case that $A= M_n({{\mathbb C}})$, the infinite tensor product is the UHF algebra of type $n^\infty$, which is known to absorb the Jiang-Su algebra $\mathcal{Z}$ (for example, see [@KW04] and [@Win10]). In the case $A=\mathcal{Z}$, the infinite tensor product is isomorphic to $\mathcal{Z}$ itself by [@JS99 Theorem 4], and in particular is Jiang-Su stable as well. Since Jiang-Su stability implies property (SI) [@MS12a Theorem 1.1], the proof is complete.
The following observation allows us to expand the class of $C^*$-algebra drastically.
\[prop:subalgebra-trick\] Let $A$ be a unital $C^*$-algebra and $B\subseteq$ A be a unital $C^*$-subalgebra (with $1_A\in B$). Let $\sigma:{{\mathbb Z}}^m\to \operatorname{Aut}(\bigotimes_{{{\mathbb Z}}^m} A)$ be the Bernoulli action. Then ${\mathrm{dim}_{\mathrm{Rok}}}(\sigma) \leq {\mathrm{dim}_{\mathrm{Rok}}}(\sigma|_{\bigotimes_{{{\mathbb Z}}^m}B})$.
Assume ${\mathrm{dim}_{\mathrm{Rok}}}(\sigma|_{\bigotimes_{{{\mathbb Z}}^m}B}) = d < \infty$ otherwise there is nothing to prove. Fix a finite subset $\{a_1,...,a_m\}\subseteq \bigotimes_{ {{\mathbb Z}}^m}A$, ${\varepsilon}> 0$, and $n\in {{\mathbb N}}$. By approximation we may assume that $\{a_1,...,a_m\}\subseteq \bigotimes_F A$ for some finite subset $F\subseteq {{\mathbb Z}}^m$. By assumption we can find positive contractions $$\{ f_v^{(\ell)} \}_{v\in B_n}^{\ell = 0,1,...,d}$$ in $\bigotimes_{{{\mathbb Z}}^m}B$ such that
1. $\| \sigma^w(f_v^{(\ell)}) - f_{v+w}^{(\ell)} \| < {\varepsilon}' \mod n \;\;\;\;\;\; (0\leq \ell \leq d,\; v\in B_n,\; w\in {{\mathbb Z}}^m)$;
2. $\| f^{(\ell)}_v f^{(\ell)}_{v'} \| < {\varepsilon}'\;\;\;\;\;\; (0\leq \ell \leq d, \;v, v'\in B_n,\; v\neq v')$.
3. $\left\| \sum_{\ell=0}^d\sum_{v\in B_n}f_v^{(\ell)} - 1_A \right\| < {\varepsilon}'$.
for some small ${\varepsilon}' > 0$ to be determined later. Find a finite subset $F'\subseteq{{\mathbb Z}}^m$ and elements $\{ g_v^{(\ell)} \}_{v\in B_n}^{\ell = 0,1,...,d}$ in $\bigotimes_{F'} B$ such that for $\ell=0,1,...,d$ and $v\in B_n$, $$\| g_v^{(\ell)} - f_v^{(\ell)} \| < {\varepsilon}'.$$ Then $$\| \sigma^w(g_v^{(\ell)}) - g_{v+w}^{(\ell)} \| < 3{\varepsilon}' \mod n,$$ $$\left\| \sum_{v,\ell}g_v^{(\ell)} - 1_A \right\| \leq \left\| \sum_{v,\ell} (g_v^{(\ell)}-f_v^{(\ell)}) \right\| + \left\| \sum_{v,\ell}f_v^{(\ell)} - 1_A \right\| < [\text{card}(B_n)(d+1)+1]{\varepsilon}'$$ and $$\begin{aligned}
\| g_v^{(\ell)} g_{v'}^{(\ell)} \| &\leq \| [g_v^{(\ell)} -f_v^{(\ell)} ] g_{v'}^{(\ell)} \| + \| f_v^{(\ell)}[g_{v'}^{(\ell)}-f_{v'}^{(\ell)}] \| + \| f_v^{(\ell)}f_{v'}^{(\ell)} \|\\
&< 3{\varepsilon}'.\end{aligned}$$ Observe that we can find $u\in {{\mathbb Z}}^m$ such that $$[\sigma^u( g_v^{(\ell)}), a_j] = 0\;\;\;\;\;\; (v\in B_n,\; 0\leq \ell\leq d,\; 1\leq j\leq m).$$ Define $$h_v^{(\ell)} := \sigma^u( g_v^{(\ell)} )\;\;\;\;\;\; (v\in B_n,\; 0\leq \ell \leq d).$$ Then $$\begin{aligned}
\| \sigma^w(h_v^{(\ell)}) - h_{v+w}^{(\ell)} \| = \| \sigma^{w+u}(g_v^{(\ell)}) - \sigma^u (g_{v+w}^{(\ell)}) \| < {\varepsilon}'\end{aligned}$$ and $$\begin{aligned}
\left\| \sum_{v,\ell}h_v^{(\ell)} - 1_A \right\| &= \left\| \sum_{v,\ell}\sigma^u (g_v^{(\ell)} ) - 1_A \right\| = \left\| \sum_{v,\ell} g_v^{(\ell)} - 1_A \right\| \\
&< [\text{card}(B_n)(d+1)+1]{\varepsilon}'.\end{aligned}$$ Finally, $$\|h_v^{(\ell)}h_{v'}^{(\ell)} \| = \| g_v^{(\ell)}g_{v'}^{(\ell)} \| < 3{\varepsilon}'.$$ If ${\varepsilon}'$ is chosen to be sufficiently small, then we have obtained the Rokhlin towers $\{h_v^{(\ell)}\}$ for $\sigma$.
Let $A$ be a unital $C^*$-algebra. Assume one of the following holds:
1. $A$ contains a unital copy of the Jiang-Su algebra $\mathcal{Z}$; for example, when $A$ is Jiang-Su stable.
2. $A$ contains a unital copy of a matrix algebra $M_n({{\mathbb C}})$ for some $n\geq 2$; for exmaple, the set of all continuous matrix-valued functions $C(X, M_n({{\mathbb C}}))$ on some compact Hausdorff space $X$.
Then the previous proposition, together with Corollary \[cor:bernoulli-shift\], implies that the Bernoulli action $\sigma:{{\mathbb Z}}^m\to \operatorname{Aut}(\bigotimes_{{{\mathbb Z}}^m} A)$ has finite Rokhlin dimension.
|
---
abstract: 'The recently released BICEP2 data detected the primordial B-mode polarization in the Cosmic Microwave Background (CMB) map which strongly supports for a large tensor-to-scalar ratio, and thus, is found to be in tension with the Planck experiment with no evidence of primordial gravitational waves. Such an observational tension, if confirmed by forthcoming measurements, would bring a theoretical challenge for the very early universe models. To address this issue, we in the present paper revisit a single field inflation model proposed in [@Wang:2002hf; @Feng:2002a] which includes a modulated potential. We show that this inflation model can give rise to a sizable negative running behavior for the spectral index of primordial curvature perturbation and a large tensor-to-scalar ratio. Applying these properties, our model can nicely explain the combined Planck and BICEP2 observations. To examine the validity of analytic calculations, we numerically confront the predicted temperature and B-mode power spectra with the latest CMB observations and explicitly show that our model is consistent with the current data.'
author:
- Youping Wan
- Siyu Li
- Mingzhe Li
- Taotao Qiu
- Yifu Cai
- Xinmin Zhang
title: Single field inflation with modulated potential in light of the Planck and BICEP2
---
Introduction
============
The inflationary hypothesis of the very early universe, since was proposed in the early 1980s [@Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi] (see also [@Starobinsky:1980te; @Fang:1980wi; @Sato:1980yn] for early works), has become the dominant paradigm for understanding the initial conditions for the hot big bang cosmology. Within this context, a well established picture of causally generating cosmological perturbations in the primordial epoch has been greatly developed theoretically and observationally in the past decades. In particular, a significant prediction of nearly scale-invariant power spectra of primordial density perturbations based on the inflationary paradigm has been verified to high precision by the CMB observations in recent years [@Ade:2013lta; @Ade:2013uln]. Inflationary cosmology also predicted a nearly scale-invariant power spectrum of primordial tensor perturbations [@Starobinsky:1979ty], which can give rise to the CMB B-mode polarization as detected by the BICEP2 collaboration [@Ade:2014xna]. Assuming that all polarization signals were contributed by inflationary gravitational waves, the BICEP2 experiment implies that a nonzero value of the ratio between the spectra of tensor and scalar modes, dubbed as the tensor-to-scalar ratio $r$, has been discovered at more than $5\sigma$ confidence level (CL) with a tight constraint as: $r=0.20^{+0.07}_{-0.05}$ at $68\%$ CL ($r=0.16^{+0.06}_{-0.05}$ with foreground subtracted). This observation, if eventually verified by other ongoing experiments, implies a large amplitude of primordial gravitational waves and hence has significant theoretical implications on various early universe models.
Such a large amplitude of primordial tensor spectrum as indicated by the BICEP2, however, is in certain tension with another CMB experiment, the Planck result with $r<0.11$ at $95\%$ CL. Thus, the combination of these two data sets leads to a critical challenge for theoretical models of the very early universe. In the literature, there are some discussions of this issue from either the perspective of new physics beyond the inflationary $\Lambda$CDM paradigm in which more degrees of freedom are introduced, see e.g. [@Miranda:2014wga; @Zhang:2014dxk; @Xia:2014tda; @Hazra:2014jka; @Smith:2014kka; @Cai:2014hja; @Liu:2014tda; @Cai:2014bea], or from the propagations of photons after decoupling, see e.g. [@Feng:2006dp]. However, it remains to be interesting to investigate the possibility of resolving such an observational challenge within the framework of single field inflation.
In order to address the issue of observational tension, we take a close look at prior assumptions made by the Planck and BICEP2. The Planck’s result of $r<0.11$ was obtained by, the assuming a constant spectral index of primordial curvature perturbations. As was pointed out in [@Li:2012ug; @Ade:2013lta; @Ade:2014xna], however, if a nonzero running of the spectral index is allowed in the data analysis, there exist reasonable degeneracies among the spectral index $n_s$, the running of the spectral index $\alpha_s$, and the tensor-to-scalar ratio $r$. For example, an allowance of the running spectral index can at most enhance the upper bound of the tensor-to-scalar ratio of the Planck data to $r<0.3$ at $95\%$ CL [@Li:2014h][@Ade:2013lta][@Ade:2014xna] (see also [@Gong:2014cqa] for theoretical discussions). Accordingly, a possibly existing running of the spectral index can efficiently circumvent the tension issue between the Planck and BICEP2 data. This phenomenological scenario, however, is not easy to be achieved in usual single field slow-roll inflation models. Therefore, we in the present paper study a notable mechanism of generating a negative running behavior for the spectral index of primordial curvature perturbations. In particular, we analyze a model of single field inflation as proposed in [@Wang:2002hf; @Feng:2002a], of which the inflaton field has a modulated potential. We will show that this model gives rise to a negative value of $\alpha_s$ and also yields a relatively large value of $r$. Therefore, our model can provide a theoretical interpretation in reconciling the tension issue existing between the Planck and BICEP2 measurements.
The present paper is organized as follows. In Sec. II we briefly review the single field inflation model with modulated potential and discuss its theoretical motivation from the physics of extra dimensions. Then, in Sec. III we perform analytic and numerical calculations on the background dynamics as well as theoretical predictions on primordial power spectra, respectively. We show that this model generally produces a large amplitude of tensor-to-scalar ratio and a negative running spectral index. Afterwards, we confront the theoretical predictions of this model with the combined Planck and BICEP2 data under a class of fixed parameter values in Sec. IV. Our numerical computation nicely demonstrate the model can explain these observations consistently. We conclude in Sec. V with a general discussion.
Single field inflation model with modulated potential
=====================================================
The single field inflation model with modulated potential was first proposed in Refs. [@Wang:2002hf; @Feng:2002a]. In these earliest papers, the modulation acts as a rapid oscillating term added to the so-called natural inflation potential [@Freese:1990rb], which can help generating large running spectral index. In this section, we will firstly take a brief review of the development of this kind of model.
Natural inflation model was motivated by the idea to connect cosmic inflation and particle physics, where the inflaton is considered as a Pseudo-Nambu-Goldstone Boson (PNGB) from the spontaneous breaking of a global symmetry [@Freese:1990rb]. In this model, shift symmetry has been introduced to make the inflaton potential flat and stable, and to maintain sufficient e-folding numbers [@Stewart:1996ey]. Similar to the axion [@Peccei:1977hh], the inflaton potential has the form \[natural\] V()=\^4(1-) , where $f$ is the scale of spontaneous symmetry breaking and the cosine term is thought to be produced by some non-perturbative effects which break the symmetry explicitly at a relative low scale $\Lambda$. So the inflaton has the mass at the order of $m\sim \Lambda^2/f$. However, the flatness condition of the inflaton potential requires the scale of spontaneous symmetry breaking $f$ to be larger than the Planck scale and it is expected that at such a high scale the global symmetries are violated explicitly by the quantum gravity effects [@quantumgravity1; @quantumgravity2]. These effects introduces a modulation to the potential, which was considered in [@Wang:2002hf] from the viewpoint of effective field theory. By considering the higher dimensional operators without derivatives (due to the global symmetry breaking at $M_p$), the model proposed in [@Wang:2002hf] has the potential \[natural2\] V()=\^4\[1--(+)\] , where $\delta$ is a small number and ${\cal N}$ is large, the phase $\beta$ is physically unrelevant and can be set to zero. It is also possible to add a constant to the above potential to make its minimum vanish. A large ${\cal N}$ will modulate the potential with rapid oscillations and superimpose a series bumps into the otherwise featureless potential. But for sufficiently small $\delta$, the amplitude of the oscillations can be controlled to be small to protect the overall picture of inflation. The slow-roll conditions are violated mildly and the predicted scalar power spectrum has strong oscillations with sizeable running but still allowed by the observations, as shown in [@Wang:2002hf; @Liu:2009nv]. Another prediction of this model is the enhanced wiggles in the matter power spectrum. These features are possible to be detected by future experiments [@Pahud:2008ae].
In 2003, Arkani-Hamed et al. proposed the extra-dimensional version of natural inflation(extranatural inflation) [@ArkaniHamed:2003wu], where a five-dimensional Abelian gauge field is considered and the fifth dimension is compactified on a circle of radius $R$. The extra component $A_5$ propagating in the bulk is considered as a scalar field from the four-dimensional view and the inflaton is identified as the gauge-invariant Wilson loop $\theta=g_5\oint dx^5 A_5$, with a 5D gauge coupling constant $g_5$. The 4D effective Lagrangian at energies below $1/R$ can be written as: =()\^2-V() , with $g_4^2=g_5^2/(2\pi R)$ the four-dimensional effective gauge coupling constant. The non-local potential $V(\theta)$ is generated in the presence of fields charged under the Abelian symmetry in the bulk [@hosotani; @Delgado:1998qr], for a massless field with charge $q$ the potential for the Wilson loop is V()=(q\_1) , where the “$+$" and “$-$" represent the bosonic and fermionic fields respectively and we have neglected higher power terms. By defining $\phi=f_{eff}\theta$ and adding a constant term, the potential has almost the same form with that natural inflation (\[natural\]) and the effective decay constant is (we have assumed $q$ is of order unity) $
f_{eff}=1/(2\pi g_4R)$. There are some advantages of this model compared with the old natural inflation based on the 4D PNGB, such as that the effective decay constant $f_{eff}$ can be naturally greater than $M_p$ for a sufficiently small coupling constant $g_4$, and gravity-induced higher-dimensional operators are generally exponentially suppressed as long as the extra dimension is larger than the Planck length due to the extra dimension nature. However, as same as the natural inflation, this model predicts a scalar spectrum with negligible running of the spectral index. In order to have large running index, in Ref.[@Feng:2002a] the authors generalized this model to the case including multiple charged fields under the Abelian symmetry. If we simply consider one massless and one massive fields coupled to $A_5$ gauge field, [*i.e.,*]{} $M_1=0$ and $M_2>R$, as studied in detail in [@Feng:2002a], the effective potential for the inflaton then becomes: \[potentialnew\] V()&=&\[1-()-()\] , where again we have neglected higher power terms, $q_1,~ q_2$ are the charges of these two fields, and $\sigma$ is related to $M_2$ as \[parameter-sigma\] =(-1)\^[F\_2]{}e\^[-2RM\_2]{}(\^2R\^2M\_2\^2+2RM\_2+1) , where the constant $F_2=0, ~1$ for the bosonic and fermionic fields respectively. For a large mass $M_2$, $\sigma$ has a small value. If the ratio of these two charges $q_2/q_1\gg 1$, we get the same potential as the model (\[natural2\]) mentioned above. It was found in [@Feng:2002a] that this model can produce a scalar spectrum with negative running as large as $10^{-2}$. In all, the model with modulations (\[natural2\]) or (\[potentialnew\]) are well-motivated, and different from the simplest slow-roll inflation models, it produces an oscillating scalar spectrum with significant running. However, the tensor-to-scalar ratio $r$ produced in [@Feng:2002a] is very small which cannot be consistent with the BICEP2 data. In this paper we will investigate whether this model can produce a large $r$ suggested by BICEP2 and at the same time a sizeable running to alleviate the tension between the Planck and BICEP2 data. Note that some related studies has been done in Ref. [@Czerny:2014wza], and here we will revisit this problem in more detail.
Inflationary dynamics with non-vanishing running spectral index
===============================================================
In this section we perform the analytical and numerical analyses of the inflationary solution described by this model. In particular, we analyze the dynamics of the slow roll parameters during inflation. Consider a canonical scalar field with the potential given by .
The inflationary dynamics can be characterized by a series of slow roll parameters, of which the expressions are given by, $$\begin{aligned}
\epsilon & \equiv \frac{M_p^2}{2} (\frac{V_\phi}{V})^2 = \frac{\mu^2}{2} \frac{(\sin\tilde\theta+\sigma\kappa\sin\kappa\tilde\theta)^2}{(1-\cos\tilde\theta-\sigma\cos\kappa\tilde\theta)^2} ~,\nonumber\\
\eta & \equiv M_p^2 \frac{V_{\phi\phi}}{V} = \mu^2 \frac{ \cos\tilde\theta+\sigma\kappa^2\cos\kappa\tilde\theta }{1-\cos\tilde\theta-\sigma\cos\kappa\tilde\theta} ~,\nonumber\\
\xi & \equiv M_p^4 \frac{V_{\phi}V_{\phi\phi\phi}}{V^2} \nonumber\\
& = -\mu^4 \frac{(\sin\tilde\theta+\sigma\kappa\sin\kappa\tilde\theta) (\sin\tilde\theta+\sigma\kappa^3\sin\kappa\tilde\theta)}{(1-\cos\tilde\theta-\sigma\cos\kappa\tilde\theta)^2}~,\end{aligned}$$ where we have introduced $$\begin{aligned}
\mu = q_1 M_p/f_{eff} ~,~ \kappa = q_2/q_1 ~,~ \tilde\theta = q_1 \theta ~.\end{aligned}$$
Note that, inflation requires the above slow roll parameters to be much less than unity. Accordingly, the inflationary e-folding number follows: $$\begin{aligned}
\label{efolding}
N & \equiv \int_{t_i}^{t_e} H dt \simeq -\frac{1}{M_p^2}\int_{\phi_i}^{\phi_e} \frac{V}{V_\phi} d\phi ~,\nonumber\\
& = -\mu^{-2}\int_{\tilde\theta_i}^{\tilde\theta_e} \frac{(1-\cos\tilde\theta -\sigma\cos\kappa\tilde\theta)}{(\sin\tilde\theta +\sigma\kappa\sin\kappa\tilde\theta)} d\tilde\theta~,\end{aligned}$$ where we have applied the approximation $\dot\phi^2\ll 2V$. Following the standard procedure of inflationary perturbation theory [@Stewart:1993bc; @Mukhanov:1990me], the power spectra of primordial curvature and tensor perturbations of this model can be expressed as: $$\begin{aligned}
\label{spectrum}
{\cal P}_{\cal S} = \frac{V(\phi)}{24\pi^2M_p^4\epsilon}\Big|_{k=aH} ~,~
{\cal P}_{\cal T}=\frac{2 V(\phi)}{3\pi^2M_p^4}\Big|_{k=aH}~,\end{aligned}$$ and correspondingly, the tensor-to-scalar ratio is defined as $$\begin{aligned}
\label{r}
r \equiv \frac{{\cal P}_{\cal T}}{{\cal P}_{\cal S}}=16\epsilon~.\end{aligned}$$ Moreover, one can define the spectral index of primordial curvature perturbations and the associated running spectral index as follows, $$\begin{aligned}
\label{index}
&& n_s-1 \equiv \frac{d\ln{\cal P}_{\cal S}}{d\ln k} \approx -6\epsilon+2\eta ~, \\
\label{index_alpha}
&& \alpha_s \equiv\frac{dn_s}{d\ln k} \approx 16\epsilon\eta-24\epsilon^2-2\xi ~.\end{aligned}$$
In regular inflation models, the slow roll parameters scale as: $\epsilon,\eta\sim N ^{-1}$ and $\xi\sim N ^{-2}$ during inflation. Moreover, the spectral index $n_s-1$ is of order $\epsilon$ and $\alpha_s$ is of order $\epsilon^2$. Thus, a large running behavior of the spectral index is difficult to be achieved due to the suppression effect by $ N ^{-2}$. However, it is interesting to notice that, in the model under consideration, the slow roll approximations can be slightly broken for a short while due to the inclusion of the rapid oscillating term in the potential.
In our model, inflation ceases when $\phi$ reaches $\phi_e$ with $\epsilon=1$, and one can have the initial value for the inflaton to be the value at the moment of Hubble-crossing. Near the Hubble-crossing, one can get the expressions for the slow roll parameters approximately, $$\begin{aligned}
& \epsilon \simeq \frac{2\mu^2}{\tilde\theta^2} ~, \nonumber\\
& \eta \simeq \frac{2\mu^2}{\tilde\theta^2}(1+\sigma\kappa^2\cos\kappa\tilde\theta) ~, \nonumber\\
& \xi \simeq -\frac{4\mu^4}{\tilde\theta^3}\sigma\kappa^3\sin\kappa\tilde\theta ~.\end{aligned}$$ When $|\mu|\leq |\tilde\theta|$, $\sigma\kappa^2\sim{\cal O}(1)$ and $\kappa\gg 1$, from the above approximation one can get the value of $\xi$ in the same order of $\epsilon$ and $\eta$, and hence a relatively large running behavior can be obtained.
In the following we perform the numerically calculation of our model. In Fig.\[Fig:slow-roll\] we plot the evolutions of the slow-roll parameters $\epsilon$, $\eta$ and $\xi$ with respect to the e-folding number $ N$. In our figure plot, inflation begins from the right side where the slow-roll parameters are small, and ends at the left side where they present some oscillatory behavior with their amplitude approaching 1. The pivot scale, which corresponds to $k\approx 0.05\text{Mpc}^{-1}$, crosses the Hubble radius at the time when $ N \approx50$, marked with black dotted line. The numerical results depends only on three parameters in the model, namely $\mu$, $\sigma$ and $\kappa$. In the numerical calculation, we take three groups of parameter choices (see the caption), and in order to have a comparison, we also plot the cases of natural inflation model. One could see that at the pivot scale $\xi$ is almost of the same order as $\epsilon$ and $\eta$ (To help see more clearly, we also plot the zoomed-in figures around the pivot scale, with the vertical coordinates of the same range.) in our model, while is negligible in natural inflation model. Therefore, as has been analyzed above, one can observe a considerable running behavior of the spectral index around this point.
![Numerical plot of the slow-roll parameters $\epsilon$, $\eta$ and $\xi$ in our model (solid lines) and the natural inflation model (dashed lines). The horizontal axis represents for the e-folding number $ N$ and $ N =0$ means the end of inflation. The parameters in our model are chosen as: $\mu=0.034$, $\sigma=1.3\times 10^{-3}$, $\kappa=51$ (cyan); $\mu=0.037$, $\sigma=5.0\times 10^{-4}$, $\kappa=68$ (red); green lines: $\mu=0.040$, $\sigma=1.8\times 10^{-3}$, $\kappa=40$ (green). The parameters in natural inflation model are chosen as: ; magenta lines: $f=2$ (magenta); $f=5$ (yellow); $f=10$ (pink). The dotted-black lines are associated with the pivot scale, which is chosen as $k_\ast=0.05~ \text{Mpc}^{-1}$.[]{data-label="Fig:slow-roll"}](slow-roll.eps)
One can directly relate the slow-roll parameters with the perturbation variables of a canonical single field inflation model. In Fig.\[Fig:f-k\] we plot the evolution of the spectral index of scalar perturbation $n_s$, the running of the spectral index $\alpha_s$, and the tensor-to-scalar ratio $r$. In the plot, we take the range from $1.0\times10^{-5}~ \text{Mpc}^{-1}$ to $1.0~ \text{Mpc}^{-1}$ which is able to cover the $l$ range ($2\leq l \leq 2500$) used in Planck and BICEP2 paper. We also marked with a vertical dotted line the pivot scale, $k_\ast\simeq 0.05~ \text{Mpc}^{-1}$ which reenters the Hubble radius and eventually can be observed by today’s experiments. From the plot we can see that both our model and natural inflation can give a large $r$, as needed by the BICEP2’s data. However, one obvious difference between the two models is that our model is able to yield a negative running spectral index for the power spectrum of scalar perturbations, roughly of the order $-0.03 \sim -0.01$ that can be applied to reconcile the Planck and BICEP2 data [@Ade:2013uln; @Ade:2014xna], while the running behavior from the model of natural inflation is negligible. Such a considerable running behavior obtained in our model arises from a large-valued parameter $\kappa$, which bring $\xi$ to the same order of $\epsilon$ and $\eta$ so that it has dominant contribution to the expression (\[index\_alpha\]) of $\alpha_s$ in comparison with other two terms.
![Numerical plot of the spectrum index $n_s$, tensor-to-scalar ratio $r$, and the running of the spectral index $\alpha_s$ of our model (solid lines) and the natural inflation model (dashed lines). The horizontal axis represents for the comoving wave-numbers $k$. The model parameters are the same as those provided in Fig. \[Fig:slow-roll\]. []{data-label="Fig:f-k"}](f-k.eps)
A sizable negative running has the possibility to make the spectral index $n_s$ vary efficiently with scales, e.g., from blue tilt to red tilt. This could lead to some observable features on the power spectrum ${\cal P}_{\cal S}$, namely, a bump would appear on the ${\cal P}_{\cal S}-k$ plot, or the amplitude on small $l$ region might get suppressed, which can be useful in the explanation of small $\ell$ anomaly. In Fig.\[Fig:ps-k\] we plot the amplitudes of scalar perturbations under various parameter choices. We can see that, although at the pivot scale the amplitudes in these cases are almost the same, which are consistent with the data, they can be very different at small $\ell$ regions.
![Numerical plot of the power spectrum of primordial scalar perturbations as a function of the comoving wave number $k$. The model parameters are the same as those provided in Fig. \[Fig:slow-roll\]. []{data-label="Fig:ps-k"}](ps-k.eps)
Fitting the cosmological data
=============================
With the analyses performed in the above section, we have shown that our model can indeed have a sizable negative running $\alpha_s$ as well as a large $r$. In this section, we directly confront our model to the observational data to see how it reconciles the Planck and BICEP2 data.
In Fig. \[Fig:r-ns\] we present our results in $n_s-r$ plot. We plot our model with two groups of parameter choices (blue and red), which can both fit the Planck+BICEP2 data very well. For each choice, we considers two cases which inflation continues for 60 (solid lines) and 50 e-foldings (dashed lines). For a comparison, we also plot natural inflation models with $N=50$ and $N=60$, presented with magenta lines. The lines grows as $f_{eff}$ grows, making its prediction of $n_s$ and $r$ more and more close to our model, and also more and more close to the allowed space by the contour. In this plot, we have chosen the pivot scale as $k_\ast=0.05\text{Mpc}^{-1}$. We have also showed the TT and BB spectrum in Fig. \[Fig:tt\] and Fig. \[Fig:bb\], with all the color lines have the same parameter-correspondence as in Fig.\[Fig:f-k\]. We see that on the large $l$ region all the lines glues together indicating a degeneracy of the parameters, and fit the data very well. On small $l$ regions, the lines deviate from each other, but since the error bars on this region are quite large, the lines are still in consistency with the data. However, one might also notice that when the line best fits the Planck temperature spectrum, i.e. the cyan one, gives smaller $BB$ auto-correlation compared with BICEP2’s data. Conversely, the line which fits BICEP2’s data much better (the green one) gives larger temperature power spectrum which fails to explain the small $l$ anomaly. This phenomenon can be easily understood: the scalar and tensor spectrum are linked by tensor-scalar ratio $r$, which won’t change too much with $k$ in our model, as can be seen in Fig.\[Fig:f-k\]. Therefore, a raising/lowering of scalar spectrum at large scales corresponds to the same behavior of tensor spectrum. Furthermore, we also plot the red line as an intermediate case, which will not deviate too much from the data points in either TT or BB spectrum. We hope the global fitting of full parameter space can provide us a better parameter choice for both Planck TT spectrum and BICEP2 BB spectrum, which we will leave for future investigations.
![The $n_s-r$ constraint of inflation models. The dark and light blue shadow regions represent for the $1\sigma$ and $2\sigma$ contours from the combined Planck+BICEP2 data [@Li:2014h], respectively. The solid and dashed lines denote our model with $60$ and $50$ e-folding numbers, respectively. The model parameters are chosen as: $\mu=0.04$, $\sigma=1.0\times 10^{-3}$ with $\kappa$ varying from 46 to 57 (blue), and $\mu=0.04$, $\sigma=5.0\times 10^{-4}$ with $\kappa$ varying from 50 to 62 (red). The magenta lines denotes natural inflation model, with $f$ changes from $0.5$ to $10$. []{data-label="Fig:r-ns"}](r-ns.eps)
![Numerical comparison of the temperature power spectrum of our model with the Planck data. The model parameters are the same as those provided in Fig. \[Fig:slow-roll\]. []{data-label="Fig:tt"}](tt.eps)
![The BB power spectrum of our model with the BICEP2 data. The model parameters are the same as those provided in Fig. \[Fig:slow-roll\]. []{data-label="Fig:bb"}](bb.eps)
Conclusions
===========
The BICEP2 group has released the results of the CMB polarization measurement, which strongly hints to an existence of a large amplitude of primordial gravitational waves. This result, however, is in tension with the Planck data released last year when they are interpreted by the standard $6$-parameter $\Lambda\text{CDM}$ (without $r$) model. One simple approach to alleviate this tension is to take into account the running of the spectral index of the curvature perturbations, which in our paper is characterized by $\alpha_s$. The inclusion of this parameter can greatly relax the observational constraint on the tensor-to-scalar ratio due to their degeneracy [@Li:2012ug]. Recently, a numerical global simulation of the $\Lambda\text{CDM}+r+\alpha_s$ model reveals that the combined Planck and BICEP2 data favor a negatively valued running spectral index of $-0.03\sim -0.01$ [@Li:2014h].
This observational implication, while puts forward a challenge to slow roll inflation models, can be nicely implemented by a single field inflation with modulated potential [@Wang:2002hf; @Feng:2002a] as demonstrated in the present paper, where we treat the modulation as a rapid oscillating term. This is because, although the whole inflationary dynamics is dominated by the regular slow roll part of the potential, this rapid oscillating term can relatively violate the slow roll approximation during some local evolutions. In the specific model considered, we explicitly show that the parameter $\xi$ which is associated with the running and a higher order slow roll parameter in the normal slow roll inflation model, can be enhanced to the value as large as the first order slow roll parameters $\epsilon$ and $\eta$. Therefore this model can give rise to a considerable and negative running spectral index. In this paper, we performed the numerical calculation of the model in detail by solving the dynamics of slow-roll parameters and perturbation variables, and then fitted them to the combined Planck and BICEP2 data. From the numerical results, one can easily see that, with a large-valued parameter $\xi$ near the pivot scale, the spectrum index $n_s$ can be changed from value larger than $1$ to value smaller than $1$ performing a negative running feature. Due to this running behavior, the power spectrum can be suppressed at small $\ell$ region. At the same time this model can also produce gravitational waves of large amplitudes as long as the effective decay constant is large enough. Hence the inflation model under consideration provides a consistent interpretation of the combined Planck and BICEP2 data.
The present model has further implications for the observations. As we have mentioned the modulation of the rapid oscillating term could amplify the wiggles of the CMB temperature spectrum [@Liu:2009nv] and the matter power spectrum and thus is of observable interest to the future experiments [@Pahud:2008ae]. Although this model is motivated by observational phenomena, it deserves mentioning that this model has interesting connections with other inflation models. For example, we have mentioned that our model can reduce to a natural inflation model when the modulation term is small enough. The extranatural inflation including higher power terms has been recently investigated in [@Kohri:2014rja]. The axion-monodromy inflation with modulations was studied in [@Higaki:2014sja; @Flauger:2009ab], while its supergravity version was discussed in [@Kallosh:2014vja]. In the literature, there are other studies on deriving a large running of the spectral index from various approaches, for instances, see [@Lidsey:2003cq; @Kawasaki:2003zv; @Feng:2003zua; @Chung:2003iu; @BasteroGil:2003bv; @Yamaguchi:2003fp; @Lee:2004ex; @Kogo:2004vt; @Paccetti:2005zm; @Ballesteros:2005eg; @Huang:2006um; @GonzalezFelipe:2007uy; @Matsuda:2008fk; @Kobayashi:2010pz; @Peloso:2014oza; @Czerny:2014wua; @Contaldi:2014zua; @Ashoorioon:2014nta].
Additionally, the wiggles in power spectrum can induce features on non-Gaussianities, especially of the squeezed shape, since from the consistency relation we roughly have $\langle{\cal R}_{k_1}{\cal R}_{k_2}{\cal
R}_{k_3}\rangle_{k_3\ll k_1,k_2}\sim f_{nl}^{squeezed} {\cal P}_{\cal
S}^2\sim (n_s-1) {\cal P}_{\cal S}(k_1) {\cal P}_{\cal S}(k_3)$ [@Maldacena:2002vr], where wiggles in $n_s$ may affect $f_{nl}^{squeezed}$ [@Gong:2014spa]. The features on non-Gaussianities are expected to be detected by the future observations. We will discuss these issues as a sequence of this work in a future project.
As a final remark, note that there may be other approaches of reconciling the tension between the Planck and BICEP2 data within the framework of inflationary cosmology, such as to suppress the scalar spectrum at large scales by a double field inflation model [@Feng:2003zua] or using a step-like process [@Contaldi:2014zua]. The existence of nontrivial tensor spectral index $n_T$ may also work, which needs to be accompanied by simulation of $\Lambda\text{CDM}+r+n_T$ model. To address this issue, we would like to numerically scan the full parameter space and check all available regions allowed by observations, which will be the future topic.
We are grateful to Junqing Xia and Hong Li for helpful discussions. YW, SL and XZ are supported by NSFC under grants Nos. 11121092, 11033005, 11375220 and also by the CAS pilotB program. ML is supported by Program for New Century Excellent Talents in University and by NSFC under Grants No. 11075074. CYF is supported in part by physics department at McGill university.
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abstract: 'Recent developments in solid state physics give a prospect to observe the parity anomaly in (2+1)D massive Dirac systems. Here we show, that the quantum anomalous Hall (QAH) state in orbital magnetic fields originates from the Dirac mass term and induces an anomalous four-current related to the parity anomaly. This differentiates the QAH from the quantum Hall (QH) state for the experimentally relevant case of an effective constant density (seen by the gate). A direct signature of QAH phase in magnetic fields is a long $\sigma_{xy}= e^2/h$ ($\sigma_{xy}= -e^2/h$) plateau in Cr$_x$(Bi$_{1-y}$Sb$_y$)$_{2-x}$Te$_3$ (HgMnTe quantum wells). Furthermore, we predict a new transition between the quantum spin Hall (QSH) and the QAH state in magnetic fields, for constant effective carrier density, without magnetic impurities but driven by effective g-factors and particle-hole asymmetry. This transition can be related to the stability of edge states in the Dirac mass gap of 2D topological insulators (TIs), even in high magnetic fields.'
author:
- 'J. Böttcher'
- 'E. M. Hankiewicz'
title: Parity anomaly driven topological transitions in magnetic field
---
*Introduction:* Condensed matter analogs of the (2+1)D and (3+1)D Dirac equation, i.e. topological insulators (TI) and Weyl semimetals, opened new directions to study high energy anomalies in the solid state lab. An anomaly in high energy physics is defined as breaking of a classical symmetry during regularization [@Bertl96] and, in particular, the parity anomaly is characterized by a broken parity symmetry in a quantized theory. For instance in case of the massless Dirac equation in (2+1)D, coupled to an electromagnetic field, parity symmetry is violated if one insists on gauge-invariance [@Deser82; @Jackiw84; @Redlich84; @Yahalom85; @Nielsen81A; @Nielsen81B]. In solid state physics on the other hand, “parity anomaly” is often understood as generation of an anomalous four-current related to a finite mass term in the Dirac equation [@Stone87; @Qi11]. Anytime when we relate to the notion of “parity anomaly”, we refer to the second definition.
At first glance, chiral currents related to the massive Dirac equation in (2+1)D, coupled to an electromagnetic field, appear similar to chiral currents associated with the QH effect (quantization of Hall conductance in a two-dimensional (2D) electron gas in an external out-of-plane magnetic field [@Thouless82; @Hatsugai93]). However, in case of the “parity anomaly” the current is induced by the mass term (violating parity), while the current in the QH phase is induced by magnetic field (no violation of parity) [@Ahmed85; @Yahalom85]. Despite a plethora of suggestions for possible realizations of the “parity anomaly” in a condensed matter system, its experimental observation is still outstanding [@Semenoff84; @Fradkin86; @Haldane88].
Recently new topological states of matter have been discovered such as 2D TIs characterized by the QSH effect where an odd number of pairs of counterpropagating (helical) edge states exist at the boundary [@Kane05A; @Kane05B; @Bernevig06; @Konig08; @Brune12]. When 2D TIs [@Liu08; @Beugeling12] or thin films of 3D TIs [@Yu10; @Chang13; @Checkelsky14] are doped with magnetic impurities, the gap for one of the spin directions can be closed due to broken time-reversal symmetry (TRS) and QAH effect forms. It is characterized by a single propagating chiral edge state existing even in the absence of magnetic field. When both TRS and particle-hole symmetry (PHS) are broken QSH, QAH and QH phases are all classified by a $Z$-topological invariant [@Schnyder08; @Schnyder16]. Outstanding questions then arise, whether any of these effects are related to the “parity anomaly” and if their experimental distinction is possible in an external magnetic field. Answering these questions is the goal of this paper.
The QSH phase is described by two copies of massive (2+1)D Dirac Hamiltonians [@Bernevig06]. As we will show in this paper, there is an anomalous four-current for each single block, indexed by $i=\left\{1,2\right\}$, if the chemical potential lies in the mass gap: $$j_\mu^{(i)}={\left(}-1{\right)}^{i+1}\frac{e^2}{4h}\left[\text{sgn}(M)+\text{sgn}(B)\right]\epsilon_{\mu\nu\tau}F^{\nu\tau},\label{eq:parityAnomaly}$$ where $F_{\nu\tau}=\partial_\nu A_\tau -\partial_\tau A_\nu$ is the electromagnetic tensor, while $M$ and $B$ are the relativistic Dirac mass and the effective mass originating from quadratic dispersion (Newtonian mass) [@Qi11], respectively.
Sum and difference of anomalous four-currents, ${j^{\pm}_\mu=j^{(1)}_\mu\pm j^{(2)}_\mu}$, can be used to distinguish between QH, QAH and QSH phases in magnetic field. Although the helical edge states of the QSH phase are not anymore protected against backscattering in magnetic fields (due to broken TRS [@Qi11; @Chen12]) as long as neither $M$ nor $B$ change their sign as a function of the magnetic field, $j_\mu^-$ remains nonzero. We denote this phase QSH-like (QSHL) phase. Similarly, the QAH state is characterized by a nonzero $j_\mu^+$ as well as $j_\mu^-$ in magnetic field and we denote this phase QAH-like (QAHL) [^1]. Chiral edge states related to the QAHL phase form in the mass gap and are therefore distinct from QH edge states which form above the mass gap and have Hall currents scaling as the sign of magnetic field [@Ahmed85].
As pointed out by Ma et al. [@Ma15], the effective carrier density and not the chemical potential can be accessed by a gate electrode in a typical experimental set-up. However, topological transitions in magnetic field have been so far mostly analyzed in terms of constant chemical potential [@Konig08]. Therefore, we predict in this paper new types of topological transitions in magnetic field at constant effective carrier density. Under this condition, we show that the “parity anomaly” allows for a unique distinction of the QAHL and QH phases in magnetic fields. In particular, we predict a long Hall plateau at $\sigma_{xy}= -e^2/h$ ($\sigma_{xy}=e^2/h$), related to the QAHL phase of HgMnTe ($\text{Cr}_x{\left(}\text{Bi}_{1-y} \text{Sb}_y{\right)}_{2-x} \text{Te}_3$) [@Chang13; @Checkelsky14]. Moreover, we predict a new topological QSH to QAHL transition, occurring without any magnetization but as a consequence of the particle-hole asymmetry and effective g-factors. This latter transition might be related to the recent observation of stable edge states of 2D TIs in the mass gap for high magnetic fields [@Ma15; @Du15].
*Model:* Let us consider the BHZ Hamiltonian, i.e. a two flavor-(2+1)D massive Dirac Hamiltonian [@Winkler15], describing 2D TIs, [@Bernevig06] $$\begin{aligned}
\mathcal{H}\left(\mathbf{k}\right)& =&
\begin{pmatrix}
h_+{\left(}\mathbf{k}{\right)}& 0 \\
0 & h_-{\left(}\mathbf{k}{\right)}\end{pmatrix}+\mathcal{H}_z{\left(}H_0 {\right)}+ \mathcal{H}_{ex}, \label{eq:BHZHamiltonian}\\
h_\pm{\left(}\mathbf{k}{\right)}&=&
\epsilon{\left(}\mathbf{k}{\right)}\sigma_0\pm M{\left(}\mathbf{k}{\right)}\sigma_3\pm A{\left(}k_x\sigma_1+k_y\sigma_2{\right)},\label{eq:ChernInsulator}\\
\mathcal{H}_z{\left(}H_0 {\right)}&=& \text{Diag} \begin{pmatrix} -g_E & -g_H &g_H& g_E \end{pmatrix}H_0,\label{eq:Zeeman}\\
\mathcal{H}_{ex} &=& \text{Diag} \begin{pmatrix} -\chi_E & -\chi_H &\chi_H& \chi_E \end{pmatrix},\label{eq:Exchange}\end{aligned}$$ written in the Dirac basis, i.e. in the low energy subbands $\left\{|E1,\downarrow\rangle,-|H1,\downarrow\rangle,-|H1,\uparrow\rangle,|E1,\uparrow\rangle\right\}$. Here, $\epsilon\left(\mathbf{k}\right)=-Dk^2$, $M\left(\mathbf{k}\right)=M-Bk^2$, $\mathbf{k}^\intercal=\begin{pmatrix} k_x & k_y\end{pmatrix}$, $k^2=k_x^2+k_y^2$, $k_\pm=k_x\pm\mathrm{i}k_y$ and $D$, $M$, $B$ and $A$ are system parameters. $\mathcal{H}_z$ denotes a Zeeman Hamiltonian and $\mathcal{H}_{ex}$ describes exchange interaction between s/p-band electrons with localized spins belonging to the magnetic impurities. For HgMnTe [@Novik05], the exchange coupling is paramagnetic [@Liu08], while it is ferromagnetic for $\text{Cr}_x{\left(}\text{Bi}_{1-y} \text{Sb}_y{\right)}_{2-x} \text{Te}_3$ [@Yu10]. In the following, we demand that the chemical potential is placed in the Dirac mass gap at zero magnetic field such that the system is insulating in the bulk.
![Spectrum of spin-up block of BHZ Hamiltonian for **(a)** QAHL $\nu=1$ and **(b)** a trivial insulator $\nu=0$ in magnetic fields. $\nu_\downarrow=0$ state is not shown and stays constant across the transition. The magnetic gap is given by $\Delta E_\uparrow^{H_0}$ and the mass gap by ${M^\star_\uparrow}$. For the trivial insulator, both the bare charge $\rho$ and the effective charge $\rho^\star$ are equal to zero (see main text). **(c)** Schematic view of charge pumping in ground state of QAHL insulator. For QAHL system an adiabatic increase of the magnetic field $H_0$ causes a circulating electric field. At the mass domain wall, indicated by the colored landscape, chiral edge states form and charges flow from the domain wall to the bulk as a response to the electric field. Due to the “parity anomaly”, an additional topological term occurs in the Chern-Simons action of a QAHL system such that the net charge/electric field is zero, $ \rho^{*} =0 $ but $\rho\neq0$. Black arrows represent schematically flux quanta attached to the electrons.[]{data-label="fig:QHvsQAH"}](./figure1.pdf){width=".49\textwidth"}
Let us first study the bulk Landau level (LL) spectrum of the spin-up block. We consider an external magnetic field $\bm{H}$ oriented perpendicular to the plane of the 2D electron gas, $\bm{H}=H_0\mathbf{e}_z$, with $H_0 > 0$. The orbital effect of the magnetic field is introduced by the Peierls substitution ${\mathbf{k}\rightarrow \mathbf{k}+e/\hbar\mathbf{A}}$ $(e>0)$, using the Landau gauge ${\mathbf{A}=-yH_0\mathbf{e}_x}$. The bulk LL spectrum is obtained using ladder operators $k_{-}=\sqrt{2}/l_H\hat{a}$ and $k_+=\sqrt{2}/l_H\hat{a}^\dagger$ fulfilling the usual commutation relation $\left[\hat{a},\hat{a}^\dagger\right]=1$, where $l_H=\sqrt{\hbar/eH_0}$ is the magnetic length. This can be used to construct an ansatz for the Schrödinger equation [@Konig08], $$\begin{aligned}
\psi^{k_x,n\neq0}_\uparrow&=&\begin{pmatrix}u_1 |n-1\rangle & u_2 |n\rangle \end{pmatrix}^\intercal,\\
\psi^{k_x,n=0}_\uparrow&=&\left(\begin{array}{cc}0&|0\rangle\end{array}\right)^\intercal,\end{aligned}$$ where $n$ is the LL index and $k_x$ is a good quantum number. The resulting spectrum is given by $$\begin{aligned}
E^{\pm}_{\uparrow,n>0} & = & \frac{g_1-\beta}{2}-n\delta\pm\sqrt{n\alpha^{2}+\left( M^\star_\uparrow{\left(}H_0{\right)}- n\beta\right)^{2}},\label{eq:LLspectrumChernN}\\
E_{\uparrow,n=0} & = & M+\frac{g_1+g_2-\beta-\delta}{2},\label{eq:LLspectrumChern0}\end{aligned}$$ where $\alpha=\sqrt{2}A/l_H$, $\beta=2B/l_H^2$ and $\delta=2D/l_H^2$, ${g_{1/2 }={\left(}\chi_E+g_EH_0{\right)}\pm {\left(}\chi_H+g_HH_0{\right)}}$ and $$M^\star_\uparrow{\left(}H_0{\right)}=M+g^\star{\left(}H_0{\right)},\label{eq:massTerm}$$ where $g^\star{\left(}H_0{\right)}={\left(}g_2{\left(}H_0{\right)}-\delta{\left(}H_0{\right)}{\right)}/2$ is the effective g-factor. Since $M^\star_\uparrow{\left(}H_0{\right)}$ transforms under parity as the usual Dirac mass M, it can be interpreted as the renormalized Dirac mass in an external magnetic field and it replaces $M$ in Eq. (\[eq:parityAnomaly\]). More details on the symmetry of the BHZ Hamiltonian are given in Appendix B. It is apparent that even for $\chi_E=\chi_H=g_E=g_H=0$ the effective g-factor is nonzero due to a broken PHS.
For $n\neq0$, all LLs come in particle/hole pairs reproducing the bulk band gap in the limit $H_0\rightarrow0$. However, the $n=0$ LL lacks a partner [@Haldane88] and its respective chiral edge state goes through the mass gap in case of the topologically non-trivial regime ${M^\star_\uparrow(0)/B>0}$.
*Trivial versus QAHL phases in orbital magnetic fields:* Here, we study the difference between trivial insulating and QAHL phases in magnetic fields. The QAHL phase is characterized by ${\nu=\nu_\uparrow+\nu_\downarrow=1}$, while the trivial insulator has $\nu=0$. Since the two blocks in Eq. (\[eq:BHZHamiltonian\]) are decoupled, we focus on the spin-up block $h_-{\left(}\mathbf{k}{\right)}$ ($\nu_\uparrow=1$) and omit the trivial ($\nu_\downarrow=0$) spin-down block. Details of the calculation are given in Appendix A.
![Landau level fan for **(a)** trivial insulator **(b)** non-trivial 2D TI, where blue and red symbolize spin-up and spin-down LLs, respectively. Both plots are calculated within BHZ model for $\text{Hg}_{0.98}\text{Mn}_{0.02}\text{Te}$ QWs with **(a)** $d=7 \text{ nm}$, $M=14 \text{ meV}$, $B=-612 \text{ meVnm}^2$, $D=-440 \text{ meVnm}^2$, $\text{A}=395\text{ meVnm}$, $g_E= 18.6\;\mu_B$, $g_H =-1.2\;\mu_B$ and with **(b)** $d=10 \text{ nm}$, $M=-8 \text{ meV}$, $B=-1070 \text{ meVnm}^2$, $D=-895 \text{ meVnm}^2$, $\text{A}=366\text{ meVnm}$, $g_E = 29.5\;\mu_B$, $g_H =-1.2\;\mu_B$. $M/B<0$ ($M/B>0$) indicates a topologically trivial (non-trivial) phase [@Qi11]. At $H_0 = 0$, Mn impurity spins are randomly oriented and the system is in the QSH phase, $\nu=0$. In a magnetic field, the gap for the spin-up block closes at $H_\uparrow^{crit}=1.2\text{ T}$ with a transition to a stable $\sigma_{xy}=-e^2/h$ QAHL plateau. The transition is indicated by crossing of the chemical potential $\mu$ with the spin-up $n=0$ LL. Numbers within plot indicate Chern numbers.[]{data-label="fig:LandauFan"}](figure2.pdf){width=".49\textwidth"}
We start the discussion from the QAH state with ${\nu=1}$, where the ground state is defined by ${|vac\rangle=\prod_{k_x,k_y}a^\dagger_{-,\uparrow}{\left(}k_x,k_y {\right)}|0\rangle}$ and the minus sign denotes that only valence band states are filled. Adiabatically switching on the magnetic field induces an azimuthal electric field, since $\nabla\times \mathbf{E}=-\partial / \partial t \;H_0$. Based on a semi-classical calculation [@Xia05; @Sundaram99; @Streda82], it is easy to show that an anomalous four-current is induced, $$j_{\mu} = -\frac{e^2}{4h} \left[ \text{sgn}{\left(}M^\star_{\uparrow}{\left(}x{\right)}{\right)}+\text{sgn}{\left(}B {\right)}\right]\epsilon_{\mu\nu\tau}F^{\nu\tau}. \label{eq:anomalousLin}$$ This means that the electric field pump current $j_x$ from the left (right) side of the sample into the bulk, $e^2 |E_y|/h$ ($-e^2 |E_y|/h$), resulting in a carrier density of ${\rho = -e^2 H_0/h=-e H_0 /\phi_0}$ with ${\phi_0=h/e}$. This corresponds to filling of the $n=0$ LL and signs of currents and charges are determined by the sign of the mass, $M^\star_{\uparrow}$ as schematically indicated in Fig. \[fig:QHvsQAH\]c. Since during the process of switching of $H_0$, the topological Dirac mass does not close, the question arises: Why does the ground state of QAHL phase change from $\rho=0$ in a zero magnetic field to a finite value for a non-zero $H_0$? Does this not break the adiabatic assumption for switching on magnetic field as well as the constraint of constant carrier density in experiments [@Ma15; @Budewitz16; @Checkelsky14]? The solution comes from the fact that QAH and QAHL phases break parity symmetry due to the topological mass, $M^\star_\uparrow{\left(}H_0{\right)}$ (see Appendix B), which enforces adding a topological Chern-Simons term in the gauge field Lagrangian, $$\Delta \mathcal{L}_C=\frac{\kappa}{2}\epsilon^{\mu\nu\tau}F_{\mu\nu}A_\tau,$$ where $\kappa=-e^2 \left[ \text{sgn} (M^\star_{\uparrow}{\left(}x{\right)})+\text{sgn} (B)\right] /h$ is the Chern-Simons coupling constant, which modifies the Maxwell equation, $$\nabla\cdot \mathbf{E}=\rho{\left(}H_0{\right)}+\kappa H_0 \equiv \rho^{\star}. \label{ren_CS_charge}$$ The term originating from the Chern-Simons action generates a flux $\kappa H_0$ which compensates charge such that the full effective charge $\rho^\star$ remains zero ($\rho\neq0$). When the chemical potential is located within the mass gap at $H_0=0$ with $\nu=1$, it remains in the mass gap for non-zero $H_0$. Using a half-space calculation, we additionally prove in Appendix D that the $n=0$ LL makes a transition from the trivial to QAHL insulator at $M^\star_\uparrow{\left(}H_0{\right)}=0$ [@Scharf12; @Tkachov10]. Even so the discussion was given for a QAHL state with $\nu=1$, it applies as well for $\nu=-1$. In comparison, a trivial insulator does not break parity symmetry and therefore no intrinsic Chern-Simons term is allowed in the gauge field Lagrangian [@Ahmed85]. This is shown in Fig. \[fig:QHvsQAH\]b, where $\rho=\rho^\star=0$. This means that QAHL phase has a non-zero Chern number but $\rho^{*} = 0$ and therefore it has to be counted differently from QH LLs forming above the magnetic gap. Therefore, the “parity anomaly” distinguishes unambiguously QAHL effect from a trivial insulator in magnetic fields.
*Stable QAHL plateau for $\nu =\pm 1$:* Let us now come back to the full BHZ Hamiltonian and compare LL fans for topological trivial and non-trivial insulators. The discussion is given for realistic parameters for HgMnTe [@Novik05; @Liu08; @Landolt]. Technical details concerning these parameters are given in Appendix C. $\text{Hg}_{0.98}\text{Mn}_{0.02}\text{Te}$ with $d=7 \text{ nm}$ is a topologically trivial insulator ($\nu=\nu_\uparrow=\nu_\downarrow=0$), the chemical potential for an effective charge $\rho=\rho^\star = 0$ lies in the mass gap (Fig. \[fig:LandauFan\] a)) and it remains to be a trivial insulator even for large magnetic fields.
$\text{Hg}_{0.98}\text{Mn}_{0.02}\text{Te}$ with $d=10 \text{ nm}$ (Fig. \[fig:LandauFan\]b) forms QSH phase with $\nu_\uparrow=-\nu_\downarrow=1$ and $\rho=\rho^* =0$. For finite $H_0$, Mn becomes polarized and the mass gap for the spin-up block closes at $M^\star_\uparrow( H^{crit}_\uparrow)=0$ signalizing formation of a massless Dirac fermion. For $H_0>H^{crit}$, it is easy to prove (Appendix A) that ${\rho=e H_0 /\phi_0}$ while ${\rho^\star=0}$. Across this transition, the system goes into a QAHL ${\nu=-1}$ phase.
In the literature, topological transitions are often discussed based on the assumption of a constant chemical potential and the QSH to QAH transition occurs at zero orbital field [@Konig08; @Beugeling12]. Our transition differs from the above mentioned in two aspects: 1) we assume a constant effective density 2) the topological transition happens in finite magnetic fields. Therefore, we claim that even at non-zero magnetic field, one can define QAHL phase with $j_\mu^{(\pm)}$ different from zero. In conclusion, we predict a stable $\nu= -1$ plateau with a Hall conductance $\sigma_{xy} = -e^2/h$ (see Fig. \[fig:QSHLtoTrivial\]) for HgMnTe above critical QW thickness. This prediction is exactly along the lines of recent experiments on HgMnTe [@Budewitz16], where one needs non-zero magnetic field to polarize Mn and observes a stable $\nu=-1$ plateau. Similarly, recent experiments on $\text{Cr}_x{\left(}\text{Bi}_{1-y} \text{Sb}_y{\right)}_{2-x} \text{Te}_3$ show a stable $\nu =1$ plateau up to $15\text{T}$ [@Chang13; @Checkelsky14]. We believe that indeed these stable plateaus are direct signatures of the “parity anomaly” in these systems and the sign of g-factors determines if the stable plateau occurs for $\nu= 1$ or $\nu = -1$. Furthermore, after crossing of the two $n=0$ LLs at $H^{cross}$ [@Scharf12], the adiabatic condition is possibly not anymore valid. We leave the discussion of this point to another paper, assuming for the moment being that the transition to the trivial phase occurs [@Konig08].
The experimental signature in case of HgMnTe is therefore reentrant behavior of the Hall conductance starting from $\sigma_{xy} =0$ in the QSH phase, changing to $\sigma_{xy}= -1$ in the QAHL state, and reentering $\sigma_{xy}=0$ plateau for $H_0 >H^{cross}$ (see Fig. \[fig:QSHLtoTrivial\]).
We have omitted off-diagonal terms connecting the two Dirac Hamiltonians such as Rashba spin-orbit interaction terms [@Rothe10; @WinklerBook] or bulk-inversion asymmetry terms [@Konig08]. They should be small in comparison to diagonal terms and should only renormalize the effective mass gap slightly [@Muhlbauer14]. Topological transitions in magnetic fields are induced by mass gaps closing and as long as these off-diagonal terms do not close the gap, we expect that only the critical magnetic field might deviate from our prediction. (see also Appendix B).
![Transition from the QSHL to the QAHL and finally to the trivial insulator as a function of the magnetic field $H_0$. Left axis corresponds to the non-local four-terminal Hall conductance, $\sigma_{xy}$ (as for example in [@Budewitz16]), while the right axis to the local two-terminal longitudinal conductance, $\sigma_{xx}$ (as for example in [@Ma15]). To highlight transitions we subdivided the plot in three areas. In the insets the blue and red lines show schematically the $n=0$ LLs and the black dashed line shows the position of the chemical potential in the ground state. $\sigma_{xy}=0$ for $H_0<H^{crit}_\uparrow$ for QSHL effect. The transition to QAHL phase with a $\sigma_{xy}=-e^2/h$ occurs at $H^{crit}_\uparrow$ by virtue of the “parity anomaly”. At the crossing point of the two $n=0$ LL, $H^{cross}$, the system might become trivial (see main text). The $\sigma_{xx} =2e^2/h$ in the QSHL phase changes to $e^2/h$ at $H^{crit}_\uparrow$ (QAHL phase), and possibly to zero at $H^{cross}$.[]{data-label="fig:QSHLtoTrivial"}](./figure3.pdf){width=".49\textwidth"}
*QSHL to QAHL transition in non-magnetic TIs:* Another hallmark of the “parity anomaly” is a transition from a QSHL into a QAHL phase in magnetic fields even without magnetic impurities. As pointed out before, for HgMnTe QWs the exchange interaction is paramagnetic and, therefore, magnetic field dependent [@Beugeling12]. Values of the critical field $H^{crit}_\uparrow$ must be therefore found numerically. However, if we omit exchange coupling ($\chi_E=\chi_H=0$), one obtains an analytical expression for a transition from the QSH phase to QAHL phase. Closing of the gap for a spin-up block corresponds to $M^\star_\uparrow( H^{crit}_\uparrow)=0$ with $H^{crit}_\uparrow=-M^\star_\uparrow{\left(}0{\right)}/g^\star$, where ${g^\star=(g_E - g_H)/2-2\pi D/\phi_0}$ is the effective g-factor without exchange interaction. Interestingly this transition can occur without any magnetization or magnetic impurities but purely from the particle-hole asymmetry and effective g-factors terms which act in magnetic field as the mass connected with the “parity anomaly”. We expect that in the QAHL phase, the Hall conductance of the edge states should be more precisely quantized in comparison to the QSHL phase and survive even in large magnetic fields (due to lack of backscattering). In particular, we predict a transition from QSHL phase, characterized by a local two-terminal conductance of $2e^2/h$, to a QAHL phase at $H_{crit}^\uparrow$, characterized by $\sigma_{xx} =e^2/h$, and finally to the trivial phase at $H^{cross}$ (see green dashed line in Fig. \[fig:QSHLtoTrivial\]). Therefore, our prediction of surviving topological edge states in high magnetic fields could be related to the recent observation in Ref. [@Ma15].
Summarizing, we have studied topological transitions in magnetic fields for a two-flavour (2+1)D massive Dirac Hamiltonian. We have studied the experimentally relevant case of an effective constant carrier density with the chemical potential located within the mass gap. We have demonstrated that the QAHL phase (QAH effect in orbital magnetic fields) can be distinguished from the QH phase since the former one is a direct consequence of the “parity anomaly” for massive Dirac systems. We have shown that a long $\nu = -1$ and $\nu = 1$ plateaus in HgMnTe and $\text{Cr}_x{\left(}\text{Bi}_{1-y} \text{Sb}_y{\right)}_{2-x} \text{Te}_3$, respectively, are direct signatures of QAHL phase. Moreover Appendix A, we have predicted a new topological transition from the QSH phase to the QAHL phase without magnetization or magnetic impurities, which can explain the stability of edge states in recent experiments [@Ma15].
*Acknowledgments.* We thank F. Wilczek, B. A. Bernevig, C. Brüne, L. W. Molenkamp, C. Kleiner, J. S. Hofmann, J. Erdmenger, C. Morais Smith and W. Beugeling for useful discussions. We acknowledge financial support from the DFG via SFB 1170 “ToCoTronics”, and the ENB Graduate School on Topological Insulators.
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\[appendix\]
Appendix A: Parity anomaly and massive topological gauge theory in (2+1)D {#sec:appendixA}
=========================================================================
In the main text, we discussed how the “parity anomaly” gives rise to a change in Landau level counting in magnetic field and discussed how this fact is related to signatures in the magnetotransport. For this purpose, we compare the continuity equation, derived from the semiclassical equations of motion, with the continuity equation, derived from Maxwell equations, and show that the QAH level has a non-zero Chern number but an effective zero charge carrier density. Therefore, in the process of adiabatically switching on magnetic field, the QAH state can be distinguished from a trivial insulator when the chemical potential is in the mass gap. The structure of this appendix is the following: in the first part, we give an explicit derivation of the “parity anomaly” based on a semi-classical formulation, while in the second part we show how an additional topological term enters the Maxwell equations. Breaking of parity symmetry due to the Dirac mass term is discussed in Appendix B.
Let us start from the spin-up block of the BHZ Hamiltonian [@Bernevig06], $h_-{\left(}\mathbf{k}{\right)}=\mathbf{d}{\left(}\mathbf{k}{\right)}\cdot\boldsymbol{\sigma}$ with $\mathbf{d}{\left(}\mathbf{k}{\right)}={\left(}k_x, -k_y, M^\uparrow{\right)}$, where we have neglected the $ k$-dependence of the mass term. This approximation is valid in the low energy limit ${A k_f \gg B k_f^2}$ which means at least up to an energy of $\approx 150 \text{ meV}$. Following, we show that the parity-breaking Dirac mass gives rise to an anomalous transverse current response to an applied electric field, as well as to a non-zero carrier density in the ground state. We demand the chemical potential to be located within the mass gap, so that the bulk is insulating and is characterized by a non-zero Chern number $$\nu^\uparrow=\intop^\mu \frac{d\mathbf{k}}{2\pi} \Omega_z^\uparrow = -\frac{\text{sgn} \left( M^\uparrow\right)}{2}\label{eqA:ChernNumber},$$ where $\mathbf{\Omega^\uparrow}{\left(\mathbf{k}\right)}= \mathrm{i}\nabla_{\mathbf{k}} \times \langle u{\left(\mathbf{k}\right)}| \nabla_{\mathbf{k}} | u{\left(\mathbf{k}\right)}\rangle$ is the Berry curvature and $|u (\mathbf{k})\rangle$ are eigenstates of the Dirac Hamiltonian. In a semi-classical approach the Berry curvature acts as magnetic field in $k$-space and induces an anomalous correction to the velocity [@Sundaram99]. In order to conserve the phase space volume in magnetic field, it must also enter as a correction in the density of states, whereby the carrier density is given by [@Xia05] $$\begin{aligned}
n_e&=&\intop^\mu \frac{d\mathbf{k}}{{\left(}2\pi{\right)}^2} \left(1+\frac{e\bm{H}\cdot\mathbf{\Omega^\uparrow}}{\hbar}{\right)}+n_{back} \nonumber \\
&=&-\frac{e \text{ sgn} {\left(}M^\uparrow {\right)}}{2h} H_0, \label{eqA:DensityResponse}\end{aligned}$$ where $n_{back}$ is a background carrier density and we demand that $n_e(H_0=0)=0$ (bulk insulator). Based on this result, it is straightforward to derive the transverse Hall conductance from Streda’s formula [@Streda82], $$\sigma_{xy}=e\frac{\partial n_e}{\partial{H_0}}\Bigr|_\mu=-\frac{e^2 \text{ sgn} {\left(}M^\uparrow {\right)}}{2h}\label{eqA:HallResponse}.$$ The result of Eq. (\[eqA:DensityResponse\]) and (\[eqA:HallResponse\]) can be combined in a covariant form $$j_\mu=-\frac{e^2}{4h}\text{sgn} {\left(}M^\uparrow{\right)}\epsilon_{\mu\nu\tau}F^{\nu\tau}\label{eqA:parityAnomaly},$$ where $j^0\equiv\rho=-en_e$ ($e>0$) is the charge carrier density and $F^{\nu\tau}\equiv\partial^\nu A^\tau-\partial^\tau A^\nu$ is the electromagnetic tensor. In comparison to the QH effect, where the current originates from the magnetic field and has a $\text{sgn} {\left(}eH_0{\right)}$-dependence, the anomalous current is hereby induced by the Dirac mass term. Since the mass term breaks parity, this effect in solid state physics is known as “parity anomaly”. If the mass term changes sign across a topological domain wall as shown in Fig. \[fig:QHvsQAH\], charge conservation is violated at the interface, corresponding to a (1+1)D system, known as chiral anomaly, $$\begin{aligned}
\partial^\mu j_\mu&=&\frac{e^2}{4h}\left[ \partial_x \text{sgn}{\left(}M^\uparrow{\left(}x{\right)}{\right)}\right]\epsilon_{\mu\nu}F^{\mu\nu}\\
&=&\eta \frac{e^2}{2h}\epsilon_{\mu\nu}F^{\mu\nu}\label{eqA:chiralAnomaly},\end{aligned}$$ where $\eta=\left[ \text{sgn} (M^\uparrow (\infty))-\text{sgn} (M^\uparrow(-\infty))\right] /2$. While the fermion-doubling theorem ensures in even spacetime dimensions that a second domain wall (fermion) must exist which cancels this anomaly in total, there is no such theorem in odd spacetime dimensions [@Nielsen81A; @Nielsen81B].
Since the Dirac mass term breaks parity, there is an intrinsic Chern-Simons term (breaks also parity) allowed in the Lagrangian giving rise to topological corrections in the Maxwell equations. The effective Lagrangian of the classical Dirac field is given by $\mathcal{L}=\mathcal{L_F}+\mathcal{L_G}+\mathcal{L_I}$, known as topological massive gauge theory [@Deser82; @Yahalom85] $$\begin{aligned}
\mathcal{L_F}&=&\mathrm{i}\overline{\psi}\slashed{\partial}\psi-M^\uparrow\overline{\psi}{\psi}, \label{eqA:fermionicL}\\
\mathcal{L_G}&=&-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{\kappa}{4}\epsilon^{\mu\nu\tau}F_{\mu\nu}A_\tau, \label{eqA:gaugeL}\\
\mathcal{L_I}&=&-j^\mu A_\mu \; \text{with} \; j^\mu=-e\overline{\psi}\gamma^\mu\psi, \label{eqA:interactionL}\end{aligned}$$ where $\mathcal{L_F}$ describes the fermionic, $\mathcal{L_G}$ the gauge field and $\mathcal{L_I}$ the interaction part, $\epsilon^{\mu\nu\tau}F_{\mu\nu}A_\tau$ is the abelian Chern-Simons term and we have used the conventional definitions ${\slashed{\partial}\equiv\gamma^\mu \partial_\mu}$ and ${\overline{\psi} \equiv \psi^\dagger \gamma^0}$. The gamma matrices fulfill a Clifford algebra $\left\{\gamma^\mu,\gamma^\nu\right\}=2\eta^{\mu\nu}I_4$, where $\eta$ is the Minkowski metric. While the following derivation is basis independent, we set $\gamma_0=\sigma_z$, $\gamma_1=\mathrm{i}\sigma_y$ and $\gamma_2=\mathrm{i}\sigma_x$ corresponding to the spin-up block of the BHZ Hamiltonian. Using Euler-Lagrange formalism, $$\partial_\mu \frac{\partial \mathcal{L}}{\partial {\left(}\partial_\mu A_\nu {\right)}} - \frac{\partial\mathcal{L}}{\partial A_\nu}=0\label{eqA:Lagrange}$$ the equations of motion for the gauge field (Maxwell equations) are derived, $$\begin{aligned}
\partial_\mu F^{\mu\nu}+\frac{1}{2}\kappa\epsilon^{\nu\alpha\beta}F_{\alpha\beta}&=&j^\nu,\label{eqA:MaxCo1}\\
\partial_\mu\tilde{F}^\mu&=&0\label{eqA:MaxCo2},\end{aligned}$$ where the second equation is the Bianchi identity and $\tilde{F}^\mu=\epsilon^{\mu\alpha\beta}F_{\alpha\beta}/2$ is the dual field tensor. For a better visualization, we write out the equation of motions in component form, $$\begin{aligned}
\nabla\cdot\mathbf{E}&=&\rho+\kappa H_0\equiv\rho^\star \label{eqA:Max1}\\
\nabla\times\mathbf{E}&=&-\frac{\partial}{\partial t}H_0\label{eqA:Max2}\\
\nabla\times\mathbf{B}&=&\mathbf{j}+\frac{\partial}{\partial t}\mathbf{E}+\kappa\mathbf{E}\times\mathbf{e}_z\label{eqA:Max3},\end{aligned}$$ where it should be noted that the curl as well as the magnetic field are pseudoscalar in two space dimensions and $\mathbf{e}_z={\left(}0,0,1{\right)}^T$. By deriving the continuity equation from Eq. (\[eqA:Max1\]) and (\[eqA:Max3\]), $$\partial^\mu j_\mu=-\frac{1}{2}\partial_x \left[ \kappa{\left(}x{\right)}\right] \epsilon_{\mu\nu}F^{\mu\nu} \label{eqA:ContiniuityMaxwell}$$ and comparing the result with Eq. (\[eqA:parityAnomaly\]), we find that $$\kappa=-\frac{e^2}{2h}\text{sgn} {\left(}M^\uparrow {\right)}\label{eqA:Kappa}.$$ Since $\rho=e^2 H_0 \text{ sgn} {\left(}M^\uparrow {\right)}/ 2h$ (see Eq. (\[eqA:parityAnomaly\])), we find that for an arbitrary magnetic field $$\nabla\cdot\mathbf{E}=\rho^\star\overset{!}{=}0 \label{eqA:rhoZero}.$$ This charge is linked to an either filled or empty ${n=0}$ LL in the ground state (noting that $n_e=H_0/\phi_0$ corresponds to the carrier density of a single Landau level), where magnetic flux is tied to the electrons.
In an experimental set-up, our theoretical demand of a zero carrier density, $n_e{\left(}H_0=0{\right)}$, is realized by tuning the carrier density via an external electric gate such that the chemical potential falls into the mass gap. However, we see in the following that instead of the bare charge carrier density $\rho$, only an effective charge carrier density $\rho^\star$ is accessible by an electric gate.
Finally, we study consequences of a $k$-dependence of the Dirac mass term as present in Eq. (\[eq:BHZHamiltonian\]) and derive results for the “parity anomaly” accordingly. By using the semi-classical approach, Eq. (\[eqA:DensityResponse\]) and (\[eqA:HallResponse\]), we find that the expression for the “parity anomaly” is renormalized by the BHZ parameter $B$ and reads $$j_\mu=-\frac{e^2}{4h}\left[\text{sgn} {\left(}M^\uparrow{\right)}+\text{sgn}{\left(}B{\right)}\right]\epsilon_{\mu\nu\tau}F^{\nu\tau}\label{eqA:fullParity}.$$ By including also the quadratic term ($Bk^2$) in the Lagrangian $\mathcal{L}$ (without writing the calculation explicitly), we find that the Maxwell equations are not changed and our main result, $\rho^\star=0$, remains valid. The discussion was presented here for the single spin-up block, however it can be analogously repeated for the spin-down block. The experimental signature is determined by the sum of both spin-blocks.
Appendix B: Symmetries of the BHZ Hamiltonian and relation to parity anomaly {#sec:appendixB}
============================================================================
The appendix deals with symmetries of the BHZ Hamiltonian, in particular with time-reversal (TRS) and parity symmetry, where the focus is on the latter since we are interested in the origin of the “parity anomaly”. Each block of the BHZ Hamiltonian $h_\pm(\mathbf{k})$ (Eq. (\[eq:BHZHamiltonian\])) can be interpreted as a massive (2+1)D Dirac equation with a $k$-dependent mass term. At first, we discuss symmetries of a single block and in the second part, we focus on the symmetries of the full BHZ Hamiltonian.
For a single block, invariance under pseudospin TRS (in the band index) implies that an anti-unitary operator $\mathcal{T}$ exists such that [@Schnyder08; @Schnyder16] $$\mathcal{T}h_\pm(\mathbf{k})\mathcal{T}^{-1}=h_\pm(-\mathbf{k}),$$ with $\mathcal{T}=\mathcal{U}\mathcal{K}$ where $\mathcal{U}$ is a unitary matrix and $\mathcal{K}$ is the operator of complex conjugation. Parity symmetry in two space dimensions amounts for e.g. $(x,y)\rightarrow(-x,y)$ [@Stone87]. Invariance under parity symmetry implies then that a unitary operator $\mathcal{P}_x$ exists such that [@Winkler15] $$\mathcal{P}_x h_\pm{\left(}k_x,k_y {\right)}\mathcal{P}_x^{-1}=h_\pm{\left(}-k_x,k_y{\right)}.$$ The mass term (including the Newtonian mass) is odd under both symmetries, e.g. in case of parity symmetry $\mathcal{P}_x=\sigma_2$ $$\begin{aligned}
\mathcal{P}_x M_\mathbf{k} \sigma_3 \mathcal{P}_x^{-1}=&-M_\mathbf{k} \sigma_3,\hspace{0,2cm} \mathcal{P}_x \epsilon_\mathbf{k} \sigma_0 \mathcal{P}_x^{-1}&=\epsilon_\mathbf{k} \sigma_0,\nonumber\\
\mathcal{P}_x k_x\sigma_1 \mathcal{P}_x^{-1}=&-k_x\sigma_1, \hspace{0,2cm}\mathcal{P}_x k_y\sigma_2 \mathcal{P}_x^{-1}&= k_y\sigma_2,\end{aligned}$$ where we have used that $\left\{\sigma_i,\sigma_j\right\}=2\delta_{ij}\sigma_0$. Therefore, pseudospin TRS as well as parity symmetry is broken by the mass term giving rise to the “parity anomaly” in odd spacetime dimensions.
However, in the full BHZ Hamiltonian the anomaly cancels since both spin blocks amount for a different sign in the induced current (compare with Eq. (\[eq:parityAnomaly\])) such that the sum must be zero. The cancellation of the anomalous four-current for the full BHZ model can be also understood in the following way: we can define a TRS as well as parity operator for the full BHZ Hamiltonian connecting both spin blocks $h_\pm$ (similar to a two flavor (2+1)D massive Dirac equation [@Winkler15]). Such a TRS operator exists and is given by ${\mathcal{T}^{BHZ}=-\mathrm{i} \sigma_y \otimes \tau_x K}$ while the parity operator is ${\mathcal{P}_x^{BHZ}=\sigma_x \otimes \tau_x}$. Consequently, there is no “parity anomaly” for the pure BHZ Hamiltonian and the total induced current must vanish.
However, in the presence of an exchange or Zeeman-like Hamiltonian, as given by Eq. (\[eq:Zeeman\]) and (\[eq:Exchange\]), parity as well as TRS are broken, since e.g. $\mathcal{H}_{ex}$ is odd under parity symmetry $$\mathcal{P}^{BHZ}_x \mathcal{H}_{ex} {\left(}\mathcal{P}^{BHZ}_x{\right)}^{-1}=-\mathcal{H}_{ex}.$$ In an external magnetic field (applying Peierls substitution), we find additionally to the Zeeman and exchange interaction that another parity breaking term arises which is given by $$\begin{gathered}
\mathcal{H}_{Z_D}=\frac{2\pi D H_0}{\phi_0} \text{Diag} \begin{pmatrix}1 & -1 & 1 & -1\end{pmatrix},\\
\mathcal{P}^{BHZ}_x \mathcal{H}_{Z_D} {\left(}\mathcal{P}^{BHZ}_x{\right)}^{-1}=-\mathcal{H}_{Z_D} .\end{gathered}$$ All in all, the exchange interaction, the Zeeman as well as the particle-hole asymmetry term break parity and renormalize the effective mass gap in magnetic fields as highlighted in Eq. (\[eq:massTerm\]). A topological transition is therefore driven by an effective g-factor consisting off all three contributions.
Starting from a QSH phase, the total four-current remains zero during an adiabatic increase in the effective g-factor as long as the mass gap of one of the spin-blocks is not closed. If one of the mass gaps is closed, the system enters a QAH (requires ferromagnetic exchange interaction without an external magnetic field) or a QAHL (QAH state in magnetic fields) phase and a non-zero anomalous four-current is observed.
Finally, one should note that a BIA term also violates parity symmetry, where a representation of the BIA term given in the Dirac basis reads [@Konig08] $$\begin{gathered}
\mathcal{H}_{BIA}=-\Delta \sigma_x \otimes \tau_z,\\
\mathcal{P}^{BHZ}_x \mathcal{H}_{BIA} {\left(}\mathcal{P}^{BHZ}_x{\right)}^{-1}=-\mathcal{H}_{BIA}.\end{gathered}$$ where $\Delta$ is a parameter. However, Rashba spin-orbit interaction in lowest order [@Rothe10; @Muhlbauer14] $$\begin{gathered}
\mathcal{H}_R=\begin{pmatrix}
0 & 0 & 0 &\mathrm{i}R_0 k_+ \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
-\mathrm{i} R_0 k_- & 0 & 0 & 0
\end{pmatrix},\\
\mathcal{P}^{BHZ}_x \mathcal{H}_R {\left(}kx, ky {\right)}{\left(}\mathcal{P}^{BHZ}_x{\right)}^{-1}=\mathcal{H}_R{\left(}-k_x, ky{\right)},\end{gathered}$$ where $R_0$ is a parameter, does not violate parity. Both BIA and Rashba spin-orbit interaction were found to be negligible (less than $1$ meV) in recent experiments on symmetric HgTe QWs [@Muhlbauer14]. Since the mass gap protects the anomalous four-current and, therefore, the “parity anomaly”, such small perturbations cannot destroy the “parity anomaly” as long as no gap closing occurs..
Appendix C: Realistic parameters for HgMnTe {#sec:appendixC}
===========================================
This appendix should elucidate how effective BHZ-parameters (see Eq. (\[eq:BHZHamiltonian\])) for $\text{Hg}_{1-y}\text{Mn}_y\text{Te}/\text{CdTe}$ quantum wells have been numerically calculated. These parameters haven been used in the main text to predict an early topological transition in magnetic fields to a $\sigma_{xy}=-e^2/h$ plateau for a $10\text{ nm}$ thick sample of $\text{Hg}_{0.98}\text{Mn}_{0.02}\text{Te}$ as depicted in Fig. \[fig:LandauFan\].
The bulk band structure in case of HgMnTe around the $\Gamma$-point is characterized by the $8\times8$ - Kane Hamiltonian, $$H{\left(}\mathbf{k}{\right)}=H_{Kane}+H_z+H_{ex}+H_s\label{eqA:KaneHamiltonian},$$ where $H_{Kane}$ is the bare Kane Hamiltonian, $H_z$ is the Zeeman term, $H_{ex}$ describes sp-d exchange interaction between the s-/p-band electrons and localized d-electrons associated to Mn atoms and $H_s$ amounts for uniaxial strain. The Hamiltonians are discussed in great detail in Ref. [@Novik05], where they are written out explicitly in Eq. (6), (14), (20) and (C3), respectively. It is worth noting that Mn does not only give rise to an exchange interaction but one must also renormalize Kane parameters for a given Mn concentration. For small fractions, it is valid to assume that only the band gap parameter $E_g {\left(}y=0.02{\right)}\approx-200 \text{ meV}$ is strongly affected [@Landolt]. This affects the critical thickness above which HgMnTe becomes a 2D TI , for instance $d_{crit}{\left(}y=0.02{\right)}\gtrsim8.5\text{ nm}$, while in pure HgTe the transition should have already emerged at $d_{crit}\gtrsim6.3\text{ nm}$ [@Konig08].
The full Hamiltonian is then mapped onto a cubic lattice and we assume periodic boundary conditions in x- and y-direction, while in z-direction the interface between HgMnTe and CdTe is taken into account by having $z$-dependent Kane parameters [@Bernevig06]. Mapping to a cubic lattice corresponds to making use of a finite difference method (also known as tight-binding method), where e.g. a single matrix element with a z-dependent Kane parameter is given by $$\begin{aligned}
k_z\gamma{\left(}z{\right)}k_z{\left(}z{\right)}&=&-\partial_z\gamma{\left(}z{\right)}\partial_z \psi {\left(}z {\right)}\nonumber \\
&\approx&-\frac{1}{a^2}\left\{\gamma{\left(}z+\frac{a}{2}{\right)}\left[\psi{\left(}z+a{\right)}-\psi{\left(}z {\right)}\right]\right.\nonumber\\
&&\left. -\gamma{\left(}z-\frac{a}{2}{\right)}\left[ \psi{\left(}z{\right)}-\psi{\left(}z -a{\right)}\right]\right\},\end{aligned}$$ where $\gamma$ is an exemplary Kane parameter, $k_z\rightarrow-\mathrm{i}\partial_z$ and in the second step we used a finite difference quotient with grid spacing $a$. The eigenvalue problem of the Schrödinger equation can then be solved numerically.
Using Löwdin perturbation theory [@WinklerBook] up to the second order, one can derive a low energy model which is valid close to the band gap at the $\Gamma$-point. This model is known as BHZ Hamiltonian [@Bernevig06], where $$H_{BHZ}=\mathcal{H}^{(0)}+\mathcal{H}^{(1)}+\mathcal{H}^{(2)}$$ and $$\begin{aligned}
\mathcal{H}^{(0)}_{i^\prime i}&=&\langle i^\prime|H_0| i \rangle \label{eqA:zeroOrder},\\
\mathcal{H}^{(1)}_{i^\prime i}&=&\langle i^\prime|V|i\rangle\label{eqA:firstOrder},\\
\mathcal{H}^{(2)}_{i^\prime i}&=&\sum_{m}\frac{1}{2}V_{i^\prime m} V_{m i}{\left(}\frac{1}{E_{i^\prime}-E_m}+\frac{1}{E_i-E_m}{\right)}\;\;\label{eqA:secondOrder},\end{aligned}$$ where $V_{i^\prime m}\equiv\langle i^\prime|V|m \rangle$ and $V_{m i}\equiv\langle m|V|i \rangle$. Moreover, the full Kane Hamiltonian was divided in two parts $H{\left(}\mathbf{k}{\right)}=H_0+V$, where $H_0=H{\left(}\mathbf{k}=0{\right)}$ is the Hamiltonian at the $\Gamma$-point and $V=H-H_0$ is treated as a perturbation. The indices $\left\{i,i^\prime\right\}$ run over all subbands at $\Gamma$ forming the basis of the low energy Hamiltonian. In case of the BHZ Hamiltonian this means that $\left\{i,i^\prime\right\}$ compose the four subbands $\left\{|E1,\downarrow\rangle,-|H1,\downarrow\rangle,-|H1,\uparrow\rangle,|E1,\uparrow\rangle\right\}$. All other subbands are indexed by $m$ and give rise to a renormalization of the bare BHZ parameters. Finally, we end up with the BHZ Hamiltonian as it was explicitly given in Eq. (\[eq:BHZHamiltonian\]). Two exemplary sets of BHZ parameters for $y=0.02$ are given in the caption of Fig. \[fig:LandauFan\] of the main text.
Finally, we want to write out the exchange Hamiltonian, ${\mathcal{H}_{ex} = \text{Diag} \begin{pmatrix} -\chi_E & -\chi_H &\chi_H& \chi_E \end{pmatrix}}$, as it was given in Eq. (\[eq:Exchange\]), since it is the exchange coupling which dominates in case of HgMnTe the effective g-factor [@Liu08; @Beugeling12] $$\begin{aligned}
\chi_E&=& -3\alpha F_1 -\beta F_4\\
\chi_H&=&-3\beta,\end{aligned}$$ where $$\begin{aligned}
\alpha&=&-\frac{1}{6} N_1 y S_0 B_{5/2} {\left(}\frac{5 g_{Mn}\mu_B H_0}{2k_B{\left(}T+T_0{\right)}}{\right)}, \\
\beta&=&-\frac{1}{6} N_2 y S_0 B_{5/2} {\left(}\frac{5 g_{Mn}\mu_B H_0}{2k_B{\left(}T+T_0{\right)}}{\right)},\end{aligned}$$ where $S_0=5/2$, $B_{5/2}$ is the Brillouin function, ${T_0=2.6 \text{ K}}$, we assume zero temperature ${T=0}$, ${N_1=400 \text{meV}}$, ${N_2=-600 \text{meV}}$ and $F_1$ and $F_4$ amounts for S- and P- band character of the $E1$ subband, respectively.
Appendix D: Half-space calculation {#sec:appendixD}
==================================
In this appendix, we highlight that the property of the $n=0$ LL being either a solution of the valence band (occupied in ground state) or being a solution of the conduction band (unoccupied in ground state) is already contained in the Landau level spectrum given by Eq. (\[eq:LLspectrumChernN\]). We focus on the spin-up block while the discussion applies analogously to the spin-down block. Therefore, it is possible to make predictions for topological transitions in magnetic field based solely on the expression for Landau level energies. A short prove is outlined in the following based on a half-space calculation [@Tkachov10; @Scharf12].
Solutions of the Schrödinger equation on the half space ($y>0$), ${h_-(\mathbf{k}+e/\hbar\mathbf{A})\psi^\eta_\uparrow=E_\uparrow (\eta)\psi^\eta_\uparrow}$, are obtained imposing the ansatz $\psi^{\eta}_\uparrow\left(\tilde{y}\right)=\left(\begin{array}{cc} \tilde{u}_1 D_{\eta-1}\left(\tilde y \right) & \tilde{u}_2 D_{\eta}\left(\tilde y \right)\end{array}\right)^T$, where $\tilde y =\sqrt{2}{\left(}y-y_k{\right)}/l_H$, $y_k=l_H^2 k_x$ and $D_\eta$ are parabolic cylindrical functions. The eigenvalue problem is solved using the recurrence relations $$\begin{aligned}
{\left(}\frac{\tilde{y}}{2}\pm\partial_{\tilde{y}}\right)D_\eta{\left(}\tilde{y}{\right)}& = &
\begin{cases}
\eta D_{\eta-1}{\left(}\tilde{y}{\right)}\\
D_{\eta+1}{\left(}\tilde{y}{\right)}\end{cases},\\
{\left(}\frac{\tilde{y}^2}{2}-\partial_{\tilde{y}}^2{\right)}D_\eta\left(\tilde{y}{\right)}&=&
{\left(}\eta+\frac{1}{2}{\right)}D_\eta\left(\tilde{y}{\right)}.\end{aligned}$$ After a straightforward calculation, it follows that the spectrum is again given by Eq. (\[eq:LLspectrumChernN\]), $$E^{\pm}_{\uparrow,\eta>0} = \frac{g_1-\beta}{2}-\eta\delta\pm\sqrt{\eta\alpha^{2}+\left( M^\star_\uparrow{\left(}H_0{\right)}- \eta\beta\right)^{2}},\label{eqA:LLspectrumChernNu}\\$$ where $\alpha=\sqrt{2}A/l_H$, $\beta=2B/l_H^2$ and $\delta=2D/l_H^2$, ${g_{1/2 }={\left(}\chi_E+g_EH_0{\right)}\pm {\left(}\chi_H+g_HH_0{\right)}}$, $M^\star_\uparrow{\left(}H_0{\right)}=M+g^\star{\left(}H_0{\right)}$ with $g^\star{\left(}H_0{\right)}={\left(}g_2{\left(}H_0{\right)}-\delta{\left(}H_0{\right)}{\right)}/2$ as in the main text. However, with the difference that $n\in\mathbb{N}$ is replaced by ${\eta\in\mathbb{Z}}$. Obviously, there are in general two solutions with $\eta=0$, where the respective “$-/+$” - sign in Eq. (\[eqA:LLspectrumChernNu\]) symbolizes that the $n=0$ LL belongs either to the valence or to the conduction band. This is in contrast to the pure bulk calculation presented in the main text, where the eigenvalue of the $n=0$ LL is comprised within a single expression (see Eq. (\[eq:LLspectrumChern0\])). This is due to the fact that the boundary conditions have already been applied in the bulk calculation by choosing the Hermite polynomials as solutions.
A solution of the eigenvalue problem is given by $$\Psi\left(\tilde{y}\right)=c_1\psi_{\eta_{1}}\left(\tilde{y}\right)+c_2\psi_{\eta_{2}}\left(\tilde{y}\right)+c_3\psi_{\eta_{1}}\left(-\tilde{y}\right)+c_4\psi_{\eta_{2}}\left(-\tilde{y}\right),$$ where we have used that $D_\eta(\tilde{y})$ and $D_\eta(-\tilde{y})$ are two independent solutions, $$\begin{aligned}
&&\eta_{1/2}=\frac{1}{2{\left(}\beta^2-\delta^2{\right)}} \left[ -\alpha^2+2\beta M^\star_\uparrow+2\delta\epsilon\right.\nonumber\\
&&\pm\left. \sqrt{\alpha^4-4\alpha^2{\left(}\beta M^\star_\uparrow+\delta\epsilon{\right)}+4{\left(}\beta\epsilon+\delta M^\star_\uparrow{\right)}^2}\right]\end{aligned}$$ and $\epsilon=E+(\beta-g_1)/2$. By applying the boundary conditions, ${\psi\left(y=0\right)=0}$ and ${\lim_{y\rightarrow\infty}\psi\left(y\right)=0}$, we find that only one $\eta=0$ solution is allowed. Depending on whether $M^\star_\uparrow/B>0$ or $M^\star_\uparrow/B<0$, the $n=0$ LL belongs either to the valence or to the conduction band, respectively. Such a topological transition in magnetic field is shown in Fig. \[figA:halfSpace\]. Here, the $n=0$ LL changes from the valence (Fig. \[figA:halfSpace\] a)) to the conduction band (Fig. \[figA:halfSpace\] b)) and the transition point is marked by a massless Dirac fermion, $M^\star_{\uparrow}=0$ as we accordingly found in Appendix A. While the discussion here was only given for the spin-up block, a transition can occur either for the spin up or the spin-down block depending on the sign of the effective g-factor as discussed in the main text.
![In **(a)** and **(b)** solutions $E_\uparrow{\left(}\eta{\right)}$ in the bulk limit $y_k=l_H^2 k_x\gg0 $ are indicated by points (toy parameters). In this limit, $\eta$ converges against integer-values. Along the topological transition (for $\text{sgn} (eH_0)>0$ and $\text{sgn} (g^\star)>0$), the $n=0$ LL changes from (a) the hole (red) to (b) the particle branch (blue). To highlight the branch change, the inset in (b) shows enlarged the $n=0$ LL. The chemical potential at constant effective carrier density $\rho^\star$ is indicated by the dashed line. During the transition the Chern number changes from $\nu_\uparrow=1$ to $\nu_\uparrow=0$. []{data-label="figA:halfSpace"}](./figure4.pdf){width=".49\textwidth"}
[^1]: The employed definitions for the QAHL phase and the QSHL phase are physically distinct from the definition used by Chen and co-workers [@Chen12].
|
---
abstract: 'A directed percolation process with two symmetric particle species exhibiting exclusion in one dimension is investigated numerically. It is shown that if the species are coupled by branching ($A\to AB$, $B\to BA$) a continuous phase transition will appear at zero branching rate limit belonging to the same universality class as that of the dynamical two-offspring (2-BARW2) model. This class persists even if the branching is biased towards one of the species. If the two systems are not coupled by branching but hard-core interaction is allowed only the transition will occur at finite branching rate belonging to the usual $1+1$ dimensional directed percolation class.'
address: |
Research Institute for Technical Physics and Materials Science,\
H-1525 Budapest, P.O.Box 49, Hungary
author:
- Géza Ódor
title: 'The universal behavior of one-dimensional, multi-species branching and annihilating random walks with exclusion'
---
[2]{}
The study of phase transitions in low dimensions is an interesting and widely investigated topic [@Dick-Mar; @Hin2000] (since the mean-field solution is not valid). The research of non-equilibrium phase transitions occurring in one dimensional coupled systems has drawn interest nowadays [@HT97; @Odo00; @HayeDP-ARW; @Park; @barw2cikk; @Lipo; @CoupledDP; @uni-PC; @dimercikk; @Frei; @Wij; @Trimp]. Several models have been found with transitions that do not belong the robust directed percolation (DP) class [@Jan81; @Gras82; @DP] or to the parity conserving (PC) class [@Gras84; @ujMeOd] which are the most prominent ones among one component systems. Particle blocking which is common in one dimension has not been taken into account in field theoretical description of these models yet [@Cardy-Tauber; @Janssen-col]. It has been known for some time that the pair contact process [@PCP] can be regarded as a coupled system that exhibits DP class static exponents while the spreading ones depend on initial densities [@MDGL]. The field theoretical investigation of Janssen [@Janssen-col] predicts that in coupled DP systems the symmetry between species is unstable and generally a phase transition belongs to the class of unidirectionally coupled DP where coupling between pairs of species is relevant in one direction only. Such systems have been shown to describe also certain surface roughening processes [@CoupledDP; @uni-PC].
Recently we have shown [@arw2cikk] that in the two-component annihilating random walk ($AA\to\emptyset$, $BB\to\emptyset$) owing to the hard-core interaction of particles dynamical exponents are non-universal. Some consequences of hard core effects for random walks in one dimension have been known for some time already [@Dhar].
Very recently simulations [@Park; @barw2cikk] gave numerical evidence that in the two-component branching and annihilating random walk (2-BARW2) the lack of particle exchange between different species results in new universality classes in contrast to widespread beliefs that bosonic field theory can well describe these systems. The critical exponents obtained numerically suggest that the location of offspring particles at branching is the relevant factor that determines the critical behavior. In particular if the parent separates the offsprings:
1\) $A\to BAB$ the steady state density will be higher than in the case when they are created on the same site:
2\) $A\to ABB$ for a given branching rate because in the former case they are unable to annihilate with each other. This results in different order parameter exponents for the symmetric (2-BARW2s) and the asymmetric (2-BARW2a) cases ($\beta_s=1/2$ vs $\beta_a=2$ for 1) and 2), respectively).
Hard-core effects are conjectured to cause a series of new universality classes in one dimension [@Park]. In this paper I point out that probably only a few universality classes emerge as the consequence of particle exclusion, because other symmetries and conservation laws (like that of the PC class) will become irrelevant.
In the present study first I show that in case of the two-component single off-spring BARW model (2-BARW1) defined as $$\begin{aligned}
A\stackrel{\sigma_A/2}{\longrightarrow} AB \ \ \ \ , \ \ \
A\stackrel{\sigma_A/2}{\longrightarrow} BA \\
B\stackrel{\sigma_B/2}{\longrightarrow} BA \ \ \ \ , \ \ \
B\stackrel{\sigma_B/2}{\longrightarrow} AB \\
AA\stackrel{\lambda}{\longrightarrow}\text{\O} \ \ \ , \ \ \
BB\stackrel{\lambda}{\longrightarrow}\text{\O} \\
A\text{\O}\stackrel{d}{\leftrightarrow}\text{\O}A \ \ \ , \ \ \
B\text{\O}\stackrel{d}{\leftrightarrow}\text{\O}B \\
AB\stackrel{0}{\leftrightarrow}BA \label{proc}\end{aligned}$$ a continuous phase transition will occur at zero branching rate limit ($\sigma=0$) like in the 2-BARW2 model where they are equivalent and therefore the exponents on the critical point must be the same as those determined in [@Park; @barw2cikk; @arw2cikk]. Furthermore I show that the order parameter exponent describing the singular behavior of the steady state density near the critical point coincides with that of the 2-BARW2s model.
The particle system was simulated on a lattice with size $L=4\times10^4$ and periodic boundary conditions for different $\sigma$-s (with $\lambda=d=1-\sigma$ condition). The initial condition was uniformly random distribution of $A$-s and $B$-s with a total concentration $0.5$. The evolution of the density was followed until steady state has been reached plus $t\sim 10^4$ Monte Carlo sweeps (throughout the whole paper $t$ is measured in units of Monte Carlo sweeps (MCS) of the lattices). As Figure \[dp2rho\] shows a phase transition occurs at $\sigma_A=\sigma_B=0$ indeed.
The order parameter exponent has been determined with the local slope analysis of the data $$\beta_{eff} (\sigma) = \frac {\ln \rho_{i} -\ln \rho_{i-1}}
{\ln \sigma_i - \ln \sigma_{i-1}} \ \ ,$$ providing an estimate for the true asymptotic behavior of the order parameter $$\beta = \lim_{\sigma\to 0} \beta_{eff}(\sigma) \,.$$ As one see on Figure \[betadp2\] $\beta_{eff}$ extrapolates to $\beta=0.50(1)$ with a strong correction to scaling like in case of the 2-BARW2s model [@barw2cikk]. The coincidence of this off-critical exponent in addition to the equivalence of processes at the critical point assures that they belong to the same universality class.
If we destroy the symmetry between species by the branching rates: $\sigma_A=\sigma_B/2$ we still get the same order parameter exponents ($\beta=0.50(1)$) for both species (Fig.\[betadp2\]). Therefore this universality class is stable with respect to coupling strengths unlike the coloured and flavoured directed percolation [@Janssen-col].
It is also insensitive whether or not the parity of particles is conserved meaning that the $A\to BAB$ process can be decomposed to a sequence of $A\to AB$, $AB\to BAB$ processes. This may seem to be quite obvious when particle exchange is not allowed, and if locality is assumed. By the choice of parameters $d=1-\sigma$ in the neighbourhood of the critical point the the diffusion is strong and the locality condition is not met. Still the two process share the same critical behavior.
If we decouple the two systems and allow hard-core exclusion only: $$\begin{aligned}
A\stackrel{\sigma}{\longrightarrow} AA \\
B\stackrel{\sigma}{\longrightarrow} BB \\
AA\stackrel{\lambda}{\longrightarrow}\text{\O} \ \ \ , \ \ \
BB\stackrel{\lambda}{\longrightarrow}\text{\O} \\
A\text{\O}\stackrel{d}{\leftrightarrow}\text{\O}A \ \ \ , \ \ \
B\text{\O}\stackrel{d}{\leftrightarrow}\text{\O}B \\
AB\stackrel{0}{\leftrightarrow}BA \label{procu}\end{aligned}$$ the critical point will be shifted to $\sigma=0.81107(1)$ and DP like density decay can be observed on the local slopes defined as $$\alpha_{eff}(t) = {- \ln \left[ \rho(t) / \rho(t/m) \right]
\over \ln(m)} \label{slopes}$$ (where we use $m=8$ usually) (see Figure\[dp2s\]).
One can not observe any relevant correction to scaling here, the most straight curve corresponding to the critical one ($\sigma=0.81107$) extrapolates to $\alpha=0.158(2)$ which agrees very well with the $\beta/\nu_{||}=0.159464(6)$ value of the $1+1$ DP class value that can be found in the literature [@IJensen99]. This is different from the case of coupled annihilating random walk, where the blocking causes marginal perturbation to the standard decay process [@arw2cikk].
One can generalize the results by taking into account that neighbouring $AA$ and $BB$ offsprings decay very quickly and therefore irrelevant for the leading scaling behavior.
[*Conjecture : In coupled, one dimensional N-component BARW systems with particle exclusion and branching processes like: $A\to BABB$, $A\to BAAA$, $A\to BAC$ ... leaving behind non-reacting neighbouring particles which block each other the universality class of a phase transition will be the same as that of 1-BARW2s.*]{} [*If the branching creates only pairs that can annihilate immediately (like: $A\to BAAB$ ... etc.) the class of transition will be the same as that of the 2-BARW2a model.*]{} We can also conclude that in case of reaction-diffusion processes where spontaneous decay is allowed: $2A\to A$, $A\to\text{\O}$ the blocking effect between dissimilar species is irrelevant.
It is very likely that the transition of a very recently introduced ladder model [@Lipo] also belongs to this class. This model is composed of two one dimensional subsystems following BARW at the critical point and coupled by ladder links. In the supercritical region by updating an active site one can create an offspring on the other subsystem or increase the inactivity level of that site. For small coupling strength ($s=1$) the very few blocking events can not introduce relevant blocking on the other subsystem and the scaling exponents agree with those of the coupled BARWe model without exclusion [@Cardy-Tauber]. For stronger coupling strength ($s=2$) there are more blocking possibility resulting in 1-BARW2s scaling exponents.
In conclusion I have shown that the one dimensional two species coupled BARW with exclusion and one offspring has the same critical transition point as that of the 2-BARW2s model investigated earlier. The hard-core interaction itself is not sufficient to cause deviation in scaling behaviour from that of DP. A conjecture is given with regard the universality classes in coupled BARW systems exhibiting particle exclusion.
[**Acknowledgements:**]{}\
The author would like to thank T. Antal for the stimulating discussions and N. Menyhárd for critically reading the manuscript. Support from Hungarian research fund OTKA (Nos. T-25286 and T-23552) and from Bólyai (No. BO/00142/99) is acknowledged.
J. Marro and R. Dickman, , Cambridge University Press, Cambridge, 1999. H. Hinrichsen, preprint, cond-mat/0001070. M. J. Howard and U. C. T[ä]{}uber, [J. Phys.]{} [**A 30**]{}, 7721 (1997). G. Ódor , Phys. Rev. E [**62**]{}, R3027 (2000). H. Hinrichsen, eprint cond-mat/0004348. S. Kwon, J. Lee and H. Park, Phys. Rev. Lett. [**85**]{}, 1682 (2000). G. Ódor, cond-mat/0008381, to be published in Phys. Rev. E. A. Lipowski, cond-mat/0007411. U.C. Täuber, M.J. Howard, and H. Hinrichsen, Phys. Rev. Lett. [**80**]{}, 2165 (1998); Y.Y. Goldschmidt, Phys. Rev. Lett. [**81**]{}, 2178 (1998); Y.Y. Goldschmidt, H. Hinrichsen, M.J. Howard, and U.C. Täuber, Phys. Rev. E [**59**]{}, 6381 (1999); H.K. Janssen, cond-mat/9901188. H. Hinrichsen and G. Ódor, Phys. Rev. Lett. [**82**]{}, 1205 (1999). H. Hinrichsen and G. Ódor, Phys. Rev. E [**60**]{}, 3842 (1999). J. E. de Freitas, L. S. Lucena, L. S. da Silva and H. Hilhorst, Phys. Rev. E [**61**]{}, 6330 (2000). F. van Wijland, K. Oerding and H. Hilhorst, Physica A [**251**]{}, 179 (1998). S. Trimper,U.C. Täuber and G.M. Schütz, cond-mat/0001387 H. K. Janssen, Z. Phys. B [**42**]{}, 151 (1981). P. Grassberger, Z. Phys. B [**47**]{}, 365 (1982). W. Kinzel, in [*Percolation Structures and Processes*]{}, ed. G. Deutscher, R. Zallen, and J. Adler, Ann. Isr. Phys. Soc. [**5**]{}.
P. Grassberger, F. Krause and T. von der Twer, J. Phys. A:Math.Gen., L105 [**17**]{} (1984). For further references see : N. Menyhárd N. and G. Ódor, cond-mat/0001101; Brazilian J. of Physics [**30**]{}, 113 (2000). J. L. Cardy and U. C. Täuber, J. Stat. Phys. [**90**]{}, 1 (1998). H. K. Janssen, preprint, cond-mat/0006129. G. Ódor and N. Menyhárd, Phys. Rev. E. [**61**]{}, 6404 (2000). G. Mennon, M. Barma and D. Dhar, J. Stat. Phys. [**86**]{}, 1237 (1997). I. Jensen and R. Dickman, Phys. Rev. E [**48**]{}, 1710 (1993). M. A. Muñoz, G. Grinstein, R. Dickman and R. Livi, Phys. Rev. Lett. [**76**]{}, 451 (1996) and Physica D [**103**]{}, 485 (1997). I. Jensen, J. Phys. A [**32**]{}, 5233 (1999).
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---
author:
- 'S. Damjanovic for the NA60 Collaboration'
date: 'Received: date / Revised version: date'
title: 'First measurement of the $\rho$ spectral function in nuclear collisions'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
Thermal dilepton production in the mass region $<$1 GeV/c$^{2}$ is largely mediated by the light vector mesons $\rho$, $\omega$ and $\phi$. Among these, the $\rho$(770 MeV/c$^{2}$) is the most important, due to its strong coupling to the $\pi\pi$ channel and its short lifetime of only 1.3 fm/c, much shorter than the lifetime of the fireball. These properties have given it a key role as [*the*]{} test particle for “in-medium modifications” of hadron properties close to the QCD phase boundary. Changes both in width and in mass were originally suggested as precursor signatures of the chiral transition [@Pisarski:mq]. There seems to be some consensus now that the [*width*]{} of the $\rho$ should increase towards the transition region, based on a number of quite different theoretical approaches [@Pisarski:mq; @Dominguez:1992dw; @Pisarski:1995xu; @Rapp:1995zy; @Rapp:1999ej]. On the other hand, no consensus exists on how the [*mass*]{} of the $\rho$ should change in approaching the transition: predictions exist for a decrease [@Pisarski:mq; @Brown:kk; @Brown:2001nh; @Hatsuda:1991ez], a constant behavior [@Rapp:1995zy; @Rapp:1999ej], and even an increase [@Pisarski:1995xu].
Experimentally, low-mass electron pair production was previously investigated at the CERN SPS by the CERES /NA45 experiment for p-Be/Au, S-Au and Pb-Au collisions [@Agakichiev:mv; @Agakichiev:1995xb; @Agakichiev:1997au]. The common feature of all results from nuclear collisions was an excess of the observed dilepton yield above the expected electromagnetic decays of neutral mesons, by a factor of 2-3, for masses above 0.2 GeV/c$^{2}$. The surplus yield has generally been attributed to direct thermal radiation from the fireball, dominated by pion annihilation $\pi^{+}\pi^{-}\rightarrow\rho\rightarrow l^{+}l^{-}$ with an intermediate $\rho$ which is strongly modified by the medium. Statistical accuracy and mass resolution of the data were, however, not sufficient to reach the sensitivity required to assess in detail the [*character*]{} of the in-medium changes. The new experiment NA60 at the CERN SPS has now achieved a decisive breakthrough in this field.
Apparatus and data analysis {#sec:1}
===========================
The apparatus is based on the muon spectrometer previously used by NA50, and a newly added telescope of radiation-tolerant silicon pixel detectors, embedded inside a 2.5 T dipole magnet in the vertex region [@Gluca:2005; @Keil:2005zq]. Matching of the muon tracks before and after the hadron absorber, both in [*angular and momentum*]{} space, improves the dimuon mass resolution in the region of the light vector mesons from $\sim$80 to $\sim$20 MeV/c$^{2}$ and also decreases the combinatorial background of muons from $\pi$ and K decays. Moreover, the additional bend by the dipole field leads to a strong increase of the detector acceptance for opposite-sign dimuons of low mass and low transverse momentum. The rapidity coverage is 3.3$<$y$<$4.3 for the $\rho$, at low p$_{T}$ (compared to 3$<$y$<$4 for the J/$\psi$). Finally, the selective dimuon trigger and the fast readout speed of the pixel telescope allow the experiment to run at very high luminosities.
The results reported here were obtained from the analysis of data taken in 2003 with a 158 AGeV indium beam, incident on a segmented indium target of seven disks with a total of 18% (In-In) interaction length. At an average beam intensity of 5$\cdot$10$^{7}$ ions per 5 s burst, about 3$\cdot$10$^{12}$ ions were delivered to the experiment, and a total of 230 million dimuon triggers were recorded on tape. The data reconstruction starts with the muon-spectrometer tracks. Next, pattern recognition and tracking in the vertex telescope are done; the interaction vertex in the target is reconstructed with a resolution of $\sim$200 $\mu$m for the z-coordinate and 10-20 $\mu$m in the transverse plane. Only events with one vertex are kept; interaction pileup and reinteractions of secondaries and fragments are thus rejected. Finally, each muon-spectrometer track is extrapolated to the vertex region and matched to the tracks from the vertex telescope.
The combinatorial background of uncorrelated muon pairs mainly originating from $\pi$ and K decays is determined using a [*mixed-event technique*]{} [@Ruben:2005qm]. Two single muons from different like-sign dimuon triggers are combined into muon pairs in such a way as to accurately account for details of the acceptance and trigger conditions. The quality of the mixed-event technique can be judged by comparing the like-sign distributions generated from mixed events with the measured like-sign distributions. It is remarkable that the two agree to within $\sim$1% over a dynamic range of 4 orders of magnitude in the steeply falling mass spectrum [@Ruben:2005qm]. After subtraction of the combinatorial background, the remaining opposite-sign pairs still contain “signal” fake matches, i.e. associations of muons to non-muon tracks in the pixel telescope. This contribution is only 7% of the combinatorial background level. It has been determined in the present analysis by an overlay Monte Carlo method. We have verified that an event-mixing technique gives the same results, both in shape and in yield, within better than 5%. More details on the experimental apparatus and data analysis will be given in a forthcoming extended paper; for now see [@Ruben:2005qm; @Andre:2006].
Results {#sec:2}
=======
A significant part of the results presented in this paper has recently been published [@Arnaldi:2006]. That part will therefore be less extensively treated than other parts published here for the first time.
Fig. \[fig1\] shows the opposite-sign, background and signal dimuon mass spectra, integrated over all collision centralities. After subtracting the combinatorial background and the signal fake matches, the resulting net spectrum contains about 360000 muon pairs in the mass range 0-2 GeV/c$^2$ of Fig. \[fig1\], roughly 50% of the total available statistics. The average charged-particle multiplicity density measured by the vertex tracker is $dN_{ch}/d\eta$ =120, the average signal-to-background ratio is 1/7. For the first time in nuclear collisions, the vector mesons $\omega$ and $\phi$ are completely resolved in the dilepton channel; even the $\eta$$\rightarrow$$\mu$$\mu$ decay is seen. The mass resolution at the $\omega$ is 20 MeV/c$^{2}$. The subsequent analysis is done in four classes of collision centrality defined through the charged-particle multiplicity density: peripheral (4-30), semiperipheral (30-110), semicentral (110-170) and central (170-240). The signal-to-background ratios associated with the individual classes are 2, 1/3, 1/8 and 1/11, respectively.
The peripheral data can essentially be described by the expected electromagnetic decays of the neutral mesons. Muon pairs produced from the 2-body decays of the $\eta$, $\rho$, $\omega$ and $\phi$ resonances and the Dalitz decays of the $\eta$, $\eta^{'}$ and $\omega$
were simulated using the improved hadron decay generator GENESIS [@genesis:2003], while GEANT was used for transport through the detectors. Four free parameters (apart from the overall normalization) were used in the fit of this “hadron decay cocktail” to the peripheral data: the cross section ratios $\eta/\omega$, $\rho/\omega$ and $\phi/\omega$, and the level of $D$ meson pair decays; the ratio $\eta^{'}/\eta$ was kept fixed at 0.12 [@Agakichiev:mv; @genesis:2003]. The fits were done without p$_{T}$ selection, but also independently in three windows of dimuon transverse momentum: p$_{T}$$<$0.5, 0.5$<$p$_{T}$$<$1 and p$_{T}$$>$1 GeV/c. Data and fits, including an illustration of the individual sources, are shown in Fig. \[fig2\] for all p$_{T}$ (upper) and for the particular selection p$_{T}$$<$0.5 GeV/c (lower). The fit quality is good throughout, even in the critical acceptance-suppressed $\eta$-Dalitz region at low mass and low p$_{T}$.
The quantitative fit results in terms of the cross section ratios, corrected for acceptance and extrapolated to full phase space (meaning here full ranges in y and p$_{T}$) are displayed in Fig. \[fig3\] (upper), where the horizontal lines (with the label $<>$) indicate the fit values without p$_{T}$ selection, and the data points (with statistical errors) the values obtained from the three different p$_{T}$ windows. The systematic errors are of order 10% in all cases, dominated by those of the branching ratios. The $\eta/\omega$ ratio agrees, within $<$10%, with the literature average for p-p, p-Be [@Agakichiev:mv]. The $\phi/\omega$ ratio is higher than the p-p, p-Be average, reflecting some $\phi$ enhancement already in peripheral nuclear collisions. Both ratios are, within 10%, independent of the pair p$_{T}$. This implies that the GENESIS input assumptions (used in the extrapolation to full p$_{T}$) as well as the acceptance corrections vs. p$_{T}$ are correct on the level of 10%. As shown in Fig. \[fig3\] (lower), the acceptance variations with p$_{T}$ are minor for the $\omega$ and the $\phi$, but very strong,
![Fits of hadron decay cocktail to the peripheral data for all p$_{T}$ (upper) and p$_{T}$$<$0.5 GeV/c (lower), showing also the individual contributions.[]{data-label="fig2"}](img/fig2_a.eps "fig:"){width="45.00000%"} ![Fits of hadron decay cocktail to the peripheral data for all p$_{T}$ (upper) and p$_{T}$$<$0.5 GeV/c (lower), showing also the individual contributions.[]{data-label="fig2"}](img/fig2_b.eps "fig:"){width="45.00000%"}
![Upper: Particle cross-section ratios for all p$_{T}$ ($<>$) and for three p$_{T}$ windows, extrapolated to full phase space. Lower: NA60 acceptance relative to 4$\pi$ for different mass windows.[]{data-label="fig3"}](img/fig3_a.eps "fig:"){width="45.00000%"} ![Upper: Particle cross-section ratios for all p$_{T}$ ($<>$) and for three p$_{T}$ windows, extrapolated to full phase space. Lower: NA60 acceptance relative to 4$\pi$ for different mass windows.[]{data-label="fig3"}](img/fig3_b.eps "fig:"){width="45.00000%"}
over two orders of magnitude, for the $\eta$ Dalitz mode (M$<$0.4 GeV/c$^{2}$). The accuracy level of 10% reached in understanding the acceptance is therefore truly remarkable.
The particle ratio $\rho/\omega$ behaves in a different way relative to the other two. It decreases with p$_{T}$, but remains significantly higher than the p-p, p-Be average [@Agakichiev:mv] throughout. This suggests that some $\pi\pi$ annihilation, enhancing the yield of the $\rho$, contributes already in peripheral collisions (see below).
In the more central bins, a fit procedure is ruled out, due to the existence of a strong excess with [*a priori unknown*]{} characteristics. We have therefore used a novel procedure as shown in Fig. \[fig4\], made possible by the high data quality. The excess is [*isolated*]{} by subtracting the cocktail, without the $\rho$, from the data. The cocktail is fixed, separately for the major sources and in each centrality bin, by a “conservative” approach. The yields of the narrow vector mesons $\omega$ and $\phi$ are fixed so as to get, after subtraction, a [*smooth*]{} underlying continuum. For the $\eta$, an upper limit is defined by “saturating” the measured data in the region close to 0.2 GeV/c$^{2}$; this implies the excess to vanish at very low mass, by construction. The $\eta$ resonance and $\omega$ Dalitz decays are now bound as well; $\eta^{'}/\eta$ is fixed as before. The [*cocktail $\rho$*]{} (shown in Figs. \[fig5\], \[fig6\], \[fig8\] and \[fig9\] for illustration purposes) is bound by the ratio $\rho/\omega$=1.2. The accuracy in the determination of the $\omega$ and $\phi$ yields by this subtraction procedure is on the level of very few %, due to the remarkable [*local*]{} sensitivity, and not much worse for the $\eta$.
The excess mass spectra for all 4 multiplicity bins, resulting from subtraction of the “conservative” hadron decay cocktail from the measured data, are shown in Figs. \[fig5\] and \[fig6\] for all p$_{T}$ and the particular selection p$_{T}$$<$0.5 GeV/c, respectively. The cocktail $\rho$ and the level of charm decays, found in the three upper centrality bins to be about 1/3 of the measured yield in the mass interval 1.2$<$M$<$1.4 GeV/c$^{2}$ [@Ruben:2005qm], are shown for comparison. The qualitative features of the spectra are striking: a peaked structure is seen in all cases, broadening strongly with centrality, but remaining essentially centered around the position of the nominal $\rho$ pole. At the same time, the total yield increases relative to the cocktail $\rho$, their ratio (for M$<$0.9 GeV/c$^{2}$) reaching values of 4 for all p$_{T}$, even close to 8 for p$_{T}$$<$0.5 GeV/c, in the most central bin. Such values are consistent with the CERES [@Agakichiev:1997au] results, if the latter are also referred
to the cocktail $\rho$ and rescaled to In-In. The errors shown are purely statistical. The systematic errors are dominantly connected to the uncertainties in the level of the combinatorial background, less so to the fake matches. For the data without p$_{T}$ selection, they are estimated to be about 3%, 12%, 25% and 25% for the 4 centralities in the broad continuum region, while the $\rho$-like structure above the continuum is much more robust. Uncertainties associated with the subtraction of the hadron decay cocktail reach locally up to 15%, dominated by those of the $\omega$-Dalitz form factor. For the data with p$_{T}$$<$0.5 GeV/c, the systematic errors
are still under investigation.
For the data without p$_{T}$ selection, a quantitative analysis of the shape of the excess mass spectra vs. centrality has been performed, using a finer subdivision into 12 centrality bins. Referring to Fig. \[fig5\], the data were subdivided into three mass windows with equal widths: 0.44$<$M$<$0.64 (L=Lower), 0.64$<$M$<$0.84 (C=Center), and 0.84$<$M$<$1.04 GeV/c$^{2}$ (U=Upper). From the yields in these windows, a peak yield R = C - 1/2(L+U) and a continuum yield 3/2(L+U) can be defined. Fig. \[fig7\] (upper) shows the ratios of peak/$\rho$ (R/$\rho$), continuum/$\rho$ (3/2(L+U)/$\rho$), and peak/continuum (RR), where $\rho$ stands for the cocktail $\rho$. The errors shown are purely statistical. The relative systematic errors between neighboring points are small compared to the statistical errors and can therefore be ignored, while the common systematic errors are related to those discussed in the previous paragraph and are of no relevance here. The ratio peak/$\rho$ is seen to decrease from the most peripheral to the most central bin by nearly a factor of 2, ruling out the naive view that the shape can simply be explained by the cocktail $\rho$ residing on a broad continuum, independent of centrality. The ratio continuum/$\rho$ shows a fast initial rise, followed by a more flat and then another more rapid rise beyond dN$_{ch}$/dy = 100; this behavior is statistically significant. The sum of the two ratios is the total enhancement factor relative to the cocktail $\rho$; it reaches about 5.5 in the most central bin. The ratio of the two ratios, peak/continuum, amplifies the two separate tendencies: a fast decay, a nearly constant part, and a decline by a further factor of 2 beyond dN$_{ch}$/dy = 100, though with large errors due to R.
A completely independent shape analysis was done by just evaluating the RMS = $\sqrt{\langle M^{2}\rangle-\langle M\rangle^{2}}$ of the mass spectra in the single total mass interval 0.44$<$M$<$1.04 GeV/c$^{2}$. The results are shown in Fig. \[fig7\] (lower), both for the full data and after subtraction of charm on the level discussed for Figs. \[fig5\], \[fig6\] (except for the most peripheral bin where the continuum yield above 1 GeV/c$^{2}$ was assumed to be 100% charm).
![Upper: Yield ratios continuum/$\rho$, peak/$\rho$, and peak/continuum; see text for definition and errors. Lower: RMS of the excess mass spectra in the window 0.44$<$M$<$1.04 GeV/c$^{2}$.[]{data-label="fig7"}](img/fig7_a.eps "fig:"){width="45.00000%"} ![Upper: Yield ratios continuum/$\rho$, peak/$\rho$, and peak/continuum; see text for definition and errors. Lower: RMS of the excess mass spectra in the window 0.44$<$M$<$1.04 GeV/c$^{2}$.[]{data-label="fig7"}](img/fig7_b.eps "fig:"){width="45.00000%"}
They are in perfect qualitative agreement with the more “microscopic” shape analysis discussed before, rising from values close to the cocktail $\rho$ all the way up to nearly a flat-continuum value. The extra rise beyond dN$_{ch}$/dy = 100 is highly significant here, due to the very small statistical and systematic errors.
Comparison to theoretical models {#sec:3}
================================
The qualitative features of the excess mass spectra shown in Figs. \[fig5\] and \[fig6\] are consistent with an interpretation as direct thermal radiation from the fireball, dominated by $\pi\pi$ annihilation. A quantitative comparison of the data to the respective theoretical models can either be done at the [*input*]{} of the experiment, requiring acceptance correction of the data, or at the [*output*]{}, requiring propagation of the theoretical results through the experimental acceptance. All the data contained in this paper have so far not been corrected for acceptance, and therefore only the second alternative is available at present (the first one being under preparation). To help intuition, Fig. \[fig8\] illustrates the effects of acceptance propagation for the particularly transparent case of $q\bar{q}$ annihilation, associated with a uniform spectral function [@rapp:nn23]. [*By coincidence*]{}, without p$_{T}$ selection, the resulting mass spectrum at the [*output*]{} is also uniform within 10% up to about 1.0 GeV/c$^{2}$, resembling the shape of the spectral function at the [*input*]{}. In other words, the always existing steep rise of the theoretical input at low masses (Fig. \[fig8\]), due to the photon propagator and a Boltzmann-like factor [@Rapp:1995zy; @Rapp:1999ej; @Brown:kk; @Brown:2001nh], is just about compensated by the falling acceptance in this region as long as no p$_{T}$ cut is applied. Variations of the input p$_{T}$ spectrum within reasonable physics limits affect the flatness of the output by at most 20%. Strong cuts in p$_{T}$ like $<$0.5 or $>$1 GeV/c, however, completely invalidate the argument.
On the basis of this discussion, the excess mass spectra shown in Fig. \[fig5\] can approximately be interpreted as spectral functions of the $\rho$, averaged over momenta and the complete space-time evolution of the fireball. The broad continuum-like part of the spectra may then reflect the early history close to the QCD boundary with a nearly divergent width, while the narrow peak on top may just be due to the late part close to thermal freeze-out, approaching the nominal width. The p$_{T}$-cut data shown in Fig. \[fig6\], on the other hand, do not allow such interpretation, due to the extreme acceptance cut on the low-mass side of the $\rho$; compare also Fig. \[fig3\] (lower).
Among the many different theoretical predictions for the properties of the intermediate $\rho$ mentioned in the introduction, only few have been brought to a level suitable for a quantitative comparison to the data. Before the first release of the present data in 2005, the in-medium broadening scenario of [@Rapp:1995zy; @Rapp:1999ej] and the moving mass scenario related to [@Brown:kk; @Brown:2001nh] were
![Propagation of thermal $q\bar{q}$ radiation, based on a uniform spectral function, through the NA60 acceptance without any p$_{T}$ selection. The resulting spectrum is, [*by coincidence*]{}, also uniform.[]{data-label="fig8"}](img/fig8.eps){width="45.00000%"}
evaluated for In-In at dN$_{ch}$/d$\eta$=140,
using exactly the same fireball evolution model extrapolated from Pb-Pb, and taking explicit account of temperature as well as of baryon density [@rapp:nn23]. In Fig. \[fig9\], these predictions (as well as the unmodified $\rho$) are confronted with the data for the semicentral bin. The integrals of the theoretical spectra are independently normalized to the data in the mass interval M$<$0.9 GeV/c$^{2}$ in order to concentrate on the spectral shapes, independent of the uncertainties of the fireball evolution. Note again that data and predictions can only be interpreted as space-time averaged spectral functions for the part without p$_{T}$ selection (Fig. \[fig9\], center), but not for the other two. Irrespective of the choice of the p$_{T}$ window, however, some general conclusions can be drawn. The unmodified $\rho$ is clearly ruled out. The moving mass scenario related to Brown/Rho scaling, which fit the CERES data [@Rapp:1999ej; @Agakichiev:1997au], is also ruled out, showing the much improved discrimination power of the present data. Only the broadening scenario appears to be in fair agreement with the data.
In the meantime, the release of the data in 2005 has triggered a number of new theoretical developments. Contrary to initial critics [@brownrho:nn], Brown/Rho scaling could not be saved by varying fireball and other parameters within extremes, including switching out the effects of temperature altogether [@rapphees:nn; @skokov:nn]. The excess of the data at M$>$0.9 GeV/c$^{2}$ may be related to the prompt dimuon excess found by NA60 in the intermediate mass region [@Ruben:2005qm]. It is not accounted for by the model results shown in Fig. \[fig9\], and it is presently (nearly quantitatively) described either by hadronic processes like 4$\pi$, 6$\pi$.. (including vector-axial-vector mixing) [@hees:nn], or by partonic processes like $q\bar{q}$ annihilation [@rr:nn]. This is a challenging theoretical ambiguity to be solved in the future. A chiral virial approach has also been able to nearly quantitatively describe the data [@zahed:nn].
Conclusions {#sec:5}
===========
The data unambiguously show that the $\rho$ primarily broadens in In-In collisions, but does not show any shift in mass. Consequently, model comparisons favor broadening scenarios, but tend to rule out moving-mass scenarios coupled directly to the chiral condensate. The issue of vector-axialvector mixing, also sensitive to chiral restoration, remains somewhat open at present. We expect that precise p$_{T}$ dependences, presently under investigation, will give more insight into the different sources operating in different mass regions.
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address:
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-
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author:
- Shaolin Liao
- Sasan Bakhtiari
- Henry Soekmadji
title: Validity of Image Theorems under Spherical Geometry
---
[**Shaolin Liao**]{}$^{1}$, [**Sasan Bakhtiari**]{}$^{1}$, [**and Henry Soekmadji**]{}$^{2}$\
$^{1}$Argonne National Laboratory, USA\
$^{2}$Hamilton Sundstrand, USA
This paper deals with different image theorems, i.e., Love’s equivalence principle, the induction equivalence principle and the physical optics equivalence principle, in the spherical geometry. The deviation of image theorem approximation is quantified by comparing the modal expansion coefficients between the electromagnetic field obtained from the image approximation and the exact electromagnetic field for the spherical geometry. Two different methods, i.e., the vector potential method through the spherical addition theorem and the dyadic Green’s function method, are used to do the analysis. Applications of the spherical imaging theorems include metal mirror design and other electrically-large object scattering.
Different image theorems have been widely used for electromagnetic modeling of mirrors and lens antenna [@cylindrical]-[@Perkins]. In [@Perkins], Rong and Perkins applied the image theorems to mirror system design for high-power gyrotrons. The author also theoretically evaluate the validity of the image theorems in the cylindrical geometry [@cylindrical]. In this article, following similar procedures in [@cylindrical], a closed-form formula for the discrepancy parameter, which is defined as the ratio of the spherical modal coefficient for image theorem to that of the exact field, has been derived for the spherical geometry.
Fig. \[IEEE\_sphere\] shows the spherical geometry for image theorem analysis.
![Image theorem in the spherical geometry: the incident field ${\bf
E}^i$ propagates onto spherical surface $S'$, then it may forward-propagate to ${\bf E}^+$ or it could be back-scattered to ${\bf E}^-$, depending on whether surface $S'$ as a fictitious surface where the equivalence theorem applies on a PEC surface. $\hat{\bf n}^+$ and $\hat{\bf n}^-$ are the outward and inward surface normals on spherical surface $S'$ respectively. ${\bf M}_s$ and ${\bf J}_s$ are equivalent surface currents for Love’s equivalence theorem. ${\bf M}_s^+$ is the image approximation of Love’s theorem and ${\bf M}_s^-$ is the image approximation for the induction theorem. []{data-label="IEEE_sphere"}](scheme.eps)
#### The Vector Potential Method
##### The spherical modal expansion
In spherical coordinates, the electrical vector potential $ { \bf F} ({\bf
r})$ for ${ \bf M}_s {\bf (r')}$ is given as [@Harrington], [@Stratton], $$\begin{aligned}
\label{FM_sph}
{ \bf F} ({\bf r}) = \epsilon_0 \int \! \! \int_{S'} dS' \ { \bf M}_s {\bf (r')}
g({\bf r- r' }) = \frac{-j k \epsilon_0}{4 \pi} \int \! \! \int_{S'} dS' \ { \bf M}_s {\bf
(r')} h_0^{(2)}\left(k[r-r'] \right)
\end{aligned}$$ where, $h_0^{(2)}$ is spherical Hankel function of the second kind of order 0. According to the spherical addition theorem [@Harrington], [@Stratton],
$$\begin{aligned}
\label{Addition_sph}
h_0^{(2)}\left(k[r-r'] \right) =
\sum_{n=0}^\infty (2n+1) j_n( k r') h_n^{(2)} ( k r) \\
\times \sum_{m=0}^n (2-\delta_m^0) \frac{(n-m)!}{(n+m)!} P_n^m
(\theta') P_n^m (\theta) \cos m(\phi-\phi') \nonumber\end{aligned}$$
where, $j_n $ is the spherical Bessel function of the first kind of integral order n; $P_n^m$ is the associated Legendre polynomial and $\delta_m^0$ is the Kronecker delta function ($\delta_m^0=1$ for m=0 and $\delta_m^0=0$ for m$\neq 0$). Substituting (\[Addition\_sph\]) into (\[FM\_sph\]), the modal expansion of ${ { \bf F} (r)}$ is obtained as,
$$\begin{aligned}
\label{ModalF_sph}
{ \bf F} ({\bf r}) & = & \sum_{n=0}^\infty \sum_{m=0}^n { \bf f}^{\hbox{\tiny{\bf M}$_s$}}_{\hbox{\tiny TE}} (n,m) h_n^{(2)} ( k r) P_n^m
(\theta) \begin{array}{c} \cos m\phi \\ \sin m\phi \end{array} \nonumber \\
{ \bf f}^{m,\hbox{\tiny{\bf M}$_s$} }_{n,\hbox{\tiny \ TE}} & = & \chi \int \! \! \int_{S'} dS' \ { \bf M}_s {\bf
(r')} j_n( k r') P_n^m (\theta') \begin{array}{c} \cos m\phi' \\
\sin m\phi'
\end{array} \nonumber
\\
\chi & = & (2-\delta_m^0) \frac{-j k \epsilon_0}{4 \pi}
\frac{(2n+1)(n-m)!}{(n+m)!} \ .\end{aligned}$$
The near field to far field transform of (\[ModalF\_sph\]) in the spherical coordinate is given as [@Yaghjian],
$$\begin{aligned}
\label{FF_sph}
{ \bf F} ( {\bf r})\vline_{r \rightarrow \infty} = \frac{j e^{-j k r}}{kr} \sum_{n=0}^\infty \sum_{m=0}^n j^n { \bf f}^{\hbox{\tiny{\bf M}$_s$}}_{\hbox{\tiny TE}} (n,m) P_n^m
(\theta) \begin{array}{c} \cos m\phi \\ \sin m\phi \end{array}\end{aligned}$$
The duality relation can be used to obtain the magnetic vector potential ${ \bf A} ({\bf r})$ for the ${\bf J}_s$ approximation as follows,
$$\begin{aligned}
{ \bf A} ({\bf r}) & = & \sum_{n=0}^\infty \sum_{m=0}^n { \bf g}^{\hbox{\tiny{\bf M}$_s$}}_{\hbox{\tiny TE}} (n,m) h_n^{(2)} ( k r) P_n^m
(\theta) \begin{array}{c} \cos m\phi \\ \sin m\phi \end{array} \nonumber \\
{ \bf g}^{m,\hbox{\tiny{\bf M}$_s$} }_{n,\hbox{\tiny \ TE}} & = & \chi' \int \! \! \int_{S'} dS' \ {\bf J}_s {\bf
(r')} j_n( k r') P_n^m (\theta') \begin{array}{c} \cos m\phi' \\
\sin m\phi'
\end{array} \nonumber
\\
\chi' & = & (2-\delta_m^0) \frac{-j k \mu_0}{4 \pi}
\frac{(2n+1)(n-m)!}{(n+m)!} \ .\end{aligned}$$
##### The back-scattered and forward-propagating waves
Similar to the cylindrical geometry, we can separate (\[ModalF\_sph\]) into back-scattered and forward-propagating waves as, $$\begin{aligned}
\label{Twoparts_sph}
j_n(k r') = \frac{1}{2} \left\{ h_n^{(1)}(k r')
+ h_n^{(2)}(k r') \right\}\end{aligned}$$ $$\begin{aligned}
{ \bf f}^{m,\hbox{\tiny{\bf M}$_s$} \pm}_{n,\hbox{\tiny \ TE}} =
\frac{\chi}{2} \int \! \! \int_{S'} dS' \ { \bf M}_s {\bf
(r')} h_n^{(1), (2)} ( k r') P_n^m (\theta') \begin{array}{c} \cos m\phi' \\
\sin m\phi'
\end{array}
\nonumber\end{aligned}$$
Since the spherical harmonics is a complete basis set, we can always express the initial incident electric field $ {\bf E}( { \bf r'} )
$ on the initial spherical surface $S'$ with radius of $r_0$ (in Figure \[IEEE\_sphere\]) as follows, $$\begin{aligned}
\label{Incident_sph}
{ \bf E} ( {\bf r_0}) & = & \sum_{n=0}^\infty \sum_{m=0}^n a_{n,o}^{m,e} {\bf M}^{m,e+}_{n,o} ( {\bf r_0}) + b_{n,o}^{m,e} {\bf N}^{m,e+}_{n,o}
({\bf r_0}) \nonumber\end{aligned}$$ $$\begin{aligned}
\label{TETM_sph}
\psi^{m,e+}_{n,o} ({\bf r_0})&=& h_m^{(2)}( k r_0)
P_n^m(\cos \theta') \begin{array}{c} \cos(m \phi') \\ \sin(m
\phi')
\end{array} \nonumber \\
{\bf L}^{m,e+}_{n,o} ({\bf r_0}) &= & \nabla \psi^{m,e+}_{n,o} ({\bf r_0}) \nonumber \\
{\bf M}^{m,e+}_{n,o} ({\bf r_0}) & = & \nabla \times \left\{ {\bf a}_r r \psi^{m,e+}_{n,o} ({\bf r_0}) \right\} \nonumber \\
{\bf N}^{m,e+}_{n,o} ({\bf r_0}) & = & \frac{1}{k} \nabla \times {\bf M}^{m,e+}_{n,o} ({\bf
r_0}) \ .\end{aligned}$$
From (\[ModalF\_sph\]) and noting that $ {\bf M}_s^+({\bf r_0})= 2 {
\bf E} ({\bf r_0}) \times {\bf a}_r $, on spherical surface $S'$ in Figure \[IEEE\_sphere\],
$$\begin{aligned}
\label{Scatter_sph}
\tilde{ \bf E} (r_0) = -\frac{1}{ \epsilon_0}
\sum_{n=0}^\infty \sum_{m=0}^n \left\{ {\bf L}^{m,e+}_{n,o}
({\bf r}) \times { \bf f}^{m,\hbox{\tiny{\bf M}$_s$}
}_{n,\hbox{\tiny \ TE}}
\right\}\end{aligned}$$
$$\begin{aligned}
{\bf L}^{m,e+}_{n,o} ({\bf r_0}) &= & \nabla \psi^{m,e+}_{n,o} ({\bf r_0}) \nonumber\end{aligned}$$
The approximate field $ \tilde{ \bf E} ({\bf r}_0)$ on the initial spherical surface $S'$ is obtained from (\[ModalF\_sph\]) through image theorem approximation, $$\begin{aligned}
\label{zeta_sph}
\tilde{ \bf E} ({\bf r}_0) = \sum_{n=0}^\infty \sum_{m=0}^n
\tilde{a}_{n,o}^{m,e} {\bf M}^{m,e}_{n,o} ( {\bf r}_0) +
\tilde{b}_{n,o}^{m,e} {\bf N}^{m,e+}_{n,o}
({\bf
r}_0)\end{aligned}$$
Now the deviation of the spherical coefficients $\tilde{a}_{n,o}^{m,e}, \tilde{b}_{n,o}^{m,e}$ in Eq. (\[zeta\_sph\]) from their exact values ${a}_{n,o}^{m,e}, {b}_{n,o}^{m,e}$ in Eq. (\[Incident\_sph\]) is defined as the discrepancy parameters $\zeta$, $$\begin{aligned}
\zeta_{\hbox{\tiny TE}}^{\hbox{\tiny \bf M}_s} =
\frac{\tilde{a}_{n,o}^{m,e}}{ {a}_{n,o}^{m,e}} = -j 2 k r_0
h_n^{(2)}(kr_0) \frac{\partial [ kr
j_n(kr)] }{\partial kr}\vline_{\ r=r_0} \nonumber \\
\zeta_{\hbox{\tiny TM}}^{\hbox{\tiny \bf M}_s} =
\frac{\tilde{b}_{n,o}^{m,e}}{ {b}_{n,o}^{m,e}} = j 2 k r_0 j_n(kr_0)
\frac{\partial [ kr h_n^{(2)}(kr) ] }{\partial kr}\vline_{\ r=r_0}
\nonumber\end{aligned}$$ and, $$\begin{aligned}
\label{zetapm_sph}
\zeta_{\hbox{\tiny TE}}^{\hbox{\tiny \bf M}_s,\pm} = -j 2 k r_0
h_n^{(2)}(kr_0) \frac{\partial [ kr h_n^{(1),(2)}(kr)] }{\partial
kr}\vline_{\ r=r_0} \\
\zeta_{\hbox{\tiny TM}}^{\hbox{\tiny \bf M}_s,\pm} = j 2 k r_0
h_n^{(1),(2)}(kr_0) \frac{\partial [ kr h_n^{(2)}(kr)] }{\partial
kr}\vline_{\ r=r_0} \ . \nonumber\end{aligned}$$
Similar expressions exist for ${\bf J}_s$ image approximation, $$\begin{aligned}
\label{J_sph}
\zeta_{\hbox{\tiny TE}}^{\hbox{\tiny \bf J}_s} = \zeta_{\hbox{\tiny
TM}}^{\hbox{\tiny \bf M}_s} , \ \ \zeta_{\hbox{\tiny
TM}}^{\hbox{\tiny \bf J}_s} = \zeta_{\hbox{\tiny TE}}^{\hbox{\tiny
\bf M}_s}\end{aligned}$$ $$\begin{aligned}
\zeta_{\hbox{\tiny TE}}^{\hbox{\tiny \bf M}_s,\pm}
=\zeta_{\hbox{\tiny TM}}^{\hbox{\tiny \bf J}_s,\pm} =
[\zeta_{\hbox{\tiny TE}}^{\hbox{\tiny \bf J}_s,\pm}]^\ast =
[\zeta_{\hbox{\tiny TM}}^{\hbox{\tiny \bf M}_s,\pm}]^\ast \ .
\nonumber\end{aligned}$$
#### The Dyadic Green’s Function Method
The magnetic dyadic Green’s function in the spherical coordinate is, $$\begin{aligned}
\label{Dyadic_sph}
\bar{\bf G}_m ({\bf r}, {\bf r'}) & = & - \frac{{\bf a}_r {\bf
a}_r}{ k^2} \delta( {\bf r - r'} ) - \sum_{n=-\infty}^\infty
\frac{ j \pi }{ 2 k n (n+1) }
\nonumber\end{aligned}$$ $$\begin{aligned}
\times \sum_{m=0}^n \frac{1}{Q_{nm}} \left\{
{\bf M}^{m,e}_{n,o} ( {\bf r'}) {\bf M}^{m,e+}_{n,o} ( {\bf r}) + {\bf N}^{m,e}_{n,o} ( {\bf r'} )
{\bf N}^{m,e+}_{n,o} ( {\bf r}) \right\} \nonumber \\
\hbox{and,} \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q_{nm} = \frac{2 \pi^2 (n+m)!}{ (2-\delta_m^0) (2n+1)
(n-m)!} \ \ \ \ \ \ \ \ \ \\end{aligned}$$ where ${\bf M}^{m,e}_{n,o} $ (${\bf
N}^{m,e}_{n,o}$) is obtained by replacing $h_n^{(2)}$ with $j_n$ in ${\bf M}^{m,e+}_{n,o}$ (${\bf N}^{m,e+}_{n,o}$). The approximate field $\tilde{\bf E} ({\bf r})$ for ${\bf M}_s^+({\bf r'})$ is given as,
$$\begin{aligned}
\label{Scatter_dyadic_sph}
\tilde{\bf E} ({\bf r}) & = & - \nabla \times \int \!\! \int_{S'} dS' \ {\bf M}_s^+({\bf r'}) {\bf
.} \bar{\bf G}_m ({\bf r}, {\bf r'})\end{aligned}$$
Substituting (\[Dyadic\_sph\]) into (\[Scatter\_dyadic\_sph\]) and using the orthogonal properties of spherical modal functions, the approximate field $ \tilde{\bf E} ( r_0 ) $ on initial spherical surface $S'$ is obtained as,
$$\begin{aligned}
\label{Scatter_dyadic_sph_1}
\tilde{\bf E} ( r_0 ) =
\sum_{n=-\infty}^{\infty} \sum_{m=0}^{n} \frac{j \pi}{n (n+1) Q_{nm}}
\begin{array}{c} {c}_{n,o}^{m,e} {\bf M}_{n,o}^{m,e+} ({\bf r}) \\
{d}_{n,o}^{m,e} {\bf N}_{n,o}^{m,e+} ({\bf r})
\end{array} \\
\times \int \! \! \int_{S'} dS' \
\begin{array}{c} \hbox{$[{\bf N}_{n,o}^{m,e} ({\bf r'})]^\ast$} \times {\bf M}_{n,o}^{m,e+} ({\bf r'})
\\ \hbox{$[{\bf M}_{n,o}^{m,e} ({\bf r'})]^\ast$} \times {\bf N}_{n,o}^{m,e+} ({\bf r'}) \end{array} \ {\bf \cdot} \ {\bf
a}_{r'} \ .
\nonumber
\end{aligned}$$
The evaluation of (\[Scatter\_dyadic\_sph\_1\]) also leads to (\[zeta\_sph\]) and (\[zetapm\_sph\]).
[cccc]{}\
TE/TM modes and ${\bf M}_s/{\bf J}_s$ & The relations & $\zeta_{\hbox{\tiny TE, TM}}^{+,-} ({\bf
M}_s , {\bf J}_s) $ & $ r_0 \rightarrow \infty$\
& & &\
\
TE & ${\bf M}_s$ / TM & ${\bf J}_s$ & & &\
Sphere: back-scattered wave & $\zeta_{\hbox{\tiny TE}}^-( {\bf M}_s )=\zeta_{\hbox{\tiny TM}}^-( {\bf J}_s )$ & $ -j k r_0 h_n^{(2)}(kr_0) \frac{\partial [ kr
h_n^{(2)}(kr)] }{\partial kr}\vline_{\ r=r_0}$ & $ (-1)^{n} e^{-j 2 k r_0}$\
Sphere: forward-propagating wave & $\zeta_{\hbox{\tiny TE}}^+( {\bf M}_s ) = \zeta_{\hbox{\tiny TM}}^+( {\bf J}_s )$ & $ -j k r_0 h_n^{(2)}(kr_0) \frac{\partial [ kr
h_n^{(1)}(kr)] }{\partial kr}\vline_{\ r=r_0} $ & 1\
& & &\
TM & ${\bf M}_s$ / TE & ${\bf J}_s$ & & &\
Sphere: back-scattered wave & $\zeta_{\hbox{\tiny TM}}^-( {\bf M}_s )=\zeta_{\hbox{\tiny TE}}^-(
{\bf J}_s )$ & $ j k r_0 h_n^{(2)}(kr_0) \frac{\partial [ kr
h_n^{(2)}(kr) ] }{\partial kr}\vline_{\ r=r_0}$ & $ - (-1)^{n} e^{-j 2 k r_0}$\
Sphere: forward-propagating wave & $\zeta_{\hbox{\tiny TM}}^+( {\bf M}_s ) = \zeta_{\hbox{\tiny TE}}^+( {\bf J}_s )$ & $ [\zeta_{\hbox{\tiny TE}}^+( {\bf M}_s )]^\ast / [\zeta_{\hbox{\tiny TM}}^+( {\bf J}_s )]^\ast$ & 1\
& & &\
\
#### The Analytical Formula for Image Theorems in the Spherical Geometry
Similar to the cylindrical geometry, $\zeta_{\hbox{\tiny
TE,TM}}^{\hbox{\tiny \bf M}_s,\hbox{\tiny \bf J}_s +} $ in (\[zetapm\_sph\]) and (\[J\_sph\]) can be considered as theoretical formulas for evaluation of the image theorems for narrow-band fields in the spherical geometry. The large argument asymptotic behaviors of $\zeta_{\hbox{\tiny TE,TM}}^{\hbox{\tiny \bf
M}_s,\hbox{\tiny \bf J}_s +} $ for $r_0 \rightarrow \infty$ can be obtained by noting that, $$\begin{aligned}
h_n^{(2)}(k r_0) = [h_n^{(1)}(k r_0)]^\ast \sim
\frac{1}{k r_0} j^{(n+1)} e^{-j k r_0}, \
k r_0 \rightarrow \infty \nonumber
\end{aligned}$$ $$\begin{aligned}
\zeta_{\hbox{\tiny TE,TM}}^{\hbox{\tiny \bf M}_s,\hbox{\tiny \bf
J}_s +} \vline_{ \ r_0 \rightarrow \infty} = 1 \ .\end{aligned}$$
![ The spherical geometry - threshold radii $r_{\hbox{\tiny th}}$ Vs. n=0 to 100, for different accuracies, from $-60$ dB to $-30$ dB (in $10$ dB increment, from bottom to top): a) the magnitudes $20 \log_{10}(|\zeta_{\hbox{\tiny TE
}}^{\hbox{\tiny \bf M}_s +}|-1)$, and b) the imaginary parts $20
\log_{10}[\Im(\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s +})]$. The inset plots in a) are used to make the display clearer. Similar to the cylindrical geometry, imaginary parts $\zeta_{\hbox{\tiny
TE }}^{\hbox{\tiny \bf M}_s +}$ require larger threshold radii $r_{th} $ for the same accuracy. []{data-label="zeta_rho_sph"}](zeta_n_.eps)
TABLE \[xi\_zeta\] summarizes the properties of $\zeta_{\hbox{\tiny TE,TM }}^{\hbox{\tiny \bf M}_s,\hbox{\tiny \bf
J}_s \pm}$, for the back-scattered and forward-propagating waves respectively. For $r_0 \rightarrow \infty$, $\zeta_{\hbox{\tiny TE,TM }}^{\hbox{\tiny \bf M}_s,\hbox{\tiny \bf
J}_s } = \zeta_{\hbox{\tiny TE,TM }}^{\hbox{\tiny \bf
M}_s,\hbox{\tiny \bf J}_s +} + \zeta_{\hbox{\tiny TE,TM
}}^{\hbox{\tiny \bf M}_s,\hbox{\tiny \bf J}_s -}$ shows fast oscillations, which can be seen from TABLE \[xi\_zeta\]. Mathematically, the oscillations only appear as modal expansion coefficients and disappear after the implementation of the double sums in (\[zeta\_sph\]). Physically, the oscillations are due to back-scattered fields, which approach 0 for $r_0 \rightarrow \infty$. For example, consider $\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s -}$ in (\[zetapm\_sph\]),
$$\begin{aligned}
\label{Eminus}
\tilde{ \bf E}^- (r_0, \phi) = \sum_{n=0}^\infty \sum_{m=0}^n
\left\{ \zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s -} \
{c}_{n,o}^{m,e} {\bf M}^{m,e+}_{n,o} ( {\bf r}_0)
+ \zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s -} \ {b}_{n,o}^{m,e} {\bf
N}^{m,e+}_{n,o} ({\bf r}_0) \right\}\end{aligned}$$
Changing the variable $\phi' = \phi - \pi$ and letting $r_0
\rightarrow \infty$, from TABLE \[xi\_zeta\], (\[Eminus\]) reduces to,
$$\begin{aligned}
\tilde{ \bf E}^- (r_0, \phi')\vline_{ \ r_0 \rightarrow \infty} =
\sum_{n=0}^\infty \sum_{m=0}^n e^{-j 2 k r_0} \left\{
{c}_{n,o}^{m,e} {\bf M}^{m,e+}_{n,o} ( {\bf r}_0)
- {b}_{n,o}^{m,e} {\bf
N}^{m,e+}_{n,o} ({\bf r}_0) \right\}.\end{aligned}$$
Now, the back-scattered field $\tilde{ \bf E}^- (r_0, \phi')\vline_{ \ r_0
\rightarrow \infty} \rightarrow 0$ due to the fast variation phase term $e^{-j 2 k r_0}$, which means that the oscillation in $\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s -}$ doesn’t appear in the actual field evaluation for $r_0 \rightarrow \infty$ .
Based on the above discussion, $\zeta_{\hbox{\tiny TE,TM }}^{\hbox{\tiny \bf M}_s,\hbox{\tiny \bf J}_s \pm}$ is the theoretical formula of interest to evaluate the validity of image theorems.
It is also helpful to plot the corresponding threshold radius $r_{\hbox{\tiny th}}$ with respect to n, for both $20 \log_{10}(|\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s
+}|-1)$ and $20 \log_{10}\{\Im[\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny
\bf M}_s +}]\}$, with different accuracies ranging from $-60$ dB to $-30$ dB (in $3$ dB increment), as in Fig. \[zeta\_rho\_sph\]. It can be seen from Fig. \[zeta\_rho\_sph\] that, in order to achieve an accuracy of $-30$ dB for $|\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s +}|$ (with respect to 1), $r_{\hbox{\tiny th}} \sim 8 \lambda$ and $r_{\hbox{\tiny th}} \sim 16 \lambda$ for $n=50$ and $n=100$ respectively. However, for the imaginary part $\Im[\zeta_{\hbox{\tiny TE
}}^{\hbox{\tiny \bf M}_s +}]$, $r_{\hbox{\tiny th}} \sim 9.5
\lambda$ and $r_{\hbox{\tiny th}} \sim 18 \lambda$ are required for $n=50$ and $n=100$ respectively, which again implies that the imaginary part $\zeta_{\hbox{\tiny TE }}^{\hbox{\tiny \bf M}_s +}$ dominates the accuracy of image theorems.
For spherical geometry, the theoretical formulas for evaluation of the image theorems (both ${\bf M}_s$ and ${\bf J}_s$ approximations) have been derived through two equivalent methods - the vector potential method and the dyadic Green’s function method, for both TE and TM modes. The ratio of the spherical modal coefficient of the image theorem to that of the exact field is used as the criterion to determine the validity of the image theorem.
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---
abstract: 'Vacuum nanodevices are devices that the electron transport through them is based on electron field emission from a nano-eimtter to another opposite electrode through a vacuum channel. Geometrically asymmetric metal-vacuum-metal structures were demonstrated to have energy conversion ability for electromagnetic waves in the optical range. Combining the ability of these structures to convert optical signals into rectified current and the ability of vacuum nanotriodes to control the field emission current can allow direct processing on converted optical signals using a single device. In this paper, a three-dimensional quantum-mechanical method, rather than the approximate Fowler-Nordheim theory, is used for modeling the field emission process in vertical-type vacuum nanotriodes consisting of an emitter, a collector and a gate. The electron transport through the device is computed using a transfer-matrix technique. The potentials of vacuum nanotriodes in the current rectification and modulation are investigated at low voltages. The effects of varying the structure geometrical parameters on the rectified current are also studied. The obtained results show that a great enhancement in the rectification properties is achievable when the gate and the collector are connected through a DC source. It is also demonstrated that a small variation in the gate voltage can be used either to modulate the rectified current or to switch the device into a resonant tunneling diode.'
author:
- 'M. S. Khalifa'
- 'A. H. Badawi'
- 'T. A. Ali'
- 'N. H. Rafat'
- 'A. A. Abouelsaood'
bibliography:
- 'References.bib'
title: 'Three-dimensional Modeling of Vacuum Field Emission Nanotriodes'
---
Introduction
============
The rapid progress in nanofabrication during the past several years allowed for realizing nanostructures with minimum features of a few nanometers. This advancement, along with the growing need for high-frequency electronics, initiated the research in the field of vacuum nanoelectronics. Unlike semiconductor devices, vacuum nanoelectronic devices do not suffer from the limitation of electron velocity saturation as they depend on ballistic electron transport, which allows operating at higher frequencies. Also, they have greater thermal tolerance and exhibit higher robustness against high levels of radiation, which make them better candidates than semiconductor devices in extreme environments such as military and space applications. On the other hand, the active research in optical rectennas (rectifying nano-antennas) has recently exhibited promising results in energy conversion and current rectification. Combining the potentials of vacuum nanodevices with optical rectennas allows for fast, local modulation of the output current of rectennas. As a result, energy conversion, current rectification and processing functions can all be implemented by a single device. This opens the area to a wide field of applications in optical computing and communication technologies, where fast logic operations can be executed directly by rectennas.
In principle, vacuum nanoelectronic devices depend in their operation on field emission (FE) from nanotips supported on cathodes. The emitted electrons are usually collected at an opposite flat electrode (the anode or collector), and a gate may be added in the electrons path to control the magnitude of the emission current. Optical rectennas as well depend in their rectification behavior on asymmetric electrons emission (or tunneling) in asymmetric metal-insulator-metal (MIM) junctions. In this work we are particularly interested in vertical (Spindt-type) FE triodes [@Spindt1968]. It was demonstrated that reducing the dimensions of the different parameters of such devices (tip radius, gate aperture diameter, and gap distance) is a key factor in enhancing their performance in terms of the current density, the applied voltages and the cutoff frequency [@Nguyen1989; @Chen2012].
Many attempts have been made in the process of miniaturizing these devices. In 1989, Brodie [@Brodie1989] discussed the physics governing FE from a conic tip to an integrated collector electrode at a separation of distance $0.5\text{ }\mbox{\ensuremath{\mu}m}$, with gate aperture of radius $0.5\text{ }\mbox{\ensuremath{\mu}m}$. In 2000, Driskill-Smith et al. [@Driskill-Smith2000] fabricated long nanopillars of radius $1\text{ }\mbox{nm}$ in a vacuum nano-chamber with dimensions of about $0.1\text{ }\mbox{\ensuremath{\mu}m}$. More recently, materials such as CNTs and Si nanowires have been used in fabrication as the emitting nanotips, for their high aspect ratio and relatively low operating voltages, with self-aligning gate around them [@Pflug2001; @Guillorn2001; @Gangloff2004; @Wong2005; @Ulisse2012]. This allowed minimizing the gate aperture down to $45\text{ }\mbox{nm}$ in radius [@Chen2012].
In $\References$ , the authors described the FE process based on Fowler-Nordheim (FN) theory, in which the current density is expressed as a function of the electric field at the emitter surface. In the original FN theory, the emitting surface is assumed to be planar, and hence a one-dimensional problem is considered. This assumption is accurate as long as the emitter radius is much greater than the potential barrier, where the electric field can be considered uniform along the emitting surface. However, when the emitter radius is comparable to or smaller than the barrier width, the one-dimensional solution is no longer valid, and tunneling through a three-dimensional potential barrier should be considered instead. Although many correcting factors were introduced to FN basic equation for considering the emitter geometry and size [@JunHe1991; @Cutler1993; @Jensen1995; @Nicolaescu2001], they were demonstrated to be inaccurate when applied to sharp emitters with radii $\apprle10\text{ }\mbox{nm}$ [@Cutler1993]. Another effective classical model, so called Quantum Corrected Model (QCM), was developed more recently for modeling electron tunneling through plasmonic nanogaps [@Esteban2012]. However, this model also assumes a potential barrier that is much smaller than the radius of the emitting surface, which is the typical case in plasmonic systems. For describing the behavior of vacuum nanotriodes without implying modeling restrictions, one can use either the Green-Function method [@Lucas1988; @Doyen1993] or the Transfer-Matrix method [@Mayer1997; @Mayer1999]. In each of these methods, Schr[ö]{}dinger’s equation is solved in three dimensions. In this work we follow the transfer-matrix method, which is more suitable in terms of memory storage when dealing with emitters with a few nanometers in height [@Mayer1999].
In this paper, the model proposed by Mayer et al. in $\Reference$ for a high-frequency rectifier is extended by introducing a metallic gate to the structure for controlling the current flow through the device, so that the device resembles Spindt-type vacuum field emission triode. We mainly investigate the effect of the gate on the I-V characteristics of the device. We also study the effect of the gate voltage on the rectification properties of the device in the limit of quasi-static bias. In $\Section$\[sec:Methodology\], we state the assumptions made in the model and present the method we follow in computing the potential energy distribution and the emission current, with emphasizing the modifications we introduced for considering the gate effect. The behavior of the device is then studied in $\Section$\[sec:Calculations-and-Results\]. First, we present and discuss the modifications introduced to the potential barrier in the device due to the effect of the gate. Next, we investigate the possibility of improving the current rectification and the output power of the device in different situations for the gate potential. Then, we investigate the effect of the thickness, the height and the aperture diameter of the gate on the output current of the device. Finally, we investigate how changing the height and the diameter of the emitting tip can be exploited for improving the behavior of the device.
![Schematic representation for the proposed structure. The rectangles at the sides represent the gate disc which surrounds the rectifier circularly, and are extended in the radial direction.\[fig:The-proposed-structure\]](\string"Fig1_Triode_Scetch\string".pdf)
Methodology\[sec:Methodology\]
==============================
Preliminaries\[sub:Preliminaries\]
----------------------------------
The structure we study in this paper is shown in $\Figure$\[fig:The-proposed-structure\]. It consists of two metallic parallel flat planes separated by a distance $D$, with the lower one supporting a metallic cylindrical nanotip of a hemispherical end whose height and diameter are $h_{t}$ and $d_{t}$, respectively. Similar structures with geometrical asymmetry were proven to exhibit rectification properties both theoretically [@Mayer2008; @Miskovsky1979] and experimentally [@Kuk1990; @Bragas1998; @Tu2006; @Dagenais2010; @Ward2010]. A gate electrode, represented by an infinite horizontal metallic disc of thickness $t_{g}$, is set at height $h_{g}$ above the lower metal, and contains a circular aperture of diameter $d_{g}$ concentric with the tip. The rest of the space between the two surfaces is assumed to be vacuum.
We consider the metallic planes as two long leads (regions I and III) of radius $R$. The leads are assumed to be perfect conducting metals, that is, electrons inside them have a uniform potential energy. The objective of this section is to study the quantum transport between region I and region III through some quantum device (region II) representing the volume enclosed by the horizontal planes $z=0$ and $z=D$, and the lateral surface defined by $\rho=R$. The vacuum cylindrical shell between the end of the leads at $\rho=R$ and the start of the gate at $\rho=d_{g}/2$ (region IV) is considered outside the device. Electrons are assumed to be confined in the leads and the device within the cylindrical space of radius $R$, that is, the leakage current in the gate is neglected in this model and left for future investigations. This approximation is acceptable in the light of the results obtained by Driskill-Smith et al. [@Driskill-Smith2000], where the calculations of the electrons trajectories showed that all electrons emitted from the tip are collected at the anode for all anode voltages except at zero volt.
We note here that $R$ should not exceed the gate aperture radius, $d_{g}/2$, for two physical reasons. The first reason is that stacking three metallic layers with two vacuum barriers in between would result in resonant tunneling between the leads at the regions where the three layers overlap [@Ricco1984]. This will then overwhelm the rectification behavior of the device as well as the current control by the gate. The second reason is to reduce the gate-cathode and the gate-anode capacitances that would decrease the cutoff frequency of the device [@Ulisse2012]. In order to treat this situation in the analysis, we solve Poisson’s equation in regions II and IV with taking the gate effect (both the potential on the gate as well as its metallic effect regardless of the metal type) as a boundary condition on the lateral edge of region IV at $\rho=d_{g}/2$ and at $z$ between $h_{g}-t_{g}/2$ and $h_{g}+t_{g}/2$. Then we use the obtained value of the potential in region II only, where electrons are assumed to be localized, to solve Schr[ö]{}dinger’s equation. The material properties of the gate, namely the workfunction and the Fermi energy, are not important in our calculations, because the leakage current is essentially assumed to be neglected, and the gate itself is taken outside the solution region which is limited by $\rho=R$.
When the structure is exposed to an external electric field directed along the axis of the device, $z$-direction, a potential difference is induced between the two leads with a magnitude that depends on the intensity of the field and the length of the device, $D$ in $\Figure$\[fig:The-proposed-structure\], where $\Delta V=-ED$. This assumption is valid when considering an electrostatic field or a quasi-static electric field that could be carried by an incident electromagnetic wave with a relatively low frequency. The quasi-static limit here applies for waves whose periods are much longer than the average time an electron would take to transport through the device. For the structure parameters we consider in $\Section$\[sec:Calculations-and-Results\], the quasi-static limit is valid for small frequencies compared to $1000\text{ }\mbox{THz}$ [@Mayer2008]. In all the upcoming analysis, we account for the external potential difference $\Delta V$ between the collector (upper lead) and the emitter (lower lead) by taking its value as a voltage applied to the collector; $V_{c}=\Delta V$, while the emitter is kept grounded. Similarly, we take the voltage applied to the gate $V_{g}$ referred to the emitter.
Potential Energy Distribution\[sub:Potential-Energy-Distribution\]
------------------------------------------------------------------
The first step in the analysis is obtaining the electron potential energy distribution in each of the four regions in $\Figure$\[fig:The-proposed-structure\] in order to solve Schr[ö]{}dinger’s equation in regions I, II, and III accordingly. Metallic leads are considered in regions I and III with a work function $W$ and a Fermi energy $E_{f}$. The electron potential energy in these regions are then $U^{I}=U^{III}=-(W+E_{f})$. When external voltage $V_{c}$ is applied to the upper lead, another term $-eV_{c}$ is added to the potential energy in region III, so that $U^{III}=-(W+E_{f})-eV_{c}$, where $e$ is the magnitude of the electron charge.
Unlike the previous two regions, regions II and IV include non-uniform potential energy distribution. Electron potential energy at any point in the vacuum in these regions consists of two parts; $U_{bias}$ and $U_{met}$. The first part, $U_{bias}$, is the potential energy induced from the externally applied voltages on the leads and the gate. The electric potential distribution due to this bias, $V_{bias}$, can be obtained by solving Poisson’s equation in the volume of regions II and IV, taking the following boundary conditions. In region II we take ground potential on the lower surface and the tip, and a constant potential $V_{c}$ on the upper surface. In region IV, we take a zero charge boundary condition on the lower and the upper surfaces. On the lateral surface at $\rho=d_{g}/2$ there are three domains; a) $0<z<h_{g}-t_{g}/2$, b) $h_{g}-t_{g}/2<z<h_{g}+t_{g}/2$, and c) $h_{g}+t_{g}/2<z<D$. The zero charge boundary condition is taken along the first and the third domains, while a constant potential $V_{g}$ is taken along the second domain which represents the gate surface. The zero charge boundary condition indicates that the normal component of the electric field is zero at the boundary, i.e. the electric potential is constant along the normal direction. The second part, $U_{met}$, is the self-induced potential energy by the tunneling electron during its transport through the device. This part represents the potential energy of the electron due to the accumulated charges on the metallic surfaces of the leads, the tip and the gate. In this model, the electron image is taken on each metal surface individually as a first approximation. This enables calculating this part of the potential by solving Poisson’s equation numerically [@Laloyaux1993; @Mayer2005], with the boundary condition [$V_{met}(\mathbf{r_{b}})=\frac{1}{4\pi\epsilon_{0}}\frac{e}{|\mathbf{r_{b}}-\mathbf{r_{e}}|}$]{} on all the metallic surfaces, including the gate, where $\mathbf{r}_{e}$ is the position of the electron and $\mathbf{r_{b}}$ is a point on the conductor at which the boundary potential is calculated. The zero charge boundary condition is taken here again on the non-metallic boundaries in region IV. The potential energy of the electron due to this electric potential is $U_{met}^{II}(\mathbf{r_{e}})=-eV_{met}(\mathbf{r}_{e})/2$, where the factor $1/2$ arises from the fact that this potential energy is self-induced by the electron[@smythe1950]. In order to avoid the divergence of this term near the metallic surfaces, we cut all the values lower than the potential energy inside the metals and set them by this value [@Murphy1956; @Modinos2001]. Since the metallic tip is supported on the grounded lead, electrons in the tip have zero $U_{bias}^{II}$ and constant $U_{met}^{II}=-(W+E_{f}^{tip})$, where $E_{f}^{tip}$ is the Fermi energy of the material of the tip.
Field Emission Current\[sub:Field-Emission-Current\]
----------------------------------------------------
Now we proceed to solving Schr[ö]{}dinger’s equation, using the obtained potential energy, in order to get the FE current. Since electrons are assumed to be confined in a cylinder of radius $R$, then their wavefunctions can be expanded in terms of complete, orthonormal eigenstates in cylindrical coordinates as follows[@Mayer1997]
$$\Psi(\rho,\phi,z)={\displaystyle \sum_{m,j}}\Phi_{mj}(z)\frac{J_{m}(k_{mj}\rho)}{\sqrt{\int_{0}^{R}\rho\left[J_{m}(k_{mj}\rho)\right]^{2}d\rho}}\frac{e^{im\phi}}{\sqrt{2\pi}}$$ where $J_{m}$ is the $m^{th}$ order of the Bessel function of the first kind, with $m$ integer, and $k_{mj}$ is the $j^{th}$ coefficient satisfying $J_{m}^{'}(k_{mj}R)=0$, with $j$ positive integer. $\Phi_{mj}(z)$ are the coefficients of the eigenstates depending on the coordinate $z$. In regions I and III, each lead has a constant potential energy and is considered semi-infinite in the $z$-direction, therefore for an electron with energy $E$ the coefficients are [$\Phi_{mj}^{I/III}(z)=\alpha e^{\pm ik_{z,mj}^{I/III}z}$]{}, where [$k_{z,mj}^{I/III}=\sqrt{\frac{2m_{e}}{\hbar^{2}}\left(E-U^{I/III}\right)-k_{mj}^{2}}$]{} and $\alpha$ is a normalization factor. The $\pm$ sign indicates the propagation direction relative to $z$-axis.
To obtain the transmission probability through the quantum device (region II), the transfer-matrix method developed in $\References$ is followed. In this technique the potential energy in region II is divided into two parts according to its coordinates-dependency; main potential $U_{0}^{II}(z)$ and local perturbing potential $U_{1}^{II}(\rho,\phi,z)$, such that $U^{II}=U_{0}^{II}(z)+U_{1}^{II}(\rho,\phi,z)$. After mathematical manipulations, a matrix equation coupling the coefficients $\Phi_{mj}(z)$ is obtained. Next, by discretizing region II into horizontal layers and assuming that $U^{II}$ is independent of $z$ inside each single layer, the matrix equation can be solved for all $\Phi_{mj}(z)$ in each layer, as illustrated in appendix A in $\Reference$. This enables calculating the scattering parameters for each layer individually. The scattering parameters of consecutive layers are then combined iteratively, using the layer addition algorithm [@Pendry1994; @Mayer1999], until one gets the total scattering parameters of region II.
Let $S^{++}$ and $S^{--}$ be the forward (from region I to III) and reverse (from region III to I) transmission matrices of the obtained scattering parameters. Then, the upward $I^{+}$ and the downward $I^{-}$ electron currents can be obtained from the following expressions, respectively [@Mayer2008]
$$I^{+}=\frac{2e}{h}{\displaystyle \int_{U_{I}}^{\infty}}f_{I}(E)\left[1-f_{III}\left(E\right)\right]{\displaystyle \sum_{mj}\sum_{m'j'}\left|S_{(m',j'),(m,j)}^{++}\right|^{2}\frac{k_{z_{m',j'}}^{III}}{k_{z_{mj}}^{I}}}dE\label{eq:UpwardCurrent}$$
$$I^{-}=\frac{2e}{h}{\displaystyle \int_{U_{III}}^{\infty}}f_{III}(E)\left[1-f_{I}\left(E\right)\right]{\displaystyle \sum_{mj}\sum_{m'j'}\left|S_{(m',j'),(m,j)}^{--}\right|^{2}\frac{k_{z_{m',j'}}^{I}}{k_{z_{mj}}^{III}}}dE\label{eq:DownwardCurrent}$$
where $f_{I}(E)$ and $f_{III}(E)$ are the Fermi functions at regions I and III, whose Fermi levels are given by $\mu_{I}=-W$ and $\mu_{III}=-W-eV_{c}$, respectively. The temperature in the Fermi functions is taken to be $300\text{ }\mbox{K}$.
For a finite number of modes and finite matrices dimensions, a finite number of values for the quantum numbers $m$ and $j$ should be considered. Ideally, all the values of $m$ and $j$ satisfying the relation
$$k_{mj}\le\sqrt{\frac{2m_{e}}{\hbar^{2}}\left(E-min\left(U^{I},U^{III}\right)\right)}$$ should be included. This condition ensures including all the modes propagating in at least one of the two leads, where these are the responsible modes for conducting current through the device. However, in the structure we study here, the existence of the tip around the $z$-axis makes the modes associated with small values of $|m|$ (the modes with high probability around the center) have higher contribution to the tunneling current than those associated with larger values of $|m|$. This is because the Bessel functions $J_{m}$ have higher values near the center for smaller values of $|m|$, [@Mayer2008a]. This allows us to consider a) all modes with $|m|\le m_{max}$, where $m_{max}$ is as high as necessary for reaching convergence, and b) all values of $j$ satisfying the above condition on $k_{mj}$ for the associated $m$. In this work we take $m_{max}=4$.
Results and Discussion\[sec:Calculations-and-Results\]
======================================================
In this section, we aim at exploring the potentials of the FE nanotriode for current rectification and modulation at low voltages. In particular, we investigate the effect of the gate in modulating the behavior of the structure and whether it can be exploited for enhancing the current rectification and the mean output power of the device. We also study how the geometrical parameters of the device affect its performance and how they can be optimized for current modulation through the gate voltage.
Potential Barrier Modulation\[sub:Barrier-Modulation\]
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In this section, we study the effect of the gate on the shape of the potential barrier. We consider cylindrical leads with radius $R=2\text{ }\mbox{nm}$, separated by distance $D=2\text{ }\mbox{nm}$, as shown in $\Figure$\[fig:The-proposed-structure\], and made of tungsten whose work function and Fermi energy are $4.5$ and $19.1\text{ }\mbox{eV}$, respectively. In region II a nanotip of height $h_{t}=1\text{ }\mbox{nm}$ and diameter $d_{t}=1\text{ }\mbox{nm}$ is set on the lower lead, and made of tungsten as well. The rest of region II is assumed to be vacuum. Outside region II we consider a gate of thickness $t_{g}=1\text{ }\mbox{nm}$ set at height $h_{g}=1\text{ }\mbox{nm}$ above the lower lead, and aperture diameter $d_{g}=4.1\mbox{\text{ }}\mbox{nm}$. The value of the gate aperture diameter is chosen to be greater than the leads diameter in order to avoid the resonant tunneling. Both the thickness and the height of the gate are chosen such that the gate is centered vertically at the end of the tip. Both the collector bias $V_{c}$ and the gate bias $V_{g}$ are taken to be $1\text{ }\mbox{V}$. As we mentioned before in $\Section$\[sub:Preliminaries\], the material properties of the gate are not important in calculations, therefore all the results in this section are applicable to a gate of any metal that can be considered a perfect conductor.
According to the model assumptions in $\Section$\[sec:Methodology\], the following results and discussion are applicable only to the limit of quasi-static fields. This limit depends on the cutoff frequency of the device, which is given by half of the reciprocal of the average time taken by an electron to travel between the emitter and the collector; the traversal time [@Bttiker1982]. In the classically forbidden region, an electron is assumed to be traveling at the Fermi velocity [@Nguyen1989; @Sullivan1989], while outside the barrier the electron propagates classically with a velocity that is proportional to its wavevector. Having the velocity and the gap distance between the emitter and the collector, the traversal time of the electrons can be calculated. For the structure parameters mentioned above, the traversal time takes a value of $0.5\text{ }\mbox{fs}$ [@Mayer2008], and the cutoff frequency is, therefore, $1000\text{ }\mbox{THz}$, which corresponds to electromagnetic wavelength of about $300\text{ }\mbox{nm}$, that is in the ultraviolet range. The quasi-static limit assumption is valid for frequencies significantly lower than this value. In case of oscillating fields with frequencies close to the cutoff frequency, photon absorption and emission should be included in the model [@Mayer2000a; @Miskovsky1994]. This problem will be treated in future work, including the gate voltage oscillation.
Due to the axial symmetry of the structure about the $z$-axis, electrons potential energy and hence their wavefunctions are independent of $\phi$. Thus, it is sufficient to calculate the potential energy at a single plane defined by a certain $\phi$. This allows calculating the bias potential $V_{bias}$ at a single $\rho z$-plane by solving Poisson’s equation in two dimensions. In contrast, when calculating the image potential $V_{met}$, Poisson’s equation is solved in three dimensions because the charge accumulation on the metallic surfaces is not axially symmetric for off-axis electron position. The axial symmetry, however, can be exploited in calculating $V_{met}$ by considering the electron positions in a definite $\rho z$-plane (e.g. the $xz$-plane). Using COMSOL simulation tool, we calculate the two terms of the potential energy in region II; $U_{bias}^{II}$ and $U_{met}^{II}$, as illustrated in $\Section$\[sub:Potential-Energy-Distribution\]. The total potential energy distribution in region II, $U_{bias}^{II}+U_{met}^{II}$, is shown in $\Figure$\[fig:Potential-Energy-Distribution\]. The corresponding potential barriers for electrons tunneling through the device at the lateral boundary ($\rho=2\text{ }\mbox{nm}$) and at the center ($\rho=0$) along the $z$-direction are drawn in solid lines in $\Figures$\[fig:Potential-Barrier-1-a\] and \[fig:Potential-Barrier-1-b\], respectively. A potential well appears at $\rho=2\text{ }\mbox{nm}$. This fast decrease in the potential energy distribution near the lateral boundary is mainly due to the image potential of the electron on the gate surface. The potential well may lead to resonant tunneling, specifically if one or more of the resonant energies are around the Fermi level of the emitter. If the minimum resonant energy is, however, significantly larger than the Fermi level of the emitter, no considerable resonant tunneling occurs because the electrons occupation for resonant energies is almost zero. The resonant tunneling effect, as mentioned before, is undesirable because it leads to high level of conduction in both directions which reduces the rectification effect of the tip.
![Potential energy distribution in $xz$-plane at equal collector and gate voltages of $1\text{ }\mbox{V}$. The gate thickness $t_{g}$, height $h_{g}$ and aperture diameter $d_{g}$ are $1$, $1$ and $4.1\text{ }\mbox{nm}$, respectively.\[fig:Potential-Energy-Distribution\]](\string"Fig2_PotDist_uniform\string".pdf)
It is interesting to see how changing the gate voltage would modulate the shape of the barrier and the well. $\FigureCap$\[fig:Potential-Barrier-1\] shows the potential energy along $z$-axis for four different values of $V_{g}$; $-1$, $1$, $3$ and $5\text{ }\mbox{V}$, at both $\rho=2\text{ }\mbox{nm}$ (a) and $\rho=0$ (b). These voltages are chosen to show the barrier modulation around the Fermi level near the edge. As the gate voltage increases the depth of the well at the edge increases, while the potential barrier at the center slightly decreases. The increasing depth of the well results in decreasing the resonant energy levels down to the vicinity of the Fermi level. Although the resonant tunneling effect is not favorable from the rectification point of view, the dependence of the well depth and the resonant levels on the gate voltage may be useful from another perspective, where the device can operate as a resonant tunneling diode with controllable potential well. The more interesting part is that with only changing the gate voltage the device can be switched between the rectification mode and the resonant tunneling mode. As shown in $\Figure$\[fig:Potential-Barrier-1-a\] at $V_{g}=-1\text{ }\mbox{V}$, the Fermi level is totally buried under the potential energy along the $z$-axis, which means that there is no chance for resonant tunneling to occur. However, a more comprehensive study for the device in the case of resonant tunneling is still needed, which is beyond the scope of this paper.
The peaks of the potential energy curves at $\rho=0$ are magnified in the inset in $\Figure$\[fig:Potential-Barrier-1-b\] in order to show the small changes in the barrier height and width at different gate voltages. As the gate voltage increases, the barrier gets slightly lower and narrower. These small changes indicate that the effect of the gate significantly decays as we go from the lateral boundary inwards until it is minimally pronounced at the center.
In order to investigate the effect of the gate aperture diameter $d_{g}$, we repeated the above calculations for $d_{g}=4.5\text{ }\mbox{nm}$. The potential energies at the lateral boundary are depicted in $\Figure$\[fig:Potential-Barrier-2\]. The image potential on the gate surface almost vanished when the gate diameter increased from $4.1$ to $4.5\text{ }\mbox{nm}$. This result caused the potential well to totally disappear at gate voltages lower than $5\text{ }\mbox{V}$, which means that the gate voltage can reach higher values without the device encountering resonant tunneling. Regarding the potential barrier at $\rho=0$, there is no significant difference due to increasing $d_{g}$ except for a slight decrease in the change of the potential barrier with changing the gate voltage.
![Potential energy along $z$-axis at $\rho=2\text{ }\mbox{nm}$ when $V_{c}=1\text{ }\mbox{V}$ and $V_{g}=-1$, $1$, $3$ and $5\text{ }\mbox{V}$. The gate thickness $t_{g}$, height $h_{g}$ and aperture diameter $d_{g}$ are $1$, $1$ and $4.5\text{ }\mbox{nm}$, respectively. The horizontal black solid line indicates the Fermi level inside the emitter.\[fig:Potential-Barrier-2\]](\string"Fig4_U_rg225_r2\string".pdf)
The current-voltage characteristics of a device of the previous parameters at the cases of $V_{g}=-1$ and $1\text{ }\mbox{V}$ are shown in $\Figure$\[fig:Current-voltage-characteristics\]. The results show the current modulation effect associated with the barrier modulation, while keeping the rectification nature of the device. This demonstrates the possibility of implementing electronic processes directly on the rectified current through applying a gate voltage of a few volts in magnitude. Both the rectification and the modulation abilities of the device are discussed in more detail in the following two sections.
![Current-voltage characteristics of the triode at $V_{g}=\pm1\text{ }\mbox{V}$. The gate thickness $t_{g}$, height $h_{g}$ and aperture diameter $d_{g}$ are $1$, $1$ and $4.5\text{ }\mbox{nm}$, respectively.\[fig:Current-voltage-characteristics\]](\string"Fig5_IV_chcs\string".pdf)
Finally, we note that the width and the position of the well along the $z$-axis at $\rho=2\text{ }\mbox{nm}$ can be controlled by the thickness and the height of the gate, respectively. Similar to the dependence of the depth of the well on the gate voltage, the dependence of the width of the well on the gate thickness fades away as the gate aperture diameter increases. A more detailed investigation for the effect of the gate thickness and height is presented in $\Section$\[sub:Current-Modulation\].
Gate Effect on Current Rectification and Modulation\[sub:Current-Rectification\]
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In this section, we investigate how the existence of the gate modifies the rectification properties of the device. We are interested here in calculating both the forward and backward currents. For a positive external bias $V_{c}$, the forward current is given by $I_{fwd}=I^{+}-I^{-}$, while for a negative external bias, the backward current is given by $I_{bwd}=I^{-}-I^{+}$, where $I^{+}$ and $I^{-}$ are the upward and downward currents whose expressions were given in $\Equations$(\[eq:UpwardCurrent\]) and (\[eq:DownwardCurrent\]), respectively. The ability of the device to deliver a higher current in the forward bias than the backward when subject to the same absolute value $V_{c}$ is measured by the rectification ratio $Rect=I_{fwd}/I_{bwd}$. If an oscillating field of a frequency significantly lower than the cutoff frequency is incident on the device, an oscillating potential difference of magnitude $V_{c}$ is induced between the collector and the emitter. The device is then supposed to deliver an asymmetric oscillating current between $I_{fwd}$ and $-I_{bwd}$. This indicates that when operating as a power source, the device is capable of delivering an output DC power given by $\left\langle P\right\rangle =\frac{1}{2}V_{c}(I_{fwd}-I_{bwd})=\frac{1}{2}V_{c}I_{fwd}(1-1/Rect)$ [@Mayer2008]. From this expression, we see that the output power can be optimized by increasing the forward current and the rectification ratio.
We now consider three cases for the gate connection; a) the gate is connected through a DC source $V_{DC}$ to the emitter, b) the gate is connected through a DC source $V_{DC}$ to the collector, and c) the gate has a floating potential. In the first two cases we consider three values for $V_{DC}$: $2\text{ }\mbox{V}$, $0$ and $-2\text{ }\mbox{V}$. This corresponds to $V_{g}=2\text{ }\mbox{V}$, $0$ and $-2\text{ }\mbox{V}$ in the first case, and $V_{g}=V_{c}+2\text{ }\mbox{V}$, $V_{c}$ and $V_{c}-2\text{ }\mbox{V}$ in the second case. Since we are targeting low voltages, the two values $2$ and $-2\text{ }\mbox{V}$ are taken as the extremes of the spanning range of $V_{DC}$. In the following results we consider the same geometrical parameters mentioned in $\Section$\[sub:Barrier-Modulation\] with $d_{g}=4.5\text{ }\mbox{nm}$. Since the gate electrode is centered at the midpoint between the collector and the emitter, therefore in the third case the floating gate voltage is driven to the mid-value between their voltages ($V_{g}\approx V_{c}/2$).
$\FiguresCap$\[fig:Emitter-Connection\] and \[fig:Collector-Connection\] show the magnitudes of the currents versus the collector voltage for the three values of $V_{DC}$ in cases (a) and (b) of the gate connection, respectively. The magnitude of the current in case (c), where the gate potential is floating, is included in both figures as a reference. The collector voltage is varied from $-4$ to $4\text{ }\mbox{V}$, which corresponds to applying electric fields of magnitudes between $0$ and $2\text{ }\mbox{Vnm\ensuremath{{}^{-1}}}$ on the device along the $z$-direction.
In the case of the floating gate potential, no difference is observed in both forward and backward currents from the original case, in which no gate existed. In order to understand the results of the other gate connections, we refer to the results of the floating gate potential as the reference results. We also refer to the self-biased gate voltage in this case, which is $V_{c}/2$, as the gate reference voltage $V_{ref}$. At $V_{g}$ greater than the reference voltage, the barrier drops below its reference height (its height when the gate voltage is floating). So, the current increases such that if $V_{c}$ is positive the forward current increases, while if it is negative the backward current increases. At $V_{g}<V_{ref}$, the barrier rises above its reference height and the current decreases. If the vertical position of the gate is changed between the emitter and the collector, the reference voltage will change. For example, if the gate is centered around $3/4$ of the distance $D$ between the emitter and the collector, then the gate reference voltage will be $\frac{3}{4}V_{c}$, above which the current increases and below which the current decreases, keeping its direction controlled by $V_{c}$ polarity.
Based on this argument the results in $\Figure$\[fig:Collector-and-Emitter\] can be explained. In the case when the gate is connected directly to the emitter ($V_{g}=0$), the gate voltage is lower than the reference voltage for a positive $V_{c}$, while it is higher than it for a negative $V_{c}$. Thus, the forward current is lower than the reference forward current, and the backward current is higher than the reference backward current. At $V_{g}=-2\text{ }\mbox{V}$, the reference voltage is greater than the gate voltage as long as $V_{c}$ is greater than $-4\text{ }\mbox{V}$ (because $V_{ref}=V_{c}/2$), and therefore the current magnitude is lower than the reference current. Exactly at $V_{c}=-4\text{ }\mbox{V}$, the two curves intersect. For lower values of $V_{c}$ the backward current of $V_{g}=-2\text{ }V$ exceeds the reference backward current. Similarly, at $V_{g}=2\text{ }\mbox{V}$ the current curve intersects with the reference current at $V_{c}=4\text{ }\mbox{V}$. However, in this case the backward current increases dramatically at collector voltages below $-2.5\text{ }\mbox{V}$. This is because the Fermi level in the emitting electrode (the collector in this case) raised above one of the vacuum barriers at $\rho=2\text{ }\mbox{nm}$ (see the inset in $\Figure$\[fig:Emitter-Connection\]). Electrons at the Fermi level, thus, encounter a single narrow barrier, which increases their tunneling probability. As the collector voltage decreases, the Fermi level inside the collector increases leading to a narrower barrier and hence a higher backward current. Such an increase in the current occurs when the gate potential is large compared to the potential of the emitting electrode. This effect can be used for switching the current on and off by varying the gate potential only, ignoring the rectification effect.
In a similar manner, the current curves when the gate is connected to the collector through a DC source in $\Figure$\[fig:Collector-Connection\] can be understood. In this case, however, all the effects are reversed. For example, at $V_{g}=V_{c}$ the forward current increases over its reference value, while the backward current is suppressed. The potential energy profile in this case is the same as if a single electrode of applied voltage $V_{c}$ is extended from the collector to the gate along the outer side of region IV. This structure represents a diode of inverted U-shaped (or concave) collector. The obtained current results then imply that such a diode shall have enhanced rectification properties over the original diode of a flat collector.
At $V_{g}=V_{c}+2\text{ }\mbox{V}$, the forward current increases dramatically at collector voltages above $2.5\text{ }\mbox{V}$ in an analogous way to the increase of the backward current at $V_{g}=2\text{ }\mbox{V}$. This time, however, the increase in the current is in favor of the rectification behavior of the device. We note that in both cases this effect appears when the difference between the gate voltage and the emitting electrode voltage is around $4.5\text{ }\mbox{V}$, which is the value of the work function of the metal.
The rectification ratio and the magnitude of the output power for the different gate connections are presented in $\Figures$ \[fig:Rectification-Ratio\] and \[fig:Output-Power\], respectively. The results show that both the rectification ratio and the output power are enhanced, compared to the floating gate situation, when the gate is connected directly to the collector, while they deteriorate when the gate is connected to the emitter through a DC source.
![Rectification ratio at the different gate connections. The ratio is calculated for the forward and backward currents in $\protect\Figure\ref{fig:Collector-and-Emitter}$.\[fig:Rectification-Ratio\]](\string"Fig7_rect\string".pdf)
![Output power at the different gate connections. The power is calculated for the corresponding forward and backward currents in $\protect\Figure\ref{fig:Collector-and-Emitter}$.\[fig:Output-Power\]](\string"Fig8_power\string".pdf)
In the conclusion of this section we can summarize how the gate electrode can be exploited to enhance the output power of the device. In order to increase the forward current, the gate voltage should be greater than the reference voltage, however this may also increase the backward current even higher and result in decreasing the rectification ratio. On the other hand, to increase the rectification ratio the gate should have a DC value referred to the collector not the emitter. This means that it should be connected to the collector with DC shift, that is $V_{g}=V_{c}+V_{DC}$ with respect to the emitter. Combining the previous two results, the output power can be enhanced by connecting the gate to the collector through DC source whose value is larger than negative the difference between $V_{c}$ and the reference voltage ($V_{DC}>-(V_{c}-V_{ref})$). As $V_{DC}$ increases, forward current, rectification ratio and the mean output power will increase. However, it should not increase much above zero to avoid resonant tunneling.
There is one remaining note regarding the validity of the model assumptions in power calculations. Since the leakage current in the gate is neglected, there is no power consumed or delivered through the gate electrode. In practice, however, this approximation is not very accurate, particularly in the case of high gate voltages, where any small current leakage in the gate would result in considerable power exchange through it. If the device is to be used in energy conversion applications, gate current should be included in power calculations, otherwise the calculated power efficiency of the device would be misleading.
Dependence of the Collector Current Modulation on the Gate Parameters\[sub:Current-Modulation\]
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In this section, we investigate the ability of the nanotriode to modulate the collector current using the gate voltage, and the modulation dependence on the geometrical parameters of the gate. The collector voltage is fixed at $1\text{ }\mbox{V}$, and only the forward current is considered in the calculations. We take values of $V_{g}$ between $-2$ and $2\text{ }\mbox{V}$. All the geometrical parameters are similar to $\Section$\[sub:Barrier-Modulation\] except that the gate aperture diameter is $4.2\text{ }\mbox{nm}$. This diameter is big enough to avoid resonant tunneling at the maximum gate voltage value used in this section ($2\text{ }\mbox{V}$).
We start with the effect of the gate thickness $t_{g}$ on the triode’s ability to alter the current through the gate voltage $V_{g}$. $\FigureCap$\[fig:Gate-Thickness-Effect\] shows the forward current as a function of the gate voltage at four different gate thicknesses. Since we are interested in studying the nanotriode in its smallest practical dimensions, we consider the smallest possible thickness for a monolayer conducting sheet. This is estimated to be in the range of $0.35\text{ }\mbox{nm}$ for a graphene layer [@Lu1997; @Jussila2016]. On the upper limit, we are restricted by the separation between the collector and the emitter. A gate with thickness approaching this separation would result in high-probability tunneling between the emitter and the collector through the gate, turning the device into a conductor. Structures with non-flat gate, such as tapered gate, around the tip will be studied in future work. $\FigureCap$\[fig:Gate-Thickness-Effect\] shows that as the gate thickness increases, the variation of the current with the gate voltage increases. At a gate thickness of $1.4\text{ }\mbox{nm}$, an increase in the gate voltage from $-2$ to $2\text{ }\mbox{V}$ causes an increase in the current by $130\%$.
![Forward current versus gate voltage at collector voltage $V_{c}$ of $1\text{ }\mbox{V}$ for four different values of gate thickness; $t_{g}=1.4$, $1$, $0.7$ and $0.35\text{ }\mbox{nm}$. The gate height $h_{g}$ and aperture diameter $d_{g}$ are $1$ and $4.2\text{ }\mbox{nm}$, respectively. \[fig:Gate-Thickness-Effect\]](\string"Fig9_tg\string".pdf)
Turning to the effect of the vertical position of the gate, we present the current results at gate thickness of $0.7\text{ }\mbox{nm}$ for three different gate positions in $\Figure$\[fig:Gate-Height-Effect\]. The three positions are chosen such that a) the gate is just above the tip end ($h_{g}=1.35\text{ }\mbox{nm}$), b) the gate is centered around the tip end ($h_{g}=1\text{ }\mbox{nm}$), and c) the gate is surrounding the highest part of the tip ($h_{g}=0.65\text{ }\mbox{nm}$). It is obvious from the results that the largest variation in the current is obtained when the gate is centered around the end of the tip. That is because the highest part in the potential barrier at any radial position within the tip range is just above its curved surface, so the gate is most effective when centered around this region. The results in $\Figure$\[fig:Gate-Height-Effect\] also show that the current variation is minimal in the third case when $h_{g}=0.65\text{ }\mbox{nm}$. That is because the gate is the furthest from the barrier between the end of the tip and the collector surface at the center of the device ($\rho=0$) where the tunneling current density is the maximum.
![Forward current versus gate voltage at collector voltage $V_{c}$ of $1\text{ }\mbox{V}$ and gate thickness $t_{g}$ of $0.7\text{ }\mbox{nm}$ for three different heights of the gate; $h_{g}=1.35$, $1$ and $0.65\text{ }\mbox{nm}$. The gate aperture diameter $d_{g}$ is $4.2\text{ }\mbox{nm}$. The chosen three heights correspond, descendingly, to the positions of the gate where it is just above, centered around, and just below the end of the tip.\[fig:Gate-Height-Effect\]](\string"Fig10_hg\string".pdf)
The effect of the gate aperture diameter on the shape of the potential barrier is discussed in $\Section$\[sub:Barrier-Modulation\]. Now we investigate how this dependency will affect the variation of the current with varying the gate voltage. $\FigureCap$\[fig:Aperture-Diameter-Effect\] shows the $I-V_{g}$ relation for gates of aperture diameters $4.2$ and $5\text{ }\mbox{nm}$. The gates considered here have a thickness and a height of $1\text{ }\mbox{nm}$. The decrease of the current variation with increasing the aperture diameter is clear from the graph. This result is quite predictable from the potential energy results in $\Figures$\[fig:Potential-Barrier-1\] and \[fig:Potential-Barrier-2\].
![Forward current versus gate voltage at collector voltage $V_{g}$ of $1\text{ }V$ and gate thickness $t_{g}$ of $1\text{ }\mbox{nm}$ for two different gate aperture diameters; $4.2$ and $5\text{ }\mbox{nm}$. The height of the gate is $1\text{ }\mbox{nm}$.\[fig:Aperture-Diameter-Effect\]](\string"Fig11_dg\string".pdf)
Dependence of Current Modulation on the Tip Parameters
------------------------------------------------------
In $\References$, Mayer et al. demonstrated that better rectification properties are for emitting tips with higher aspect ratio. With adding the gate electrode to the structure, we aim in this section at investigating the effect of the aspect ratio on current modulation. In this section we use gate thickness, height and aperture diameter of $1$, $1$ and $4.2\text{ }\mbox{nm}$, respectively. As shown in $\Figure$\[fig:Tip-Effect\], by increasing the tip diameter $d_{t}$ from $1$ to $1.5\text{ }\mbox{nm}$ at the same tip height ($h_{t}=1\text{ }\mbox{nm}$), an increase in both the average current and the slope of the $I-V_{g}$ curve is observed. The increase in the average current value is due to the increase of the emitting area represented by the tip surface. Such an increase in the forward current is necessarily accompanied by a higher increase in the backward current, due to the decreasing field enhancement, resulting at the end in reducing the rectification ratio. The increase in the slope is mainly because the emitting tip extended to a closer region to the gate, where the modulating effect is more significant on the potential energy and the emitted current.
![Forward current versus gate voltage at collector voltage $V_{c}$ of $1\text{ }\mbox{V}$ for different tip parameters. At the two cases where $h_{t}=1\text{ }\mbox{nm}$, the separation distance between the leads $D$ is $2\text{ }\mbox{nm}$. At $h_{t}=3\text{ }\mbox{nm}$, $D=3\text{ }\mbox{nm}$. In the three cases, the gate thickness $t_{g}$ and aperture diameter $d_{g}$ are $1$ and $4.2\text{ }\mbox{nm}$, respectively, and $h_{g}=h_{t}$. \[fig:Tip-Effect\]](\string"Fig12_tip\string".pdf)
Finally, we investigate the gate modulation effect on tips of different heights. A tip of height $3\text{ }\mbox{nm}$ and diameter $1\text{ }\mbox{nm}$ is examined. The separation between the leads is $4\text{ }\mbox{nm}$, so that the width of the potential barrier between the tip and the collector is $1\text{ }\mbox{nm}$ as in the previous cases. Also the gate is set at height of $3\text{ }\mbox{nm}$, in order to be centered around the end of the tip as well. A better current modulation is observed at this higher tip (dotted curve in $\Figure$\[fig:Tip-Effect\]). This result exhibits a favorable behavior for the device due to its compatibility with the results obtained in $\References$, where optimized rectification properties were also demonstrated for higher tips.
Conclusion
==========
In this paper, a transfer matrix method is used to model quantum tunneling through vacuum nanotriodes at low applied voltages. The structure consists of a metal-vacuum-metal junction with a nanotip supported on one of the metals and surrounded by a thin gate electrode. The device resembles Spindt-type vacuum triodes with a collector-emitter separation and a gate aperture diameter of a few nanometers. The details of calculating the potential energy distribution and the field emission current are presented with considering the gate effect. The behavior of the device is then investigated as a current rectifier and modulator at different electric and geometric parameters.
The results show a significant enhancement in the rectification properties of the geometrically asymmetric metal-vacuum-metal diode when a gate electrode is connected to the collector through a DC source. This finding suggests a better rectification for vacuum nanodiodes with concave collectors instead of flat ones. It is also demonstrated that the output current of the device can be modulated by the gate voltage. The effect of the geometrical parameters of the gate and the tip on current modulation are investigated. It is shown that a gate centered around the end of the tip with a narrow aperture and a large thickness gives the best current modulation. It is also shown that the currents emitted from tips of higher aspect ratios are better controlled through the gate potential. In order to achieve electric currents of higher magnitudes at the same applied potentials, we have to either decrease the gap distance between the emitting tip and the collecting surface, or increase the emitting surface area. The second solution can be realized by adding more than one nanotip or a protruding ring. Such structures can be designed and characterized following the same approach we present in this work. However, in order for this model to be fully reliable in the design and the characterization processes, experimental verification on similar dimensions is needed.
Based on this analysis, the vacuum nanotriode we study in this paper represents an excellent candidate for high-frequency applications. It can be optimized to do the basic functions of transistors, in addition to its rectification effect, at high frequencies up to the infrared range. Thus, it is a possible alternative to semiconductor transistors as a basic unit in electronic circuits. It can also be used for implementing local processing operations on rectennas, avoiding the transmission problems of high-frequency signals. This shall open the door to a new era of fast electronics, where communication systems and processing units operating at optical frequencies are achievable. It can also be used as an electronic interface for plasmonic circuits.
M. Khalifa would like to thank A. Mayer for his valuable comments and suggestions.
|
---
author:
- 'I.F. Mirabel'
title: 'Microquasars: summary and outlook'
---
Introduction {#sec:1}
============
Microquasars are binary stellar systems where the remnant of a star that has collapsed to form a dark and compact object (such as a neutron star or a black hole) is gravitationally linked to a star that still produces light, and around which it makes a closed orbital movement. In this cosmic dance of a dead star with a living one, the first sucks matter from the second, producing radiation and very high energetic particles (Fig. \[fig:1\]). These binary star systems in our galaxy are known under the name of “microquasars” because they are miniature versions of the quasars (‘quasi-stellar-radio-source’), that are the nuclei of distant galaxies harboring a super massive black hole, and are able to produce in a region as compact as the solar system, the luminosity of 100 galaxies like the Milky Way. Nowadays the study of microquasars is one of the main scientific motivations of the space observatories that probe the X-ray and $\gamma$-ray Universe.
Despite of the differences in the involved masses and in the time and length scales, the physical processes in microquasars are similar to those found in quasars. That is why the study of microquasars in our galaxy has enabled a better understanding of what happens in the distant quasars and AGN. Moreover, the study of microquasars may provide clues for the understanding of the class of gamma-ray bursts that are associated to the collapse of massive stars leading to the formation of stellar black holes, which are the most energetic phenomena in the Universe after the Big-Bang.
![ In our galaxy there exist binary stellar systems where an ordinary star gravitates around a black hole that sucks the outer layers of the star’s atmosphere. When falling out to the dense star, the matter warms and emits huge amounts of energy as X- and $\gamma$-rays. The accretion disk that emits this radiation also produces relativistic plasma jets all along the axis of rotation of the black hole. The physical mechanisms of accretion and ejection of matter are similar to those found in quasars, but in million times smaller scales. Those miniature versions of quasars are known under the name of ‘microquasars’. []{data-label="fig:1"}](chap01_figure1.eps){height="9cm"}
Discovery of microquasars {#sec:2}
=========================
During the second half of the 18th century, John Michell and Pierre-Simon Laplace first imagined compact and dark objects in the context of the classical concept of gravitation. In the 20th century in the context of Einstein’s General Relativity theory of gravitation, those compact and dark objects were named black holes. They were then identified in the sky in the 1960s as X-ray binaries. Indeed, those compact objects, when associated to other stars, are activated by the accretion of very hot gas that emits X and $\gamma$-rays. In 2002, Riccardo Giacconi was awarded the Nobel Prize for the development of the X-ray Space Astronomy that led to the discovery of the first X-ray binaries [@Giacconi]. Later, Margon et al. [@Margon] found that a compact binary known as SS 433 was able to produce jets of matter. However, for a long time, people believed that SS 433 was a very rare object of the Milky Way and its relation with quasars was not clear since the jets of this object move only at 26% of the speed of light, whereas the jets of quasars can move at speeds close to the speed of light.
In the 1990s, after the launch of the Franco–Soviet satellite GRANAT, growing evidences of the relation between relativistic jets and X-ray binaries began to appear. The on-board telescope SIGMA was able to take X-ray and $\gamma$-ray images. It detected numerous black holes in the Milky Way. Moreover, thanks to the coded-mask-optics, it became possible for the first time to determine the position of $\gamma$-ray sources with arcmin precision. This is not a very high precision for astronomers who are used to dealing with other observing techniques. However, in high-energy astrophysics it represented a gain of at least one order of magnitude. It consequently made possible the systematic identification of compact $\gamma$-ray sources at radio, infrared and visible wavelengths.
With SIGMA/GRANAT it was possible to localize with an unprecedented precision the hard X-ray and $\gamma$-ray sources. In order to determine the nature of those X-ray binaries, a precision of a few tens of arc-seconds was needed. Sources that produce high energy photons should also produce high energy particles, that should then produce synchrotron radiation when accelerated in magnetic fields. Then, with Luis Felipe Rodríguez, we performed a systematic search of synchrotron emissions from X-ray binaries with the Very Large Array (VLA) of the National Radio Astronomy Observatory of the USA.
In 1992, using quasi-simultaneous observations from space with GRANAT and from the ground with the VLA, we determined the position of the radio counterpart of an X-ray source named 1E 1740.7-2942 with a precision of sub-arc-seconds. With GRANAT this object was identified as the most luminous, persistent source of soft $\gamma$-rays in the Galactic center region. Moreover, its luminosity, variability and the X-ray spectrum were consistent with those of an accretion disk gravitating around a stellar mass black hole, like in Cygnus X-1. The most surprising finding with the VLA was the existence of well collimated two-sided jets that seem to arise from the compact radio counterpart of the X-ray source [@Mirabel1740]. These jets of magnetized plasma had the same morphology as the jets observed in quasars and radio galaxies. When we published those results, we employed the term microquasar to define this new X-ray source with relativistic jets in our Galaxy. This term appeared on the front page of the British journal Nature (see Fig. \[fig:2\]), which provoked multiple debates. Today the concept of microquasar is universally accepted and used widely in scientific publications.
Before the discovery of its radio counterpart, 1E 1740.7-2942 was suspected to be a prominent source of 511 keV electron-positron annihilation radiation observed from the centre of our Galaxy [@Leventhal], and for that reason it was nicknamed as the “Great Annihilator”. It is interesting that recently it was reported [@Weiden] that the distribution in the Galactic disk of the 511 keV emission, due to positron-electron annihilation, exhibit similar asymmetric distribution as that of the hard low mass X-ray binaries, where the compact objects are believed to be stellar black holes. This finding suggests that black hole binaries may be important sources of positrons that would annihilate with electrons in the interstellar medium. Therefore, positron-electron pairs may be produced by $\gamma$–$\gamma$ photon interactions in the inner accretion disks, and microquasar jets would contain positrons as well as electrons. If this recent report is confirmed, 1E 1740.7-2942 would be the most prominent compact source of anti-matter in the Galactic Centre region.
![ The British journal Nature announced the 16th of July, 1992 the discovery of a microquasar in the galactic centre region [@Mirabel1740]. The image shows the synchrotron emission at a radio wavelength of 6cm produced by relativistic particles jets ejected from some tens of kilometers to light-years of distance from the black hole binary which is located inside the small white ellipse. []{data-label="fig:2"}](chap01_figure2.eps){height="9cm"}
Discovery of superluminal motions {#sec:3}
=================================
If the proposed analogy [@MirabelRodriguez1998] between microquasars and quasars was correct, it should be possible to observe superluminal apparent motions in Galactic sources. However, superluminal apparent motions had been observed only in the neighborhood of super-massive black holes in quasars. In 1E 1740.7-2942 we could not be able to discern motions, as in that persistent source of $\gamma$-rays the flow of particles is semi-continuous. The only possibility of knowing if superluminal apparent movements exist in microquasars was through the observation of a discreet and very intense ejection in an X-ray binary. This would allow us to follow the displacement in the firmament of discrete plasma clouds. Indeed, with the GRANAT satellite was discovered [@Castro1915] a new source of X-rays with such characteristics denominated GRS 1915+105. Then with Rodríguez we began with the VLA a systematic campaign of observations of that new object in the radio domain, and in collaboration with Pierre-Alain Duc (CNRS-France) and Sylvain Chaty (Paris University) we performed the follow-up of this source in the infrared with telescopes of the Southern European Observatory, and telescopes at Mauna Kea, Hawaii.
Since the beginning, GRS 1915+105 exhibited unusual properties. The observations in the optical and the infrared showed that this X-ray binary was very absorbed by the interstellar dust along the line of sight in the Milky Way, and that the infrared counterpart was varying rapidly as a function of time. Moreover, the radio counterpart seemed to change its position in the sky, so that at the beginning we did not know if those changes were due to radiation reflection or refraction in an inhomogeneous circumstellar medium (“Christmas tree effect”), or rather due to the movement at very high speeds of jets of matter. For two years we kept on watching this X-ray binary without exactly understanding its behavior. However, in March 1994, GRS 1915+105 produced a violent eruption of X and $\gamma$-rays, followed by a bipolar ejection of unusually bright plasma clouds, whose displacement in the sky could be followed during 2 months. From the amount of atomic hydrogen absorbed in the strong continuum radiation we could infer that the X-ray binary stands at about 30000 light years from the Earth. This enabled us to know that the movement in the sky of the ejected clouds implies apparent speeds higher than the speed of light.
The discovery of these superluminal apparent movements in the Milky Way was announced in Nature [@MirabelRodriguez1994] (Fig. \[fig:3\]). This constituted a full confirmation of the hypothesis, that we had proposed two years before, on the analogy between microquasars and quasars. With Rodríguez we formulated and solved the system of equations that describe the observed phenomenon. The apparent asymmetries in the brightness and the displacement of the two plasma clouds could naturally be explained in terms of the relativistic aberration in the radiation of twin plasma clouds ejected in an antisymmetric way at 98% of the speed of light [@MirabelRodriguez1999]. The super-luminal motions observed in 1994 with the VLA [@MirabelRodriguez1994] were a few years later re-observed with higher angular resolution using the MERLIN array [@Fender1999].
Using the Very Large Telescope of the European Southern Observatory, it was possible to determine the orbital parameters of GRS 1915+105, concluding that it is a binary system constituted by a black hole of $\sim$14 solar masses accompanied by a star of 1 solar mass [@Greiner]. The latter has become a red giant from which the black hole sucks matter under the form of an accretion disk (see Fig. \[fig:1\]).
![ The journal Nature announces the 1st of September, 1994 the discovery of the first Galactic source of superluminal apparent motions [@MirabelRodriguez1994]. The sequence of images shows the temporal evolution in radio waves at a wavelength of 3.6 cm of a pair of plasma clouds ejected from black hole surroundings at a velocity of 98% the speed of the light. []{data-label="fig:3"}](chap01_figure3.eps){height="9cm"}
Disk–jet coupling in microquasars {#sec:4}
=================================
The association of bipolar jets and accretion disks seems to be a universal phenomenon in quasars and microquasars. The predominant idea is that matter jets are driven by the enormous rotation energy of the compact object and accretion disk that surrounds it. Through magneto-hydrodynamic mechanisms, the rotation energy is evacuated through the poles by means of jets, as the rest can fall towards the gravitational attraction centre. In spite of the apparent universality of this relationship between accretion disks and bipolar, highly collimated jets, the temporal sequence of the phenomena had never been observed in real time.
Since the scales of time of the phenomena around black holes are proportional to their mass, the accretion-ejection coupling in stellar-mass black holes can be observed in intervals of time that are millions of time smaller than in AGN and quasars. Because of the proximity, the frequency and the rapid variability of energetic eruptions, GRS 1915+105 became the most adequate object to study the connection between instabilities in the accretion disks and the genesis of bipolar jets.
After several attempts, finally in 1997 we could observe [@Mirabel1998] on an interval of time shorter than an hour, a sudden fall in the luminosity in X and soft $\gamma$-rays, followed by the ejection of jets, first observed in the infrared, then at radio frequencies (see Fig. \[fig:4\]). The abrupt fall in X-ray luminosity could be interpreted as the silent disappearance of the warmer inner part of the accretion disk beyond the horizon of the black hole. A few minutes later, fresh matter coming from the companion star come to feed again the accretion disk, which must evacuate part of its kinetic energy under the form of bipolar jets. When moving away, the plasma clouds expand adiabatically, becoming more transparent to its own radiation, first in the infrared and then in radio frequency. The observed interval of time between the infrared and radio peaks is consistent with that predicted by van der Laan [@vdlaan] for extragalactic radio sources.
Based on the observations of GRS 1915+105 and other X-ray binaries, it was proposed [@fbg] proposed a unified semiquantitative model for disk-jet coupling in black hole X-ray binary systems that relate different X-ray sates with radio states, including the compact, steady jets associated to low-hard X-ray states, that had been imaged [@Dhawan] using the Very Long Baseline Array of the National Radio Astronomy Observatory.
After three years of multi-wavelength monitoring an analogous sequence of X-ray emission dips followed by the ejection of bright super-luminal knots in radio jets was reported [@Marscher] in the active galactic nucleus of the galaxy 3C 120. The mean time between X-ray dips was of the order of years, as expected from scaling with the mass of the black hole.
![ Temporal sequence of accretion disk – jet coupling observed for first time in real time simultaneously in the X-rays, the infrared and radio wavelengths in the microquasar GRS 1915+105 [@Mirabel1998]. The ejection of relativistic jets takes place after the evacuation and/or dissipation of matter and energy, at the time of the reconstruction of the inner side of the accretion disk, corona or base of the jet. A similar process has been observed years later in quasars [@Marscher], but on time scales of years. As expected in the context of the analogy between quasars and microquasars [@MirabelRodriguez1998], the time scale of physical processes in the surroundings of black holes is proportional to their masses. []{data-label="fig:4"}](chap01_figure4.eps){height="9cm"}
Can we prove the existence of black holes? {#sec:5}
==========================================
Horizon is the basic concept that defines a black hole: a massive object that consequently produces a gravitational attraction in the surrounding environment, but that has no material border. In fact, an invisible border in the space-time, which is predicted by general relativity, surrounds it. This way, matter could go through this border without being rejected, and without losing a fraction of its kinetic energy in a thermonuclear explosion, as sometimes is observed as x-ray bursts of type I when the compact object is a neutron star instead of a black hole. In fact, as shown in Fig. \[fig:4\], the interval of time between the sudden drop of the flux and the spike in the X-ray light curve that marks the onset of the jet signaled by the starting rise of the infrared synchrotron emission is of a few minutes, orders of magnitude larger than the dynamical time of the plasma in the inner accretion disk. Although the drop of the X-ray luminosity could be interpreted as dissipation of matter and energy, the most popular interpretation is that the hot gas that was producing the X-ray emission falls into the black hole, leaving the observable Universe.
So, have we proved with such observations the existence of black holes? Indeed, we do not find any evidence of material borders around the compact object that creates gravitational attraction. However, the fact that we do not find any evidence for the existence of a material surface does not imply that it does not exist. In fact, such type I x-ray bursts are only observed in certain range of neutron star mass accretion rates. That means that it is not possible to prove the existence of black holes using the horizon definition. According to Saint Paul, *“faith is the substance of hope for: the evidence of the not seen”*. That is why for some physicists, black holes are just objects of faith. Perhaps the intellectual attraction of these objects comes from the desire of discovering the limits of the Universe. In this context, studying the physical phenomena near the horizon of a black hole is a way of approaching the ultimate frontiers of the observable Universe.
The rotation of black holes {#sec:6}
===========================
For an external observer, black holes are the simplest objects in physics since they can be fully described by only three parameters: mass, rotation and charge. Although black holes could be born with net electrical charge, it is believed that because of interaction with environmental matter, astrophysical black holes rapidly become electrically neutral. The masses of black holes gravitating in binary systems can be estimated with Newtonian physics. However, the rotation is much more difficult to estimate despite it being probably the main driver in the production of relativistic jets.
There is now the possibility of measuring the rotation of black holes by at least three different methods: a) X-ray continuum fitting [@Zhang; @McClintock], b) asymmetry of the broad component of the Fe K$_\alpha$ line from the inner accretion disk [@Tanaka], and c) quasi-periodic oscillations with a maximum fix frequency observed in the X-rays [@RemillardMcClintock]. The main source of errors in the estimates of the angular momentum resides in the uncertainties of the methods employed.
The side of the accretion disk that is closer to the black hole is hotter and produces huge amounts of thermal X and $\gamma$ radiations and is also affected by the strange configuration of space-time. Indeed, next to the black hole, space-time is curved by the black hole mass and dragged by its rotation. This produces vibrations that modulate the X-ray emission. Studies of those X-ray continuum and vibrations suggest that the microquasars that produce the most powerful jets are indeed those that are rotating fastest. It has been proposed that these pseudo-periodic oscillations in microquasars are, moreover, one of the best methods today to probe by means of observations general relativity theory in the limit of the strongest gravitational fields.
Analogous oscillations in the infrared range, may have been observed in the super massive black hole at the centre of the Milky Way. The quasi-periods of the oscillations (a few milliseconds for the microquasars X-ray emission and a few tens of minutes for the galactic centre black hole infrared emission) are proportionally related to the masses of the objects, as expected from the physical analogy between quasars and microquasars. Comparing the phenomenology observed in microquasars to that in black holes of all mass scales, several correlations among observables such as among the radiated fluxes in the low hard X-ray state, quasi-periodic oscillations, flickering frequencies, etc., are being found and used to derive the mass and angular momentum, which are the fundamental parameters that describe astrophysical black holes.
Extragalactic microquasars, microblazars and ultraluminous X-ray sources {#sec:7}
========================================================================
Have microquasars been observed beyond the Milky Way galaxy? X-ray satellites are detecting far away from the centers of external galaxies large numbers of compact sources called ‘ultraluminous X-ray sources’, because their luminosities seem to be greater than the Eddington limit for a stellar-mass black hole [@Fabbiano]. Although a few of these sources could be black holes of intermediate masses of hundreds to thousands solar masses, it is believed that the large majority are stellar-mass black hole binaries.
Since the discovery of quasars in 1963, it was known that some quasars could be extremely bright and produce high energetic emissions in a short time. These particular quasars are called blazars and it is thought that they are simply quasars whose jets point close to the Earth’s direction. The Doppler effect produces thus an amplification of the signal and a shift into higher frequencies. With Rodríguez we imagined in 1999 the existence of microblazars, that is to say X-ray binaries where the emission is also in the Earth’s direction [@MirabelRodriguez1999]. Microblazars may have been already observed but the fast variations caused by the contraction of the time scale in the relativistic jets, make their study very difficult. In fact, one question at the time of writing this chapter is whether microblazars could have been already detected as “fast black hole X-ray novae” [@Kasliwal]. In fact, the so called “fast black hole X-ray novae” Swift J195509.6+261406 (which is the possible source of GRB 070610 [@Kasliwal]), and V4541 Sgr [@Orosz] are compact binaries that appeared as high energy sources with fast and intense variations of flux, as expected in microblazars [@MirabelRodriguez1999].
Although some fast variable ultraluminous X-ray sources could be microblazars, the vast majority do not exhibit the intense, fast variations of flux expected in relativistic beaming. Therefore, it has been proposed [@King] that the large majority are stellar black hole binaries where the X-ray radiation is –as the particle outflows– anisotropic, but not necessarily relativistically boosted. In fact, the jets in the Galactic microquasar SS 433, which are directed close to the plane of the sky, have kinetic luminosities of more than a few times 10$^{39}$ erg/sec, which would be super-Eddington for a black hole of 10 solar masses.
An alternative model is that ultraluminous X-ray sources may be compact binaries with black holes of more than 30 solar masses that emit largely isotropically with no beaming into the line of sight, either geometrically or relativistically [@Pakull]. This conclusion is based on the formation, evolution and overall energetics of the ionized nebulae of several 100 pc diameter in which some ultraluminous X-ray sources are found embedded. The recent discoveries of high mass binaries with black holes of 15.7 solar masses in M 33 [@Orosz2007] and 23-34 solar masses in IC 10 [@Prestwich] support this idea. Apparently, black holes of several tens of solar masses could be formed in starburst galaxies of relative low metal content.
Very energetic $\gamma$-ray emission from compact binaries {#sec:8}
==========================================================
Very energetic $\gamma$-rays with energies greater than 100 gigaelectron volts have recently been detected with ground based telescopes from four high mass compact binaries [@Mirabel2006]. These have been interpreted by models proposed in the contexts represented in Fig. \[fig:5\]. In two of the four sources the $\gamma$ radiation seems to be correlated with the orbital phase of the binary, and therefore may be consistent with the idea that the very high energy radiation is produced by the interaction of pulsar winds with the mass outflow from the massive companion star [@Dubus; @DhawanMQW]. The detection of TeV emission from the black hole binary Cygnus X-1 [@Albert] and the TeV intraday variability in M 87 [@Aharonian] provided support to the jet models [@Romero], which do not require relativistic Doppler boosting as in blazars and microblazars . It remains an open question whether the $\gamma$-ray binaries LS 5039 and LS I +61 303 could be microquasars where the $\gamma$ radiation is produced by the interaction of the outflow from the massive donor star with jets [@Romero] or pulsar winds [@Dubus].
![ Alternative contexts for very energetic $\gamma$-ray binaries [@Mirabel2006]. Left: microquasars are powered by compact objects (neutron stars or stellar-mass black holes) via mass accretion from a companion star. The interaction of collimated jets with the massive outflow from the donor star can produce very energetic $\gamma$-rays by different alternative physical mechanisms [@Romero], depending on whether the jets are baryonic or purely leptonic. Right: pulsar winds are powered by rotation of neutron stars; the wind flows away to large distances in a comet-shape tail. Interaction of this wind with the companion-star outflow may produce very energetic $\gamma$-rays [@Dubus]. []{data-label="fig:5"}](chap01_figure5.eps){height="6cm"}
Microquasars and gamma-ray bursts {#sec:9}
=================================
It is believed that gamma-ray bursts of long duration (t$>$1 sec) mark the birth of black holes by core collapse of massive stars. In this context, microquasars that contain black holes would be fossils of gamma-ray burst sources of long duration, and their study in the Milky Way and nearby galaxies can be used to gain observational inside into the physics of the much more distant sources of gamma-ray bursts. Questions of topical interest are: a) do all black hole progenitors explode as very energetic hypernovae of type Ib/c? ; b) what are the birth places and nature of the progenitors of stellar black holes?
The kinematics of microquasars provide clues to answer these questions. When a binary system of massive stars is still gravitationally linked after the explosion of one of its components, the mass centre of the system acquires an impulse, whatever matter ejection is, symmetric or asymmetric. Then according to the microquasar movement we can investigate the origin and the formation mechanism of the compact object. Knowing the distance, proper motion, and radial velocity of the centre of mass of the binary, the space velocity and past trajectory can be determined. Using multi-wavelength data obtained with a diversity of observational techniques, the kinematics of eight microquasars have so far been determined.
One interesting case is the black hole wandering in the Galactic halo, which is moving at high speed, like globular clusters [@Mirabel2001] (Fig. \[fig:6\]). It remains an open question whether this particular halo black hole was kick out from the Galactic plane by a natal explosion, or is the fossil of a star that was formed more than 7 billions of years ago, before the spiral disk of stars, gas and dust of the Milky Way was formed. In this context, the study of these stellar fossils may represent the beginning of what could be called ‘Galactic Archaeology’. Like archaeologists, studying these stellar fossils, astrophysicists can infer what was the history of the Galactic halo.
![ A wandering black hole in the Galactic halo [@Mirabel2001]. The trajectory of the black hole for the last 230 million years is represented in red. The bright dot on the left represents the Sun. []{data-label="fig:6"}](chap01_figure6.eps){height="6cm"}
The microquasars LS 5039 [@Ribo] and GRO J1655-40 [@Mirabel2002] which contain compact objects with less that $\sim$7 solar masses were ejected from their birth place at high speeds, and therefore the formation of these compact objects with relative small masses must have been associated with energetic supernovae. On the contrary, the binaries Cygnus X-1 [@Mirabel2003] and GRS 1915+105 [@Dhawan2007] which contain black holes of at least 10 solar masses do not seem to have received a sudden impulse. Preliminary results on the kinematics of the X-ray binaries suggest that low mass black holes are formed by a delayed collapse of a neutron star with energetic supernovae, whereas stellar black holes with masses equal or greater than 10 solar masses are the result of the direct collapse of massive stars, namely, they are formed in the dark. This is consistent with the recent finding of gamma-ray bursts of long duration in the near universe without associated luminous supernovae [@DellaValle].
There are indications that the mass of the resulting black hole may be a function of the metal content of the progenitor star. In fact, the black holes with 16 solar masses in M 33 [@Orosz2007] and more than 23 solar masses in IC 10 [@Prestwich], are in small galaxies of low metal content. This is consistent with the fact that the majority of the gamma-ray bursts of long duration take place in small starburst galaxies at high redshift, namely, in galactic hosts of low metal content [@LeFloch]. Since the power and redshift of gamma-ray bursts seem to be correlated this would imply a correlation between the mass of the collapsing stellar core and the power of the $\gamma$-ray jets.
Gamma-ray bursts of long duration are believed to be produced by ultra relativistic jets generated in a massive star nucleus when it catastrophically collapses to form a black hole. Gamma-ray bursts are highly collimated jets and it has been proposed [@MirabelSky] that there may be a unique universal mechanism to produce relativistic jets in the Universe, suggesting that the analogy between microquasars and quasars can be extended to the gamma-ray bursts sources, as illustrated in the diagram of Fig. \[fig:7\].
![ The same physical mechanism can be responsible for three different types of objects: microquasars, quasars and massive stars that collapse (‘collapsars’) to form a black hole producing gamma-ray bursts. Each one of these objects contains a black hole, an accretion disk and relativistic particles jets. Quasars and microquasars can eject matter several times, whereas the collapsars form jets only once. When the jets are aligned with the line of sight of the observer these objects appear as microblazars, blazars and gamma-ray bursts, respectively. Reproduced from [@MirabelSky]. []{data-label="fig:7"}](chap01_figure7.eps){height="7.5cm"}
Conclusions {#sec:10}
===========
Black-hole astrophysics is presently in an analogous situation as was stellar astrophysics in the first decades of the 20th century. At that time, well before the physical understanding of the interior of stars and the way by which they produce and radiate energy, empirical correlations such as the HR diagram were found and used to derive fundamental properties of the stars, such as the mass. Similarly, at present before a comprehensive understanding of black hole physics, empirical correlationsbetween X-ray and radio luminosities and characteristic time scales are being used to derive the mass and spin of black holes of all mass scales, which are the fundamental parameters that describe astrophysical black holes. Therefore, there are historical and epistemological analogies between black hole astrophysics and stellar astrophysics. The research area on microquasars has become one of the most important areas in high energy astrophysics. In the last 14 years there have been seven international workshops on microquasars: 4 in Europe, 1 in America and 2 in Asia. They are currently attended by 100–200 young scientists who, with their work on microquasars, are contributing to open new horizons in the common ground of high energy physics and modern astronomy.
[**Apologies:**]{} This manuscript is based on short courses given at international schools for graduate students, intended to give an introduction to this area of research. It is biased by my own personal choice, and hence it is by no means a comprehensive review. Because references had to be minimized I apologize for incompleteness to colleagues working in the field. Part of this work was written while the author was staff member of the European Southern Observatory in Chile.
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abstract: 'Spatially resolved observations of the planetary nebula M2-42 (PNG008.2$-$04.8) obtained with the Wide Field Spectrograph on the Australian National University 2.3 m telescope have revealed the remarkable features of bipolar collimated jets emerging from its main structure. Velocity-resolved channel maps derived from the \[N[ii]{}\] $\lambda$6584 emission line disentangle different morphological components of the nebula. This information is used to develop a three-dimensional morpho-kinematic model, which consists of an equatorial dense torus and a pair of asymmetric bipolar outflows. The expansion velocity of about 20 kms$^{-1}$ is measured from the spectrum integrated over the main shell. However, the deprojected velocities of the jets are found to be in the range of $80$–160 kms$^{-1}$ with respect to the nebular center. It is found that the mean density of the collimated outflows, $595 \pm 125$cm$^{-3}$, is five times lower than that of the main shell, $3150$cm$^{-3}$, whereas their singly ionized nitrogen and sulfur abundances are about three times higher than those determined from the dense shell. The results indicate that the features of the collimated jets are typical of fast, low-ionization emission regions.'
author:
- 'A. Danehkar, Q. A. Parker, and W. Steffen'
title: 'Fast, low-ionization emission regions of the planetary nebula M2-42'
---
Introduction {#m2_42:sec:introduction}
============
M2-42 (=PNG008.2$-$04.8 = Hen2-393 = VV177 = Sa2-331) was discovered as a planetary nebula (PN) by @Minkowski1947. The H$\alpha$ image, Fig. \[fig:m2\_42:shs\] (top panel), obtained from the AAO/UKST SuperCOSMOS H$\alpha$ Sky Survey [SHS; @Parker2005] revealed an elliptical morphological structure with a clear extension to the north east, suggesting the presence of bipolar outflows. The long-slit data from the San Pedro Mártir kinematic catalog [SPM; @Lopez2012a] disclosed the presence of a dense torus-like component and collimated bipolar outflows [@Akras2012]. The *JHK*$_{\rm s}$ image, Fig. \[fig:m2\_42:shs\] (bottom panel), obtained from the VISTA variables in the Vía Láctea Survey [VVV; @Saito2012] also shows the presence of a compact dusty torus embedded in the main shell.
@Wang2007 carried out plasma diagnostics and abundance analysis of M2-42 using deep long-slit optical spectroscopy. They derived a mean electron density of $N_{\rm e}\simeq3 \times 10^3$cm$^{-3}$, and an electron temperature of $T_{\rm e}$=9350 K from the \[N[ii]{}\] line ratio, which is consistent with those of other PNe [see e.g. @Kingsburgh1994]. The oxygen abundance of O/H = $5.62 \times 10^{-4}$ derived by @Wang2007 is slightly above the solar metallicity, while N/O=0.32 corresponds to a non-Type I PN [based on N/O $< 0.8$; @Kingsburgh1994].
The central star of M2-42 depicts weak emission-line star characteristics [*wels* defined by @Tylenda1993] dominated by nitrogen and helium [@Depew2011]. The nebular spectrum of moderate excitation, $I$(5007$)=807$ on a scale where $I$(H$\beta)=100$ [@Wang2007], is related to an excitation class of 3.6 [@Dopita1990], and a stellar temperature of 74kK [@Dopita1991] or 69kK [@Reid2010]. Based on the Energy-Balance method, @Preite-Martinez1989 estimated a stellar temperature of 74.9 kK. According to @Tylenda1991, the central star has a B magnitude of 18.2. Using the H[i]{} Zanstra method, @Tylenda1991a derived a stellar temperature of 56 kK and a luminosity of $\log L/$L$_{\bigodot} = 2.87$, which correspond to a current core mass of 0.62M$_{\bigodot}$.
Based on its angular diameter and radio brightness (6 cm), @Acker1991a suggested that M2-42 is most likely located in the Galactic bulge. @Cahn1992 estimated a distance of 8754pc to the PN, which places it near to the Galactic center. The most recent distance estimation by @Stanghellini2008 yielded a distance of 9444pc. Moreover, we estimate a distance of 7400$^{+570}_{-550}$ from the H$\alpha$ surface brightness-radius relation for a sample of 332 PNe [@Frew2016], total flux value of $\log F($H$\alpha)= -11.39$ ergcm$^{-2}$s$^{-1}$ [@Frew2013], $c$([H$\beta$]{})$=0.99$ [@Wang2007], and angular radius of 2 arcsec [@Stanghellini2008]. Therefore, it could be a Galactic Bulge PN (GBPN).
In this paper, we present our integral field spectroscopy of M2-42, from which we determine ionization and kinematic properties of the nebula and its collimated outflows. In Section \[m2\_42:sec:observations\], we present the observations together with the physical and chemical conditions, stellar characteristics, and kinematic results derived from our data. Section \[m2\_42:sec:model\] describes the morpho-kinematic model of M2-42 and, finally, in Section \[m2\_42:sec:discussion\] we draw our conclusion.
![Top panel: the H$\alpha$ image obtained from the SHS [@Parker2005] with the morphological features labeled. The rectangle shows the $25\arcsec \times 38\arcsec$ WiFeS field of view observed using the ANU 2.3-m telescope in 2010 April. The image scale is shown by a solid line. North is up and east is toward the left-hand side. Bottom panel: the *JHK*$_{\rm s}$ image obtained from the VVV Survey [@Saito2012] with the compact dusty torus labeled. The red, green, and blue colors are assigned to the *K*$_{\rm s}$, *H*, and *J*, respectively. \[fig:m2\_42:shs\]](figures/fig1_m2-42a.eps "fig:"){width="2.4in"}\
![Top panel: the H$\alpha$ image obtained from the SHS [@Parker2005] with the morphological features labeled. The rectangle shows the $25\arcsec \times 38\arcsec$ WiFeS field of view observed using the ANU 2.3-m telescope in 2010 April. The image scale is shown by a solid line. North is up and east is toward the left-hand side. Bottom panel: the *JHK*$_{\rm s}$ image obtained from the VVV Survey [@Saito2012] with the compact dusty torus labeled. The red, green, and blue colors are assigned to the *K*$_{\rm s}$, *H*, and *J*, respectively. \[fig:m2\_42:shs\]](figures/fig1_m2-42b.eps "fig:"){width="2.4in"}
Observations {#m2_42:sec:observations}
============
Moderate resolution, integral field observations were obtained on 2010 April 22 under program number 1100147 (PI: Q.A. Parker) with the Wide Field Spectrograph [WiFeS; @Dopita2007; @Dopita2010] mounted on the Australian National University (ANU) 2.3 m telescope at Siding Spring Observatory. CCD chips with $4096 \times 4096$ pixels are used as detectors. The spectrograph samples $0\farcs5$ along each of twenty five $38\arcsec \times 1\arcsec$ slitlets, which provides a field of view of $25\arcsec \times 38\arcsec$ and a spatial resolution element of $1\farcs0\times0\farcs5$. Each slitlet is designed to project to 2 pixels on the CCD chips, yielding a reconstructed point-spread function with a full width at half maximum (FWHM) of $\sim 2\arcsec$.
Figure \[fig:m2\_42:shs\] shows the WiFeS areal footprint used for our study. The main shell, the northeast (NE) jet and the southwest (SW) jet are also labeled on the figure. We used the spectral resolution of $R\sim 7000$, covering $\lambda\lambda$4415–5589[Å]{} in the blue channel and $\lambda\lambda$5222–7070[Å]{} in the red channel. The red spectrum has a linear wavelength dispersion per pixel of $0.45$ [Å]{}, which yields a resolution of $\sim 20$ kms${}^{-1}$ in velocity channels. The exposure time of 20minutes used for our observation yields a signal-to-noise ratio of S/N$\gtrsim 10$ for the \[N[ii]{}\] emission line. Data reduction was performed with the <span style="font-variant:small-caps;">iraf</span> package [described by @Danehkar2013; @Danehkar2014a].
![From left to right, spatial distribution maps of flux intensity and LSR velocity of \[N[ii]{}\] $\lambda$6584. Flux unit is in logarithm of $10^{-15}$ ergs${}^{-1}$cm${}^{-2}$spaxel${}^{-1}$ and velocity in kms${}^{-1}$. The rectangles show apertures used to extract fluxes across the main shell ($6\arcsec \times 7\arcsec$), and the NE and SW jets ($4\arcsec \times 7\arcsec$). The white/black contour lines show the distribution of the narrow-band emission of H$\alpha$ in arbitrary unit obtained from the SHS. North is up and east is toward the left-hand side. \[fig:m2\_42:ifu\_map\]](figures/fig2_6584_flux_log.eps "fig:"){width="1.70in"}![From left to right, spatial distribution maps of flux intensity and LSR velocity of \[N[ii]{}\] $\lambda$6584. Flux unit is in logarithm of $10^{-15}$ ergs${}^{-1}$cm${}^{-2}$spaxel${}^{-1}$ and velocity in kms${}^{-1}$. The rectangles show apertures used to extract fluxes across the main shell ($6\arcsec \times 7\arcsec$), and the NE and SW jets ($4\arcsec \times 7\arcsec$). The white/black contour lines show the distribution of the narrow-band emission of H$\alpha$ in arbitrary unit obtained from the SHS. North is up and east is toward the left-hand side. \[fig:m2\_42:ifu\_map\]](figures/fig2_6584_vel.eps "fig:"){width="1.70in"}\
[lccccccccc]{} $\lambda_{0}$([Å]{}) &ID &Mult &&&\
& & &$F(\lambda)$&$I(\lambda)$&$F(\lambda)$&$I(\lambda)$&$F(\lambda)$&$I(\lambda)$\
4471.50 &[He[i]{}]{}&V14 &4.56 &5.67 &4.07 &4.97 &3.75 &4.58\
4609.44 &[O[ii]{}]{}&V92a &0.03 &0.04 &– &– &– &–\
4634.14 &[N[iii]{}]{}&V2 &0.13 &0.15 &– &– &– &–\
4649.13 &[O[ii]{}]{}&V1 &0.35 &0.39 &– &– &– &–\
4676.23 &[O[ii]{}]{}&V1 &0.09 &0.10 &– &– &– &–\
4685.68 &[He[ii]{}]{}&3.4 &0.30 &0.34 &0.34 &0.37 &– &–\
4740.17 &[\[Ar[iv]{}\]]{}&F1 &0.51 &0.54 &– &– &– &–\
4861.33 &[H[i]{}]{}4-2 &H4 &100.00 &100.00 &100.00 &100.00 &100.00 &100.00\
4881.11 &[\[Fe[iii]{}\]]{}&F2 &0.05 &0.05 &– &– &– &–\
4921.93 &[He[i]{}]{}&V48 &1.45 &1.41 &– &– &1.68 &1.63\
4958.91 &[\[O[iii]{}\]]{}&F1 &226.94 &214.89 &172.83 &164.37 &193.45 &184.01\
5006.84 &[\[O[iii]{}\]]{}&F1 &728.99 &672.08 &529.92 &491.74 &589.62 &547.27\
5666.63 &[N[ii]{}]{}&V3 &0.07 &0.04 &– &– &– &–\
5679.56 &[N[ii]{}]{}&V3 &0.11 &0.07 &– &– &– &–\
5754.60 &[\[N[ii]{}\]]{}&F3 &1.03 &0.66 &2.21 &1.47 &1.35 &0.90\
5875.66 &[He[i]{}]{}&V11 &24.78 &15.20 &23.82 &15.20 &25.96 &16.58\
6101.83 &[\[K[iv]{}\]]{}&F1 &0.05 &0.03 &– &– &– &–\
6312.10 &[\[S[iii]{}\]]{}&F3 &2.24 &1.18 &– &– &– &–\
6461.95 &[C[ii]{}]{}&V17.04 &0.12 &0.06 &– &– &– &–\
6548.10 &[\[N[ii]{}\]]{}&F1 &26.55 &12.87 &82.73 &42.49 &56.74 &29.21\
6562.77 &[H[i]{}]{}3-2 &H3 &– &286.00 &564.29 &288.58 &564.65 &289.38\
6583.50 &[\[N[ii]{}\]]{}&F1 &81.59 &39.10 &258.01 &131.14 &174.03 &88.64\
6678.16 &[He[i]{}]{}&V46 &8.60 &4.00 &8.59 &4.25 &8.81 &4.36\
6716.44 &[\[S[ii]{}\]]{}&F2 &8.16 &3.75 &39.09 &19.11 &28.82 &14.12\
6730.82 &[\[S[ii]{}\]]{}&F2 &12.97 &5.93 &36.51 &17.78 &30.02 &14.65\
& &0.989 & &0.910 & &0.907\
Table \[m2\_42:tab:obs\_lines\] presents a full list of observed line fluxes measured from three different apertures shown in Fig.\[fig:m2\_42:ifu\_map\]: the main shell ($6\arcsec \times 7\arcsec$), the NE jet ($4\arcsec \times 7\arcsec$) and the SW jet ($4\arcsec \times 7\arcsec$). The laboratory wavelength, emission line identification and multiplet number are given in columns 1–3, respectively. Columns 4–9 present the observed line fluxes $F(\lambda)$ and the dereddened fluxes $I(\lambda)$ after correction for interstellar extinction for the three different regions, respectively. All fluxes are given relative to H$\beta$, on a scale where ${\rm H}\beta=100$. To extract the observed line fluxes, we applied a single Gaussian profile to each line. The logarithmic extinction $c({\rm H}\beta)$ was calculated from the Balmer flux ratio H$\alpha$/H$\beta$. However, we adopted the extinction $c$([H$\beta$]{})$=0.989$ derived by @Wang2007 for the main shell since the H$\alpha$ emission line was saturated over the main shell area.
Physical and chemical conditions
--------------------------------
Electron temperature $T_{\rm e}$ and electron density $N_{\rm e}$ for the different regions of M2-42 are presented in Table \[m2\_42:tab:te\_ne\_abund\]. The electron temperatures and densities were obtained using the <span style="font-variant:small-caps;">equib</span> code [@Howarth1981] from the [\[N[ii]{}\]]{} nebular to auroral line ratio and the [\[S[ii]{}\]]{} doublet line ratio, respectively. The electron temperature [${\it T}_{\rm e}$]{}([\[N[ii]{}\]]{})$_{\rm corr}$ was corrected for recombination contribution to the auroral line using the formula given by @Liu2000 and the ionic abundance ${\rm N^{++}}/{\rm H^{+}}$ derived from the N[ii]{} lines. The values of $N_{\rm e}($[\[S[ii]{}\]]{}$)=3150$cm$^{-3}$ and $T_{\rm e}($[\[N[ii]{}\]]{}$)_{\rm corr}=9600$K are in agreement with $N_{\rm e}($[\[S[ii]{}\]]{}$)= 3240$cm$^{-3}$ and $T_{\rm e}($[\[N[ii]{}\]]{}$)=9350$K derived by @Wang2007. Additionally, we determined the physical conditions of the NE and SW jets. The jets show a mean electron temperature of $8840 \pm 180$K, which is 760K lower than that of the main shell, whereas their mean electron density of $595 \pm 125$cm$^{-3}$ is by a factor of five lower than that of the main shell.
Table \[m2\_42:tab:te\_ne\_abund\] also lists the ionic abundances X${}^{i+}$/H${}^{+}$ derived from collisionally excited lines (CELs) and optical recombination lines (ORLs). We used the <span style="font-variant:small-caps;">equib</span> code to calculate the ionic abundances. We adopted the physical conditions, $T_{\rm e}$ ([${\it T}_{\rm e}$]{}$_{\rm corr}$ for the main shell) and $N_{\rm e}$, derived from CELs. The atomic data sets used for plasma diagnostics and abundances analysis are the same as those used by @Danehkar2014b [Chapter 3].
Our value of He$^{+}$/H$^{+}=0.105$ for the main shell is in good agreement with He$^{+}$/H$^{+}$ = 0.107 derived by @Wang2007. However, they derived O$^{++}$/H$^{+}$ = $5.27 \times 10^{-4}$, which is twice our value. This could be due to the different atomic data used by them. Our values of N$^{+}$/H$^{+}$, S$^{+}$/H$^{+}$ and Ar$^{3+}$/H$^{+}$ are in reasonable agreement with N$^{+}$/H$^{+}$ = $1.03 \times 10^{-4}$, S$^{+}$/H$^{+}$ = $4.96\times 10^{-7}$ and Ar$^{3+}$/H$^{+}$ = $1.59 \times 10^{-7}$ obtained by @Wang2007. Note that a slit with a width of $2\arcsec$ used by @Wang2007 is not completely related to the main shell. We see that the abundance discrepancy factor for O$^{++}$, ${\rm ADF}({\rm O}^{++}) \equiv ({\rm O}^{++}/{\rm H}^{+})_{\rm ORL} / ({\rm O}^{++}/{\rm H}^{+})_{\rm CEL}= 3.14$, is in agreement with ${\rm ADF}({\rm O}^{++})=2.09$ [@Wang2007]. Moreover, our abundance ratio of (N$^{++}$/O$^{++})_{\rm ORL}=0.388$ derived from ORLs is in excellent agreement with (N$^{++}$/O$^{++})_{\rm ORL}=0.399$ obtained by @Wang2007. Although He$^{+}$/H$^{+}$ and O$^{++}$/H$^{+}$ derived from the jets are similar to those of the main shell, N$^{+}$/H$^{+}$ and S$^{+}$/H$^{+}$ derived from the jets are about three times higher than those of the main shell. These ionization features of the bipolar collimated jets are typical of fast, low-ionization emission regions [FLIERs; @Balick1993; @Balick1994; @Balick1998].
[lccc]{} Parameter &[Main Shell]{}&[NE Jet]{}&[SW Jet]{}\
&10270 &9020 &8660\
&9600 &– &–\
&3150 &470 &720\
&0.105 &0.107 &0.110\
&0.764&2.912 &2.236\
&2.606 &2.469&3.208\
&0.347 &1.150 &1.040\
&3.116 &– &–\
&1.871&– &–\
&3.175 &– &–\
&8.185 &– &–\
Comments on stellar characteristics
-----------------------------------
The stellar emission-line fluxes presented in Table \[m242:tab:cspn\], are measured from a spectrum integrated over an aperture ($3\arcsec \times 3\arcsec$) covering the central star in the WiFeS field. The emission line identification, wavelength, dereddened flux corrected for reddening using $c$([H$\beta$]{})$=0.99$, equivalent width $W_{\lambda}$ ([Å]{}), and FWHM ([Å]{}) are given in columns 1–5, respectively. All fluxes are given relative to C[iv]{} 5805, on a scale where C[iv]{}5805=100. We note that the width of C[iv]{} $\lambda$5805 is narrower than typical Wolf–Rayet central starts of PNe with the same stellar temperature [see e.g., @Crowther1998; @Acker2003], so it could be a *wels* as identified by @Depew2011. Following the method used by @Acker2003, a terminal wind velocity of 640kms$^{-1}$ is deduced from FWHM$($C[iv]{}$\lambda$5805$)=10.27$[Å]{}. However, we get a terminal velocity of 1560kms$^{-1}$ from FWHM$($C[iv]{}$\lambda$5805$)=25$[Å]{} reported by @Depew2011. Although the C[iii]{} $\lambda$5696 line is not detected, the C[iii]{}/[iv]{} $\lambda$4650 is possibly identified. The He[ii]{} $\lambda$4686 line is fairly strong, but it could have a nebular origin. We see the presence of strong N[iii]{} $\lambda\lambda$4634,4641 lines and weak N[v]{} $\lambda\lambda$4603,4932 lines. Assuming that the He[ii]{} $\lambda$4686 line is a stellar emission line, M2-42 could have a stellar characteristics similar to WN8-type stars of @vanderHucht2001 based on $I($N[iii]{}$\lambda$4641$)\gtrsim I($He[ii]{}$\lambda$4686) and $I($N[iii]{}$)\gg I($N[v]{}$)$. However, N[ii]{} $\lambda$3995 and N[iv]{} $\lambda\lambda$3479–3484, 4058 lines are not in the wavelength coverage of our WiFeS observations, so we cannot certainly classify it as one of nitrogen sequences of Wolf–Rayet central stars of PNe.
[lcrrc]{} Line &$\lambda$([Å]{})& $I(\lambda)$& $W_{\lambda}$([Å]{}) & [FWHM([Å]{})]{}\
N[v]{} &4603 &$33.58$: &$-1.16$ &$3.14$\
N[iii]{} &4634 &$77.56$ &$-3.15$ &$1.25$\
N[iii]{} &4641 &$283.41$ &$-10.91$ &$3.08$\
C[iii]{}/[iv]{} &4650 &$220.38$ &$-9.49$ &$2.97$\
C[iii]{} &4655 &$42.05$ &$-1.94$ &$3.55$\
C[iv]{} &4659 &$63.19$ &$-2.81$ &$2.84$\
He[ii]{} &4686 &$139.46$ &$-8.17$ &$1.23$\
N[v]{} &4932 &$57.20$: &$-2.93$ &$2.47$\
C[iv]{} &5805 &$100.00$ &$-5.62$ &$10.27$\
C[ii]{} &6462 &$19.28$ &$-1.11$ &$2.09$\
C[iii]{} &7037 &$16.29$: &$-1.04$ &$5.01$\
Kinematic results
-----------------
We derived an expansion velocity of $V_{\rm HWHM}=20.2\pm1.3$ kms$^{-1}$ from the half width at half maximum (HWHM) for the \[N[ii]{}\] $\lambda\lambda$6548,6584 and \[S[ii]{}\] $\lambda\lambda$6716,6731 emission-line profiles integrated over the main shell ($6 \arcsec \times 7 \arcsec$). The local standard of rest (LSR) systemic velocity of the whole nebula was estimated to be at $122.9\pm 12$kms$^{-1}$, which is in agreement with $V_{\rm LSR}=133.1 \pm 13.3$kms$^{-1}$ measured by @Durand1998. The LSR velocity is defined as the line of sight radial velocity, transferred to the local standard of rest by correcting for the motions of the Earth and Sun.
Figure \[fig:m2\_42:ifu\_map\] shows spatially resolved flux and velocity maps of M2-42 extracted from the \[N[ii]{}\] $\lambda$6584 emission line across the WiFeS field. The observed radial velocity map was transferred to the LSR radial velocity. The white/black contour lines in the figures depict the 2D distribution of the H$\alpha$ emission obtained from the SHS, which can aid us in distinguishing the nebular border. As seen in Fig.\[fig:m2\_42:ifu\_map\], the kinematic map depicts an elliptical structure with a pair of collimated bipolar outflows, which is easily noticeable in the channel maps (see Fig. \[fig:m2\_42:vmap\]) and discussed below.
Figure \[fig:m2\_42:vmap\] presents the flux intensity maps of the \[N[ii]{}\] $\lambda$6584 emission lines on a logarithmic scale observed in a sequence of 6 velocity channels with a resolution of $\sim 20$ kms$^{-1}$, which can be used to identify different morphological components of the nebula. We subtracted the systemic velocity $v_{\rm sys}=123$ kms$^{-1}$ from the central velocity value given at the top of each channel. The stellar continuum map was also subtracted from the flux intensity maps. While there is a dense torus in the center, a pair of collimated bipolar outflows can be also identified in the velocity channels. The torus has a radius of $3\arcsec \pm 1 \arcsec$. This torus is clearly evident in the VVV *J*, *H*, and *K*$_{\rm s}$ color combined image of M2-42 presented in Figure. \[fig:m2\_42:shs\] (bottom panel). We notice that the bipolar outflows are highly asymmetric, the SW jet apparently having a bow shock structure. While the NE jet reaches a distance of $12\arcsec \pm 2\arcsec$ from the nebular center, the distance of the SW jet from the central star is about 25% shorter than the NE jet. Interaction with the interstellar medium (ISM) can lead to the formation of asymmetric bipolar outflows [see e.g. @Wareing2007]. Both the jet components have similar brightness in the velocity channels. Brightness discontinuities are seen in the channels, where the bipolar outflows emerge from the main shell.
![Velocity slices of M2-42 along the \[N[ii]{}\] $\lambda$6584 emission-line profiles. The slices have a $\sim 20$ kms$^{-1}$ width, the central velocity is given at the top of each slice, and the LSR systemic velocity is $v_{\rm sys}=123$ kms$^{-1}$. The color bars show flux measurements in logarithm of $10^{-15}$ ergs${}^{-1}$cm${}^{-2}$spaxel${}^{-1}$. Velocity channels are in kms${}^{-1}$. The contours in the channel maps are the narrow-band H$\alpha$ emission in arbitrary unit obtained from the SHS. North is up and east is toward the left-hand side. \[fig:m2\_42:vmap\]](figures/fig3_6584_vmap.eps){width="3.5in"}
Morpho-kinematic model {#m2_42:sec:model}
======================
We have used the morpho-kinematic modeling program <span style="font-variant:small-caps;">shape</span> (version 5.0) described in detail by @Steffen2006 and @Steffen2011. This program has been used for modeling many PNe, such as NGC 2392 [@Garcia-Diaz2012], NGC 3242 [@Gomez-Munoz2015], Hen 2-113 and Hen 3-1333 [@Danehkar2015]. It uses interactively molded geometrical polygon meshes to generate three-dimensional structures of gaseous nebulae. The program produces several outputs that can be directly compared with observations, namely position-velocity diagrams, velocity channels and synthetic images. The modeling procedure consists of defining the geometry, assigning a density distribution and defining a velocity law. Geometrical and kinematic parameters are modified in a manual interactive process until a satisfactorily fitting model has been constructed.
Figure \[m2\_42:shape\] (a) shows the morpho-kinematic model before rendering at two different orientations (inclination: 0$^{\circ}$ and 90$^{\circ}$), and their best-fitting inclination, together with the result of the rendered model. The morpho-kinematic model consists of an equatorial dense torus (main shell) and a pair of asymmetric bipolar outflows. The values of the parameters of the final model are summarized in Table \[m2\_42:parameters\]. For the velocity field, we assume a Hubble-type flow [@Steffen2009].
The velocity-channel maps of the final model are shown in Figure \[m2\_42:shape\] (b), where they can be directly compared with the observed velocity-resolved channel maps presented in Figure \[fig:m2\_42:vmap\]. The model maps are a good match to the observational maps. The model successfully produces two kinematic components of the jets moving in opposite directions on both sides of the torus. From the morpho-kinematic model, we derived an inclination of $i=-82^{\circ}\pm 4^{\circ}$ with respect to the line of sight. Taking the inclination derived by the best-fitting model, we estimated a “jet” expansion velocity of $ 120 \pm 40$ kms$^{-1}$ with respect to the central star.
[(a) <span style="font-variant:small-caps;">shape</span> model]{}\
![Top panels: <span style="font-variant:small-caps;">shape</span> mesh model of M2-42 before rendering at two different orientations (inclination: 0$^{\circ}$ and 90$^{\circ}$), the best-fitting inclination, and the corresponding rendered image, respectively. Bottom panels: synthetic images at different velocity channels obtained from the best-fitting <span style="font-variant:small-caps;">shape</span> model. \[m2\_42:shape\]](figures/fig4_Shape_model.eps "fig:"){width="3.5in"}\
[(b) Velocity channels]{}\
![Top panels: <span style="font-variant:small-caps;">shape</span> mesh model of M2-42 before rendering at two different orientations (inclination: 0$^{\circ}$ and 90$^{\circ}$), the best-fitting inclination, and the corresponding rendered image, respectively. Bottom panels: synthetic images at different velocity channels obtained from the best-fitting <span style="font-variant:small-caps;">shape</span> model. \[m2\_42:shape\]](figures/fig4_shape_vmap.eps "fig:"){width="3.4in"}
[lc]{} Parameter & Value\
Inclination of major axis, $i$& $-82^{\circ} \pm 4^{\circ}$\
Position angle of major axis, PA & $50^{\circ} \pm 5^{\circ}$\
Galactic position angle of major axis, GPA & $112^{\circ}24\arcmin \pm 5^{\circ}$\
Outer radius of the main shell & $3\pm1$ arcsec\
NE Jet distance from the center & $12\pm2$ arcsec\
SW Jet distance from the center & $9\pm2$ arcsec\
Jet velocity from the center & $120\pm40$ kms${}^{-1}$\
As seen in Table \[m2\_42:parameters\], the symmetric axis of the bipolar outflows has a position angle (PA) of $50^{\circ} \pm 5^{\circ}$ measured from the north toward the east in the equatorial coordinate system (ECS). This leads to a Galactic position angle (GPA) of $112 \fdg 4$. The GPA is the position angle of the nebular symmetric axis projected on to the sky plane, measured from the North Galactic Pole toward the Galactic east. Note that ${\rm GPA}=90^{\circ}$ describes an alignment with the Galactic plane, whereas ${\rm GPA}=0^{\circ}$ is perpendicular to the Galactic plane. Therefore, the symmetric axis of M2-42 is roughly aligned with the Galactic plane. This alignment could have some implications for other studies of GBPNe [see e.g. @Rees2013; @Falceta-Gonccalves2014; @Danehkar2016].
Summary and discussions {#m2_42:sec:discussion}
=======================
In this paper, we present the spatially resolved observations of M2-42 obtained with the WiFeS on the ANU 2.3 m telescope. Using the velocity-resolved channel maps derived from the \[N[ii]{}\] $\lambda$6584 emission line, a morpho-kinematic model has been developed which includes different morphological components of the nebula: a dense torus and a pair of asymmetric bipolar outflows in opposite directions. From the HWHM method, the torus is found to expand slowly at $20$kms$^{-1}$, almost in agreement with $15$kms$^{-1}$ derived by @Akras2012. From the reconstruction model, the trail of bipolar outflows was found to go along the direction of (GPA, $i$) $=$ ($112^{\circ}$, $-82^{\circ}$), which is very similar to the inclination of $i=77^{\circ}$ derived by @Akras2012 based on the SPM long-slit data. We find a “jet” expansion velocity of $ 120 \pm 40$kms$^{-1}$ with respect to the nebular center, which is higher than the value of $70$kms$^{-1}$ estimated by @Akras2012. Moreover, we found that the SW jet, which moves toward us, has possibly a bow shock structure relating to the interaction with ISM [see e.g. @Wareing2007].
An empirical analysis of the nebular spectra separately integrated over the three different regions shows that the mean density of the jets is a factor of five lower than that in the main shell. Although the abundances of singly ionized helium and doubly ionized oxygen are almost the same in both the shell and the jets, the singly ionized nitrogen and sulfur abundances derived from the jets are about three times higher than those obtained from the main shell. The similar ionization characteristics have been found in collimated jets emerged from other PNe [see e.g. @Balick1993; @Balick1994].
Nearly 10% of Galactic PNe have been found to have the small-scale low-ionization structures in opposite directions on both sides of their central stars. Around half of them are fast, highly collimated outflows with velocities of 30–200 kms$^{-1}$ relative to the main bodies, so called FLIERs [@Balick1993; @Balick1994; @Balick1998]. Previously, @Balick1994 claimed the presence of nitrogen enrichment by factors of 2–5 in the FLIERs of some PNe. However, @Gonccalves2003 suggested that empirically derived nitrogen overabundance seen in FLIERs are a result of inaccurate ionization correction factors applied in the empirical analysis. @Gonccalves2006 constructed a chemically homogeneous photoionization model of NGC7009, which can reproduce the ionization characteristics of its shell and FLIERs. Similarly, the enhancement of N$^{+}$/H$^{+}$ and S$^{+}$/H$^{+}$ in the FLIERs of M2-42 could be attributed to the geometry and density distribution rather than chemical inhomogeneities.
The previous observations of M2-42 showed that its central star is of *wels* type [@Depew2011]. Moreover, we found that its stellar spectrum might be similar to the WN8 subclass of @vanderHucht2001 based on $I($N[iii]{}$)\gtrsim I($He[ii]{}). The terminal wind velocity was also estimated to be about 640 kms$^{-1}$. However, our observations did not cover the N[ii]{} and N[iv]{} lines, which are necessary for the WN classification. This typical stellar characteristics and its point-symmetric morphology could be a result of a common-envelope evolutionary phase [see e.g. @Nordhaus2006]. Currently, there is no evidence for binarity in M2-42. We believe that further observations of its central star will help develop a better stellar classification and also shed light on the mechanism producing its FLIERs.
A.D. acknowledges the award of a Research Excellence Scholarship from Macquarie University. Q.A.P. acknowledges support from Macquarie University and the Australian Astronomical Observatory (AAO). W.S. acknowledges support from grant UNAM-PAPIIT 101014. We would like to thank the staff at the ANU Siding Spring Observatory for their support. We acknowledge use of data from the VISTA telescope under ESO Survey programme ID 179.B-2002. We thank the anonymous referee whose suggestions and comments have greatly improved the paper.
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|
---
abstract: 'To explore and react to their environment, living micro-swimmers have developed sophisticated strategies for locomotion — in particular, motility with multiple gaits. To understand the physical principles associated with such a behavioural variability, synthetic model systems capable of mimicking it are needed. Here, we demonstrate bimodal gait switching in autophoretic droplet swimmers. This minimal experimental system is isotropic at rest, a symmetry that can be spontaneously broken due to the nonlinear coupling between hydrodynamic and chemical fields, inducing a variety of flow patterns that lead to different propulsive modes. We report a dynamical transition from quasi-ballistic to bimodal chaotic motion, controlled by the viscosity of the swimming medium. By simultaneous visualisation of the chemical and hydrodynamic fields, supported quantitatively by an advection-diffusion model, we show that higher hydrodynamic modes become excitable with increasing viscosity, while the recurrent mode-switching is driven by the droplet’s interaction with self-generated chemical gradients. We further demonstrate that this gradient interaction results in anomalous diffusive swimming akin to [self-avoiding spatial]{} exploration strategies observed in nature.'
author:
- 'Babak Vajdi Hokmabad [[](https://orcid.org/0000-0001-5075-6357)]{}'
- 'Ranabir Dey [[](https://orcid.org/0000-0002-0514-7357)]{}'
- 'Maziyar Jalaal [[](https://orcid.org/0000-0002-5654-8505)]{}'
- 'Devaditya Mohanty [[](https://orcid.org/0000-0001-6797-5206)]{}'
- 'Madina Almukambetova [[](https://orcid.org/0000-0003-3223-5392)]{}'
- 'Kyle A. Baldwin [[](https://orcid.org/0000-0001-9168-6412)]{}'
- 'Detlef Lohse [[](https://orcid.org/0000-0003-4138-2255)]{}'
- 'Corinna C. Maass [[](https://orcid.org/0000-0001-6287-4107)]{}'
bibliography:
- 'stopandgo.bib'
title: 'Stop-and-go droplet swimmers'
---
In response to physical constraints in nature, microorganisms have adapted and developed various locomotion strategies. Depending on cues from the environment, these strategies include helical swimming [@bearon2013_helical; @rossi2017_kinematics], run-and-tumble and switch-and-flick motility [@stocker2011_reverse], but also more sophisticated transient behaviours, e.g. peritrichous bacteria switching poles in response to a steric stress [@cisneros2006_reversal], octoflagellate microalga exhibiting run-stop-shock motility with enhanced mechanosensitivity [@wan2018_time], and starfish larvae maximising fluid mixing, and thereby nutrition uptake, through rapid changes of ciliary beating patterns [@gilpin2017_vortex]. Such intricate gait-switching dynamics [@tsang2018_polygonal; @son2013_bacteria] [can]{} enable organisms to navigate in external flows [@mathijssen2019_oscillatory; @figueroa-morales2020_e], follow gradients [@wadhams2004_making] or efficiently explore their environment [@perezipina2019_bacteria; @guadayol2017_cell].
To understand the physical mechanisms behind the locomotion strategies of biological swimmers, one requires motile synthetic protocells which mimic these sophisticated behaviours. This is ideally done in a minimal model system based on fundamental principles of non-equilibrium physics, rather than building on intricate biochemical machinery. Examples include autophoretic microswimmers powered by chemical activity which leave long-living chemical traces in their wake [@liebchen2018_synthetic].
[We show]{} that self-interaction with the swimmer’s own slowly diffusing chemical field can lead to nonlinear feedback triggering unsteady swimming and chaotic dynamics. [Specifically]{}, we report the emergence of such unsteady dynamics for a self-propelled isotropic droplet, in the absence of any bio-complexity. The droplet, driven by micellar solubilization [@maass2016_swimming], adapts to an increase in the viscosity of the swimming medium with increasingly unsteady motion — a counter-intuitive response given that increasing viscosity generally tends to stabilise non-inertial dynamics. Using time-resolved *in situ* visualisation of the chemical and the hydrodynamic fields around the droplet interface, we have found that the emergent unsteady dynamics of the droplet correlate with the onset of higher hydrodynamic modes. The experiments agree quantitatively with theoretical predictions of dominant higher hydrodynamic modes for large Péclet numbers, which characterise the underlying advection-diffusion-driven transport of the chemical species necessary for droplet activity. Once these higher modes prevail, the droplet exhibits an unsteady bimodal exploration of space triggered by its interaction with self-generated chemical gradients, which is reminiscent of gait-switching dynamics in biological organisms. The visualisation technique and the findings presented here lay the groundwork for future investigations of emergent dynamics in active phoretic matter, from the individual to the collective scale.
Droplets propelled by micellar solubilisation {#droplets-propelled-by-micellar-solubilisation .unnumbered}
=============================================
Oil droplets that are slowly dissolving in supramicellar aqueous surfactant solutions can spontaneously develop self-sustaining gradients in interfacial surfactant coverage, resulting in Marangoni stresses which lead to self-propulsion [@herminghaus2014_interfacial] This interfacial instability may be understood as follows([Fig. \[fig:fillingcartoon\]a,b]{}): During the solubilisation of the droplet, oil molecules migrate into surfactant micelles in a boundary layer around the droplet interface, causing the micelles to swell and take up additional surfactant molecules from the aqueous phase. This depletes the interfacial surfactant concentration, unless there are empty micelles present to replenish it by disintegration. The interfacial tension therefore increases with the local ratio of filled to empty micelles. Following an advective perturbation in the vicinity of the droplet, the radial symmetry of the filled micelle distribution is spontaneously broken; the resulting fore-aft asymmetry generates a surface tension gradient which drives the droplet towards more empty micelles leading to sustained self-propulsion, while leaving behind a trail of swollen micelles. Such spontaneous self-propulsion stemming from the advection-diffusion driven interfacial instability arises only if the Péclet number, $Pe$, which characterises the ratio of advective to diffusive transport, exceeds a critical threshold [@michelin2013_spontaneous; @izri2014_self-propulsion; @morozov2019_nonlinear; @morozov2019_self-propulsion]. For the active droplet system, $Pe$ can be shown to be a monotonically increasing function of the swimming medium ([*outer*]{}) viscosity $\mu^o$, here non-dimensionalised as $\mu=\mu^o/\mu^i$ using the constant [*inner*]{} viscosity $\mu^i$: $$\label{eqn:pe}
Pe=\frac{V_t R_d}{D} \approx \frac{18 \pi^2}{k_BT} q_s r_s^2 \zeta R_d^2 \mu^i \left[\mu \left(\frac{2\mu+3 \zeta/R_d}{2 \mu+3}\right)\right],$$ where $V_t$ is the theoretical terminal droplet velocity in an external surfactant gradient [@anderson1989_colloid; @morozov2019_nonlinear], $R_d=30\,\mu$m the droplet radius, $D=\frac{k_BT}{6 \pi r_s \mu^o}$ the diffusion coefficient for the surfactant monomer (length scale $r_s \sim 10^{-10}$m), $q_s$ the isotropic interfacial surfactant consumption rate per area, and $\zeta \sim 10$nm the characteristic length scale over which the surfactants interact with the droplet [@anderson1989_colloid; @izri2014_self-propulsion] (see Supplementary section \[sisec:peclet\] for details). In experiments, we controlled $\mu^o$ by using water/glycerol mixtures as the swimming medium (see Supplementary [Fig. \[sifig:Viscosity\]]{}). Henceforth, we represent an increase in $\mu^o$ by the corresponding increase in $Pe$.
![\[fig:fillingcartoon\] [**Droplet propulsion mechanism and visualisation technique.**]{} [**a**]{}, Top: Schematic illustration of the micellar solubilization of oil at the droplet interface leading to self-propulsion. Bottom: Streaks of tracers following the flow inside and outside of the droplet during 2 seconds, with streamlines of the external flow from PIV analysis (droplet reference frame). Data from double channel fluorescence microscopy, with illumination at 561nm (Nile Red doped oil, red emission) and 488nm (tracer colloids, green emission). [**b**]{}, Sketch of the filling and growth of micelles travelling in a boundary layer along the interface, causing a propulsive Marangoni flow. [**c**]{}, Microscopy set-up schematic with the droplet (radius 30$\mu$m) swimming in a Hele–Shaw cell (height 60$\mu$m). [**d**]{}, Sample micrograph, with the droplet’s centroid trajectory traced in white.](figures/Fig1.png){width="\columnwidth"}
Simultaneous visualisation of chemical and hydrodynamic fields {#simultaneous-visualisation-of-chemical-and-hydrodynamic-fields .unnumbered}
==============================================================
To visualise the chemical and hydrodynamic fields involved in the droplet activity, we directly imaged the [chemical field of swollen micelles]{} by adding the hydrophobic dye Nile Red to the oil phase ([Fig. \[fig:fillingcartoon\]c,d]{}). The dye co-migrates with the oil molecules into the filled micelles, which fluoresce when illuminated. We seeded the surrounding medium, a supramicellar aqueous surfactant solution, with green fluorescent tracer colloids ([Fig. \[fig:fillingcartoon\]c,d]{}; see also Supplementary Section \[sisec:DCFM\] and Video S1), and measured the flow field using particle image velocimetry (PIV). The emission spectra of dye and colloids are sufficiently non-overlapping to be separately detected in dual channel fluorescence microscopy. Consequently, both fields can be simultaneously observed and analysed; we provide an example micrograph with an overlay of the extracted droplet trajectory in [Fig. \[fig:fillingcartoon\]d]{}. Due to the large size ($O$(10nm)) of the filled micelles, the time scale of their diffusive relaxation exceeds that of the droplet motion; thus, there is a persistent fluorescent trail in the wake of the droplet.
{width="75.00000%"}
Destabilised motion with increasing Péclet number {#destabilised-motion-with-increasing-péclet-number .unnumbered}
=================================================
We begin, however, with an overview of the droplet dynamics using trajectory plots and statistical analyses of speed and orientational persistence taken from bright-field microscopy ([Fig. \[fig:statistics\]]{}). With increasing $Pe$, the droplet propulsion changes from uniform speeds and persistent motion to unsteady [motion]{} with abrupt reorientations ([Fig. \[fig:statistics\]a-d]{}). We define $P(|\delta\theta(t)|)$ as the distribution of the reorientation angle $\delta\theta$ of the 2D droplet velocity ${\mathbf}{V}(t)$ during a fixed time step $\delta t$ [@bhattacharjee2019_bacterial], $$\delta\theta(t) = \arctan\left(\frac{{\mathbf}{V}(t)\times {\mathbf}{V}(t+\delta t)}{{\mathbf}{V}(t) \cdot {\mathbf}{V}(t+\delta t)}\right).\label{eqn:dtheta}$$
$P(|\delta\theta(t)|)$ broadens [significantly]{}, [corresponding to]{} more frequent and sharper reorientation events [([Fig. \[fig:statistics\]e]{})]{}. The faster decay of the angular velocity autocorrelation function, $$C_{VV}(t)=\left\langle\frac{{\mathbf}{V}(t_0 + t)\cdot{\mathbf}{V}(t_0)}{|{\mathbf}{V}(t_0 + t)| |{\mathbf}{V}(t_0)|}\right\rangle_{t_0},\label{eqn:cvv}$$ illustrates the loss of directionality with increasing $Pe$ ([Fig. \[fig:statistics\]e]{}, inset). [Fig. \[fig:statistics\]f]{} shows that at sufficiently large $Pe$, the speed distribution $P(V)$ includes values as small as zero (stopping events) and, surprisingly, as large as 70$\mu$m/s, much greater than the uniform speed of $30\,\mu$m/s observed for low $Pe\approx 4$. While the mean speed barely changes with $Pe$, the standard deviation of $V$ grows by over one order of magnitude ([Fig. \[fig:statistics\]f]{}, inset). Hence, both the rotational and the translational motion of the swimmer are destabilised with increasing $Pe$, similar to recent numerical studies of solid phoretic particles [@hu2019_chaotic]. Note that the thermal fluctuations ($O(k_bT/2 R_d)\sim 10^{-16}$N) are negligible compared to the hydrodynamic drag force ($O(6\pi \mu^o R_d V)\,{\gtrsim}\, 10^{-10}$N), such that thermal noise is an unlikely cause for the unsteady swimming.
{width="\textwidth"}
Signatures of unsteady dynamics in the time evolution of chemical and hydrodynamic fields {#signatures-of-unsteady-dynamics-in-the-time-evolution-of-chemical-and-hydrodynamic-fields .unnumbered}
=========================================================================================
To investigate the origin of this unsteady behaviour, we studied the evolution of chemical and hydrodynamic fields around the droplet. We extracted the tangential flow velocity $u_\theta(\theta)$ and the red fluorescence intensity $I(\theta)$ of the chemical field close to the interface ([Fig. \[fig:kymographs\]d]{}[, Supplementary section \[sisec:improc\]]{}), and mapped them in kymographs $I(\theta, t)$ and $u_\theta(\theta, t)$.
For low $Pe\approx 4$, at persistent propulsion, $I(\theta,t)$ shows a single fixed-orientation band marking the origin of the filled micelle trail at the rear stagnation point of the droplet ([Fig. \[fig:kymographs\]a]{} and video S6). The two bands in $u_\theta(\theta,t)$ correspond to a steady flow field with dipolar symmetry that is consistent with the $I(\theta,t)$ profile. On the right side of [Fig. \[fig:kymographs\]a]{} we have superimposed the streamlines of this dipolar flow field on the corresponding chemical micrograph at the time marked by I in the $I(\theta,t)$ kymograph.
For intermediate $Pe \approx 28$ ([Fig. \[fig:kymographs\]b]{}, Video S7), $I(\theta,t)$ shows secondary branches forming at the anterior stagnation point of the droplet and subsequently merging with the main filled micelle trail. This coincides with a transient second hydrodynamic mode with quadrupolar symmetry ([Fig. \[fig:kymographs\]b,II]{}), causing the accumulation of an additional aggregate of filled micelles at the droplet anterior (see also Supplementary [Fig. \[sifig:ComplementaryFig4\]]{}).
The ratio of the diffusive $(R_d^2/D_\text{{\it fm}})$ to advective $(R_d/V)$ time scales for the migration of filled micelles is $\frac{VR_d}{D_\text{{\it fm}}}\gg1$ for all experiments, assuming a diffusion coefficient $D_\text{{\it fm}}=k_BT/6\pi\mu^or_\text{{\it fm}}$, with a micellar radius of $O(r_\text{{\it fm}})\sim 10$nm. Therefore, the aggregate is unlikely to dissipate by diffusion, and will continue to grow as long as the quadrupolar mode exists. However, this mode is not stable. Eventually, the dipolar mode dominates and advects the secondary aggregate towards the main trail ([Fig. \[fig:kymographs\]b,III]{}; also see [Fig. \[sifig:ComplementaryFig4\]]{}). The transport of the aggregate along one side of the droplet locally disturbs the interfacial flow, leading to an abrupt reorientation of the swimming direction ([Fig. \[fig:kymographs\]b,I-III]{}). As shown in the trajectories in [Fig. \[fig:statistics\]b and c]{}, these reorientation events become more frequent with increasing $Pe$; accordingly, $u_\theta$ in [Fig. \[fig:kymographs\]b]{} exhibits quasi-periodic reorientation patterns.
For high $Pe\approx 190$ ([Fig. \[fig:kymographs\]c]{}, Video S8), the quadrupolar mode eventually prevails, resulting in a predominantly symmetric extensile flow around the droplet ([Fig. \[fig:kymographs\]c,I]{}), as shown by a pronounced fourfold [pattern]{} in the additional kymograph $u_r(\theta,t)$ of the radial velocity. In the absence of [the dipolar mode]{}, the droplet is trapped in place. The gradual accumulation of filled micelles at the two stagnation points with radially outward flow manifests in two stable branches in the chemical kymograph (marked by I in [Fig. \[fig:kymographs\]c]{}). The growth of the two micellar aggregates locally generates a lateral chemical gradient, which eventually pushes the droplet out of its self-made trap. Concomitantly, the two points of filled micelle emission move along the droplet interface and merge on the new rear side of the droplet into a single filled micelle trail ([Fig. \[fig:kymographs\]c,II and III]{}). [The chemorepulsion from the local field micelle gradient induces an apparent dipolar mode which gradually decays as the droplet leaves the self-made trap.]{} Now, the dominant quadrupolar mode re-saturates, with an aggregate growing at the droplet anterior, until the droplet is trapped again and a new bimodal ‘stop-and-go’ cycle begins. Since the escape direction is always lateral, consecutive runs are approximately perpendicular, resulting in the sharp reorientation events apparent in the trajectories in [Fig. \[fig:kymographs\]c]{} and [Fig. \[fig:statistics\]d]{}, as well as the broadening $|\delta\theta|$ distribution in [Fig. \[fig:statistics\]e]{}.
{width=".9\textwidth"}
Dependence of hydrodynamic modes on the Péclet number {#dependence-of-hydrodynamic-modes-on-the-péclet-number .unnumbered}
=====================================================
In order to understand the dependence of the dominance of the hydrodynamic modes on $Pe$, we analysed the underlying advection-diffusion problem for the active droplet within the framework of an axisymmetric Stokes flow (see [Fig. \[fig:modestab\]]{}, Supplementary section \[sisec:hdmodel\] and [@morozov2019_nonlinear; @morozov2019_self-propulsion; @leal2007_advanced]). At the smallest value of $\mu$, $Pe$ is approximately equal to the critical value of 4 necessary for the onset of the first hydrodynamic mode ($n=1$), i.e. the mode with dipolar flow symmetry [@michelin2013_spontaneous; @morozov2019_nonlinear; @morozov2019_self-propulsion]. With increasing $\mu$, $Pe$ (markers in [Fig. \[fig:modestab\]a]{}) eventually exceeds the critical values necessary for the onset of the higher hydrodynamic modes (lines in [Fig. \[fig:modestab\]a]{}), specifically the second hydrodynamic mode ($n=2$), i.e. the mode with quadrupolar symmetry. A linear stability analysis around a motionless base state (see Supplementary section \[sisec:linstab\] and [@michelin2013_spontaneous; @morozov2019_self-propulsion]) shows that for small to moderate $Pe$, the non-dimensionalised instability growth rate $\lambda$ for $n=1$ exceeds that for $n=2$ ([Fig. \[fig:modestab\]c]{}). Accordingly, for lower $Pe$, $n=1$ dominates, representing steady self-propulsion stemming from the fore-aft asymmetry of the surfactant distribution ([Fig. \[fig:modestab\]b,I]{}). Consequently, the active droplet exhibits persistent steady translation (trajectories in [Fig. \[fig:statistics\]a-b]{}) with a dominant dipolar flow field ([Fig. \[fig:modestab\]b,II]{} and [Fig. \[fig:kymographs\]a]{}). However, for $Pe\gtrsim92$, $n=2$ ([Fig. \[fig:modestab\]b,III]{}) has a faster instability growth rate, thereby becoming the dominant mode ([Fig. \[fig:modestab\]c]{}). Consequently, for experiments with $Pe>92$, the active droplet experiences periods of dynamical arrest during which it remains stationary with a surrounding extensile flow ([Fig. \[fig:modestab\]b,IV]{} and [Fig. \[fig:kymographs\]c]{}). As discussed above, the subsequent genesis of the bimodal ‘stop-and-go’ motion of the droplet for moderate to higher $Pe$ (trajectories in [Fig. \[fig:statistics\]c-d]{}) is due to the synergy between the $n=2$ mode and the transiently-growing filled micelle field. Note that we restrict our analysis to the first two hydrodynamic modes since these two are solely responsible for the droplet propulsion and the associated far-field hydrodynamic disturbance.
{width="90.00000%"}
Interactions with self-generated chemical gradients cause speed bursts {#interactions-with-self-generated-chemical-gradients-cause-speed-bursts .unnumbered}
======================================================================
It remains to explain the broadening of $P(V)$ with increasing $Pe$ ([Fig. \[fig:statistics\]e]{}), particularly the remarkable bursts in speed for high $Pe$. While the dipolar mode is propulsive, the quadrupolar mode is not. Hence, the growth and decay of the respective modes will affect the droplet speed. As shown in [Fig. \[fig:kymographs\]]{}, recurrent transitions between the two hydrodynamic modes lead to abrupt reorientation events; we therefore investigated the correlation between changes in speed and reorientation angle $|\delta \theta|$.
In a typical trajectory for intermediate $Pe=28$, each sharp turn is preceded by a deceleration and followed by an acceleration, as shown in the plot of the positional data colour-coded by speed in [Fig. \[fig:correlation\]c]{}. Signatures of these correlations in the droplet dynamics appear in the conditional averages of $|\delta\theta|$, $V$ and tangential acceleration $a_t$ for all sharp reorientation events in the trajectory, centered at $t=0$ of maximum $|\delta \theta|$ ([Fig. \[fig:correlation\]a]{}); the events were identified by choosing a threshold value of $|\delta\theta|>0.2$ (see [Fig. \[sifig:Cutoff\]]{}). We can now directly compare these dynamics to the higher resolution fluorescence data taken at $Pe=28$ presented in the kymographs in [Fig. \[fig:kymographs\]b]{}. [Fig. \[fig:correlation\]b]{} shows a series of micrographs of the chemical field, with arrows marking the droplet velocity vector (black) and the position of the secondary filled micelle aggregate (white). The aggregate accumulates, is then entrained and finally merges with the posterior trail, corresponding to the creation and merging of a secondary chemical branch in the kymograph.
For $t<0$ the droplet decelerates while the secondary aggregate is accumulating. $t=0$ marks the point in time where $V$ is minimal and the aggregate is on the cusp of leaving the anterior stagnation point. For $t>0$, the aggregate is advected to the droplet posterior and the droplet accelerates due to the re-saturation of the dipolar mode. $V$ [peaks]{} once the aggregate has merged with the main trail — creating an amplified fore-aft gradient — at $t\approx 1\,$s, which is comparable to the advective timescale $R_d/V\approx1\,$s. In the wide-field data analysis in [Fig. \[fig:correlation\]a]{}, this is the time $\tau_1$ it takes the droplet to reach maximum speed after a reorientation.
We now use the correlation function between $V$ and $|\delta\theta|$, $C_{|\delta\theta|,V}(\Delta t)=\big \langle {|\delta\theta(t)|\cdot V(t+\Delta t)}\big\rangle_{t}$, plotted in [Fig. \[fig:correlation\]d]{}, to estimate the growth times of the second mode from our data for $Pe>10$. Since $V$ is minimal at maximum $|\delta\theta(t)|$ ([Fig. \[fig:correlation\]a]{}), $C_{|\delta\theta|,V}(\Delta t)$ dips at $\Delta t=0$. It subsequently peaks at the point of maximum $V$ with a time delay $\Delta t=\tau_1$, when the contribution of the propulsive dipolar flow is maximal. The next dip at a time $\tau_2>\tau_1$ marks the next reorientation event; based on the discussion pertaining to [Fig. \[fig:kymographs\]]{} and [Fig. \[fig:modestab\]c]{}, for moderate to high $Pe$, $\tau_2-\tau_1$ should correspond to the time scale for the growth and re-saturation of the $n=2$ mode. To test this, we compare the experimentally obtained $\tau_2-\tau_1$ with the theoretical growth times for the $n=2$ mode, $\lambda^{-1}_{n=2}R_d/V_t$ ([Fig. \[fig:modestab\]c]{}), for different values of $Pe$. [Fig. \[fig:correlation\]e]{} shows that indeed these two are of the same order of magnitude and decrease with increasing $Pe$. We note that the growth time of the dipolar flow above $Pe\approx100$ cannot be used for comparison to $\lambda_{n=1}$, since this flow is imposed by the lateral chemical gradient. However, we can assume that this gradient increases with $Pe$, resulting in faster acceleration, markedly higher swimming speeds, and hence, reduced $\tau_1$, as observed experimentally ([Fig. \[fig:correlation\]d]{}).
![\[fig:SAW\][**Anomalous diffusive swimming.**]{} [**a**]{}, Mean squared displacement profiles of experimental trajectories for different $Pe$. Dashed lines mark the predicted scaling for ballistic motion, $\propto t^2$, 2D self-avoiding walk (SAW), $\propto t^{3/2}$, and random walk (RW), $\propto t$. For higher $Pe$, there is a transition from ballistic to 2D SAW. [**b**]{}, A segment of the trajectory associated with the SAW and schematics of the droplet exhibiting bimodal swimming which is responsible for the SAW. Also see [Fig. \[sifig:BimodalSignal\]]{}.](figures/Fig6.png){width="\columnwidth"}
Consequences for spatial exploration {#consequences-for-spatial-exploration .unnumbered}
====================================
Reminiscent of gait switching dynamics in biological locomotion, we have demonstrated the emergence of complex swimming behaviour in a minimal active droplet system by tuning the Péclet number. We found a transition from persistent swimming at low $Pe$ to chaotic bimodal swimming at high $Pe$ — the latter results from the excitation of higher hydrodynamic modes beyond critical $Pe$ values, while the continuous switching between them is caused by the self-generated chemical gradient in the environment.
This gradient sensitivity causes trail avoidance [@jin2017_chemotaxis], which in turn affects the way these droplet swimmers explore their environment. With increasing reorientation frequency, we find a transition from quasi-ballistic propulsion to a 2D self-avoiding walk ([2D SAW]{}). [This effect is illustrated by the trajectories in [Fig. \[fig:statistics\]a-d]{}, and also by the fact that $C_{VV}$ in [Fig. \[fig:statistics\]e]{} does not decay to zero.]{} For a statistical analysis we have plotted mean squared displacements for selected $Pe$ values in [Fig. \[fig:SAW\]a]{}, which reproduce the expected scaling with $t^2$ (ballistic) for $Pe=4$ and a transition to $t^{3/2}$ ([2D SAW]{}, [@slade1994_self-avoiding]) for $Pe\ge28$, with the crossover time decreasing with increasing $Pe$. While transitions to random walks governed by run-and-tumble gait switching are common in bioswimmers [@najafi2018_flagellar], self-avoidance requires chemical self-interaction [@golestanian2009_anomalous]. Examples of anomalous diffusion driven by repulsive biochemical signalling have been found in the spreading of slime molds [@reid2012_slime; @cherstvy2018_non-gaussianity] — active droplets can show analogous behaviour based on purely physicochemical mechanisms.
Outlook {#outlook .unnumbered}
=======
As demonstrated above, the manner in which hydrodynamic and self-generated chemical fields are coupled significantly determines the dynamics of autophoretic micro-swimmers. The fluorescence-based analysis technique used to simultaneously probe these can provide insight into many recent autophoretic models [@hokmabad2019_topological; @izzet2020_tunable; @maass2016_swimming; @izri2014_self-propulsion; @michelin2013_spontaneous; @schmitt2013_swimming; @meredith2019_predator-prey]. We propose this minimal droplet paradigm as a testing bed to investigate a number of current problems in biomimetic swimmers and active matter, where research is primarily theoretical, or experiments have not yielded a clear picture yet.
For example, the capability to approach each other or form bound states is vital to nutrient entrainment, food uptake and mating in bioswimmers. Extensive theoretical studies [@jabbarzadeh2018_viscous; @nasouri2020_exact; @lippera2020_bouncing; @yang2019_autophoresis] have shown the importance of quantifying far-field and near-field contributions, coupling to chemical fields and the effects of confinement in these interactions.
Beyond binary interactions, chemo-hydrodynamic interactions mediate collective behaviours, leading to a variety of clustering and swarming states that can diverge widely even in closely related systems [@thutupalli2018_flow-induced; @kruger2016_dimensionality; @varma2018_clustering-induced; @pohl2014_dynamic].
Unlike many micro-swimmer models which incorporate unsteady dynamics via stochastic fluctuations, the interplay of nonlinear dynamics and interaction with the history of motion allows for the emergence of memory-driven chaotic behaviour. A particularly interesting example are droplet walkers on a vibrated bath [@couder2005_walking], which show a transition from persistent to a bimodal, stop-and-go motion based on an effective ‘system memory’ parameter [@hubert2019_tunable; @valani2019_superwalking]. The corresponding theoretical framework [@hubert2019_tunable] is general enough to also apply to bimodal chaotic motion in droplet swimmers.
We acknowledge fruitful discussions with Stephan Herminghaus, Arnold Mathijssen and Prashanth Ramesh, as well as financial and organisational support from the DFG SPP1726 “Microswimmers” (CCM, RD, BVH), the ERC-Advanced Grant “DDD” (DL, MJ), and the Max Planck Center for Complex Fluid Dynamics.
**Stop-and-go droplet swimmers: Supplementary information**
|
---
abstract: 'We show that the flux-field expansion derived by for the Rothman-Keller immiscible fluid model can be derived in a simpler and more general way in terms of the completely symmetric tensor kernels introduced by those authors. Using this generalised flux-field expansion we show that the more complex amphiphilic model of can also be derived from an underlying model of particle interactions. The consequences of this derivation are discussed in the context of previous equilibrium Ising-like lattice models and other non-equilibrium mesoscale models.'
author:
- |
Peter J. Love\
[Centre for Computational Science,]{}\
[Queen Mary, University of London,]{}\
[Mile End Road, London E1 4NS, U.K.]{}\
\
title: ' **A particulate basis for a lattice-gas model of amphiphilic fluids**'
---
[**Keywords**]{}: Rothman-Keller model, immiscible fluids, surfactants, lattice gases.
Introduction
============
Modelling and simulation of amphiphilic fluids remains a challenging area of physical and computational science. In equilibrium, ternary amphiphilic fluids self-assemble into a multitude of mesophases. The morphology of each mesophase depends on the molecular character of the amphiphile as well as on temperature, concentration, and solvent salinity. In turn, the mesophase morphology influences the nonequilibrium properties of the fluid. Wormlike micellar and sponge phases may exhibit visco-elastic rheological properties, shear induced phase transitions or demixing under flow. For a comprehensive review of amphiphilic behaviour and (mostly equilibrium) modelling techniques see .
These examples illustrate the central challenge posed by the study of amphiphilic fluids: how does one construct models which are sufficiently computationally tractable to reach the time and length scales of interest ($\mu$m - cm) while including sufficient detail from the molecular scale (nm) to describe the behaviour of specific compounds?
The present article goes some way to addressing this issue. In particular, it shows how a previously phenomenological lattice-gas model of amphiphile dynamics can be related to an underlying model of particle interactions. In section \[sec:hlga\] we describe previous hydrodynamic lattice-gas automata models used to simulate binary immiscible and ternary amphiphilic behaviour. In section \[sec:BCE\] we describe the particular collision rules used in the Boghosian, Coveney and Emerton (1996) model. In section \[sec:ce\] we specify the form of the underlying model of particle interactions and obtain an expression for the change in system energy due to both collision and propagation. In section \[sec:ff\] we obtain the generalised form of the flux-field expansion and use it to show that the Boghosian, Coveney and Emerton (BCE) model is a truncation of this expansion at first order.
Hydrodynamic Lattice-Gas Automata {#sec:hlga}
=================================
Lattice-gas automata have been used extensively for modelling hydrodynamics since Frisch, Hasslacher, and Pomeau [@bib:fchc], and Wolfram [@bib:w4] showed that it is possible to simulate the incompressible Navier-Stokes equations using discrete Boolean elements on a lattice. The dynamics of all hydrodynamic lattice gases take place in two substeps: all the particles simultaneously [*propagate*]{} along their velocity vectors to arrive at a new lattice site (retaining their velocity vectors as they do so), and then [*collide*]{} with the other arriving particles. The collisions are required to conserve mass and momentum. Even under the constraints imposed by conservation laws, a multiplicity of collision outcomes are usually possible.
The simplest adaptation of the single phase lattice-gas which enables one to simulate multiphase phenomena is the introduction of an internal degree of freedom (termed ‘colour’) for the particles. For example, if the particle colour takes values $(+1,-1)$, one has two phases, which are typically referred to as red and blue. In order to extend the range of multiphase behaviour captured by the lattice-gas, it is necessary to include colour dependent collisions. This was first done by Rothman and Keller in order to allow simulation of immiscible fluids [@bib:rk]. Rothman and Keller introduced a [*colour field*]{}, which gives information about the composition of the neighbourhood of a site, and a colour flux. Those authors introduced a simple collision rule, choosing collision outcomes based on the colour flux and colour field which have the effect of inducing cohesion in the red and blue phases.
The Rothman and Keller model has been used to investigate surface tension and interfacial fluctuations [@bib:adler], spinodal decomposition in both two and three dimensions [@bib:rk; @bib:appert], and the effect of shear on phase separation in three dimensions [@bib:olroth]. generalised the Rothman-Keller model by including point surfactant particles, possessing a colour dipole. Those authors introduced three additional terms into the Rothman-Keller model. The form of these terms was based on the dipole-charge, charge-dipole and dipole-dipole interactions. The terms were then combined in a local Hamiltonian with arbitrary parameters, which were subsequently fixed by an extensive parameter search for canonical amphiphilic behaviour (micellisation). The derivation of their local amphiphilic Hamiltonian is described in [@bib:bce] and its form is given below. This model has been extensively studied in both two and three dimensions, and has been very successful in capturing a wide range of both equilibrium and non-equilibrium surfactant behaviour [@bib:em1; @bib:em2; @bib:em3; @bib:em4; @bib:emshear; @bib:bcp; @bib:molsim; @bib:LCB; @bib:LCB2; @bib:LCB3]. However, the model lacks any clear connection to a molecular description of surfactants, and as such the connection between different parameterisations of the model and different surfactant compounds remains unclear. Additionally, there exist many equilibrium lattice models (see for an overview) which are superficially very similar to the Boghosian, Coveney and Emerton (BCE) model, and it is unclear how the equilibrium phase diagram of the lattice gas model is related to the equilibrium phase diagram of those models.
Recently Boghosian and Coveney derived the Rothman-Keller model from an underlying model of particle interactions [@bib:micrork]. In the limit where the ratio of the mean free path to the interaction range (denoted by $\epsilon$) is small, those authors showed that an arbitrary potential may be expanded in terms of fluxes and fields. The Rothman Keller model is a truncation of such an expansion at first order in $\epsilon$. This work enables one to reinterpret previous phenomenological lattice gas models in terms of an underlying potential. This has two consequences. Firstly, the interaction potential will have a specific range, which in turn associates a characteristic scale for one lattice spacing. Secondly, the parameterisation of the interaction potential enables one in principle to properly introduce molecular specificity. However, the derivation of the flux-field expansion from the underlying potential obtained for the Rothman and Keller model was of considerable complexity, making the direct extension of this work to more complex local Hamiltonians daunting. In fact, those authors noted that the simplicity of the final result obtained implied that a simpler derivation might exist.
In this paper we show that there is indeed a simpler derivation of Boghosian and Coveney’s result. We write our initial potential in terms of the symmetric tensor kernels introduced by Boghosian and Coveney. By directly considering the Taylor expansion of the $mth$ rank symmetric tensor kernel, and utilising a recursion relation between the tensor kernels, we show that the flux-field expansion may be obtained much more directly. In addition, this derivation allows us to naturally generalise to the higher order tensor fluxes required for the amphiphilic lattice gas. The local amphiphilic Hamiltonian of the BCE model is obtained from this flux-field expansion to first order, in exactly the same way as the Rothman-Keller model was obtained for the binary fluid case.
Lattice-gas amphiphilic fluid dynamics {#sec:BCE}
======================================
In this section we outline the original phenomenological BCE model. In single-phase lattice-gas dynamics a given incoming state $s$ has a set of possible outgoing states $\{s'\}$ which conserve mass and momentum. This set is referred to as the [*equivalence class*]{} of the state $s$. The hydrodynamics will be identical for any choice of $s'$ from $\{s'\}$, but in a multiphase model, the colour dynamics will be different. In the Rothman-Keller model, for each member of $\{s'\}$, a colour flux is computed, and the collision process is modified such that the notional colour work done by the flux against the field is minimised. Chan and Liang introduced a sampling procedure over the equivalence class members such that $s'$ is sampled from a probability density ${{\cal P}}(s')$, which is modelled as the Gibbsian equilibrium corresponding to a Hamiltonian $H(s')$: $${{\cal P}}(s')
=
\frac{1}{{{\cal Z}}}\exp\left[-\beta H(s')\right], \label{eq:beta_defn}$$ where $\beta$ is an inverse temperature, $H(s')$ is the energy associated with collision outcome $s'$, and ${{\cal Z}}$ is the equivalence-class partition function. Whereas the Rothman-Keller model had a single term in the Hamiltonian, the BCE model has three additional terms, capturing the interaction of surfactant particles at a site with neighbouring colour charges, the interaction of colour charges at a site with neighbouring surfactant particles, and the interaction of surfactants with neighbouring surfactants. It is these four terms in the Hamiltonian which we systematically obtain in the remainder of the paper.
Collisional Energetics {#sec:ce}
======================
We denote the postcollision charge-like and dipolar attribute with velocity ${{\bf c}}_i$ at site ${{\bf x}}$ by $q'_i({{\bf x}})$ and ${{\bf p}}'_i({{\bf x}})$. Upon subsequent propagation, the charges and dipoles $q'_i({{\bf x}})$, ${{\bf p}}'_i({{\bf x}})$ will be at position ${{\bf x}}+{{\bf c}}_i$ and the charge $q'_j({{\bf x}}+{{\bf y}})$ and dipole ${{\bf p}}'_j({{\bf x}}+ {{\bf y}})$ will be at position ${{\bf x}}+{{\bf y}}+{{\bf c}}_j$. This is illustrated in Fig. \[fig:dp\]. We take as our starting point the BCE potential energy for a set of point charges and point dipoles: $$\begin{aligned}
V
&=&
\sum_{{{\bf x}},{{\bf y}}}\sum_{i,j}{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\left(q_i({{\bf x}})q_j({{\bf x}}+{{\bf y}})\phi(|{{\bf y}}|)\right)\nonumber\\
&+& {\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\left[q_i({{\bf x}}) {{\bf p}}_j({{\bf x}}+{{\bf y}}) \cdot {{\bf y}}\phi_1(|{{\bf y}}|) - q_j({{\bf x}}+{{\bf y}}) {{\bf p}}_i({{\bf x}}) \cdot {{\bf y}}\phi_1(|{{\bf y}}|)\right]\nonumber\\
&-& {\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\left[{{\bf p}}_j({{\bf x}}){{\bf p}}_j({{\bf x}}+{{\bf y}}):{{\bf y}}{{\bf y}}\phi_2(|{{\bf y}}|)+ {{\bf p}}_j({{\bf x}}){{\bf p}}_j({{\bf x}}+{{\bf y}}):{\bf 1}\phi_1(|{{\bf y}}|)\right]\nonumber.\\
\label{eq:pot}\end{aligned}$$ Where we have defined the following functions related to the derivatives of $\phi (y)$: $$\phi_m(y)\equiv
\left(\frac{1}{y}\frac{d}{dy}\right)^m
\phi(y).$$ In this case there are three contributions to the potential energy: the first term is the charge-charge interaction of the RK model, the second is the charge dipole interaction and the third term is the dipolar-dipolar interaction. We have explicitly separated the charge-dipole contribution into two terms, to make it clear that vector $i$ at site ${{\bf x}}$ may be occupied by either a charge or a dipole. The factor of $1/2$ premultiplying this term compensates for double counting. We now introduce the completely symmetric $m$th rank kernel defined by Boghosian and Coveney: $$\begin{aligned}
{{\cal K}}_n ({{\bf y}}) & = &
\sum_{m=\lceil n/2 \rceil}^n
\frac{\phi_m(y)}{(n-m)!}per
\left[
\left(\bigotimes^{2m-n}{{\bf y}}\right)
\otimes
\left(\bigotimes^{n-m}{{\bf 1}}\right)
\right].\end{aligned}$$ where “per” indicates a summation over all distinct permutations of indices. This enables us to write the BCE potential energy in a particularly compact form: $$\begin{aligned}
V
&=&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}},{{\bf y}}}\sum_{i,j}
q_i({{\bf x}})q_j({{\bf x}}+{{\bf y}}){{\cal K}}_0 ({{\bf y}})\nonumber\\
&+& \left[q_j({{\bf x}}+ {{\bf y}}) {{\bf p}}_i({{\bf x}}) - q_i({{\bf x}}) {{\bf p}}_j({{\bf x}}+{{\bf y}})\right] \cdot {{\cal K}}_1({{\bf y}})\nonumber\\
&-& {{\bf p}}_j({{\bf x}}){{\bf p}}_j({{\bf x}}+{{\bf y}}): {{\cal K}}_2({{\bf y}})\nonumber\\
\label{eq:pot2}\end{aligned}$$ The change in the potential energy due to [*both*]{} collision and propagation is then given by: $$\begin{aligned}
\Delta V_{tot}
&=&
\Delta V_{n} + \Delta V_{c}\nonumber\\
\Delta V_{n}
&=&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}},{{\bf y}}}\sum_{i,j}\left[q'_i({{\bf x}})\Delta {{\bf p}}'_j({{\bf x}}+{{\bf y}})- q'_j({{\bf x}}+ {{\bf y}})\Delta {{\bf p}}'_i({{\bf x}})\right]\cdot {{\cal K}}_1 ({{\bf y}})\nonumber\\
&-&
\Delta\left[{{\bf p}}'_j({{\bf x}}){{\bf p}}'_j({{\bf x}}+{{\bf y}})\right]:{{\cal K}}_2({{\bf y}})\nonumber\\
\Delta V_{c}
&=&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}},{{\bf y}}}\sum_{i,j}q'_i({{\bf x}})q'_j({{\bf x}}+{{\bf y}})\Delta {{\cal K}}_0 ({{\bf y}})\nonumber\\
&+&
\left[q'_i({{\bf x}}){{\bf p}}'_j({{\bf x}}+{{\bf y}}) - q'_j({{\bf x}}+{{\bf y}}){{\bf p}}'_i({{\bf x}})\right]\cdot \Delta {{\cal K}}_1 ({{\bf y}})\nonumber\\
&+&
{{\bf p}}'_j({{\bf x}}){{\bf p}}'_j({{\bf x}}+{{\bf y}}) : \Delta{{\cal K}}_2({{\bf y}})\nonumber,\\
\label{eq:dv}\end{aligned}$$ where we have separated the contribution to $\Delta V$ due to the movement of the interacting particles, $\Delta V_c$, and that due to nonconservation of the dipole direction in collisions, $\Delta V_n$.
(210,225)(50,25) (100,60) (60,120) (180,180) (260,180) (100,60)[(4,3)[160]{}]{} (100,60)[(-2,3)[40]{}]{} (60,120)[(2,1)[120]{}]{} (260,180)[(-1,0)[80]{}]{} (100,60)[( 1, 0)[20]{}]{} (100,60)[( 2, 3)[10]{}]{} (100,60)[(-2, 3)[10]{}]{} (100,60)[(-1, 0)[20]{}]{} (100,60)[(-2,-3)[10]{}]{} (100,60)[( 2,-3)[10]{}]{} (60,120)[( 1, 0)[20]{}]{} (60,120)[( 2, 3)[10]{}]{} (60,120)[(-2, 3)[10]{}]{} (60,120)[(-1, 0)[20]{}]{} (60,120)[(-2,-3)[10]{}]{} (60,120)[( 2,-3)[10]{}]{} (180,180)[( 1, 0)[20]{}]{} (180,180)[( 2, 3)[10]{}]{} (180,180)[(-2, 3)[10]{}]{} (180,180)[(-1, 0)[20]{}]{} (180,180)[(-2,-3)[10]{}]{} (180,180)[( 2,-3)[10]{}]{} (260,180)[( 1, 0)[20]{}]{} (260,180)[( 2, 3)[10]{}]{} (260,180)[(-2, 3)[10]{}]{} (260,180)[(-1, 0)[20]{}]{} (260,180)[(-2,-3)[10]{}]{} (260,180)[( 2,-3)[10]{}]{} (98,70)[${{\bf x}}$]{} (250,158)[${{\bf x}}+{{\bf y}}$]{} (160,200)[${{\bf x}}+{{\bf y}}+{{\bf c}}_j$]{} (50,140)[${{\bf x}}+{{\bf c}}_i$]{} (175,123)[${{\bf y}}$]{} (80,95)[${{\bf c}}_i$]{} (215,172)[${{\bf c}}_j$]{} (120,142)[${{\bf y}}+{{\bf c}}_j-{{\bf c}}_i$]{}
The Flux-Field Decomposition {#sec:ff}
============================
Boghosian and Coveney proceeded by expanding $\Delta {{\cal K}}_0({{\bf y}})$ in a Taylor series in the ratio of the characteristic lattice spacing $c$ to the characteristic interaction range $y$. We wish to expand $\Delta {{\cal K}}_0({{\bf y}})$, $\Delta {{\cal K}}_1({{\bf y}})$ and $\Delta {{\cal K}}_2({{\bf y}})$ in a similar way. These first and second rank tensors are functions of the vector ${{\bf y}}$, not only of its magnitude. The Taylor expansion of the $m$th rank kernel is given by: $$\label{eq:taylordef}
{{\cal K}}_m({{\bf y}}+ {{\bf c}}) = \sum_{n=1}^\infty \frac{1}{n!} ({{\bf c}}\cdot \nabla)^n {{\cal K}}_m({{\bf y}}).$$ This may be written as a sum over $n$th rank inner products of $n$th rank outer products: $$\label{eq:taylordef2}
{{\cal K}}_m({{\bf y}}+ {{\bf c}}) = \sum_{n=0}^\infty \frac{1}{n!} \biggl( \bigotimes^n \nabla \biggr) {{\cal K}}_m({{\bf y}}) \bigodot^n \bigotimes^n {{\bf c}},$$ enabling one to use the property of the $n$th tensor derivative of the $m$th rank kernel (which is proved in Appendix \[app:proof\]): $$\label{eqn:recursion}
\biggl (\bigotimes^n \nabla \biggr) {{\cal K}}_m({{\bf y}}) = {{\cal K}}_{m+n}({{\bf y}}).$$ Letting ${{\bf c}}= {{\bf c}}_j - {{\bf c}}_i$ in equation (\[eq:taylordef\]): $$\begin{aligned}
{{\cal K}}_m({{\bf y}}+ {{\bf c}}_j - {{\bf c}}_i)
&=&
\sum_{n=1}^\infty \frac{1}{n!} \left[({{\bf c}}_j - {{\bf c}}_i)\cdot \nabla\right]^n {{\cal K}}_m({{\bf y}})\end{aligned}$$ and binomially expanding $\left[({{\bf c}}_j - {{\bf c}}_i)\cdot \nabla\right]^n$: $$\left[({{\bf c}}_j - {{\bf c}}_i)\cdot \nabla\right]^n = \sum_{k=0}^n {{n\choose k}} (-1)^k ({{\bf c}}_j \cdot \nabla)^{n-k}({{\bf c}}_i \cdot \nabla)^{k}{{\cal K}}_m({{\bf y}}),$$ writing $({{\bf c}}_j \cdot \nabla)^{n-k}$ and $({{\bf c}}_i \cdot \nabla)^{k}$ in terms of $n$th rank inner products of $n$th rank outer products as in equation (\[eq:taylordef2\]) gives: $$\begin{aligned}
\label{eq:kexp}
{{\cal K}}_m({{\bf y}}+ {{\bf c}}_j - {{\bf c}}_i)
&=&
\sum_{n=0}^\infty \frac{1}{n!}\sum_{k=0}^n {{n\choose k}} (-1)^k \left( \bigotimes^k {{\bf c}}_i \right) \bigodot^k {{\cal K}}_{m+n}({{\bf y}}) \bigodot^{n-k} \left(\bigotimes^{n-k} {{\bf c}}_j\right).\end{aligned}$$ Using (\[eq:kexp\]) to substitute for $\Delta {{\cal K}}_m({{\bf y}})$ in $\Delta V_c$ give the charge-charge interaction as: $$\begin{aligned}
\Delta V_{cc}
&=&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}}}\sum_{n=1}^\infty \frac{1}{n!}\sum_{k=0}^n {\cal J}^q_k({{\bf x}}) \bigodot^k {{n\choose k}} (-1)^k \sum_{{{\bf y}}} {{\cal K}}_{n}({{\bf y}}) \bigodot^{n-k} {\cal J}^q_{n-k}({{\bf x}}+{{\bf y}}),\nonumber\\
\nonumber\\\end{aligned}$$ and the charge-dipole interaction as: $$\begin{aligned}
\Delta V_{cd}
&=&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}}}\sum_{n=1}^\infty \frac{1}{n!}\sum_{k=0}^n{\cal J}^q_k({{\bf x}})\bigodot^k {{n\choose k}} (-1)^k \sum_{{{\bf y}}} {{\cal K}}_{n+1}({{\bf y}}) \bigodot^{n-k+1} {\cal J}^p_{n-k+1}({{\bf x}}+ {{\bf y}})\nonumber\\
&-&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}}}\sum_{n=1}^\infty \frac{1}{n!}\sum_{k=0}^n{\cal J}^p_{k+1}({{\bf x}})\bigodot^{k+1} {{n\choose k}} (-1)^k \sum_{{{\bf y}}} {{\cal K}}_{n+1}({{\bf y}}) \bigodot^{n-k} {\cal J}^q_{n-k}({{\bf x}}+{{\bf y}})\nonumber\\
\nonumber\\\end{aligned}$$ and the dipole-dipole interaction as: $$\begin{aligned}
\Delta V_{dd}
&=&
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}}}\sum_{n=1}^\infty \frac{1}{n!}\sum_{k=0}^n {\cal J}^p_{k+1}\bigodot^{k+1} {{n\choose k}} (-1)^k \sum_{{{\bf y}}} {{\cal K}}_{n+2}({{\bf y}}) \bigodot^{n-k+1}{\cal J}^p_{n-k+1} ,\nonumber\\
\nonumber\\\end{aligned}$$ where we have defined: $$\begin{aligned}
{\cal J}^p_{k+1}({{\bf x}})
&=&
\sum_{i}{{\bf p}}'_i({{\bf x}})\bigotimes^{k}{{\bf c}}_i\nonumber\\
{\cal J}^q_k({{\bf x}})
&=&
\sum_{i}q'_i({{\bf x}})\bigotimes^k{{\bf c}}_i\nonumber\\\end{aligned}$$ Because these equations are summed over the whole lattice we have a freedom of choice in labelling the particles. This means that the above expressions are invariant under the interchange: $$\begin{aligned}
i &\leftarrow& j\nonumber\\
j &\leftarrow& i\nonumber\\
{{\bf x}}&\leftarrow&{{\bf x}}+{{\bf y}}\nonumber\\
{{\bf y}}&\leftarrow&-{{\bf y}}\label{eq:symsa}\end{aligned}$$ In the first and third terms this means that the $kth$ term and the $n-kth$ term are equal. For the charge dipole terms the $kth$ term in the $q_i {{\bf p}}_j$ series is equal to the $n-kth$ term in the $q_j {{\bf p}}_i$ series. We can therefore drop half of the sum over $k$. To first order in epsilon the change in potential energy due to the motion of the particles is therefore: $$\begin{aligned}
\label{eqn:firstorder}
\Delta V_c^{(1)}
&=&
-{\cal J}^q_1({{\bf x}}) \bigodot^1 \sum_{{{\bf y}}} {{\cal K}}_{1}({{\bf y}}){\cal J}^q_{0}({{\bf x}}+{{\bf y}})\nonumber\\
&-&
{\cal J}^q_1({{\bf x}})\bigodot^1 \sum_{{{\bf y}}} {{\cal K}}_{2}({{\bf y}}) \bigodot^{1} {\cal J}^p_{1}({{\bf x}}+ {{\bf y}})\nonumber\\
&+&
{\cal J}^p_{2}({{\bf x}})\bigodot^{2}\sum_{{{\bf y}}} {{\cal K}}_{2}({{\bf y}}) {\cal J}^q_{0}({{\bf x}}+{{\bf y}})\nonumber\\
&-&
{\cal J}^p_{2}\bigodot^{2}\sum_{{{\bf y}}} {{\cal K}}_{3}({{\bf y}}) \bigodot^{1}{\cal J}^p_{1}.\end{aligned}$$ Introducing the definitions of colour flux: $${\bf J} = {\cal J}^q_1,$$ colour field: $${\bf E} = -\sum_{{{\bf y}}} {{\cal K}}_{1}({{\bf y}}){\cal J}^q_{0}({{\bf x}}+{{\bf y}}),$$ dipolar field: $${\bf P} = -\sum_{{{\bf y}}} {{\cal K}}_{2}({{\bf y}})\cdot{\cal J}^p_{1}({{\bf x}}+{{\bf y}}),$$ dipolar flux tensor: $${\cal J} = {\cal J}^p_2 = \sum_{i}{{\bf p}}'_i({{\bf x}}){{\bf c}}_i,$$ colour field gradient tensor: $${\cal E} = \sum_{{{\bf y}}} {{\cal K}}_{2}({{\bf y}}){\cal J}^q_{0}({{\bf x}}+{{\bf y}}),$$ and dipolar field gradient tensor: $${\cal P} = -\sum_{{{\bf y}}} {{\cal K}}_{3}({{\bf y}})\cdot{\cal J}^q_{1}({{\bf x}}+{{\bf y}}).$$ from the BCE model. Substituting these definitions into the first order expansion (\[eqn:firstorder\]) yields: $$\Delta V_c^{(1)} = {\bf J}\cdot({\bf E} + {\bf P}) + {\cal J}:({\cal E}+{\cal P})$$ which is identically the BCE local Hamiltonian without the terms due to collisional non-conservation of dipolar moment.
It only remains to evaluate $\Delta V_n$. Firstly we retain only those parts of $\Delta V_n$ which distinguish between outgoing states, writing: $$\Delta V_n = \sum_{{{\bf x}},{{\bf y}}}\sum_{i,j} q'_j({{\bf x}}+ {{\bf y}}){{\bf p}}'_i({{\bf x}})\cdot {{\cal K}}_1 ({{\bf y}}) +
{\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\left[{{\bf p}}'_j({{\bf x}}){{\bf p}}'_j({{\bf x}}+{{\bf y}})\right]:{{\cal K}}_2({{\bf y}})$$ Using the definitions (18-23) and defining the total outgoing dipole vector: $${{\bf p}}'({{\bf x}}) = \sum_{i}{{\bf p}}'_i({{\bf x}})$$ we obtain: $$\label{collchange}
\Delta V_n' = {\frac{\mbox{\footnotesize $1$}}{\mbox{\footnotesize $2$}}}\sum_{{{\bf x}}} {{\bf p}}'({{\bf x}}) \cdot \left( 2{\bf E} + {\bf P}\right)$$ This has exactly the form used in the BCE model. The factor of two above, which was not included in subsequent utilisations of the model, arises because they derived the various terms in a local Hamiltonian piecewise, without taking account of double counting considerations. These terms in the local Hamiltonian were then combined with arbitrary coefficients.
Discussion and Conclusions
==========================
Having obtained the BCE model Hamiltonian from an underlying potential, it is now possible to obtain parameterisations of this model for a given form of interaction potential. Equally, it is possible to attempt the inverse problem and use the much studied two- and three-dimensional parameterisations of the model to obtain a form for the interaction potential. Any computational implementation of the model will utilise particular stencils to evaluate the tensor fields and gradients over a finite range. These stencils, together with the form of the Hamiltonian defined here and a given model parameterisation will yield a set of differential equations and boundary conditions for $\phi(y)$. Once obtained, this form of $\phi(y)$ could be utilised in other mesoscale models. Most notably, the amphiphilic Malevanets-Kapral described in [@bib:anatolytern1] is very similar to the BCE model. This raises the interesting possibility of studying the same underlying model of particle interactions in both lattice and off-lattice models. However, previous studies of the BCE used nearest neighbour stencils. The interaction range is therefore comparable to the mean-free path (epsilon is close to one) and so additional studies of the BCE model with longer-range stencils may be necessary to make this comparison.
Additionally, many equilibrium lattice models of amphiphilic systems exist, whose phase diagrams have been obtained by Monte-Carlo simulation. One in particular, that of Matsen and Sullivan utilises the same form for the potential of a set of interacting charges and dipoles as that used here [@bib:matsen1; @bib:matsen2]. A given set of parameters for this equilibrium model can therefore be directly translated into a set of parameters for our non-equilibrium model. This enables the equilibrium phase diagram of a given set of parameters to be calculated by Monte-Carlo simulation, or equivalently the dynamic behaviour calculated for a set of parameters for which the equilibrium phase diagram is already known.
By giving previously phenomenological models an underlying theoretical foundation, the conceptual problems inherent in the lattice-gas approach, and in mesoscale modelling in general, are thrown into sharp relief. Although we have referred to a ‘local Hamiltonian’, in fact the lattice gas algorithm does not pay any attention to energy conservation. One could consider following the same approach as that used for energy-conserving DPD, and attach an internal degree of freedom to the lattice gas particles [@bib:consdpd]. However, such an internal energy changes the nature of the particles from microscopic to thermodynamic entities. Their dynamics should then be required not only to satisfy energy conservation, but also guarantee positive entropy production.
The lattice-gas models studied in [@bib:em1; @bib:em2; @bib:em3; @bib:em4; @bib:emshear; @bib:bcp; @bib:molsim; @bib:LCB; @bib:LCB2; @bib:LCB3] have elements of both interpretations. On the one hand lattice gas surfactant particles are regarded as being genuinely molecular in character, whereas in the collision step outgoing states are sampled over Boltzmann weights constructed from the local Hamiltonian. This sampling procedure effectively puts each lattice site in contact with a heat bath, thereby introducing a thermodynamic aspect to the simulation. This thermodynamics is flawed in three ways. Firstly, as was shown by Chan and Liang, there is a net heat flow between the heat bath and the simulation, despite the fact that they are at the same notional temperature [@bib:chli]. Secondly, the total energy of the heat bath plus system is not conserved, and thirdly the system does not possess a H-function. Put more simply, the thermodynamic aspects of these lattice-gas simulations violate the zeroth, first and second laws of thermodynamics.
Thermodynamically consistent mesoscale models remain elusive. At present only the GENERIC formulation of DPD can claim complete thermodynamic consistency, at the expense of abandoning any molecular level information [@bib:serpep]. Such a completely top-down approach can only be useful for materials for which a widely accepted macroscopic description exists. Such materials include liquid-gas mixtures or nematic liquids, but no commonly agreed upon continuum description exists for many materials, including amphiphiles.
The thermodynamic difficulties of the lattice-gas algorithm described above arise partly from the mixture of microcanonical and canonical elements. In the context of equilibrium Monte-Carlo simulation an approach by Creutz shows how to perform equilibrium Monte-Carlo simulations in which microcanonical and canonical elements are incorporated in a systematic way [@bib:creutz]. It is possible that similar approaches to the non-equilibrium simulations considered here could also yield useful modifications.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Peter Coveney, Bruce Boghosian and Jonathan Chin for productive discussions and CINECA and the EU human mobility program for a MINOS grant during which part of this work was undertaken. Figure 1 is reproduced with permission from .
Proof of recursion relation eqn (\[eqn:recursion\]). {#app:proof}
====================================================
We now prove the recursion relation (\[eqn:recursion\]) by induction. We start by noting that (\[eqn:recursion\]) is trivially true for $n=1, m=0$ and therefore the result $\nabla {{\cal K}}_n({{\bf y}}) = {{\cal K}}_{n+1}({{\bf y}})$ completes our proof of (\[eqn:recursion\]). Writing out the left hand side of this expression and using $\nabla \phi_m(y) = {{\bf y}}\phi_{m+1}(y)$ gives: $$\begin{aligned}
\nabla {{\cal K}}_n({{\bf y}})
&=&
\sum_{m=\lceil n/2 \rceil}^n
\frac{\phi_{m+1}(y)}{(n-m)!}{{\bf y}}per\left[2m-n,n-m \right]\nonumber\\
&+&
\sum_{m=\lceil n/2 \rceil}^n
\frac{\phi_m(y)}{(n-m)!}\nabla per\left[2m-n,n-m \right],\end{aligned}$$ where we have introduced the further shorthand: $$per \left[ \left(\bigotimes^{2m-n}{{\bf y}}\right) \otimes \left(\bigotimes^{n-m}{{\bf 1}}\right) \right] = per \left[2m-n,n-m \right].$$ By substituting $m' = m-1$ in the first term here and considering series for odd and even $n$ separately, we obtain: $$\begin{aligned}
\nabla {{\cal K}}_n({{\bf y}})
&=& \phantom{\sum_{m=\lceil (n+1)/2 \rceil}^{n}}\frac{\phi_{n/2}(y)}{(n/2)!} \nabla per \left[0,m \right]\nonumber\\
&+& \sum_{m=\lceil n/2 + 1 \rceil}^{n}
\frac{\phi_{m}(y)}{(n+1-m)!}\biggl({{\bf y}}per\left[2m-2-n,n+1-m \right]\nonumber\\
&+& (n+1-m)\nabla per\left[2m-n,n-m \right]\biggr)\nonumber\\
&+&
\phantom{\sum_{m=\lceil (n+1)/2 \rceil}^{n}}\phi_n+1(y) {{\bf y}}per\left[n,0 \right].\end{aligned}$$ for even $n$ and: $$\begin{aligned}
\nabla {{\cal K}}_n({{\bf y}})
&=&
\sum_{m=\lceil n/2 + 1 \rceil}^{n}
\frac{\phi_{m}(y)}{(n+1-m)!}\biggl({{\bf y}}per\left[2m-2-n,n+1-m \right]\nonumber\\
&+& (n+1-m)\nabla per\left[2m-n,n-m \right]\biggr)\nonumber\\
&+&
\phantom{\sum_{m=\lceil (n+1)/2 \rceil}^{n}}\phi_n+1(y) {{\bf y}}per\left[n,0 \right].\end{aligned}$$ for odd $n$. Using ${{\bf y}}per\left[n,0 \right] = per\left[n+1,0 \right]$ and $\nabla per \left[0,m \right] = 0$ enables us to reunite these two series, giving: $$\begin{aligned}
\label{nearlythere}
\nabla {{\cal K}}_n({{\bf y}})
&=&
\sum_{m=\lceil (n+1)/2 \rceil}^n \frac{\phi_{m}(y)}{(n+1-m)!} {{\bf y}}per\left[2m+2-n,n+1-m \right] \nonumber\\
&+&
\sum_{m=\lceil (n+1)/2 \rceil}^n \frac{\phi_{m}(y)}{(n+1-m)!}(n+1-m) \nabla per\left[2m-n,n-m \right]\nonumber\\
&+&
\phantom{\sum_{m=\lceil (n+1)/2 \rceil}^n} \phi_{n+1}(y) per\left[n+1,0 \right].\end{aligned}$$
We can simplify this expression if we consider the types of additional permutations of indices in $per \left[2m-n-1,n+1-m \right]$ as compared with those in $per \left[2m-n,n-m \right]$. Clearly, the $(n+1)th$ rank tensor has one additional index, which can be attached either to a ${{\bf y}}$ or to a Krönecker delta. If the new index is attached to a ${{\bf y}}$ there is only one additional permutation, which is taken care of by the first term in (\[nearlythere\]). The second term will result in $2m-n$ terms on differentiation, the new index being combined in each successive term in a Krönecker delta with each of the $2m-n$ indices labelling the ${{\bf y}}$’s in the original expression. These terms capture all the unique ways of assigning the new index to a Krönecker delta.
The prefactor $(n+1-m)$ is necessary as the whole expression is divided by $(n+1-m)!$, which arises as this factor is exactly the number of ways of permuting $n+1-m$ Krönecker deltas, this corresponds to $(n+1-m)!$ equivalent permutations of the indices. These permutations are not generated by the outer derivative, but this factor captures their effect on the resulting tensor, enabling us to write: $$\begin{aligned}
\biggl({{\bf y}}per\left[2m-2+n,n+1-m \right]+ \nabla per\left[2m-n,n-m \right]\biggr)=per \left[2m-n-1,n+1-m \right]\end{aligned}$$ Substitution of this result into (\[nearlythere\]) immediately yields the result $\nabla {{\cal K}}_n({{\bf y}}) = {{\cal K}}_{n+1}({{\bf y}})$ and so completes our proof.
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|
---
abstract: 'We introduce a method called [`TrackIn`]{} that computes the influence of a training example on a prediction made by the model, by tracking how the loss on the test point changes during the training process whenever the training example of interest was utilized. We provide a scalable implementation of [`TrackIn`]{} via a combination of a few key ideas: (a) a first-order approximation to the exact computation, (b) using random projections to speed up the computation of the first-order approximation for large models, (c) using saved checkpoints of standard training procedures, and (d) cherry-picking layers of a deep neural network. An experimental evaluation shows that [`TrackIn`]{} is more effective in identifying mislabelled training examples than other related methods such as influence functions and representer points. We also discuss insights from applying the method on vision, regression and natural language tasks.'
bibliography:
- 'main.bib'
---
Motivation
==========
Deep learning has been used to solve a variety of real-world problems. For instance, the prediction of disease from medical imaging, recommending and ranking content, questions-answering over databases and text corpora, etc. A common form of machine learning is supervised learning, where the model is trained on *labelled* data. For instance, there are models that are used to predict diabetic retinopathy, a degenerative eye condition, from fundus images of the retina (cf. [@jama-dr]). These models are trained on images labelled by ophthalmologists. Labelling processes are prone to error. Furthermore, they can also be somewhat subjective. For diabetic retinopathy, while there as some widely accepted diagnosis guidelines, there is also disagreement across doctors (see for instance eFigure3 in the supplement of [@jama-dr]). Controlling the training data input to the model is one of the main quality knobs to improve the quality of the deep learning model. For instance, such a technique could be used to identify and fix mislabelled data using the workflow described in Section \[sec:approach\].
Our main motivation is to identify practical techniques to improve the analysis of the training data. Specifically, we study the problem of *identifying the **influence** of training examples on the prediction of a test example*.
Related Work
============
[@bien2011; @kim] approach the problem as the problem of identifying small set of prototypical training points, and use those to perform prediction. [@bien2011] uses a set-cover algorithm to pick a few points that cover the rest. [@kim] adopts a clustering approach to learn prototypes during the training process. [@khanna] additionally emphasizes criticisms, i.e., examples that are not adequately predicted by the prototypes. These works don’t study deep learning.
[@Liang; @representer] tackle influential training examples in the context of deep learning. [@Liang], uses a classic technique from robust statistics called *influence functions*. Influence functions mimic the process of tracking the change in an individual prediction when you *drop* an individual training point and retrain. Implementing this directly would be prohibitively expensive. Therefore, the influence functions approach approximates this by using the first and second order optimality conditions. Unfortunately this involves inversion of a Hessian matrix that has a size that is quadratic in the number of model parameters, making the approach costly. [@representer] computes the influence of training point using the representer theorem, which posits that when only the top layer of a neural network is trained with $\ell_2$ regularization, the obtained model parameters can be specified as a linear combination of the post-activation values of the training points at the last layer.
There are related notions of influence used to explain deep learning models that differ in either the target of the explanation or the choice of influencer or both. For instance, [@STY17; @Lundberg2017AUA; @Lime] identify the influence of features on an individual prediction. [@Owen2; @Owen1] identify the influence of features on the overall accuracy (loss) of the model. [@DataValuation; @DataShapley] identify the influence of training examples on the overall accuracy of the model.
Techniques that compute an example similarity measure could also be used instead of influence methods. (One commonly used model-based similarity measure is the distance between activation vectors of the two examples.) The premise is that a training example that is influential for specific test example’s prediction is also likely to be similar to it. However, it is also possible that influential examples do not resemble the test example, but nevertheless affect its prediction via certain model parameters. Furthermore, similarity is a symmetric concept, whereas all influence measures (including [`TrackIn`]{}, the influence functions approach and the representer point approach) are asymmetric. Thus similarity and influence are conceptually different.
The Method
==========
In this section we define [`TrackIn`]{}. We start with an idealized definition to clarify the idea, but this definition will be impractical because it would require that the test examples (the ones to be explained) to be specified at training time. We will then develop practical approximations that resolve this constraint.
Idealized Notion of Influence {#subsection:idealized}
-----------------------------
Let $Z$ represent the space of examples, and we represent training or test examples in $Z$ by the notation $z, z'$ etc. We train predictors parameterized by a weight vector $w \in \mathbb{R}^p$. We measure the performance of a predictor via a loss function $\ell: \mathbb{R}^p \times Z \rightarrow \mathbb{R}$; thus, the loss of a predictor parameterized by $w$ on an example $z$ is given by $\ell(w, z)$.
Given a set of $n$ training points $S = \{z_1, z_2, \ldots, z_n \in Z\}$, we train the predictor by finding parameters $w$ that minimize the training loss $\sum_{i=1}^n \ell(w, z_i)$, via an iterative optimization procedure (such as stochastic gradient descent) which utilizes *one* training example $z_t \in S$ in iteration $t$, updating the parameter vector from $w_t$ to $w_{t+1}$. Then the idealized notion of influence of a particular **training example $z \in S$** on a given **test example[^1] $z' \in Z$** is defined as the total reduction in loss on the test example $z'$ that is induced by the training process whenever the training example $z$ is utilized, i.e. $$\begin{aligned}
{\texttt{TrackInIdeal}}(z, z') = \sum_{t:\ z_t = z} \ell(w_t, z') - \ell(w_{t+1}, z') .\end{aligned}$$
Idealized influence has the appealing property that the sum of the influences of all training examples on a fixed test point $z'$ is exactly the total reduction in loss on $z'$ in the training process:
\[lem:budget-balance\] Suppose the initial parameter vector before starting the training process is $w_0$, and the final parameter vector is $w_T$. Then $$\sum_{i=1}^n {\texttt{TrackInIdeal}}(z_i, z') = \ell(w_{0}, z') - \ell(w_T, z')$$
Our treatment above assumes that the iterative optimization technique operates on one training example at a time. Practical gradient descent algorithms almost always operate with a group of training examples, i.e., a *minibatch*. We cannot extend the definition of idealized influence to this setting, because there is no obvious way to redistribute the loss change across members of the minibatch. In Section \[subsec:mini\], we will define an approximate version for minibatches.
We will term training examples that have a positive value of influence score as **proponents**, because they serve to reduce loss, and examples that have a negative value of influence score as **opponents**, because they increase loss. In [@Liang], proponents are called ’helpful’ examples, and opponents called ’harmful’ examples. We chose more neutral terms to make the discussions around mislabelled test examples more natural. [@representer] uses the terms ’excitory’ and ’inhibitory’, which can be interpreted as proponents and opponents for test examples that are correctly classified, and the reverse if they are misclassified. The distinction arises because the representer approach explains the prediction score and not the loss.
First-order Approximation to Idealized Influence {#subsec:first-order}
------------------------------------------------
Since the step-sizes used in updating the parameters in the training process are typically quite small, we can approximate the change in the loss of a test example in a given iteration $t$ via a simple first-order approximation: $$\begin{aligned}
\ell(w_{t+1}, z') &= \ell(w_t, z') + \nabla \ell(w_t, z') \cdot (w_{t+1} - w_t) \notag \\
& \qquad + O(\|w_{t+1} - w_t\|^2). \label{eq:first-order-apx}\end{aligned}$$ Here, the gradient is with respect to the parameters and is evaluated at $w_t$. Now, if stochastic gradient descent is utilized in training the model, using the training point $z_t$ at iteration $t$, then the change in parameters is $$\label{eq:weight-change-gd}
w_{t+1} - w_t = -\eta_t \nabla \ell(w_t, z_t),$$ where $\eta_t$ is the step size in iteration $t$. Note that this formula should be changed appropriately if other optimization methods (such as AdaGrad, Adam, or Newton’s method) are used to update the parameters. The first-order approximation remains valid, however, as long as a small step-size is used in the update.
For the rest of this section we restrict to gradient descent for concreteness. Substituting the formula in , and ignoring the higher-order term (which is of the order of $O(\eta_t^2)$), we arrive at the following first-order approximation for the change in the loss: $$\ell(w_{t}, z') - \ell(w_{t+1}, z') \approx \eta_t \nabla \ell(w_t, z') \cdot \nabla \ell(w_t, z_t).$$ For a particular training example $z$, we can approximate the idealized influence by summing up this approximation in all the iterations in which $z$ was used to update the parameters. We call this first-order approximation [`TrackIn`]{}, our primary notion of influence. $$\begin{aligned}
{\texttt{TrackIn}}(z, z') = \sum_{t:\ z_t = z} \eta_t \nabla \ell(w_t, z') \cdot \nabla \ell(w_t, z).
\label{eq:trackin}\end{aligned}$$
Extension to Mini-batches {#subsec:mini}
-------------------------
It is common to use mini-batches comprising a number of training data points in each iteration of the optimization process. We can extend the above derivation to mini-batches of size $b \geq 1$. We compute the influence of a mini-batch on the test point $z'$, mimicking the derivation in Section \[subsection:idealized\], and then take its first-order approximation as follows: $$\begin{aligned}
&\text{First-Order Approximation}(B_t, z') \\
&= \frac{1}{b}\sum_{z \in B_t} \eta_t \nabla \ell(w_t, z') \cdot \nabla \ell(w_t, z), \end{aligned}$$
because the gradient for the mini-batch $B_t$ is $\frac{1}{b}\sum_{z \in B_t} \nabla \ell(w_t, z)$. Then, for each training point $z \in B_t$, we attribute the $\frac{1}{b} \cdot \eta_t \nabla \ell(w_t, z') \cdot \nabla \ell(w_t, z)$ portion of the influence of $B_t$ on the test point $z'$. Summing up over all iterations $t$ in which a particular training point $z$ was chosen in $B_t$, we arrive at the following definition of [`TrackIn`]{} when mini-batches are used in training:
$$\begin{aligned}
{\texttt{TrackIn}}(z, z') = \frac{1}{b}\sum_{t:\ z \in B_t} \eta_t \nabla \ell(w_t, z') \cdot \nabla \ell(w_t, z). \end{aligned}$$
\[rem:approximation\] The derivation suggests a way to measure the goodness of the approximation for a given step: We can check that the change in loss for a step $\ell(w_t, z') - \ell(w_{t+1}, z')$ is approximately equal to $\textrm{First-Order Approximation}(B_t, z')$.
Random Projection Approximation {#subsection:projection}
-------------------------------
Modern deep learning models frequently have a huge number of parameters, making the inner product computations in the first-order approximation of the influence expensive, especially in the case where the influence on a number of different test points needs to be computed. In this situation, we can speed up the computations significantly by using the the technique of random projections. This method allows us to pre-compute low-memory sketches of the loss gradients of the training points which can then be used to compute randomized unbiased estimators of the influence on a given test point. The same sketches can be re-used for multiple test points, leading to computational savings. This is a well-known technique (see for example [@random-proj]) and here we give a brief description of how this is done. Choose a random matrix $G \in \mathbb{R}^{d \times p}$, where $d \ll p$ is a user-defined dimension for the random projections (larger $d$ leads to lower variance in the estimators), whose entries are sampled i.i.d. from $\mathcal{N}(0, \frac{1}{d})$, so that $\mathbb{E}[G^\top G] = I$. We compute the following sketch: in iteration $t$, compute and save $\eta_t G\nabla \ell(w_t, z_t)$. Then given a test point $z'$, the dot product $(\eta_t G\nabla \ell(w_t, z_t)) \cdot (G\nabla \ell(w_t, z'))$ is an unbiased estimator of $\eta_t\nabla \ell(w_t, z_t)) \cdot (\nabla \ell(w_t, z'))$, and can thus be substituted in all influence computations.
Practical Heuristic Influence via Checkpoints {#subsection:practical}
---------------------------------------------
The method described so far does not scale to typically used long training processes since it involves keeping track of the parameters, as well as training points used, at each iteration: effectively, in order to compute the influence, we need to replay the training process, which is obviously impractical. In order to make the method practical, we employ the following heuristic. It is common to store checkpoints (i.e. the current parameters) during the training process at regular intervals. Suppose we have $k$ checkpoints $w_{t_1}, w_{t_2}, \ldots, w_{t_k}$ corresponding to iterations $t_1, t_2, \ldots, t_k$. We assume that between checkpoints each training example is visited exactly once. (This assumption is only needed for an approximate version of Lemma \[lem:budget-balance\]; even without this, [`TrackInCP`]{} is a useful measure of influence.) Furthermore, we assume that the step size is kept constant between checkpoints, and we use the notation $\eta_i$ to denote the step size used between checkpoints $i-1$ and $i$. While the first-order approximation of the influence needs the parameter vector at the specific iteration where a given training example is visited, since we don’t have access to the parameter vector, we simply approximate it with the first checkpoint parameter vector after it. Thus, this heuristic results in the following formula: $$\begin{aligned}
{\texttt{TrackInCP}}(z, z') = \sum_{i=1}^k \eta_i \nabla \ell(w_{t_i}, z) \cdot \nabla \ell(w_{t_i}, z')
\label{eq:heuristic}\end{aligned}$$
We can use an approximate nearest neighbors technique to quickly identify influential examples for a specific prediction. The idea is to pre-compute the training loss gradients at the various checkpoints (possibly using the random projection trick to reduce space). Then, we concatenate the loss gradients for a given training point $z$ (i.e., $\ell(w_{t_1}, z), \ell(w_{t_2}, z) \ldots \ell(w_{t_k}, z)$) together into one vector. This can be then loaded into an approximate nearest neighbor library(e.g. [@Github:annoy]). During analysis, we can do the same for a test example—the gradient calls for the different checkpoints can be done in parallel. We then invoke nearest neighbor search. The nearest neighbor library then performs the computation implicit in Equation \[eq:heuristic\].
In our derivation of [`TrackIn`]{} we have assumed a certain form of training. In practice, there are likely to be differences in optimizers, learning rate schedules, the handling of mini-batches etc. It should be possible to redo the derivation of [`TrackIn`]{} to handle these differences. Also, we expect the practical form of [`TrackInCP`]{} to remain the same across these variations.
\[rem:selecting\] In the application of ${\texttt{TrackInCP}}$, we choose checkpoints at epoch boundaries, i.e., between checkpoints, each training example is visited exactly once. However, it is possible to be smarter about how checkpoints are chosen: Generally, it makes sense to sample checkpoints at points in the training process where there is a steady decrease in loss, and to sample more frequently when the rate of decrease is higher. It is worth avoiding checkpoints at the beginning of training when loss fluctuates. Also, checkpoints that are selected after training has converged add little to the result, because the loss gradients here are tiny. Relatedly, computing ${\texttt{TrackInCP}}$ with *just* the final model could result in noisy results.
Evaluations
===========
In this section we compare [`TrackIn`]{} with influence functions [@Liang] and the representer point selection method [@representer]. Brief descriptions of these two methods can be found in the supplementary material. We also compare practical implementations of [`TrackIn`]{} against an idealized version.
Conceptual Comparison of the Methods
------------------------------------
Beyond the simplicity of [`TrackIn`]{} compared to the influence functions or representer methods, a primary point of distinction [`TrackIn`]{} is that unlike the other two methods, it makes no optimality assumptions on the trained classifier. The other two methods are only defined if the parameters of the trained classifer satisfy, at the very least, some local optimality conditions. This point is crucial to meaningfully deploy any method in practice, since modern deep learning models are rarely, if ever, trained to even moderate-precision convergence. Regardless, we do compare our approach against both of these approaches in the subsequent sections.
Evaluation Approach {#sec:approach}
-------------------
Real world datasets can contain mislabelled examples, that can harm model performance. One application of any training data influence computation technique is to identify mislabelled examples automatically. This has been used as a form of evaluation (see Section 4.1 [@representer] and Section 5.4 of [@Liang]). The idea is to measure self-influence, i.e., the influence of a training point on its own loss, i.e., the training point $z$ and the test point $z'$ in Equation \[eq:trackin\] are identical.
Incorrectly labelled examples are likely to be strong proponents (recall terminology in Section \[subsection:idealized\]) for themselves. Strong, because they are outliers, and proponents because they would tend to reduce loss (with respect to the incorrect label). Therefore, when we sort training examples by decreasing self-influence, an effective influence computation method would tend to rank mislabelled examples in the beginning of the ranking. We use the fraction of mislabelled data recovered for different prefixes of the rank order as our evaluation metric; higher is better.
To simulate the real world mislabelling errors, we first trained a model on correct data. Then, for 10% of the training data, we changed the label to the highest scoring incorrect label. We then attempt to identify mislabelled examples as discussed above.
CIFAR-10
--------
In this section, we work with ResNet-56 [@he2016deep] trained on the CIFAR-10 [@krizhevsky2009learning]. The model on the original dataset has 93.4% test accuracy. [^2]
### Identifying Mislabelled Examples {#subsection:dataset-debugging}
Recall the evaluation set up and metric in Section \[sec:approach\]. Training on the mislabelled data reduces test accuracy from 93.4% to 87.0% (train accuracy is 99.6%). We compare [`TrackIn`]{} with influence functions (Section \[sec:influence\])and the representer point selection method (Section \[sec:representer\]).
For influence functions, it is prohibitively expensive to compute the Hessian for the whole model, so work with parameters in the last layer, essentially considering the layers below the last as frozen. This mimics the set up in Section 5.1 of [@Liang]. Given that CIFAR-10 only has 50K training examples, we directly compute inverse hessian by definition.
For representer points, we fine-tuned the last layer with line-search, which requires the full batch to find the stationary point and use $|\alpha_{ij}|$ as described in Section 4.1 of [@representer] to compare with self-influence.
We use [`TrackInCP`]{} with only the last layer. We sample every 30 checkpoints starting from the 30th checkpoint; every checkpoint was at a epoch boundary. The right hand side of Figure \[fig:cifar-eval\] shows that [`TrackInCP`]{} identifies a larger fraction of the mislabeled training data (y-axis) regardless of the fraction of the training data set that is examined (x-axis). For instance, [`TrackIn`]{} recovers more than 80% of the mislabelled data in the first 20% of the ranking, whereas the other methods recover less than 50% at the same point. Furthermore, we show that *fixing* the mislabelled data found within a certain fraction of the training data, results in a larger improvement in test accuracy for [`TrackIn`]{} compared to the other methods (see the plot on the left hand side of Figure \[fig:cifar-eval\]). We also show that weighting checkpoints equally yield similar results. This provides support to ignore learning rate for implementation simplification.
### Effect of different checkpoints on TrackIn scores {#sec:effect-different-checkpoints}
Next, we discuss the contributions of the different checkpoints to the scores produced by [`TrackIn`]{}; recall that [`TrackIn`]{} computes a weighted average across checkpoints (see Equation \[eq:heuristic\]). We find that different checkpoints contain different information. We identify the number of mislabelled examples from each class (the true class, not the mislabelled class) within the first 10% of the training data in Fig. \[fig:cifar-ckpt-compare\] (in the supplementary material), we show results for the 30th, 150th and 270th checkpoint. We find that the mix of classes is different between the checkpoints. The 30th checkpoint has a larger fraction (and absolute number) of mislabelled deer and frogs, while the 150th emphasizes trucks. This is likely because the model learns to identify different classes at different points in training process, highlighting the importance of sampling checkpoints.
MNIST {#sec:mnist}
-----
In this section, we work on the MNIST digit classification task. We use a model with 3 hidden layers and 240K parameters. This model has 97.55% test accuracy. Because the model is smaller than the Resnet model we used for CIFAR-10, we can perform a slightly different set of comparisons. First, we are able to compute approximate influence for each training step (Section \[subsec:mini\]), and not just heuristic influence using checkpoints. Second, we can apply [`TrackIn`]{} and the influence functions method to all the model parameters, not just the last layer.
Since we have a large number of parameters, we resort to a randomized sketching based estimator of the influence whose description can be found in the supplementary material. In our experiments, this model would sometimes not converge, and there was significant noise in the influence scores, which are estimating a tiny effect of excluding one training point at a time. To mitigate these issues, we pick lower learning rates, and use larger batches to reduce variance, making the method time-intensive.
### Visual inspection of Proponents and Opponents
We eyeball proponents and opponents of a random sample of test images from MNIST test set. We observe that [`TrackInCP`]{} and representer consistently find proponents visually similar to test examples. Although, the opponents picked by representer are not always visually similar to test example (the opponent ’7’ in Figure \[fig:mnist-correct-3\] and ’5’ and ’8’s in Figure \[fig:mnist-incorrect\]). In contrast, [`TrackInCP`]{} seems to pick pixel-wise similar opponents.
### Identifying mislabelled examples {#identifying-mislabelled-examples}
Recall the evaluation set up and metric in Section \[sec:approach\]. We train on MNIST mislabelled data as described there. After 140 epochs, it achieves accuracy of 89.94% on mislabelled train set, and 89.95% on test set. As in Section \[subsection:dataset-debugging\] we compare against the influence functions method and the representer point method. Similar to CIFAR-10, [`TrackIn`]{} outperforms the other two methods and retrieves a larger fraction of mislabelled examples for different fractions of training data inspected (Figure \[fig:mnist-mislabelled\]). Furthermore, as expected, batch approximate [`TrackIn`]{} is able to recover mislabelled examples faster than heuristic [`TrackInCP`]{} (we use every 30th checkpoint, starting from 20th checkpoint), but not by a large margin.
Next, we evaluate the effects of our various approximations. We use the same 3-layer model architecture, but with the correct MNIST dataset. The model has 97.55% test set accuracy on test set, and 99.30% train accuracy.
### Effect of the First-Order Approximation {#sec:effect-first-order}
We now evaluate the effect of the first-order approximation (described in Sections \[subsec:mini\] and \[subsec:first-order\]). By Remark \[rem:approximation\], we would like the total first-order influence at a step $\textrm{First-Order Approximate Influence}(B_t, z')$ to approximate the change in loss at the step $\ell(w_t, z') - \ell(w_t, z')$. Figure \[fig:delta-loss-influence\] (in the supplementary material) shows the relationship between the two quantities; every point corresponds to one parameter update step for one test point. We consider 100 random test points. The overall Pearson correlation between the two quantities is 0.978, which is sufficiently high.
![Analysis of effect of approximations with Pearson correlation of first order approximate TrackIn influences with heuristic influences over multiple checkpoints and with projections of different sizes.[]{data-label="fig:correlations"}](./figures/influence_correlations_all){width="1.0\linewidth" height="0.15\textheight"}
### Effect of checkpoints {#subsec:ckpts-effect}
We now discuss the approximation from Section \[subsection:practical\], i.e., the effect of using checkpoints. We compute the correlation of the influence scores of 100 test points using [`TrackInCP`]{} with different checkpoints against the scores from the first-order approximation [`TrackIn`]{}. As discussed in Remark \[rem:selecting\], we find that selecting checkpoints with high loss reduction, are more informational than selecting same number of evenly spaced checkpoints. This is because in later checkpoints the loss flattens, hence, the loss gradients are small. Figure \[fig:correlations\] shows [`TrackInCP`]{} with just one checkpoint from middle correlates more than the last checkpoint with [`TrackIn`]{} scores. Consequently, [`TrackInCP`]{} with more checkpoints improves the correlation, more so if the checkpoints picked had high loss reduction rates.
### Effect of Random Projections
As mentioned in Section \[subsection:projection\], gradient matrices can be projected onto much smaller dimensions to speed-up computation and to save space. We study the effect of this approximation, by layering this approximation on top of the checkpoint approximation [`TrackInCP`]{} (we use six checkpoints). Figure \[fig:correlations\] also shows correlations for random projections of different numbers of dimensions. No projection has a 0.889 median correlation, while using projection of dimension 1000 shifts the median to 0.880 while providing 240:1 compression of the gradient matrices.
Applications
============
We apply [`TrackIn`]{} to a regression problem (Section \[sec:houses\]) a text problem (Section \[sec:text\]) and an computer vision problem (Section \[sec:images\]) to demonstrate its versatility. The last of these use cases is on a ResNet-50 model trained on the (large) Imagenet dataset, demonstrating that [`TrackIn`]{} scales.
\[fig:california\]
\[tab:text-samples\]
California Housing Prices {#sec:houses}
-------------------------
We study [`TrackIn`]{} on a regression problem using California housing prices dataset [@california]. We used a 80:20 train-test split and trained a regression model with 3 hidden layers with 168K parameters, using Adam optimizer minimizing MSE for 200 epochs. The model achieves explained variance of 0.70 on test set, and 0.72 on train set. We use every 20th checkpoint to get [`TrackIn`]{} influences.
The notion of comparables in real estate refers to recently sold houses that are similar to a home in location, size, condition and features, and are therefore indicative of the home’s market value. We can use [`TrackInCP`]{} to identify model-based comparables, by examining the proponents for certain predictions. For instance, we could study proponents for houses in the city of Palo Alto, a city in the Bay Area known for expensive housing. We find that the proponents are drawn from other areas in the Bay Area, and the cities of Sacramento, San Francisco and Los Angeles (Figure \[fig:palo-alto\]). One of the influential examples lies on the island of Santa Catalina, also known for expensive housing.
We also study self-influences of training examples (see Section \[sec:approach\] for the definition of self-influence). High self-influence is more likely to be indicative of memorization. We find that the high self influence examples come from densely populated locations, where memorization is reasonable, and conversely, low self-influence ones comes from sparsely populated areas,where memorization would hurt model performance (Figure \[fig:cal-self-influences\]).
Text Classification {#sec:text}
-------------------
We apply [`TrackIn`]{} on the DBPedia ontology dataset introduced in [@zhang2015character]. The task is to predict the ontology with title and abstract from Wikipedia. The dataset consists of 560K training examples and 70K test examples equally distributed among 14 classes. We train a Simple Word-Embedding Model (SWEM) [@shen2018baseline] for 60 epochs and use the default parameters of sentencepeice library as tokenizer [@kudo2018sentencepiece] and achieve 95.5% on both training and test. We apply [`TrackInCP`]{}and sample 6 evenly spaced checkpoints and the gradients are taken with respect to the last fully connected layer.
Table \[tab:text-samples\] shows the top 3 opponents for one test example (Manuel Azana); we filter misclassified training examples from the list to find a clearer pattern. (Misclassified examples have high loss, and therefore high training loss gradient, and are strong proponents/opponents for different test examples, and are thus not very disciminative.) The list of opponents provide insight about data introducing correlation between politicians and artists.
Imagenet Classification {#sec:images}
-----------------------
Real world applications tend to be large in data size. Methods approximating influence of training data should be able to scale at least linearly as the data size increases. It seems difficult to scale influence functions or representer methods to ImageNet. However, using a special random projection idea it becomes possible to apply [`TrackIn`]{} on the fully connected layer of ResNet-50 trained on Imagenet [@deng2009imagenet][^3], which consists of 1.28M training examples with 1000 classes. The 30th, 60th, and 90th checkpoints are used for [`TrackInCP`]{} and we project the gradients to a vector of size 1472. The random projection idea relies on the fact that for fully-connected layers, the gradient of the loss w.r.t. the weights for the layer is a rank 1 matrix. Thus, [`TrackIn`]{} involves computing the dot (Hadamard) product of two rank 1 matrices, for which much faster random projection based estimators than the ones described in Section \[subsection:projection\] exist. Details are in the supplementary material.
We show three proponents and three opponents for five examples in figure \[fig:imagenet\]. We filtered out misclassified examples as we did for text classification. A few quick observations: (i) The proponents are mostly images from the same label. (ii) In the first row of figure \[fig:imagenet\], the style of the microphone in the test example is different from the top proponents, perhaps augmenting the data with more images that resemble the test one can fix the misclassification. (iii) For the correctly classified test examples, the opponents give us an idea which examples would confuse the model (for the church, there are castles, for the bostonbull there are french bulldogs, for the wheel there are loupes and spotlights, and for the chameleon there is a closely related animal (agama) but there are also broccoli and jackfruits.
Description of Influence Functions and Representer Point Methods
================================================================
Influence Functions {#sec:influence}
-------------------
@Liang proposed using the idea of Influence functions [@cook1982residuals] to measure the influence of a training point on a test example. Specifically, they use optimality conditions for the model parameters to mimic the effect of perturbing single training example:
$$\begin{aligned}
\textrm{Inf}&(z, z') = - \nabla_{\hat{w}} \ell(\hat{w}, z') \cdot H^{-1}_{\hat{w}} \cdot \nabla_{\hat{w}} \ell(\hat{w}, z).
\label{eq:influence-functions}\end{aligned}$$
Here, $H_{\hat{w}} = \frac{1}{n} \sum_{1}^{n} \nabla^2 \ell(\hat{w}, z_i)$ is the Hessian. As pointed out by @Liang, for large deep learning models with massive training sets, the inverse Hessian computation is costly and complex. This technique also assumes that the model is at convergence so that the optimality conditions hold.
### Scalable implementation via randomized sketching
It becomes infeasible to compute the inverse Hessian when the number of parameters is very large, as is common in modern deep learning models. To mitigate this issue we compute randomized estimators of $H^{-1}_{\hat{w}}\nabla_{\hat{w}} \ell(\hat{w}, z')$ via a [*sketch*]{} of the inverse Hessian in the form of the product $H^{-1}_{\hat{w}}G^\top$ where $G$ is the same kind of random matrix as in Section \[subsection:projection\]. The product $[H^{-1}_{\hat{w}}G^\top][G\nabla_{\hat{w}} \ell(\hat{w}, z')]$ is then an unbiased estimator of $H^{-1}_{\hat{w}}\nabla_{\hat{w}} \ell(\hat{w}, z')$. Note that the sketch takes only $O(dp)$ memory rather than $O(p^2)$ that the inverse Hessian would take. We compute the sketch by solving the optimization problem $\min_S \|H_{\hat{w}}S - G^\top\|_F^2$, via a customized stochastic gradient descent procedure based on the formula $$\nabla_S \|H_{\hat{w}}S - G^\top\|_F^2 = 2H_{\hat{w}}(H_{\hat{w}}S - G^\top).$$ This customized stochastic gradient descent procedure uses the following stochastic gradient computed using [*two*]{} indepdently chosen minibatches of examples $B_1, B_2$ instead of the customary one: $$\label{eq:sketch-gradient}
2[\tfrac{1}{|B_1|}\textstyle\sum_{z \in B_1} \nabla^2 \ell(\hat{w}, z)][\tfrac{1}{|B_2|}\textstyle\sum_{z \in B_2} \nabla^2 \ell(\hat{w}, z)S - G^\top].$$ Note that $\mathbb{E}[\tfrac{1}{B_1}\textstyle\sum_{z \in B_1} \nabla^2 \ell(\hat{w}, z)] = H_{\hat{w}}$ and $\mathbb{E}[\tfrac{1}{|B_2|}\textstyle\sum_{z \in B_2} \nabla^2 \ell(\hat{w}, z)] = H_{\hat{w}}$, and since $B_1$ and $B_2$ are independently chosen, we conclude that the expectation of the quantity in is indeed $2H_{\hat{w}}(H_{\hat{w}}S - G^\top)$ as required. Note that can be computed using Hessian-vector products, which can be computed easily using the Pearlmutter trick [@Pearlmutter94fastexact].
Representer Point Selection {#sec:representer}
---------------------------
The second method is proposed in [@representer] and is based on the representer point theorem [@representer_theorem]. The method decomposes the logits for any test point into a weighted combination of dot products between the representation of the test point at the top layer of a neural network and those of the training points; this is effectively a kernel method. The weights in the decomposition capture the influences of that training points.
Specifically, consider a neural network model with fitted parameters into $\{w_1, w_2\}$, where $w_2$ is the matrix of parameters that produces the logits from the input representation (i.e. the top layer weights) and $w_1$ are the remaining parameters. To meet the conditions of the representer theorem, the final layer of the model is tuned by adding a term L2 regularization term $\lambda {\left\lVertw_2\right\rVert}^2$ to the loss and training the model to convergence. This optimization produces a new set of parameters $w'_2$ for the last layer, resulting in a new model with parameters $w' = \{w_1, w'_2\}$. Then the influence of a training example $z$ on a test example $z'$ is a $k$-dimensional vector (one element per class) given by $$\begin{aligned}
\textrm{Rep}&(z, z') = \notag \\ &-\frac{1}{2\lambda n} (f(w_1, z) \cdot f(w_1, z')) \partial_{\phi(w', z')} \ell(w', z').
\label{eq:influence-functions}\end{aligned}$$
Here, $f(w_1, z)$ is the input representation, i.e. the outputs of the last hidden layer, and $\phi(w', z) = w_2'f(w_1, z)$ are the logits. The L2 regularization requires a complex, memory-intensive line search, and results in a model different from the original one, possibly resulting in influences that are unfaithful to the original model. Conceptually, it is also not clear how to study the influence that flows via the parameters in lower layers—computing a stationary point is harder in this situation. Furthermore, both the influence functions approach and [`TrackIn`]{} could be used to explain the influence of a training example on the loss of a test example or its prediction score. In contrast, it is unclear how to use the representer point method to explain loss on a test example.
A Visual Inspection of Proponents and Opponents for CIFAR {#sec:cifar-proponents}
=========================================================
We now consider the same training procedure in Section 4.1 but on the regular CIFAR-10 dataset. We show the top 5 proponent and opponent examples of an image from the test set and compare the three methods qualitatively in Figures \[fig:correct\] and \[fig:incorrect\]. All three methods retrieved mostly cats as positive examples and dogs as negative examples, but [`TrackIn`]{} seems more consistent on the types of cats and dogs. For the mis-classified automobile, proponents of [`TrackIn`]{} pick up automobiles of a similar variety type.
Fast Random Projections for Gradients of Fully-Connected Layers
===============================================================
Suppose we have a fully connected layer in the neural network with a weight matrix $W \in \mathbb{R}^{m \times n}$, where $m$ is the number of units in the input to that layer, and the $n$ is the number of units in the output of the layer. For the purpose of [`TrackIn`]{} computations, it is possible to obtain a random projection of the gradient w.r.t. $W$ into $d$ dimensions with time and space complexity $O((m + n) \cdot \sqrt{d})$ rather than the naive $O(mnd)$ complexity that the standard random projection needs.
To formalize this, let us represent the layer as performing the following computation: $y := Wx$ where $x \in \mathbb{R}^n$ is the input to the layer, and $y$ is the vector of pre-activations (i.e. the value fed into the activation function). Now suppose we want to compute the gradient of some function $f$ (e.g. loss, or prediction score) of the output of the layer, i.e. we want to compute $\nabla_W(f(Wx))$. A simple application of the chain rule shows gives the following formula for the gradient: $$\nabla_W(f(Wx)) = \nabla_y f(y) x^\top.$$ In particular, note that the gradient w.r.t. $W$ is rank 1. This property is very useful for [`TrackIn`]{} since it involves computations of the form $\nabla_W(f(Wx)) \cdot \nabla_W(f'(Wx'))$, where $f'$ is another function and $x'$ is another input. Note that for $y' = Wx'$, we have $$\begin{aligned}
&\nabla_W(f(Wx)) \cdot \nabla_W(f'(Wx')) \\
&= (\nabla_y f(y) x^\top) \cdot (\nabla_{y'} f'(y') {x'}^\top)\\
&= (\nabla_y f(y) \cdot \nabla_{y'} f'(y')) (x \cdot x'). \end{aligned}$$ The final expression can be computed in $O(m + n)$ time by computing the two dot products $(\nabla_y f(y) \cdot \nabla_{y'} f'(y'))$ and $(x \cdot x')$ separately and then multiplying them. This is much faster than the naive dot product of the gradients, which takes $O(mn)$ time.
This can already speed up [`TrackIn`]{} computations. We can also save on space by randomly projecting $\nabla_y f(y)$ and $x$ separately, but unfortunately this doesn’t seem to be amenable to fast nearest-neighbor search. If we want to use fast nearest-neighbor search, we will need to use random projections in the following manner which also exploits the rank-1 property. To project into $d$ dimensions, we can use two independently chosen random projection matrices $G_1 \in \mathbb{R}^{\sqrt{d} \times m}$ and $G_2 \in \mathbb{R}^{\sqrt{d} \times n}$, with $\mathbb{E}[G_1G_1^\top] = \mathbb{E}[G_2G_2^\top] = I$, and compute $$G_1 \nabla_y f(y) x^\top G_2^\top \in \mathbb{R}^{\sqrt{d} \times \sqrt{d}},$$ which can be flattened to a $d$-dimensional vector. Note that this computation requires time and space complexity $O((m + n) \cdot \sqrt{d})$. Furthermore, since $G_1$ and $G_2$ are chosen independently, it is easy to check that $$\begin{aligned}
&\mathbb{E}[(G_1 \nabla_y f(y) x^\top G_2^\top) \cdot (G_1 \nabla_{y'} f'(y') {x'}^\top G_2^\top)] \\
&= (\nabla_y f(y) x^\top) \cdot (\nabla_{y'} f'(y') {x'}^\top), \end{aligned}$$ so the randomized dot-product is unbiased.
Additional Results
==================
This section contains charts and images that support discussions in the main body of the paper.
{width="40.00000%"}
\[tab:text-samples-proponents\]
[^1]: By test example, we simply mean an example whose prediction is being explained. It doesn’t have to be in the test set.
[^2]: All model for CIFAR-10 are trained with 270 epochs with a batch size of 1000 and 0.1 as initial learning rate and the following schedule (1.0, 15), (0.1, 90), (0.01, 180), (0.001, 240) where we apply learning rate warm up in the first 15 epochs.
[^3]: The model is trained for 90 epochs and achieves 73% top-1 accuracy.
|
---
abstract: |
In this paper, we establish a weight identity for stochastic beam equation by means of the multiplier method. Based on this identity, we first establish the global Carleman estimate for the special system with zero initial value and end value, then a revised Carleman estimate for stochastic beam equation is established through a cutoff technique. Finally, we use the revised Carleman estimate to get the required boundary observability estimate.\
**Keywords:** Stochastic beam equation; Carleman estimate; Observability inequality
---
[**Carleman and observability estimates for stochastic beam equation**]{}\
[Maoding Zhen$^{1,2}$,Jinchun He,$^{1,2}$Haoyuan Xu$^{1,2}$ and Meihua Yang$^{1,2,*}$ ]{}\
[1) School of Mathematics and Statistics, Huazhong University of Science and Technology,\
Wuhan 430074, China]{}\
[2) Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology,]{} [Wuhan, 430074, China]{}
Introduction
============
Let $I=[a,b]$ be a closed interval. We are concerned with the following stochastic beam equation $$\label{int1}
\begin{cases}
dy_{t}+y_{xxxx}dt =fdt+gdB(t) &\text{in} \ \ Q=(0,T)\times I,\\
y(a,t)=0,\ y(b,t)=0&\text{in} \ (0,T),\\
y_{x}(a,t)=0,\ y_{x}(b,t)=0 & \text{in}\ (0,T),\\
y(x,0)=y_{0}(x),\ y_{t}(x,0)=y_{1}(x) & \text{in}\ \ I ,
\end{cases}$$ where $\{B(t)\}_{t\ge 0}$ is a one dimensional standard Brownian motion and $y_t=\frac{dy}{dt}$.
The beam equation is one of the main models used to describe the fluid-structure interactions, composite laminates in smart materials, structural-acoustic system and so on (see [@HBW; @WG; @Kr; @ZZ1; @ZZ2] and references therein). Hence, the stabilization and controllability problems for beam equations attracts many authors’ attentions. Note that, observability estimate is an important tool for studying the stabilization and controllability problems for partial differential equations.
For deterministic partial differential equations, the observability problems have been studied by many authors, and there are many approaches to establish the observability inequality, such as the multiplier techniques, the nonharmonic Fourier series techniques, the method based on the microlocal analysis and the global Carleman estimate, which can be regarded as a more developed version of the classical multiplier technique. See [@L; @KP; @BGJ; @ZUA] and references therein for more details.
Compared to the observability problems for deterministic partial differential equations, the stochastic counterparts are more challenging and need further understanding. Because of the time irreversible property of stochastic equations, the global Carleman estimate becomes the main technique to derive observability and controllability for stochastic evolution equations. See [@L2; @L5; @Zh1; @GCL; @TZ1; @G2; @Li] and reference therein.
The main aim of this paper is to provide a boundary observability estimate for the equation . Due to the complexity of the 4th order equation, the global Carleman estimate for the stochastic beam equation is difficult to establish directly. Hence, the method to establish observability estimate by global Carleman estimate is invalid in our case. To overcome this difficulty, we first establish the global Carleman estimate for the special system with zero initial value and end value(see below). Then a revised Carleman estimate(see Theorem \[Th4\] below), can be established through a cutoff technique [@L5], which is still enough for us to get the required observability estimate. As far as we know, our result is new about the stochastic beam equation.
Before we state our main results, we present our notations.
Let $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq0},P )$ be a complete filtered probability space, on which a one dimensional standard Brownian motion $\{B(t)\}_{t\geq0}$ is defined such that $\{\mathcal{F}_{t}\}_{t\geq0}$ is the nature filtration generated by $\{B(t)\}_{t\geq0}$ augmented by all the $P$-null sets in $\mathcal{F}$.
Let $H$ be a Banach space and $C([0,T]; H)$ be the Banach space of all $H$-valued continuous functions defined on $[0,T]$.
We denote by $L^{2}_{\mathcal{F}}(0, T; H)$ the Banach space consisting of all $H$-valued $\{\mathcal{F}_{t}\}_{t\geq0}$ adapted processes $X(\cdot)$ such that $\mathbb{E}(\|X(\cdot)\|^{2}_{L^{2}(0, T; H)})<+\infty$, with the canonical norm; by $L^{\infty}_{\mathcal{F}}(0, T; H)$ the Banach space consisting of all $H$-valued $\{\mathcal{F}_{t}\}_{t\geq0}$ adapted bounded processes; by $L^{2}_{\mathcal{F}}(\Omega;L^{\infty}([0,T];H ))$ the Banach space consisting of all adapted $H$-valued bounded processes such that $\mathbb{E}(\sup\limits_{t\in[0, T]}\|X(t)\|^{2}_{H})<+\infty$ and by $L^{2}_{\mathcal{F}}(\Omega;C([0,T];H ))$ the Banach space consisting of all $H$-valued $\{\mathcal{F}_{t}\}_{t\geq0}$ adapted continuous processes $X(\cdot)$ such that $\mathbb{E}(\|X(\cdot)\|^{2}_{C([0,T ]; H)})<+\infty$ with the canonical norm. See more details in [@L2]. We will use $C$, $C(I)$ or $C(I,T)$ to denote a generic positive constant, a generic positive constant depending only on interval $I=[a,b]$ or on $I$ and $T$ seperateley, which may vary from line to line.
Assume $$\label{con1}
f\in L^{2}_{\mathcal{F}}(0, T; H^{2}(I)),\ g\in L^{\infty}_{\mathcal{F}}(0, T; H^{4}(I)).$$
The main result in this paper is the following:
\[Th1\] Let holds, then there exists a constant $C(I,T)>0$ such that the solution of system satisfies the following observability inequality: $$\begin{aligned}
&\|(y(T),y_{t}(T))\|_{L^{2}(\Omega,\mathcal{F}_{T},P;H_0^{2}(I)\times L^{2}(I))}\\
&\leq C(I,T)\left(\mathbb{E}\int^{T}_{0}(y^2_{xx}(b,t)+y^2_{xxx}(b,t))dt
+\|f\|_{L^{2}_{\mathcal{F}}(0,T;H^{2}(I))}+\|g\|_{L^{\infty}_{\mathcal{F}}(0,T;H^{4}(I))}\right),\end{aligned}$$ $$\forall \ (y_{0},y_{1})\in L^{2}(\Omega, \mathcal{F}_{0},P;(H^2_0(I)\cap H^{4}(I))\times H^{2}(I)).$$
Theorem \[Th1\], together with the energy estimates of (Theorem \[Th3\] in next section), gives the following $$\begin{aligned}
&\|(y_{0},y_{1})\|_{L^{2}(\Omega,\mathcal{F}_{0},P;H_0^{2}(I)\times L^{2}(I))}\\
&\leq C(I,T)\left(\mathbb{E}\int^{T}_{0}(y^2_{xx}(b,t)+y^2_{xxx}(b,t))dt
+\|f\|_{L^{2}_{\mathcal{F}}(0,T;H^{2}(I))}+\|g\|_{L^{\infty}_{\mathcal{F}}(0,T;H^{4}(I))}\right),\end{aligned}$$ $$\forall \ (y_{0},y_{1})\in L^{2}(\Omega, \mathcal{F}_{0},P;(H^2_0(I)\cap H^{4}(I))\times H^{2}(I)).$$
The solution of is defined as follows:
\[def\] A stochastic process $y$ is said to be a solution of , if $y\in H_{T}$ satisfies the initial conditions and $$\begin{aligned}
(y_{t}(t),v)_{L^{2}(I)}&=(y_{t}(0),v)_{L^{2}(I)}-\int^{t}_{0}(y_{xx}(s),v_{xx})_{L^{2}(I)}ds\\
&+\int^{t}_{0}(f(s),v)_{L^{2}(I)}ds+\int^{t}_{0}(g(s),v)_{L^{2}(I)}dB(s),\end{aligned}$$
holds for all $t\in[0,T]$ and all $v\in H_{0}^{2}(I)$ for almost $\omega \in \Omega$. Where $$H_{T}:=L^{2}_{\mathcal{F}}(\Omega;C([0,T];H_{0}^{2}(I)))\cap L^{2}_{\mathcal{F}}(\Omega;C^{1}([0,T];L^{2}(I))).$$ Note that, $H_{T}$ is a Banach space with the canonical norm.
In order to obtain the observation inequality, we first establish a global Carleman estimate for the following special case $$\label{int50}
\begin{cases}
dy_{t}+y_{xxxx}dt =fdt+gdB(t) &\text{in} \ \ Q=(0,T)\times I,\\
y(a,t)=0,\ y(b,t)=0&\text{in} \ (0,T),\\
y_{x}(a,t)=0,\ y_{x}(b,t)=0 & \text{in}\ (0,T),\\
y(x,0)=y(x,T)=0,\ y_{t}(x,0)=y_{t}(x,T)=0 & \text{in}\ \ I.
\end{cases}$$
Let $I=[a,b], a>0$, for any $x_0\in {\mathbb R}\setminus [a,b]$, we define the following weight function, for any $t\in(0,T)$, define $$\label{l}
l(t,x)=\lambda[(x-x_0)^{2}+(t-T)^{2}t^{2}], \qquad
\theta=e^{l}$$ for simplicity, we just choose $x_0=0$.
The global Carleman estimate for system is the following:
\[Th2\] Let holds, then there exist a constant\
$ C(I,T)>0$ and a constant $\lambda_0>0$ sufficiently large, such that for every $\lambda>\lambda_0$, the solution of system satisfies the following Carleman inequality $$\begin{aligned}
\label{int40}
\mathbb{E}&\int^{T}_{0}\int^{b}_{a}\theta^2(\lambda y^{2}_{xxx}+\lambda^{3} y^{2}_{xx}+\lambda^{5} y^{2}_{x}+\lambda^{7} y^{2}+\lambda^{3}y^{2}_{t})dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(b,t)+\lambda y^2_{xxx}(b,t))dt+\int_{Q}\theta^{2}\lambda^{2}(f^{2}+g^{2})dxdt\right\},\end{aligned}$$ $$\forall\ (y_{0},y_{1})\in L^{2}(\Omega, \mathcal{F}_{0},P;(H^2_0(I)\cap H^{4}(I))\times H^{2}(I)).$$
We choose $a>0$ and $x_0=0$ just for convenience. For any $x_0<a$, still holds with the constant $C=C(I,T,x_0)$. When we choose $x_0>b$, will be modified with the RHS estimate depending on $\mathbb{E}\left(\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(a,t)+\lambda y^2_{xxx}(a,t))dt\right)$.
Due to the complexity of the 4th order equation, the global Carleman estimate for the stochastic beam equation is difficult to establish directly. However, when considering the special case that $y(x,0)=y(x,T)=0,\ y_{t}(x,0)=y_{t}(x,T)=0$, we could overcome the difficulties to establish the global Carleman estimates. Similar conditions are also used by P.Gao where they derive the Carleman estimates for the deterministic beam equation [@G1].
In order to obtain the revised Carleman estimate for , we choose a cutoff function $\chi\in C^{\infty}_{0}[0,T]$ satisfying: for $\epsilon>0$ small, $$\chi(t)=\left\{\begin{array}{ll}
1,& t\in [\epsilon,T-\epsilon]\\
0,& t\in [0,\epsilon/2]\cup [T-\epsilon/2,T]\\
\in (0,1),&\mbox{ otherwise}.
\end{array}\right.$$
Let $z=\chi y$ where $y$ solves . Applying Theorem \[Th2\] on $z$, we have the following Carleman estimate for system :
\[Th4\] Let holds, then there exist a constant $ C(I,T)>0$, a constant $\lambda_0>0$ sufficiently large, such that for every $\lambda>\lambda_0$ and $0<\epsilon<\frac{T}{2}$, the solution of system satisfies the following Carleman inequality $$\begin{aligned}
\label{int41}
\mathbb{E}&\int^{T-\epsilon}_{\epsilon}\int^{b}_{a}\theta^2(\lambda y^{2}_{xxx}+\lambda^{3} y^{2}_{xx}+\lambda^{5} y^{2}_{x}+\lambda^{7} y^{2}+\lambda^{3}y^{2}_{t})dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(b,t)+\lambda y^2_{xxx}(b,t))dt+\int_{Q}\theta^{2}\lambda^{2}(f^{2}+g^{2})dxdt\right\}\\\nonumber
&+\frac{C(I,T)}{\varepsilon^{4}}\lambda^{2}\left[\mathbb{E}\int^{\epsilon}_{0}\int^{b}_{a}\theta^2(y^{2}_{t}+y^{2})dxdt+\mathbb{E}\int^{T}_{T-\epsilon }\int^{b}_{a}\theta^2(y^{2}_{t}+y^{2})dxdt\right].\end{aligned}$$
The paper is organized as follows. In section \[sec2\], we show the existence and regularity results of the solution to . In section \[sec3\], we establish an identity for stochastic beam equation. In section \[sec4\], we prove Theorem \[Th2\] and Theorem \[Th4\]. Finally, the proof of Theorem \[Th1\] is given in section \[sec5\].
Existence and Regularity {#sec2}
========================
In this section, we give the existence and regularity results for the solution of , which will be used in the following sections. First, we give a special case of the $It$ô formula, which is enough for our purpose. The general form can be found in \[[@Pa],Chapter 1\].
Let $X(\cdot)\in L^{2}_{\mathcal{F}}(0,T; H^{2}_{0}(I))$ be a continuous process with values in $H^{-2}(I)$. Suppose for $X_{0}\in L^{2}(\Omega, \mathcal{F}_{0}, P; L^{2}(I))$, $\Phi (\cdot)\in L^{2}_{\mathcal{F}}(0, T; H^{-2}(I))$, $\Psi (\cdot)\in L^{2}_{\mathcal{F}}(0, T; L^{2}(I))$ and any $t\in [0,T]$, it holds that $$X(t)=X_{0}+\int^{t}_{0}\Phi(s)ds+\int^{t}_{0}\Psi(s)dB(s),\quad P-a.s.$$ in $H^{-2}(I)$. Then we have $$\begin{aligned}
\|X(t)\|^{2}_{L^{2}(I)}=&\|X(0)\|^{2}_{L^{2}(I)}+2\int^{t}_{0}(X(s),\Phi(s))_{H_{0}^{2}(I),\ H^{-2}(I)}ds\\
&+2\int^{t}_{0}(X(s),\Psi(s))_{ L^{2}(I)}dB(s)+\int^{t}_{0}\|\Psi(s)\|^{2}_{L^{2}(I)}ds,\end{aligned}$$ for arbitrary $t\in [0,T]$.
Due to the classical theory of stochastic partial differential equations [@PLC], system has a unique mild solution. In order to establish the Carleman estimate, we give the following well-posedness and regularity results of the solution. Here we borrow the idea from [@Ki].
\[Th3\] Under the following condition, the system has a unique solution $y\in H_{T}$ which satisfies:
For $ \forall \ 0\leq s, t\leq T$, if $(y_{0},y_{1})\in L^{2}(\Omega, \mathcal{F}_{0},P;H_0^2(I)\times L^{2}(I))$, $f\in L^{2}_{\mathcal{F}}(0, T; L^{2}(I))$, $g\in L^{2}_{\mathcal{F}}(0, T; L^{2}(I))$, then $$\begin{aligned}
\label{th3-eq2}
&\|(y(t),y_{t}(t))\|_{L^2(\Omega,\mathcal{F}_t,P;H_0^2(I)\times L^2(I))}\\ \nonumber
&\leq C\left(\|(y(s),y_{t}(s))\|_{L^2(\Omega,\mathcal{F}_s,P;H_0^2(I)\times L^2(I))}+\|f\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}\right).\end{aligned}$$ Moreover, if $(y_{0},y_1)\in L^{2}(\Omega, \mathcal{F}_{0},P; (H^2_0(I)\cap H^{4}(I))\times H^{2}(I))$, $f\in L^{2}_{\mathcal{F}}(0, T; H^{2}(I))$ and $g\in L^{\infty}_{\mathcal{F}}(0, T; H^{4}(I))$, then $(y,y_t)\in L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H^{4}(I)\cap H^2_0(I)\times H^2(I)\cap H_0^1(I)))$ and satisfies $$\begin{aligned}
\label{th3-eq3}
&\|y\|_{L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H^{4}(I)))}+\|y_{t}\|_{L^{2}_{\mathcal{F}}(\Omega ;C([0, T]; H^{2}(I)))}\\ \nonumber
&\leq C\left(\|(y_{0},y_1)\|_{L^2(\Omega,\mathcal{F}_0,P;H^4(I)\times H^2(I))}+\|f\|_{L^{2}_{\mathcal{F}}(0,T ;H^{2}(I))}+\|g\|_{L^{\infty}_{\mathcal{F}}(0,T ;H^{4}(I))}\right).\end{aligned}$$
Let us consider the one-dimensional fourth order elliptic operator $\Lambda$ on $L^{2}(I)$ $$\begin{cases}
D(\Lambda)=H^{2}_{0}(I)\cap H^{4}(I)\\
\Lambda y=y_{xxxx}\quad \forall\ y\in D(\Lambda).
\end{cases}$$
Let $\{v_{k}\}^{+\infty}_{k=1}$ be the eigenfunctions of $\Lambda$ corresponding to the eigenvalues $\{\lambda_{k}\}^{+\infty}_{k=1}$ such that $\|v_{k}\|_{L^{2}(I)}=1$ (k=1,2,3...), which serves as an orthonormal basis of $L^{2}(I)$(see[@RR], Theorem 8\]). That is $$\begin{cases}
\Lambda v_{k}=\lambda_{k}v_{k}&\text{in} \ I, \\
v_{k}=0& \text{on} \ \partial I,\\
\frac{d}{dx}v_k(x)=0 & \text{on} \ \partial I.
\end{cases}$$
Let $\{c_k\},\ k= 1,2,...,$ satisfy the following stochastic differential equation, $$\begin{cases}
dc'_{k}=-\lambda_{k}c_{k}dt+\langle f,v_{k}\rangle dt+\langle g,v_{k}\rangle dB(t), \ \ k=1,2,...\\
c_{k}(0)=\langle y_{0},v_{k}\rangle, \ c'_{k}(0)=\langle y_{1},v_{k}\rangle,\\
(y_{0},y_{1})\in H_{0}^{2}(I)\times L^{2}(I),
\end{cases}$$ for almost all $\omega\in \Omega$. Due to the classical theory of stochastic differential equations, we know that there is a pathwise unique solution $c_k$ adapted to $\{\mathcal{F}_{t\ge 0}\}$, such that $c_k\in C^{1}([0,T])$ for almost all $\omega\in \Omega$. By $It$ô formula, we have $$\begin{aligned}
d(c'_{k})^{2}&=2c'_{k}dc'_{k}+(dc'_{k})^{2}\\
&=-2c'_{k}\lambda_{k}c_{k}(t)dt+2c'_{k}\langle f,v_{k}\rangle dt\\
&+2c'_{k}\langle g,v_{k}\rangle dw+|\langle g,v_{k}\rangle |^{2}dt.\end{aligned}$$ Which implies that $$\begin{aligned}
\label{int4}
|c'_{k}|^{2}+\lambda_{k}&|c_{k}(t)|^{2}=|c'_{k}(0)|^{2}+\lambda_{k}|c_{k}(0)|^{2}+2\int^{t}_{0}c'_{k}(s)\langle f(s),v_{k}\rangle ds\\\nonumber
&+2\int^{t}_{0}c'_{k}(s)\langle g(s),v_{k}\rangle dB(s)+\int^{t}_{0}|\langle g(s),v_{k}\rangle|^{2}ds,\end{aligned}$$ for all $t\in [0,T]$, for almost all $\omega\in \Omega.$
Define $$\begin{aligned}
y^{m}=\sum^{m}_{k=1}c_{k}v_{k}.\end{aligned}$$
If we multiply $v^{2}_{k}$ on both side of and sum about $k$ from 1 to $m$, then integrate over $I$, we have $$\begin{aligned}
\label{int5}
\|&y_{t}^{m}(t)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(t)\|^{2}_{L^{2}(I)}=\|y_{t}^{m}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(0)\|^{2}_{L^{2}(I)}\\\nonumber
&+2\int^{t}_{0}\langle f(s),y_{t}^{m}(s)\rangle ds+2\int^{t}_{0}\langle g(s),y_{t}^{m}(s)\rangle dB(s)+\sum^{m}_{k=1}\int^{t}_{0}|\langle g(s),v_{k}\rangle|^{2}ds\end{aligned}$$ for all $t \in [0,T], \ \omega\in\Omega.$
Next, we fix $m\geq1$ and any positive integer $L$ and define a stopping time $$\tau_{L}=\left\{
\begin{array}{ccl}
0& & if\ {\|y_{t}^{m}(0)\|_{L^{2}(I)}\geq L}\\
\inf\{t\in[0,T]:\|y_{t}^{m}(t)\|_{L^{2}(I)}\geq L\} & &if\ {\|y_{t}^{m}(0)\|_{L^{2}(I)}< L}\\
T & & if\ {\|y_{t}^{m}(0)\|_{L^{2}(I)}< L\ and \ the \ set }\\
& &\ {\{t\in[0,T]:\|y_{t}^{m}(t)\|_{L^{2}(I)}\geq L\} \ is \ empty.}
\end{array} \right.$$
From , it is easy to obtain the following inequality
$$\begin{aligned}
\label{aaa}
&\mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}(\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(s)\|^{2}_{L^{2}(I)}))\\\nonumber
&\leq \mathbb{E}(\|y_{t}^{m}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(0)\|^{2}_{L^{2}(I)})+2\mathbb{E}(\sup_{\eta\in[0, t\wedge\tau_{L}]}|\int^{\eta}_{0}\langle g(s),y_{t}^{m}(s)\rangle dB(s)|)\\\nonumber
&+2\mathbb{E}(\sup_{\eta\in[0, t\wedge\tau_{L}]}|\int^{\eta}_{0}\langle f(s),y_{t}^{m}(s)\rangle ds|)+\mathbb{E}\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds.\end{aligned}$$
By the Burkholder-Davis-Gundy inequality, we have $$\begin{aligned}
\label{int6}
&\mathbb{E}(\sup_{\eta\in[0, t\wedge\tau_{L}]}|\int^{\eta}_{0}\langle g(s),y_{t}^{m}(s)\rangle dB(s)|)\leq C\mathbb{E}(\int^{t\wedge\tau_{L}}_{0}|\langle g(s),y_{t}^{m}(s)\rangle |^{2}ds)^{\frac{1}{2}}\\\nonumber
&\leq C\mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|_{L^{2}(I)} (\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds)^{\frac{1}{2}})\\\nonumber
&\leq C(\mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)}))^{\frac{1}{2}}(\mathbb{E}(\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds))^{\frac{1}{2}}\\\nonumber
&\leq C\epsilon \mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)})+C(\epsilon)\mathbb{E}(\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds),\end{aligned}$$ for any $\epsilon>0$, where $C$ denotes a positive constant independent of $m, L$, and $T$. We also have,
$$\begin{aligned}
\label{int7}
&\mathbb{E}(\sup_{\eta\in[0, t\wedge\tau_{L}]}|\int^{\eta}_{0}\langle f(s),y_{t}^{m}(s)\rangle ds|)\\\nonumber
&\leq \mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|_{L^{2}(I)})\int^{t\wedge\tau_{L}}_{0}(\sum^{m}_{k=1}|\langle f(s),v_{k}\rangle|^{2})^{\frac{1}{2}}ds)\\\nonumber
&\leq C\epsilon \mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)})+C(\epsilon)\mathbb{E}(\int^{t\wedge\tau_{L}}_{0}(\sum^{m}_{k=1}|\langle f(s),v_{k}\rangle|^{2})^{\frac{1}{2}}ds)^{2} \\\nonumber
&\leq C\epsilon \mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)})+C(\epsilon)\mathbb{E}(\int^{t\wedge\tau_{L}}_{0}(\sum^{m}_{k=1}|\langle f(s),v_{k}\rangle|^{2})ds)\end{aligned}$$
for any $\epsilon>0$. Since
$$\label{bbb}
\mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)})\leq \mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}(\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(s)\|^{2}_{L^{2}(I)})).$$
Combining , and with and choose a $\epsilon >0$ sufficiently small ($C\epsilon<\frac{1}{2}$), we have
$$\begin{aligned}
& \mathbb{E}(\sup_{s\in[0, t\wedge\tau_{L}]}(\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(s)\|^{2}_{L^{2}(I)}))\\
&\leq \mathbb{E}(\|y_{t}^{m}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(0)\|^{2}_{L^{2}(I)})+C\mathbb{E}(\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds)\\
&+C\left(\mathbb{E}\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle f(s),v_{k}\rangle|^{2}ds\right)+\mathbb{E}\int^{t\wedge\tau_{L}}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds,\end{aligned}$$
for all $t\in[0,T]$, where $C$ denotes a positive constant independent of $m, L$. By passing $L\rightarrow+\infty$, we obtain
$$\begin{aligned}
\label{int8}
&\mathbb{E}(\sup_{s\in[0, t]}(\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(s)\|^{2}_{L^{2}(I)}))\\\nonumber
&\leq \mathbb{E}(\|y_{t}^{m}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(0)\|^{2}_{L^{2}(I)})+C\mathbb{E}(\int^{t}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds) \\\nonumber
&+C\left(\mathbb{E}\int^{t}_{0}(\sum^{m}_{k=1}|\langle f(s),v_{k}\rangle|^{2})ds\right)+\mathbb{E}\int^{t}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds,\end{aligned}$$
for all $t\in[0,T]$.
Choose $t=T$, $$\begin{aligned}
&\mathbb{E}(\sup_{s\in[0, T]}(\|y_{t}^{m}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(s)\|^{2}_{L^{2}(I)}))\\
&\leq \mathbb{E}(\|y_{t}^{m}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(0)\|^{2}_{L^{2}(I)})+C\mathbb{E}(\int^{T}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds)\\
&+C\left(\mathbb{E}\int^{T}_{0}(\sum^{m}_{k=1}|\langle f(s),v_{k}\rangle|^{2})ds\right)+\mathbb{E}\int^{T}_{0}\sum^{m}_{k=1}|\langle g(s),v_{k}\rangle|^{2}ds.\end{aligned}$$
By the same argument, we also have, for $m\geq n\geq1$, $$\begin{aligned}
\label{int9}
&\mathbb{E}(\sup_{s\in[0, T]}(\|y_{t}^{m}(s)-y_{t}^{n}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(s)-y_{xx}^{n}(s)\|^{2}_{L^{2}(I)})ds\\\nonumber
&\leq \mathbb{E}(\|y_{t}^{m}(0)-y_{t}^{n}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}^{m}(0)-y_{xx}^{n}(0)\|^{2}_{L^{2}(I)})\\\nonumber
&+C\mathbb{E}(\int^{T}_{0}\sum^{m}_{k=n+1}|\langle g(s),v_{k}\rangle|^{2}ds)\\\nonumber
&+C\mathbb{E}(\int^{T}_{0}\sum^{m}_{k=n+1}|\langle f(s),v_{k}\rangle|^{2}ds)\\\nonumber
&+\mathbb{E}\int^{T}_{0}\sum^{m}_{k=n+1}|\langle g(s),v_{k}\rangle|^{2}ds,\end{aligned}$$ where $C$ denotes a positive constant independent of $m, n$. Next, we observe that the right hand side of converges to zero as $m, n\rightarrow+\infty$. Hence $(y^{m}, y_t^{m})$ is a Cauchy sequence that converges strongly in $L^{2}_{\mathcal{F}}(\Omega ; L^{\infty}([0,T]; H_{0}^{2}(I)\times L^{2}_{\mathcal{F}}(\Omega ; L^{\infty}([0,T]; L^{2}(I))$. By the standard semigroup theory, we can show that $(y^{m}, y_t^{m})$ is also a Cauchy sequence that converges strongly in $L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H_{0}^{2}(I)\times L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; L^{2}(I))$. Let $(y,y_t)$ be the limit of $(y^{m},y^m_t)$ as $m\rightarrow+\infty$, then it solves .
To prove the uniqueness of solution, assume $y_{1}$ and $y_{2}$ be solutions of and let $h=y_{1}-y_{2}$, then $h$ satisfies $$\begin{cases}
dh_{t}+h_{xxxx}dt =0 &\text{in} \ \ Q=(0,T)\times I,\\
h(a,t)=0,\ h(b,t)=0&\text{in} \ (0,T),\\
h_{x}(a,t)=0,\ h_{x}(b,t)=0 & \text{in}\ (0,T),\\
h(x,0)=0,\ h_{t}(x,0)=0 & \text{in}\ \ I.
\end{cases}$$ Consequently $$\begin{aligned}
\mathbb{E}(\sup_{s\in[0, T]}(\|h_{xx}(s)\|^{2}_{L^{2}(I)}+\|h_{t}(s)\|^{2}_{L^{2}(I)})ds)=0,\end{aligned}$$ therefore, $y_{1}=y_{2}$. By and let $m\rightarrow +\infty$, we have $$\begin{aligned}
&\|y\|_{L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H_{0}^{2}(I))}+\|y_{t}\|_{L^{2}_{\mathcal{F}}(\Omega ;C([0, T]; L^{2}(I))}\\
&\leq C\left(\|(y_{0},y_{1})\|_{L^2(\Omega,\mathcal{F}_0,P;H_0^2(I)\times L^2(I))}+\|f\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}\right).\end{aligned}$$ Now, let $m\rightarrow +\infty$, without loss of generality, we assume that $s\leq t$, by , we have $$\begin{aligned}
\label{int57}
\|&y_{t}(t)\|^{2}_{L^{2}(I)}+\|y_{xx}(t)\|^{2}_{L^{2}(I)}=\|y_{t}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}(0)\|^{2}_{L^{2}(I)}\\\nonumber
&+2\int^{t}_{0}\langle f(s),y_{t}(s)\rangle ds+2\int^{t}_{0}\langle g(s),y_{t}(s)\rangle dB(s)+\int^{t}_{0}\| g(s)\|_{L^{2}(I)}^{2}ds,\end{aligned}$$ similarly, we have $$\begin{aligned}
\label{int56}
\|&y_{t}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}(s)\|^{2}_{L^{2}(I)}=\|y_{t}(0)\|^{2}_{L^{2}(I)}+\|y_{xx}(0)\|^{2}_{L^{2}(I)}\\\nonumber
&+2\int^{s}_{0}\langle f(\tau),y_{t}(\tau)\rangle d\tau+2\int^{s}_{0}\langle g(\tau),y_{t}(\tau)\rangle dB(\tau)+\int^{s}_{0}\| g(\tau)\|_{L^{2}(I)}^{2}d\tau\end{aligned}$$ then, if subtract , we have $$\begin{aligned}
\label{int58}
\|&y_{t}(t)\|^{2}_{L^{2}(I)}+\|y_{xx}(t)\|^{2}_{L^{2}(I)}=\|y_{t}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}(s)\|^{2}_{L^{2}(I)}\\\nonumber
&+2\int^{t}_{s}\langle f(\tau),y_{t}(\tau)\rangle d\tau+2\int^{t}_{s}\langle g(\tau),y_{t}(\tau)\rangle dB(\tau)+\int^{t}_{s}\| g(\tau)\|_{L^{2}(I)}^{2}d\tau.\end{aligned}$$ By , we have $$\begin{aligned}
\|&y_{t}(t)\|^{2}_{L^{2}(I)}+\|y_{xx}(t)\|^{2}_{L^{2}(I)}\leq\|y_{t}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}(s)\|^{2}_{L^{2}(I)}\\\nonumber
&+2\left|\int^{t}_{s}\langle f(\tau),y_{t}(\tau)\rangle d\tau\right|+2\left|\int^{t}_{s}\langle g(\tau),y_{t}(\tau)\rangle dB(\tau)\right|+\left|\int^{t}_{s}\| g(\tau)\|_{L^{2}(I)}^{2}d\tau\right|\end{aligned}$$ and $$\begin{aligned}
\|&y_{t}(s)\|^{2}_{L^{2}(I)}+\|y_{xx}(s)\|^{2}_{L^{2}(I)}\leq\|y_{t}(t)\|^{2}_{L^{2}(I)}+\|y_{xx}(t)\|^{2}_{L^{2}(I)}\\\nonumber
&+2\left|\int^{t}_{s}\langle f(\tau),y_{t}(\tau)\rangle d\tau\right|+2\left|\int^{t}_{s}\langle g(\tau),y_{t}(\tau)\rangle dB(\tau)\right|+\left|\int^{t}_{s}\| g(\tau)\|_{L^{2}(I)}^{2}d\tau\right|.\end{aligned}$$ By the same arguments as above, we have $$\begin{aligned}
&\|y\|_{L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H_{0}^{2}(I))}+\|y_{t}\|_{L^{2}_{\mathcal{F}}(\Omega ;C([0, T]; L^{2}(I))}\\
&\leq C\left(\|(y(s),y_{t}(s))\|_{L^2(\Omega,\mathcal{F}_s,P;H_0^2(I)\times L^2(I))}+\|f\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}\right).\end{aligned}$$ That is $$\begin{aligned}
&\|(y(t),y_{t}(t))\|_{L^2(\Omega,\mathcal{F}_t,P;H_0^2(I)\times L^2(I))}\\
&\leq C\left(\|(y(s),y_{t}(s))\|_{L^2(\Omega,\mathcal{F}_s,P;H_0^2(I)\times L^2(I))}+\|f\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}\right).\end{aligned}$$ This completes the proof of $\eqref{th3-eq2}$.
Under the conditions $(y_{0},y_1)\in L^{2}(\Omega, \mathcal{F}_{0},P; (H^2_0(I)\cap H^{4}(I))\times H^{2}(I))$, $f\in L^{2}_{\mathcal{F}}(0, T; H^{2}(I))$ and $g\in L^{\infty}_{\mathcal{F}}(0, T; H^{4}(I))$, we multiply $\lambda_{k}v^{2}_{k}$ on both side of and sum about k from 1 to m, then integrate over I, we have $$\begin{aligned}
\label{int10}
&\|y_{xxt}^{m}(t)\|^{2}_{L^{2}(I)}+\|\Lambda y^{m}(t)\|^{2}_{L^{2}(I)}=\|y_{xxt}^{m}(0)\|^{2}_{L^{2}(I)}+\|\Lambda y^{m}(0)\|^{2}_{L^{2}(I)}+ \\\nonumber
&2\int^{t}_{0}\langle\Lambda f(s),y_{t}^{m}(s)\rangle ds+2\int^{t}_{0}\langle\Lambda g(s),y_{t}^{m}(s)\rangle dB(s)+\sum^{m}_{k=1}\int^{t}_{0}\lambda_{k}|\langle g(s),v_{k}\rangle |^{2}ds.\end{aligned}$$ Integrating by parts, we get that $$\begin{aligned}
\label{int11}
&\|y_{xxt}^{m}(t)\|^{2}_{L^{2}(I)}+\|\Lambda y^{m}(t)\|^{2}_{L^{2}(I)}=\|y_{xxt}^{m}(0)\|^{2}_{L^{2}(I)}+\|\Lambda y^{m}(0)\|^{2}_{L^{2}(I)}+ \\\nonumber
&2\int^{t}_{0}\langle f_{xx}(s),y_{xxs}^{m}(s)\rangle ds+2\int^{t}_{0}\langle g_{xx}(s),y_{xxs}^{m}(s)\rangle dB(s)+\sum^{m}_{k=1}\int^{t}_{0}\lambda_{k}|\langle g(s),v_{k}\rangle |^{2}ds.\end{aligned}$$
If we use Burkholder-Davis-Gundy inequality, Cauchy inequality on , similarly we have $$\begin{aligned}
&\mathbb{E}(\sup_{s\in[0, T]}(\|\Lambda y(s)\|^{2}_{L^{2}(I)}+\|y_{xxt}(s)\|^{2}_{L^{2}(I)})ds)\\
&\leq C\mathbb{E}(\|\Lambda y(0)\|^{2}_{L^{2}(I)}+\|y_{xxt}(0)\|^{2}_{L^{2}(I)})+C\mathbb{E}(\int^{T}_{0}\|\Lambda g(s)\|_{L^{2}(I)}^{2}ds)\\
&+C\mathbb{E}(\int^{T}_{0}\|f_{xx}(s)\|_{L^{2}(I)}^{2})ds.\end{aligned}$$
That is $y\in L^{2}_{\mathcal{F}}(\Omega ; L^{\infty}([0,T]; H^{4}(I)\cap H^2_0(I)))$ and due to the boundary condition of , $y_t\in L^{2}_{\mathcal{F}}(\Omega ; L^{\infty}([0,T]; H^2(I)\cap H^1_0(I)))$. Due to Sobolev embedding theorem, it is easy to see that $\|y_t\|_{H^2(I)}$ can be controlled by $\|y_{xxt}\|_{L^2(I)}$. As $I$ is an interval, $H^4(I)\hookrightarrow C^3(I)$. If $y\in H^4(I)\cap H^2_0(I)$, we can show that $\|y\|_{H^4(I)}$ can be controlled by $\|y_{xxxx}\|_{L^2(I)}$. Hence, by the same arguments as $(1)$, we conclude that $$\begin{aligned}
&\|y\|_{L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H^{4}(I))}+\|y_{t}\|_{L^{2}_{\mathcal{F}}(\Omega ;C([0, T]; H^{2}(I))}\\
&\leq C\{\|(y_{0},y_1)\|_{L^2(\Omega,\mathcal{F}_0,P;H^4(I)\times H^2(I))}+\|f\|_{L^{2}_{\mathcal{F}}(0,T ;H^{2}(I))}+\|g\|_{L^{\infty}_{\mathcal{F}}(0,T ;H^{4}(I))}\}.\end{aligned}$$
Identity for stochastic beam equation {#sec3}
=====================================
We show the following fundamental identity for stochastic beam equation.
Assume $y$ is a ${{H^{2}(I)}}$-value $\{\mathcal{F}_{t}\}_{t\geq0} $-adopted processes such that $y_{t}$ is a $L^{2}(I)$-valued semi-martingale. Set $\theta=e^{l}$ and $u=\theta y$. Then for a.e. $x\in I$ and $P$-a.s $\omega\in\Omega $ we have $$\begin{aligned}
\label{lem6}
&2\{-2l_{t}u_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}\}\theta(dy_{t}+y_{xxxx}dt)\\\nonumber
&+2d\{l_{t}u^{2}_{t}-\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}u_{t}+\frac{\Phi_{t}}{2}u^{2}\}\\\nonumber
&=\{\cdots\}_{xxx}dt+\{\cdots\}_{xx}dt+\{\cdots\}_{x}dt+u^{2}\{\cdots\}dt\\\nonumber
&+u_{x}^{2}\{\cdots\}dt+u_{xx}^{2}\{\cdots\}dt+u_{xxx}^{2}\{\cdots\}dt+2(l_{tt}-\Phi)u^{2}_{t}dt\\\nonumber
&-2\{u_{t}[[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}]_{t}+2[l_{t}u_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{x}\}dt\\\nonumber
&-2\{2[l_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{t}u_{x}-2l_{tx}u_{t}(u_{xxx}+(G-\Phi_{1})u_{x})\}dxdt+2p^{2}dt\\\nonumber
&-2\{l_{t}(A-\Phi)u^{2}-2l_{t}u_{x}(u_{xxx}+(G-\Phi_{1})u_{x})\}_{t}dxdt+2l_{t}(du_{t})^{2}.\end{aligned}$$ Where $$A=l^{4}_{x}+4l_{x}l_{xxx}-l_{xxxx}-6l^{2}_{x}l_{xx}+3l^{2}_{xx}+l^{2}_{t}-l_{tt},$$ $$G=6l^{2}_{x}-6l_{xx},\ B=12l_{x}l_{xx}-4l^{3}_{x}-4l_{xxx},\ D=-4l_{x},$$ $$p=-2l_{t}u_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\},$$ $$\{\cdots\}_{xxx}=\{[B-(G-\Phi_{1})_{x}]u^{2}_{x}-\Phi_{x}u^{2}+(A-\Phi)Du^{2}\}_{xxx},$$ $$\begin{aligned}
\{\cdots\}_{xx} &=\{-3[B-(G-\Phi_{1})_{x}]_{x}u^{2}_{x}+\Phi_{1}u^{2}_{xx}-\Phi u^{2}_{x}+[(G-\Phi_{1})_{x}]Du^{2}_{x}\\
&+3\Phi_{xx}u^{2}+(G-\Phi_{1})\Phi u^{2}+(A-\Phi)\Phi_{1}u^{2}-3[(A-\Phi)D]_{x}u^{2}\}_{xx},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
\{\cdots\}_{x}&=\{3[B-(G-\Phi_{1})_{x}]_{xx}u^{2}_{x}-3[B-(G-\Phi_{1})_{x}]u^{2}_{xx}-2\Phi_{1x}u^{2}_{xx}+2u_{xxx}\Phi u\\
&+Du^{2}_{xxx}+5\Phi_{x}u^{2}_{x}-3\Phi_{xxx}u^{2}+(G-\Phi_{1})[B-(G-\Phi_{1})_{x}]u^{2}_{x}\\
&+(G-\Phi_{1})_{x}\Phi_{1}u^{2}_{x}-2[(G-\Phi_{1})_{x}D]_{x}u^{2}_{x}+(G-\Phi_{1})Du^{2}_{xx}\\
&+(G-\Phi_{1})_{x}\Phi u^{2}-2[(G-\Phi_{1})\Phi]_{x}u^{2}-2[(A-\Phi)\Phi_{1}]_{x}u^{2}\\
&+(A-\Phi)[B-(G-\Phi_{1})_{x}]u^{2}+3[(A-\Phi)D]_{xx}u^{2}-3(A-\Phi)Du^{2}_{x}\}_{x},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\end{aligned}$$ $$\begin{aligned}
u^{2}\{\cdots\}&=u^{2}\{-[(G-\Phi_{1})_{x}\Phi]_{x}-[(A-\Phi)[B-(G-\Phi_{1})_{x}]]_{x}+\Phi_{tt}+[(G-\Phi_{1})\Phi]_{xx}\\
&+\Phi_{xxxx}+[(A-\Phi)\Phi_{1}]_{xx}-[(A-\Phi)D]_{xxx}+2(A-\Phi)\Phi+2[l_{t}(A-\Phi)]_{t}\}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\end{aligned}$$ $$\begin{aligned}
u^{2}_{x}\{\cdots\}&=u^{2}_{x}\{-[B-(G-\Phi_{1})_{x}]_{xx}-4\Phi_{xx}+2(G-\Phi_{1})_{x}[B-(G-\Phi_{1})_{x}]\\
&-[(G-\Phi_{1})[B-(G-\Phi_{1})_{x}]]_{x}-[(G-\Phi_{1})_{x}\Phi_{1}]_{x}+[(G-\Phi_{1})_{x}D]_{xx}\\
&-2(G-\Phi_{1})\Phi-2(A-\Phi)\Phi_{1}+3[(A-\Phi)D]_{x}\}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\end{aligned}$$ $$\begin{aligned}
u^{2}_{xx}\{\cdots\}&=u^{2}_{xx}\{3[B-(G-\Phi_{1})_{x}]_{x}+\Phi_{1xx}+2\Phi+2(G-\Phi_{1})\Phi_{1}-2(G-\Phi_{1})_{x}D\\
&-[(G-\Phi_{1})D]_{x}\},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\end{aligned}$$ $u^{2}_{xxx}\{\cdots\}=u^{2}_{xxx}\{-D_{x}-2\Phi_{1}\}.$
Let $u=\theta y, \ \theta=e^{l}$. Direct computation shows that\
$\theta(dy_{t}+y_{xxxx}dt)=du_{t}-2l_{t}u_{t}dt+Audt+Bu_{x}dt+Gu_{xx}dt+Du_{xxx}dt+u_{xxxx}dt$.\
Where $$A=l^{4}_{x}+4l_{x}l_{xxx}-l_{xxxx}-6l^{2}_{x}l_{xx}+3l^{2}_{xx}+l^{2}_{t}-l_{tt},$$ $$G=6l^{2}_{x}-6l_{xx},\ B=12l_{x}l_{xx}-4l^{3}_{x}-4l_{xxx},\ D=-4l_{x}.$$ Let $$\begin{aligned}
&p=-2l_{t}u_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\},\\
&p_{1}=\{u_{xxxx}+[(G-\Phi_{1})u_{x}]_{x}+(A-\Phi)u\}dt,\\
&p_{2}=du_{t}-2l_{t}u_{t}dt+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}dt.\\\end{aligned}$$ Then $$\begin{aligned}
\label{ccc}
2p\theta(dy_{t}+y_{xxxx}dt)=2p(p_{1}+p_{2})=2p(p_{1}+du_{t}+pdt).\end{aligned}$$ Since $$\begin{aligned}
2pp_{1}&=-4l_{t}u_{t}\{u_{xxxx}+[(G-\Phi_{1})u_{x}]_{x}+(A-\Phi)u\}dt\\
&+2u_{xxxx}\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}\\
&+2[(G-\Phi_{1})u_{x}]_{x}\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}dt\\
&+2(A-\Phi)u\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}dt\end{aligned}$$ and $$\begin{aligned}
2u_{xxxx}[B-&(G-\Phi_{1})_{x}]u_{x}l_{t}\ dt\\
&=\{\{[B-(G-\Phi_{1})_{x}]u^{2}_{x}\}_{xxx}-3\{[B-(G-\Phi_{1})_{x}]_{x}u^{2}_{x}\}_{xx}\\
&+3\{[B-(G-\Phi_{1})_{x}]_{xx}u^{2}_{x}-[B-(G-\Phi_{1})_{x}]u^{2}_{xx}\}_{x}\\
&+3[B-(G-\Phi_{1})_{x}]_{x}u^{2}_{xx}-[B-(G-\Phi_{1})_{x}]_{xxx}u^{2}_{x}\}\ dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2u_{xxxx}\Phi_{1}u_{xx}dt=[(\Phi_{1}u^{2}_{xx})_{xx}-2(\Phi_{1x}u^{2}_{xx})_{x}-2\Phi_{1}u^{2}_{xxx}+\Phi_{1xx}u^{2}_{xx}]dt, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2u_{xxxx}Du_{xxx}dt=[(Du^{2}_{xxx})_{x}-D_{x}u^{2}_{xxx}]\ dt, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2u_{xxxx}\Phi udt&=2[(u_{xxx}\Phi u)_{x}-u_{xxx}(\Phi u)_{x}]\ dt\\
&=\{2(u_{xxx}\Phi u)_{x}-(\Phi u^{2}_{x})_{xx}+2(\Phi_{x}u^{2}_{x})_{x}+2\Phi u^{2}_{xx}-\Phi_{xx}u^{2}_{x}+\Phi_{xxxx}u^{2}\\
&-(\Phi_{x}u^{2})_{xxx}+3(\Phi_{xx}u^{2})_{xx}-3(\Phi_{xxx}u^{2}-\Phi_{x}u^{2}_{x})_{x}-3\Phi_{xx}u^{2}_{x}\}dt, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2[(G-\Phi_{1})&u_{x}]_{x}[B-(G-\Phi_{1})_{x}]u_{x}\ dt\\
&=\{2(G-\Phi_{1})_{x}[B-(G-\Phi_{1})_{x}]u^{2}_{x}+\{(G-\Phi_{1})[B-(G-\Phi_{1})_{x}]u^{2}_{x}\}_{x}\\
&-\{(G-\Phi_{1})[B-(G-\Phi_{1})_{x}]\}_{x}u^{2}_{x}\}\ dt, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2[(G-\Phi_{1})&u_{x}]_{x}\Phi_{1}u_{xx}\ dt\\
&=\{2(G-\Phi_{1})\Phi_{1}u^{2}_{xx}+[(G-\Phi_{1})_{x}\Phi_{1}u^{2}_{x}]_{x}-[(G-\Phi_{1})_{x}\Phi_{1}]_{x}u^{2}_{x}\}dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2[(G-\Phi_{1})&u_{x}]_{x}Du_{xxx}\ dt\\
&=\{[(G-\Phi_{1})_{x}Du^{2}_{x}]_{xx}-2[[(G-\Phi_{1})_{x}D]_{x}u^{2}_{x}]_{x}-2(G-\Phi_{1})_{x}Du^{2}_{xx}\\
&+[(G-\Phi_{1})_{x}D]_{xx}u^{2}_{x}+[(G-\Phi_{1})_{x}Du^{2}_{xx}]_{x}-[(G-\Phi_{1})_{x}D]_{x}u^{2}_{xx}\}dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2[(G-\Phi&_{1})u_{x}]_{x}\Phi u\ dt\\
&=\{[(G-\Phi_{1})_{x}\Phi u^{2}]_{x}-[(G-\Phi_{1})_{x}\Phi ]_{x}u^{2}+[(G-\Phi_{1})\Phi u^{2}]_{xx}\\
&-2[[(G-\Phi_{1})\Phi ]_{x}u^{2}]_{x}-2(G-\Phi_{1})\Phi u^{2}_{x}+[(G-\Phi_{1})\Phi ]_{xx}u^{2}\}dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $$\begin{aligned}
2(A-\Phi)u\Phi_{1}u_{xx}\ dt&=\{[(A-\Phi)\Phi_{1}u^{2}]_{xx}-2[[(A-\Phi)\Phi_{1}]_{x}u^{2}]_{x}\\
&-2(A-\Phi)\Phi_{1}u^{2}_{x}+[(A-\Phi)\Phi_{1}]_{xx}u^{2}\}dt,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\end{aligned}$$ $$\begin{aligned}
2(A-\Phi)u&[B-(G-\Phi_{1})_{x}]u_{x}\ dt\\
&=\{[(A-\Phi)[B-(G-\Phi_{1})_{x}]u^{2}]_{x}-[(A-\Phi)[B-(G-\Phi_{1})_{x}]]_{x}u^{2}\}\ dt, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\\end{aligned}$$ $$\begin{aligned}
2(A-\Phi)u&Du_{xxx}\ dt\\
&=\{[(A-\Phi)Du^{2}]_{xxx}-3[[(A-\Phi)D]_{x}u^{2}]_{xx}+3[(A-\Phi)D]_{x}u^{2}_{x}\\
&-[(A-\Phi)D]_{xxx}u^{2}+3[[(A-\Phi)D]_{xx}u^{2}-(A-\Phi)Du^{2}_{x}]_{x}\ dt, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\end{aligned}$$ $2(A-\Phi)u\Phi u\ dt=2(A-\Phi)\Phi u^{2}dt.$\
On the one hand, $$\begin{aligned}
&-4l_{t}u_{t}\{u_{xxxx}+[(G-\Phi_{1})u_{x}]_{x}+(A-\Phi)u\}dt\\
&=-4\{[l_{t}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{x}-(l_{t}u_{t})_{x}[u_{xxx}+[(G-\Phi_{1})u_{x}]]\\
&+l_{t}u_{t}(A-\Phi)u\}\ dt\\
&=-4\{[l_{t}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{x}-l_{tx}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]\\
&-l_{t}u_{tx}[u_{xxx}+[(G-\Phi_{1})u_{x}]]+l_{t}u_{t}(A-\Phi)u\}\ dt\\
&=-4\{[l_{t}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{x}+\frac{1}{2}l_{t}(A-\Phi)(u^{2})_{t}\\
&-l_{tx}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]+[l_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{t}u_{x}\\
&-[l_{t}u_{x}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{t}\}\ dt\\
&=-4\{[l_{t}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{x}-[\frac{1}{2}l_{t}(A-\Phi)]_{t}u^{2}\\
&+[l_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{t}u_{x}-l_{tx}u_{t}[u_{xxx}+[(G-\Phi_{1})u_{x}]]\\
&+[\frac{1}{2}l_{t}(A-\Phi)u^{2}-l_{t}u_{x}[u_{xxx}+[(G-\Phi_{1})u_{x}]]]_{t}\}\ dt.\\\end{aligned}$$ On the other hand, $$\begin{aligned}
&2\{-2l_{t}u_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}\}du_{t}\\
&=2d\{-l_{t}u^{2}_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}u_{t}\}\\
&-2\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}_{t}u_{t}dt\\
&+2l_{t}(du_{t})^{2}dx+2l_{tt}u^{2}_{t}dt\\
&=2d\{-l_{t}u^{2}_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}u_{t}-\frac{\Phi_{t}}{2}u^{2}\}\\
&-2\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}\}_{t}u_{t}dxdt+2l_{t}(du_{t})^{2}\\
&+2(l_{tt}u^{2}_{t}+\frac{\Phi_{tt}}{2}u^{2}-u^{2}_{t}\Phi )dt.\end{aligned}$$ Combining the above equalities into , we can obtain the weight identity .
Proof of Carleman estimate (Theorem \[Th2\] and Theorem \[Th4\]) {#sec4}
================================================================
We choose $l=\lambda[x^{2}+(t-T)^{2}t^{2}]$, $\Phi_{1}=-6l_{xx}$ and $\Phi=-8l^{2}_{x}l_{xx}$ in identity , then we integrate over Q on both side of and taking expectation on both sides of the identity , by regularity result in Theorem \[Th3\], $l_{tx}=0,l_{t}(0)=l_{t}(T)=0$, the fact that $\mathbb{E}(\int^{T}_{0}X(t)dB(t))=0$ if $\mathbb{E}(\int^{T}_{0}X^{2}(t)dt)<+\infty$ and the equation $$dy_{t}+y_{xxxx}dt = fdt+ gdB(t),$$ we can obtain the following identity. $$\begin{aligned}
\label{lem9}
&2\mathbb{E}\int_{Q}\{-2l_{t}u_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}\}\theta fdtdx\\\nonumber
&+2\int_{Q} d\{l_{t}u^{2}_{t}-\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}u_{t}+\frac{\Phi_{t}}{2}u^{2}\}dx\\\nonumber
&=\mathbb{E}\int_{Q}\{\cdots\}_{xxx}dxdt+\mathbb{E}\int_{Q}\{\cdots\}_{xx}dxdt+\mathbb{E}\int_{Q}\{\cdots\}_{x}dxdt+\mathbb{E}\int_{Q} u^{2}\{\cdots\}dxdt \\\nonumber
&+\mathbb{E}\int_{Q} u_{x}^{2}\{\cdots\}dxdt+\mathbb{E}\int_{Q} u_{xx}^{2}\{\cdots\}dxdt+\mathbb{E}\int_{Q} u_{xxx}^{2}\{\cdots\}dxdt+\mathbb{E}\int_{Q}2(l_{tt}-\Phi)u^{2}_{t}dxdt\\\nonumber
&-2\mathbb{E}\int_{Q} \left\{u_{t}[[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}]_{t}+2[l_{t}u_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{x}\right\}dxdt\\\nonumber
&-2\mathbb{E}\int_{Q}2[l_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{t}u_{x}dxdt+2\mathbb{E}\int_{Q} p^{2}dxdt+2\mathbb{E}\int_{Q} l_{t}(du_{t})^{2}dx\\\nonumber
&-2\mathbb{E}\int_{Q}\{l_{t}(A-\Phi)u^{2}-2l_{t}u_{x}(u_{xxx}+(G-\Phi_{1})u_{x})\}_{t}dxdt.\end{aligned}$$
Next, we estimate each integral term on both side of . Define $$\begin{aligned}
A_{1}:=&\mathbb{E}\int_{Q}\{\cdots\}_{xxx}dxdt+\mathbb{E}\int_{Q}\{\cdots\}_{xx}dxdt+\mathbb{E}\int_{Q}\{\cdots\}_{x}dxdt,\\
A_{2}:=&\mathbb{E}\int_{Q} u^{2}\{\cdots\}dxdt+\mathbb{E}\int_{Q} u_{x}^{2}\{\cdots\}dxdt+\mathbb{E}\int_{Q} u_{xx}^{2}\{\cdots\}dxdt\\
&+\mathbb{E}\int_{Q} u_{xxx}^{2}\{\cdots\}dxdt+\mathbb{E}\int_{Q}2(l_{tt}-\Phi)u^{2}_{t}dxdt,\\
A_{3}:&=2\mathbb{E}\int_{Q} u_{t}[[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}]_{t}dxdt,\\
A_{4}:=&2\mathbb{E}\int_{Q} \{2[l_{t}u_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{x}\}dxdt\\
&+2\mathbb{E}\int_{Q} \{2[l_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{t}u_{x}\}dxdt,\\
A_{5}:=&2\mathbb{E}\int_{Q} [l_{t}(A-\Phi)u^{2}-2l_{t}u_{x}(u_{xxx}+(G-\Phi_{1})u_{x})]_{t}dxdt.\end{aligned}$$ Since by Theorem \[Th3\], $$y\in L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H^4(I)\cap H^2_0(I))) \mbox{ and } y_t\in L^{2}_{\mathcal{F}}(\Omega ; C([0,T]; H^2(I)\cap H^1_0(I))),$$so is $u$. We have $$u_{t}(a,t)=0,\ u_{t}(b,t)=0,\ u_{xt}(a,t)=0,\ u_{xt}(b,t)=0.$$ Since $u=\theta y, u_{t}=\theta_{t}y+\theta y_{t}$, we have $u(x,0)=u(x,T)=0, u_{t}(x,0)=u_{t}(x,T)=0$. Consequently, $$2\int_{Q} d\{l_{t}u^{2}_{t}-\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}u_{t}+\frac{\Phi_{t}}{2}u^{2}\}dx=0.$$ Since $$\begin{aligned}
A_{1}=&\mathbb{E}\int_{Q}\{\cdots\}_{xxx}dxdt+\mathbb{E}\int_{Q}\{\cdots\}_{xx}dxdt+\mathbb{E}\int_{Q}\{\cdots\}_{x}dxdt\\
&=\mathbb{E}\int^{T}_{0}2[B-(G-\Phi_{1})_{x}]u^{2}_{xx}|^{b}_{a}dt+\mathbb{E}\int^{T}_{0}(\Phi_{1x}u^{2}_{xx}+2\Phi_{1}u_{xx}u_{xxx}+Du^{2}_{xxx})|^{b}_{a}dt\\
&+\mathbb{E}\int^{T}_{0}-3[B-(G-\Phi_{1})_{x}]u^{2}_{xx}|^{b}_{a}dt+\mathbb{E}\int^{T}_{0}(-2\Phi_{1x}u^{2}_{xx}+(G-\Phi_{1})Du^{2}_{xx})|^{b}_{a}dt\\
&=\mathbb{E}\int^{T}_{0}[-20l^{3}_{x}u^{2}_{xx}-12l_{xx}u_{xx}u_{xxx}-4l_{x}u^{2}_{xxx}]|^{b}_{a}dt,\end{aligned}$$ when $\lambda$ is large enough, we can use the high order terms of $\lambda$ to absorb the low order terms, by direct computation and Cauchy inequality, we have $$\begin{aligned}
&\mathbb{E}\int^{T}_{0}[-20l^{3}_{x}u^{2}_{xx}-12l_{xx}u_{xx}u_{xxx}-4l_{x}u^{2}_{xxx}]|^{b}_{a}dt\\
&\geq \mathbb{E}\int^{T}_{0}[(-160\lambda^{3}b^{3}u^{2}_{xx}(b,t)+160\lambda^{3}a^{3}u^{2}_{xx}(a,t)]dt-\mathbb{E}\int^{T}_{0}(12\lambda^{2}u^{2}_{xx}(b,t)+12u^{2}_{xxx}(b,t) )dt\\
&+ \mathbb{E}\int^{T}_{0}( -8\lambda bu^{2}_{xxx}(b,t)+8\lambda au^{2}_{xxx}(a,t))dt+\mathbb{E}\int^{T}_{0}(12\lambda^{2}u^{2}_{xx}(a,t)+12u^{2}_{xxx}(a,t) )dt\\
&\geq -C \mathbb{E}\int^{T}_{0}(\lambda^{3}u^{2}_{xx}(b,t)+\lambda u^{2}_{xxx}(b,t))dt.\end{aligned}$$ That is $$\begin{aligned}
\label{LLL}
A_{1}\geq -C \mathbb{E}\int^{T}_{0}(\lambda^{3}u^{2}_{xx}(b,t)+\lambda u^{2}_{xxx}(b,t))dt.\end{aligned}$$ Direct computation shows that $$\begin{aligned}
& u^{2}_{xxx}\{\cdots\}=F_{1} u^{2}_{xxx},\ \ u^{2}_{xx}\{\cdots\}=F_{2} u^{2}_{xx}, \\
& u^{2}_{x}\{\cdots\}=F_{3} u^{2}_{x},\ \ u^{2}\{\cdots\}=F_{4} u^{2},\end{aligned}$$ where $$\begin{aligned}
F_{1}=&32\lambda,\ \ F_{2}=352\lambda^{3}x^{2},\\
F_{3}=&2304\lambda^{5}x^{4}-768\lambda^{4}x^{2}+192\lambda^{3}x+320\lambda^{3},\\
F_{4}=&1536\lambda^{7}x^{6}+512\lambda^{6}x^{4}-4224\lambda^{5}x^{2}+384\lambda^{4}\\
&-32\lambda^{3}x^{2}(l^{2}_{t}-l_{tt})+2l_{tt}(A-\Phi)+2l_{t}(A-\Phi)_{t}.\end{aligned}$$ Consequently, we have $$\begin{aligned}
\label{abc}
A_{2}:&=\mathbb{E}\int_{Q} F_{4}u^{2}dxdt+\mathbb{E}\int_{Q}F_{3} u_{x}^{2}dxdt+\mathbb{E}\int_{Q} F_{2}u_{xx}^{2}dxdt\\\nonumber
&+\mathbb{E}\int_{Q}F_{1} u_{xxx}^{2}dxdt+\mathbb{E}\int_{Q}2(l_{tt}-\Phi)u^{2}_{t}dxdt.\end{aligned}$$ Since $$\begin{aligned}
A_{3}=&2\mathbb{E}\int_{Q} u_{t}[[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}]_{t}dxdt\\
&=2\mathbb{E}\int_{Q} [u_{t}(-4l^{3}_{x}u_{xt}-6l_{xx}u_{xxt}-4l_{x}u_{xxxt})]dxdt.\end{aligned}$$ Integral by parts, we have $$\begin{aligned}
&\mathbb{E}\int_{Q} u_{t}(-4l^{3}_{x})u_{xt}dxdt
=\mathbb{E}\int_{Q} 6l^{2}_{x}l_{xx}u^{2}_{t}dxdt,\end{aligned}$$ $$\begin{aligned}
\mathbb{E}\int_{Q} u_{t}(-6l_{xx})u_{xxt}dxdt
=\mathbb{E}\int_{Q}6 l_{xx}u^{2}_{xt}dxdt\end{aligned}$$ and $$\begin{aligned}
&\mathbb{E}\int_{Q}(-4l_{x})u_{t}u_{xxxt}dxdt
=-\mathbb{E}\int_{Q}2 l_{xx}u^{2}_{xt}dxdt+\mathbb{E}\int_{Q}4 l_{xx}u_{t}u_{xxt}dxdt\\
&=-\mathbb{E}\int_{Q}2 l_{xx}u^{2}_{xt}dxdt-4\mathbb{E}\int_{Q} l_{xx}u^{2}_{xt}dxdt=-6\mathbb{E}\int_{Q} l_{xx}u^{2}_{xt}dxdt.\end{aligned}$$ Consequently, we have $$\begin{aligned}
\label{lem13}
&A_{3}=2\mathbb{E}\int_{Q}6l^{2}_{x}l_{xx}u^{2}_{t}dxdt.\end{aligned}$$ Since $l_{t}(T)=l_{t}(0)=0,\ u_{t}(a,t)=u_{t}(b,t)=0,\ u_{x}(a,t)=u_{x}(b,t)=0,\ u_{xt}(a,t)=u_{xt}(b,t)=0, $ integral by parts, we obtain $$\begin{aligned}
\label{lem14}
A_{4}:=&2\mathbb{E}\int_{Q} \{2[l_{t}u_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{x}\}dxdt\\\nonumber
&+2\mathbb{E}\int_{Q} \{2[l_{t}(u_{xxx}+(G-\Phi_{1})u_{x})]_{t}u_{x}\}dxdt\\\nonumber
&=2\mathbb{E}\int_{Q} [ 2l_{tt}(u_{xxx}+6l^{2}_{x}u_{x})u_{x}+2l_{t}u_{x}(u_{xxxt}+6l^{2}_{x}u_{xt})]dxdt.\end{aligned}$$ Next, we compute each integral term in , define $$\begin{aligned}
&B_{1}=2\mathbb{E}\int_{Q}12 l_{t}l^{2}_{x}u_{x}u_{xt}dxdt,\ \ \ B_{2}=2\mathbb{E}\int_{Q}12 l_{tt}l^{2}_{x}u^{2}_{x}dxdt,\\
&B_{3}=2\mathbb{E}\int_{Q}2 l_{t}u_{xxxt}u_{x}dxdt,\ \ \ B_{4}=2\mathbb{E}\int_{Q}2 l_{tt}u_{xxx}u_{x}dxdt,\end{aligned}$$ then $$A_{4}=B_{1}+B_{2}+B_{3}+B_{4}.$$ Since $$B_{1}=-\mathbb{E}\int_{Q}12 l_{tt}l^{2}_{x}u^{2}_{x}dxdt,$$ $$\begin{aligned}
B_{3}=\mathbb{E}\int_{Q}2 l_{tt}u^{2}_{xx}dxdt,\ \ \ B_{4}=-2\mathbb{E}\int_{Q}2 l_{tt}u^{2}_{xx}dxdt.\end{aligned}$$ We have $$\begin{aligned}
A_{4}&=\mathbb{E}\int_{Q}12 l_{tt}l^{2}_{x}u^{2}_{x}dxdt-\mathbb{E}\int_{Q}2l_{tt}u^{2}_{xx}dxdt.
\end{aligned}$$ Since $l_{t}(0)=l_{t}(T)=0$, we have $$\begin{aligned}
A_{5}&=2\mathbb{E}\int_{Q} [l_{t}(A-\Phi)u^{2}-2l_{t}u_{x}(u_{xxx}+(G-\Phi_{1})u_{x})]_{t}dxdt=0.
\end{aligned}$$ Consequently, we have $$\begin{aligned}
\label{efg}
A_{4}+ A_{5}=\mathbb{E}\int_{Q}12 l_{tt}l^{2}_{x}u^{2}_{x}dxdt-\mathbb{E}\int_{Q}2l_{tt}u^{2}_{xx}dxdt.
\end{aligned}$$ On the one hand, $$\begin{aligned}
\label{lem20}
&2\mathbb{E}\int_{Q}\{-2l_{t}u_{t}+\{[B-(G-\Phi_{1})_{x}]u_{x}+\Phi_{1}u_{xx}+Du_{xxx}+\Phi u\}\}\theta fdtdx\\\nonumber
&\leq \mathbb{E}\int_{Q} p^{2}dxdt+\lambda^{2}\mathbb{E}\int_{Q} \theta^{2}f^{2}dxdt.\end{aligned}$$ On the other hand, $$\begin{aligned}
\label{lem23}
\mathbb{E}\int_{Q}l_{t}(du_{t})^{2}dx=E\int_{Q} l_{t}\theta^{2}g^{2}dtdx\leq C\lambda \mathbb{E}\int_{Q} \theta^{2}g^{2}dtdx.\end{aligned}$$ Combining , , , - with , we can obtain $$\begin{aligned}
\label{lem24}
&\mathbb{E}\int_{Q} H_{1}u^{2}_{t}dxdt
+\mathbb{E}\int_{Q} H_{2}u^{2}dxdt
+\mathbb{E}\int_{Q} H_{3}u_{x}^{2}dxdt\\\nonumber
&+\mathbb{E}\int_{Q} H_{4} u_{xx}^{2}dxdt
+\mathbb{E}\int_{Q} H_{5}u_{xxx}^{2}dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}(\lambda^{3}u^2_{xx}(b,t)+\lambda u^2_{xxx}(b,t))dt+\int_{Q}\lambda^{2}(f^{2}+g^{2})dxdt\right\}\\\end{aligned}$$ where $$\begin{aligned}
&H_{1}=2l_{tt}+4l^{2}_{x}l_{xx},\ \ H_{2}=F_{4},\ \ H_{3}=F_{3}-12l_{tt}l^{2}_{x} \\
&H_{4}=F_{2}+2l_{tt},\ \ H_{5}=F_{1}.\end{aligned}$$ When $\lambda$ large enough, we can use the high order terms of $\lambda$ to absorb the low order terms, it is easy to obtain, $$\begin{aligned}
\label{lem25}
H_{1}\geq C_{1}\lambda^{3},\ \ H_{2}\geq C_{2}\lambda^{7},\ \ H_{3}\geq C_{3}\lambda^{5},\ H_{4}\geq C_{4}\lambda^{3},\ \ H_{5}\geq C_{5}\lambda,\end{aligned}$$ where $C_{1},C_{2},C_{3},C_{4},C_{5}$ only depended on $a,b$.
Consequently, from and , there exists a constant $C(I,T)$ and a constant $\lambda_0>0$ sufficiently large, such that for every $\lambda>\lambda_0$, it holds that $$\begin{aligned}
&\mathbb{E}\int_{Q}(\lambda u^{2}_{xxx}+\lambda^{3} u^{2}_{xx}+\lambda^{5} u^{2}_{x}+\lambda^{7} u^{2}+\lambda^{3}u^{2}_{t})dxdt\\
&\leq C(I,T)\mathbb{E}\int^{T}_{0}(\lambda^{3}u^{2}_{xx}(b,t)+\lambda u^{2}_{xxx}(b,t))dt+C(I,T)\mathbb{E}\int_{Q}(\lambda^{2}f^{2}+\lambda^{2}g^{2})dxdt.\end{aligned}$$ Substituting $u$ to $\theta y$, we can obtain $$\begin{aligned}
\mathbb{E}&\int_{Q}\theta^2(\lambda y^{2}_{xxx}+\lambda^{3} y^{2}_{xx}+\lambda^{5} y^{2}_{x}+\lambda^{7} y^{2}+\lambda^{3}y^{2}_{t})dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(b,t)+\lambda y^2_{xxx}(b,t))dt+\int_{Q}\theta^{2}\lambda^{2}(f^{2}+g^{2})dxdt\right\}.\end{aligned}$$ This completes the proof of Theorem \[Th2\].
Next, we prove Theorem \[Th4\].
Now we choose a $\chi\in C^{\infty}_{0}[0,T]$ satisfying $$\label{lem38}
\chi(t)=\left\{\begin{array}{ll}
1,& t\in [\epsilon,T-\epsilon],\\
0,& t\in [0,\epsilon/2]\cup [T-\epsilon/2,T],\\
\in (0,1),&\mbox{ otherwise},
\end{array}\right.$$ such that $|\chi'(t)|\leq\frac{c_{1}}{\epsilon},\ \ |\chi''(t)|\leq\frac{c_{2}}{\epsilon^{2}}$, for some positive constants $c_{1},c_{2}$ independent of $\epsilon$ see[@YDW].
Let $z=\chi y$ for $y$ solving equation , then we know that $z$ is the solution to the following equation $$\label{int55}
\begin{cases}
dz_{t}+z_{xxxx}dt =(\chi f+\alpha)dt+\chi gdB(t) &\text{in} \ \ Q=(0,T)\times I,\\
z(a,t)=0,\ z(b,t)=0&\text{in} \ (0,T),\\
z_{x}(a,t)=0,\ z_{x}(b,t)=0 & \text{in}\ (0,T),\\
z(x,0)=z(x,T)=0,\ z_{t}(x,0)=z_{t}(x,T)=0 & \text{in}\ \ I ,
\end{cases}$$ where $\alpha=\chi_{tt}y+2\chi_{t}y_{t}$.
Then, using the result of Theorem \[Th2\] for system , we obtain $$\begin{aligned}
\label{int37}
\mathbb{E}&\int_{Q}\theta^2(\lambda z^{2}_{xxx}+\lambda^{3} z^{2}_{xx}+\lambda^{5} z^{2}_{x}+\lambda^{7} z^{2}+\lambda^{3}z^{2}_{t})dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}\theta^{2}(\lambda^{3}z^2_{xx}(b,t)+\lambda z^2_{xxx}(b,t))dt+\int_{Q}\theta^{2}\lambda^{2}(f^{2}+\alpha^{2}+g^{2})dxdt\right\}.\end{aligned}$$ Recalling the property of $\chi$(see) and $z=\chi y$, from , we obtain $$\begin{aligned}
\mathbb{E}&\int^{T-\epsilon}_{\epsilon}\int^{b}_{a}\theta^2(\lambda y^{2}_{xxx}+\lambda^{3} y^{2}_{xx}+\lambda^{5} y^{2}_{x}+\lambda^{7} y^{2}+\lambda^{3}y^{2}_{t})dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(b,t)+\lambda y^2_{xxx}(b,t))dt+\int_{Q}\theta^{2}\lambda^{2}(f^{2}+g^{2})dxdt\right\}\\\nonumber
&+\frac{C(I,T)}{\epsilon^{4}}\lambda^{2}\left[\mathbb{E}\int^{\epsilon}_{0}\int^{b}_{a}\theta^2(y^{2}_{t}+y^{2})dxdt+\mathbb{E}\int^{T}_{T-\epsilon}\int^{b}_{a}\theta^2(y^{2}_{t}+y^{2})dxdt\right].\end{aligned}$$ This completes the proof of Theorem \[Th4\].
Proof of Observability inequality (Theorem \[Th1\]) {#sec5}
===================================================
In this section, we prove Theorem \[Th1\].
By Theorem \[Th4\], fix $0<\epsilon<\frac{T}{2}$, we have $$\begin{aligned}
\label{int59}
\mathbb{E}&\int^{T-\epsilon}_{\epsilon}\int^{b}_{a}\theta^2(\lambda^{3} y^{2}_{xx}+\lambda^{5} y^{2}_{x}+\lambda^{7} y^{2}+\lambda^{3}y^{2}_{t})dxdt\\\nonumber
&\leq C(I,T)\mathbb{E}\left\{\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(b,t)+\lambda y^2_{xxx}(b,t))dt+\int_{Q}\theta^{2}\lambda^{2}(f^{2}+g^{2})dxdt\right\}\\\nonumber
&+\frac{C(I,T)}{\epsilon^{4}}\lambda^{2}\left[\mathbb{E}\int^{\epsilon}_{0}\int^{b}_{a}\theta^2(y^{2}_{t}+y^{2}+y^{2}_{x}+y^{2}_{xx})dxdt+\mathbb{E}\int^{T}_{T-\epsilon}\int^{b}_{a}\theta^2(y^{2}_{t}+y^{2}+y^{2}_{x}+y^{2}_{xx})dxdt\right].\end{aligned}$$ By in Theorem \[Th3\], we have $$\begin{aligned}
\label{int60}
\lambda^{2}\mathbb{E}\int^{\epsilon}_{0}\int^{b}_{a}(y^{2}_{t}+y^{2}+y^{2}_{x}+y^{2}_{xx})dxdt&\leq \epsilon C\lambda^{2}\mathbb{E}(\|y(T)\|^{2}_{H_{0}^{2}(I)}+\|y_{t}(T)\|^{2}_{L^{2}(I)})\\\nonumber
&+\epsilon C\lambda^{2}\left(\|f\|_{L^{2}_{\mathcal{F}}(0,T;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T;L^{2}(I))}\right)\\\nonumber
\end{aligned}$$ $$\begin{aligned}
\label{int66}
\lambda^{2}\mathbb{E}\int^{T}_{T-\epsilon}\int^{b}_{a}(y^{2}_{t}+y^{2}+y^{2}_{x}+y^{2}_{xx})dxdt&\leq \epsilon C\lambda^{2}\mathbb{E}(\|y(T)\|^{2}_{H_{0}^{2}(I)}+\|y_{t}(T)\|^{2}_{L^{2}(I)})\\\nonumber
&+\epsilon C\lambda^{2}\left(\|f\|_{L^{2}_{\mathcal{F}}(0,T;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T;L^{2}(I))}\right)\\\nonumber
\end{aligned}$$ $$\begin{aligned}
\label{int61}
&\mathbb{E}\int^{T-\epsilon}_{\epsilon}\int^{b}_{a}\theta^2(\lambda^{3} y^{2}_{xx}+\lambda^{5} y^{2}_{x}+\lambda^{7} y^{2}+\lambda^{3}y^{2}_{t})dxdt\\\nonumber
&\geq(T-2\epsilon)C\lambda^{3}\mathbb{E}(\|y(T)\|^{2}_{H_{0}^{2}(I)}+\|y_{t}(T)\|^{2}_{L^{2}(I)})
-C\left(\|f\|_{L^{2}_{\mathcal{F}}(0,T;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T;L^{2}(I))}\right)\end{aligned}$$ Combining , and ,we know there is a $\lambda_{1}=\lambda(\epsilon)$ large enough, such that for all $\lambda\geq \max\{\lambda_{0},\lambda_{1}\}$, if we chose $\lambda$ such that $(T-2\epsilon)C\lambda-\frac{2\epsilon C(I,T)}{\epsilon^{4}}>\frac{1}{2}$, we obtain $$\begin{aligned}
\mathbb{E}(\|y(T)\|^{2}_{H_{0}^{2}(I)}+\|y_{t}(T)\|^{2}_{L^{2}(I)})&\leq
C(I,T)\left\{\mathbb{E}\int^{T}_{0}\theta^{2}(\lambda^{3}y^2_{xx}(b,t)+\lambda y^2_{xxx}(b,t))dt\right\}\\
&+C(I,T)\left(\|f\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}+\|g\|_{L^{2}_{\mathcal{F}}(0,T ;L^{2}(I))}\right).\end{aligned}$$ That is $$\begin{aligned}
&\|(y(T),y_{t}(T))\|_{L^{2}(\Omega,\mathcal{F}_{T},P;H_0^{2}(I)\times L^{2}(I))}\\
&\leq C(I,T)\left(\mathbb{E}\int^{T}_{0}(y^2_{xx}(b,t)+y^2_{xxx}(b,t))dt
+\|f\|_{L^{2}_{\mathcal{F}}(0,T;H^{2}(I))}+\|g\|_{L^{\infty}_{\mathcal{F}}(0,T;H^{4}(I))}\right),\end{aligned}$$ $$\forall \ (y_{0},y_{1})\in L^{2}(\Omega, \mathcal{F}_{0},P;(H^2_0(I)\cap H^{4}(I))\times H^{2}(I)).$$
This completes the proof of Theorem \[Th1\].
[99]{}
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abstract: 'This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite differences schemes for solving the system of inequalities is proposed and the convergence of the numerical solution towards the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application, namely, that of range extender electric vehicles. This class of electric vehicles uses an additional module – the range extender – as an extra source of energy in addition to its main source – a high voltage battery. The reachability study of this system is used to establish the maximum range of a simple vehicle model.'
author:
- 'G. Granato[^1]'
- 'H. Zidani[^2]'
bibliography:
- 'journal\_reachability.bib'
title: 'Level-set approach for Reachability Analysis of Hybrid Systems under Lag Constraints[^3]'
---
Optimal control, Quasi-variational Hamilton-Jacobi equation, Hybrid systems, Reachability analysis
49LXX, 34K35, 34A38, 65M12
Introduction
============
This paper deals with the characterization of a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set.
The term hybrid system refers to a general framework that can be used to model a large class of systems. Broadly speaking, they arise whenever a collection continuous- and discrete-time dynamics are put together in a single model. In that sense, the discrete dynamics may dictate switching between the continuous dynamics, jumps in the system trajectory or both. Moreover, they can contain specificities, as for instance, autonomous jumps and/or switches, time delay between discrete decisions, switching/jumping costs. This work considers a particular class of hybrid system where only switching between continuous dynamics are operated by the discrete logic, with no jumps in the trajectory, and there are no switching costs. In addition, switch decisions are constrained to be separated in time by a non-zero interval, fact which is referred to as switching lag.
Before referring to the reachability problem in the hybrid setting, the main ideas are introduced in the non-hybrid framework. Given a time $t>0$, a closed target set $X_0$ and a closed admissible set $K$, considering a controlled dynamical system $$\label{eq.classique}
\dot y(\tau) = f(\tau,y(\tau),u(\tau)), ~~ \text{a.e. } \tau \in [0,t],$$• where $f: {{\mathrm I\! \textsc{R} }}^+ \times {{\mathrm I\! \textsc{R} }}^d \times U \to {{\mathrm I\! \textsc{R} }}^d$ and $u: {{\mathrm I\! \textsc{R} }}^+ \rightarrow U $ is a measurable function, the reachable set $R_{X_0}$ at time $t$ is defined as the set of all initial states $x$ for which there exists a trajectory that stays inside $K$ on $[0,t]$ and arrives at the target: $$R_{X_0}(t) := \{ x ~|~ (y,u) \text{ satisfies \eqref{eq.classique}}, \mbox{ with } y(0)=x \text{ and } y(t) \in X_0 \text{ and } y(s)\in K \mbox{ on } [0,t] \}.$$• It is a known fact that the reachable set can be characterized by the the negative region of the value function of an optimal control problem. For this, following the idea introduced by Osher [@OsherSethian88], one can consider the control problem defined by: $$\label{eq.pbclassique}
v(x,t):=\inf\{v_0(y(t))\mid (y,u) \text{ satisfies \eqref{eq.classique}}, \mbox{ with } y(0)=x
\mbox{ and } y(s)\in K \mbox{ on } [0,t]\},$$ where $v_0$ is a Lipschitz continuous function satisfying $v_0(x)\leq 0 \Longleftrightarrow x\in X_0$ (for instance,$v_0$ can be the signed distance $d_{X_0}$ to $X_0$). Under classical assumptions on the vector-field $f$, one can prove that the reachable set is given by $$R_{X_0}(t)=\{x\in K, v(x,t)\leq 0 \}.$$ Moreover, when $K$ is equal to ${{\mathrm I\! \textsc{R} }}^d$, the value function has been shown to be the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation [@Bardi97optimalcontrol]: $$\partial_t v+\sup_{u\in U}(-f(s,x,u)\cdot \nabla v)=0 \quad \mbox{ on } {{\mathrm I\! \textsc{R} }}^d \times (0,t],$$ with the initial condition $v(x,0)=v_0(x)$.
When the set $K$ is a subset of ${{\mathrm I\! \textsc{R} }}^d$ ($K\neq {{\mathrm I\! \textsc{R} }}^d$), the characterization of $v$ by means of a HJB equation becomes a more delicate matter and usually requires some additional controllability properties [@CardQuincampoix; @BFZ]. However, it was pointed out in [@BFZ] that in case of state-constraints, the auxiliary control problem should be introduced as $$\label{eq.pbcontraint}
V(x,t):=\inf\left\{v_0(y(t)) \bigvee \max_{\theta\in[0,t]}g(y(\theta))
\mid (y,u) \mbox{ satisfies \eqref{eq.classique}}, \mbox{ with } y(0)=x \right\},$$ where $g$ is any Lipschitz continuous function satisfying $g(x)\leq 0 \Longleftrightarrow x\in K$ (again, $g$ can be the signed distance $d_{K}$ to $K$). This new control problem involves a supremum cost but does not include any state constraints. The reachability set is still given by $$R_{X_0}(t)=\{x\in K, V(x,t)\leq 0 \},$$ and the value function $V$ is the unique viscosity solution of a variational inequality $$\min( \partial_t V+\sup_{u\in U}(-f(s,x,u)\cdot \nabla V), V-g)=0 \quad \mbox{ on } (0,t]\times {{\mathrm I\! \textsc{R} }}^d,$$ with the initial condition $v(x,0)=\min(v_0(x),g(x))$, but no controllability assumption is needed.
In this paper, we are interested in the extension of the reachability framework to some class of control problems of Hybrid systems.
Let us recall that a hybrid dynamical system is a collection of controlled continuous-time processes selected through a high-level discrete control logic. A general framework for the (optimal) control hybrid dynamical systems was introduced in [@Branicky]. Several papers deal with the optimal control problem of hybrid systems, let us just mention here the papers of [@Garavello; @Arutyunov; @Vinter-Caines; @Sussman] where the optimality conditions in the form of Pontryagin principle are studied and [@Zhang; @Dolcetta:143; @Dharmatti2] where the HJB approach is analyzed.
A feature of the hybrid system used in our work is a time lag between two consecutive switching decisions. From the mathematical viewpoint this remove the particularities linked to Zeno-like phenomena [@zenoYuBarbot]. Indeed, the collection of state spaces is divided in subsets labeled in three categories according to whether they characterize discrete decisions as optional, required (autonomous) or forbidden. Landing conditions ensure that whatever the region of the state space the state vector lands after a switch no other switch is possible by requiring a positive distance (in the Hausdorff sense) between the landing sets and the optional/autonomous switch sets. In the other hand, when allowed to switch freely without any costs, when no time interval is imposed between discrete transitions, a controller with a possibly infinite number of instantaneous switches may become admissible. Switch costs can be introduced in order to rule out this kind of strategy by the controller as it becomes over-expensive to switch to a particular mode using superfluous transitions. However, such costs do not make sense in the level-set approach used in this paper.
Our study is motivated by an industrial application, namely, that of range extender electric vehicles. This class of electric vehicles uses an additional module – the range extender – as an extra source of energy in addition to its main energy source (a high voltage battery). For that matter, an adequate class of hybrid systems considers only mode switching, i.e. switching between different continuous time dynamics, without any trajectory discontinuities. Moreover, no transition costs are taken into account and the switching decisions can be done freely without any penalty. Also motivated by the application, the discrete control must respect a time interval of at least $\delta >0$ between two consecutive switching decisions. This decision lag condition can be viewed as replacing landing conditions and positive switch costs requirements [@Dharmatti2].
Diffusion processes with impulse controls including switch lags are studied in [@pham], where it is considered the idea of introducing a state variable to keep track of the time since the last discrete control decision. There, in addition, discrete decisions also suffer from a time delay before they can manifest in the continuous-time process. In that case, one has the possibility of scheduling discrete orders whenever the time for a decision to take place may be longer than that of deciding again. Then, the analysis also includes keeping record of the nature of this scheduled orders. This work inspired the idea of a state variable locking possible transitions used here.
To study the reachability sets for our system, we follow the level set approach and adapt the ideas developed in [@BFZ] to hybrid systems by proposing a suitable control problem which allows us to handle in a convenient way the state constraints and the decision lag. It is proven that $$R_{X^0}(s) = \{ x ~|~ \exists (q,p) \in Q\times P, ~v(x,q,p,s) \leq 0 \}.$$ where $v$ denotes the value function of a hybrid optimal control problem. Thus, through a characterization of $v$, one obtains the desired reachable set, defined in the hybrid context, $x$ is in the (physical) state of the system, $q$ is the discrete variable and $p$ is a switch lock variable (defined further below). Here the main difficulty is to characterize the value function associated to the control problem. It turns out that this value function satisfies a quasi-variational HJB inequality system (in the viscosity sense) $$\begin{aligned}
\min(\partial_s u + \partial_p u + H(s,x,q,\nabla_x u) , u-\varphi(x) ) = 0 &,& ~(x,q,p,s) \in \Omega, \label{HJBinside}\\
u(x,q,p,s) -(Mu)(x,q,p,s)= 0 &,&~ p=0 \\
u(x,q,p,0) = \max(\phi(x),\varphi(x)).&&
\label{HJBmain}
\end{aligned}$$ where $t>0$, $d>0$, $X = {{\mathrm I\! \textsc{R} }}^d$, $Q = \{0,1\}$, $P=(0,t]$, $T=(0,t]$, $\Omega = X \times Q \times P \times T$ and where $\phi,\varphi : X \to {{\mathrm I\! \textsc{R} }}$ are target and obstacle indicator functions (defined properly further below). Here, $M$ is a non-local switch operator that acts whenever the state variable $p$ touches the boundary $p=0$. Moreover, we give a comparison principle of this system. Usually, the proof of the comparison principle requires some transversality assumptions that do not make sense in the kind of applications we are interested in. However, while the decision lag complicates the structure of the problem it also plays a role in the proof of comparison principle (the same role that the transversality assumed in [@Zhang; @Dharmatti]) acting as a kind of landing condition in the sense explained above. The proof of the comparison principle is close to the one given in [@Dharmatti2; @BarlesSheetal] and adapts the idea of using friendly giants-type functions.
This work is motivated by an application in the automobile industry, namely, calculating the autonomy of a class of electric vehicles (EVs), the range extender electric vehicle (REEV) class, which posses two distinct sources of energy from which we can use to tract the vehicle. In this setting, the study aims at finding the control sequence of the two energy sources that allows the vehicle the reach the furthest possible point of a given route. The REEV is modeled as a hybrid dynamical system in which the state vector represents the energy capacities of the two different energy sources embedded in the vehicle.
This paper is organized as follows: firstly, it describes the industrial application motivating the study and states the associated hybrid optimal control problem (in a slightly more general setting than that required for the application). Then, the reachable set and the value function are defined and a dynamic programming principle for the value function is obtained. The value function is shown to be Lipschitz continuous and also a solution of an system of quasi-variational inequalities (SQVI). It follows with the proof of a comparison principle for the SQVI that ensures uniqueness of its solution and is used to show the convergence of a class of numerical schemes for the computation of the value function. Lastly results of some numerical simulations evaluating the autonomy of a REEV toy model and illustrating the convergence of a discretization scheme are presented.
Motivation and Problem Settings
===============================
Range Extender Electric Vehicles
--------------------------------
A range extender (RE) electric vehicle is an electric vehicle that disposes of an additional source of energy besides the main high voltage (HV) traction battery. Both the vehicle’s energy sources are considered to have normalized energetic capacities – thus valued between $0$ and $1$.
The controls available are the RE’s state – on or off – and the power produced in the RE (and delivered into the powertrain). The power delivered into the powertrain is a non-negative piecewise continuous time function. The RE’s state is controlled by a discrete sequence of switching orders decided and executed at discrete times. An important feature of the REEV model is a time interval $\delta > 0$ imposed between two consecutive decisions times. From the physical viewpoint, this assumption incorporates the fact that frequent switching of the RE is undesirable in order to avoid mechanical wear off and acoustic nuisance for the driver.
The model considers that the vehicle’s traction capability is conditioned to the existence of some electric energy in the battery. Since the vehicle must halt whenever there is no charge available in the battery, the objective of finding the vehicle autonomy is summarized into finding the furthest point away from the vehicle geographic starting point where the battery is depleted for the first time.
Hybrid Dynamical System
-----------------------
Hybrid systems have some supervision logic that intervenes punctually between two or more continuous functions. The main elements of the class of hybrid dynamical systems considered in this work are a family of continuous dynamics (vector fields) $f$ and continuous state spaces $X$, indexed by a discrete state $q$ valued in a discrete set $Q$. Each continuous dynamic system $f_q$ valued in $X_q$ models a physical process controlled by a continuous control function $u$, from which the system is free to switch to another process $f_{q\rq{}}$ using a specified discrete control $w$ and a discrete dynamics $g$.
More precisely, the continuous state variable is denoted $y$ and it is valued in the state space $X = {{\mathrm I\! \textsc{R} }}^d$. The discrete variable is $q \in Q = \{0,1,\cdots,d_q \}$, where $d_q$ is the number of possible dynamics that can operate the system. (for simplicity of the presentation, through this paper, we consider that $X_q= X$ for all $q\in Q$). Each of these dynamics models a different mode of operation of the system or a different physical process that the system is undergoing. Moreover, define a compact set $K \subset X$ as the hybrid system admissible set, i.e., a set inside which the state must remain.
The continuous control is supposed to be a measurable function $u$ valued in a set that depends on the mode that is currently active $U(q)$. The discrete control is a sequence of switching decisions $$w =\{ (w_1,s_1), \cdots, (w_i,s_i), (w_{i+1},s_{i+1}),\cdots\},$$• where each $s_i \in [0,\infty[$ and $w_i \in W(q) \subset \{0,1, \cdots,d_q\}$. The sequence of discrete switching decisions $\{w_i\}_{i>0}$ (designating the new mode of operation) is associated with a sequence of switching times $\{s_i\}_{i>0}$ where each decision $w_i$ is exerted at time $s_i$. The set of available discrete decisions, at time $s$, $W(q(s))$, depends on the discrete state variable and it corresponds to a decision of switching the system to another process $w_i$.
The lag condition between switches is included by demanding that two switch orders must be separated by a time interval of $\delta >0$, i.e., $$s_{i+1}-s_i \geq \delta.$$
Regarding the vehicle application, the vehicle’s energy state is a two-dimensional vector $y \in X = {{\mathrm I\! \textsc{R} }}^2$, where $y=(y_1,y_2)$ denotes the state of charge of the battery and the fuel available in the range extender module. Each of these quantities are the image of the remaining energy in the battery and the RE respectively. It is clear here that the state variables have to be constrained to remain in the compact set $K = [0,1]^2$, where the energies quantities are normalized. $q \in Q = \{0,1\}$ is the RE state, indicating whether the RE is off ($q=0$) or on ($q=1$). The power output is a measurable function $u(\cdot) \in U(q(\cdot))$ where $U(\cdot)$ is the admissible control set, compact subset of ${{\mathrm I\! \textsc{R} }}$ dependent naturally on the RE state.
In this setting, given a discrete state $q$, the continuous control $u$ steers the continuous system $$\dot{y}(\tau) = f(\tau,y(\tau),u(\tau),q_i), \text{ for a.e. } \tau \in [0,t] \label{sysContinuous}$$ where some continuous dynamics $f(\cdot,\cdot,\cdot,q_i)$ is activated. $f$ is a family of vector fields indexed by the discrete variable $q$. When $q=q_i$, the corresponding vector field is active and dictates the evolution of the continuous state. At some isolated times $\{s_i\}_{i>0}$, given by the discrete switching control sequence, the discrete dynamic $g$ is activated $$q_{i-1} = g(w_i,q_i)$$ and the continuous state follows another vector field $f(\cdot,\cdot,\cdot,q_{i-1})$. In the considered system, the discrete decisions only switch the continuous dynamics and introduce no discontinuities on the trajectory. In more general frameworks, we can also include jumps in the continuous state vector that can be used to model an instant change in the value of the state following a discrete decisions [@Branicky; @BM].
Assume controlled continuous dynamics $f$ and the discrete dynamic $g$ satisfy the following:
(H0)
: The continuous control is a measurable function $u: [0,\infty[ \rightarrow {{\mathrm I\! \textsc{R} }}^m$ such that $$u(\tau) \in U(q(\tau)) \text{ for a.e. } \tau\in [0,t].$$
(H1)
: There exists $L_f > 0$ such that, for all $s\geq 0$, $y,y' \in X$, $q \in Q$ and $u \in U(q)$, $$\nonumber \| f(s,y,u,q) - f(s,y',u,q) \| \leq L_f\|y-y'\|, ~~ \|f(s,y,u,q) \| \leq L_f.$$
(H2)
: For all $q\in Q$, $f(\cdot,\cdot,\cdot,q) : [0,\infty[ \times X \times U \rightarrow X$ is continuous and for all $s \in [0,\infty[, x\in X, u \in U $, $f(s,x,u,\cdot) : Q \to X$ is continuous with respect to the discrete topology.
(H3)
: For all $s\in [0,t]$, $y \in X$ and $q \in Q$, $f(s,y,U,q)$ is a convex subset of $X$.
(H4)
: There exists $L_g > 0$ such that, for all $q \in Q$ and $w \in W(q)$, $$\nonumber \| g(w,q) \| \leq L_g$$
(H5)
: $g(\cdot,\cdot) $ is continuous with respect to the discrete topology.
Assumption (H1) ensures that a trajectory exists and that it is unique. Assumptions (H2)-(H5) are used to prove the Lipschitz continuity of the value function and (H3) is needed in order to observe the compactness of the trajectory space.
Denote $A$ the space of hybrid controls $a = (u,w)$. We precise the class of admissible controls ${{\mathcal{A} }}\subset A$ in the following definition:
For a fixed $t\geq 0$ a hybrid control $a = (u,w) \in {{\mathcal{A} }}$ is said to be admissible if the continuous control verifies (H0) and the discrete control sequence $w=\{w_i,s_i \}_{i >0}$ has increasing decision times $$s_1 \leq s_2 \leq \cdots \leq s_i \leq s_{i+1} \leq \cdots \leq t,
\label{discrete1}$$ admissible decisions $$\forall i > 0 ,~ w_i \in W(q(s_i)) \subset Q,
\label{discrete2}$$ and verifies a decision lag $$s_{i+1}-s_i \geq \delta,
\label{discrete3}$$ where $\delta >0$.
An important consequence in the definition of admissible control is the finiteness of the number of switch orders:
Fix $s \geq 0$. Let $a \in A$ be an admissible hybrid control. Then, the discrete control sequence has at most $N= \lfloor s / \delta \rfloor$ switch decisions. \[finiteJumps\]
Fix $t>0$. Given a hybrid control $a \in A$ with $N$ switch orders and given $x\in X$, $q\in Q$, the hybrid dynamical system is $$\begin{aligned}
\dot{y}(\tau) &=& f(\tau,y(\tau),u(\tau),q_i),~~ \tau \in [0,t],~~~~~~~~~~ y(t) = x \label{HybridSystemA} \\
q_{i-1} &=& g(w_i,q_i),~~ i=1,\cdots,N,~~~~~~~~~~~~~~~~~\, q_N = q \label{HybridSystemB}
\end{aligned}$$
Denote the solutions of - with final conditions $x,q$ by $y_{x,q;t}$ and $q_{x,q;t}$. As pointed out, not all discrete control sequences are admissible. Only admissible control sequences engender admissible trajectories. Thus, given $t >0$, $x \in X$ and $q \in Q$, the admissible trajectory set $Y^{x,q}_{[0,t]} $ is defined as $$Y^{x,q}_{[0,t]} = \{ y(\cdot) ~|~ a \in {{\mathcal{A} }}\text{ and }y_{x,q;t} \text{ solution of \eqref{HybridSystemA}-\eqref{HybridSystemB} }\}
\label{admTrajAux}$$• A consequence of proposition \[finiteJumps\] and the above definition is the finiteness of the number of discrete decisions in any admissible trajectory. Observe that the admissible trajectories set does not include the discrete trajectory.
The hybrid control admissibility condition formulated as in conditions - is not well adapted to a dynamic programming principle formulation, needed later on. In order to include the admissibility condition in the optimal control problem in a more suitable form, we introduce a new state variable $\pi$. Recall that the decision lag conditions implies that new switch orders are not available up to a time $\delta$ since the last switch. The new variable is constructed such that at a given time $\tau \in [0,t]$, the value of $\pi(\tau)$ measures the time since the last switch. The idea is to impose constraints on this new state variable and treat them more easily in the dynamic programming principle. Thus, if $\pi(\tau)<\delta$ all switch decisions are blocked and if, conversely, $\pi(\tau) \geq \delta$ the system is free to switch. For that reason, this variable can be seen as a switch lock.
Now given $t >0$, $\tau \in [0,t]$ and a discrete control $w=\{w_i,s_i \}_{i > 0}$, the switch lock dynamics is defined by $$\pi^w(\tau) = \pi(\tau) = \left\{ \begin{array}{lll}
\delta + \tau & \text{if} & \tau < s_1 \\
\inf_{s_i \leq s} \tau-s_i & \text{if} & \tau \geq s_1 \\
\end{array} \right.
\label{pDynamic}$$ Indeed, once the discrete control is given, the trajectory $\pi(\cdot)$ can be determined. Proceeding with the idea of adapting the admissibility condition in order to manipulate it in a dynamic programming principle, we wish to consider $\pi(t)=p$, with $p \in P := (0,t]$, the final value of the switch lock variable trajectory and impose the lag condition under the form $\pi(s_i^-) \geq \delta$ for all $s_i$, where $s_i^-$ denotes the limit to the left at the switching times $s_i$ (notice that $\pi(s_i^+)=0$ by construction). Then, since these conditions suffice to define an admissible discrete control set, while optimizing with respect to admissible functions, one needs only look within the set of hybrid controls that engenders a trajectory $\pi(\cdot)$ with the appropriate structure. In other words, given $t >0$, $x \in X$, $q \in Q$ and $p \in P$, define a admissible trajectory set $ S_{[0,t]}^{x,q,p} $ as $$\begin{aligned}
\nonumber S_{[0,t]}^{x,q,p} &=& \{ y(\cdot) ~|~ a = (u,\{w_i,s_i \}_{i =1}^{N}) \in A, ~y_{x,q,p;t} \text{ solution of \eqref{HybridSystemA}-\eqref{HybridSystemB}}, ~~~~~~~~~~ \\
&& ~~~~~~~~\pi(\cdot) \text{ solution of \eqref{pDynamic}}, ~\pi(t) = p, ~\pi(s_i^-) \geq \delta, ~ i=1,\cdots,N \}.
\label{admTrajSet}
\end{aligned}$$
The next lemma states a relation between sets $Y$ and $S$:
Following the above definitions, sets and satisfy $$Y^{x,q}_{[0,t]} = \bigcup_{p \in P} S_{[0,t]}^{x,q,p}$$•
The equivalence between $Y^{x,q}_{[0,t]}$ and $\bigcup_{p \in P} S_{[0,t]}^{x,q,p}$ is obtained by construction.
In the following of the paper, whenever we wish to call attention to the fact that the final conditions of , and are fixed, we denote their solutions respectively by $y_{x,q,p;t},q_{x,q,p;t},\pi_{x,q,p;t}$.
Reachability of Hybrid Dynamical Systems and Optimal Control Problem
--------------------------------------------------------------------
Let $X_0 \subset X$ be the set of allowed initial states, i.e. the set of states from which the system - is allowed to start. Define the reachable set as the set of all points attainable by $y$ after a time $s$ starting within the set of allowed initial states $X_0$ to be $$\begin{aligned}
\nonumber R_{X^0}(s) &=& \{ x ~|~ \exists q \in Q,~ y_{x,q;s} \in Y^{x,q}_{[0,s]}, ~y_{x,q;s}(0) \in X^0, \text{ and } y_{x,q;s}(\theta) \in K,~ \forall \theta \in [0,s] \} \\
\nonumber &=& \{ x ~|~ \exists (q,p) \in Q\times P, ~ y_{x,q,p;s} \in S^{x,q,p}_{[0,s]}, ~y_{x,q,p;s}(0) \in X^0 \\
& & \hspace{5cm} \text{ and } y_{x,q,p;s}(\theta) \in K,~ \forall \theta \in [0,s] \}.
\label{reachForward}
\end{aligned}$$
In other words, the reachable set $R_{X^0}(s)$ contains the values of $y_{x,q;s}(s)$, regardless of the final discrete state, for all admissible trajectories – i.e., trajectories obtained through an admissible hybrid control – starting within the set of possible initial states $X_0$, that never leave set $K$.
Observe that defines the reachable set $R_{X^0}(s) $ in terms of both admissible trajectory sets $Y$ and $S$.
In particular, the information contained in allows one to determine the first time where the reachable set is empty. More precisely, given $X_0 \subset X$, define $s^* \geq 0$ to be $$s^* = \inf \{ s ~|~ R_{X^0}(s) \subset \emptyset \}.
\label{hybridAutonomy}$$• The time is the autonomy of the hybrid system -. Indeed, one can readily see that if no more admissible energy states are attainable after $s^*$, any admissible trajectory must come to a stop beyond this time. Therefore, $s^*$ is the longest time during which the state remains inside $K$.
The following proposition ensures that the space of admissible trajectories is a compact set.
Given $T' >0$, the admissible trajectory set $Y^{x,q}_{[0,T']}$ is a compact set in $C([0,T\rq{}])$ endowed with the topology $W^{1,1}$. \[lemmaINF\]
Fix $q \in Q$ and $0 \leq s < t \leq T\rq{}$. Consider a bounded admissible continuous control sequence $u_n \in L^1([s,t])$. Since $u_n$ is bounded, there exists a subsequence $u_{n_j}$ such that $u_{n_j} \rightharpoonup u$ in $ L^1([s,t])$. Invoking $(H1)-(H2)$, we have $y^{u_n} = y_n \rightharpoonup y$ in $W^{1,1}([s,t])$. Since $W^{1,1}([s,t])$ is compactly embedded in $C^0([0,t])$, we get the strong convergence of the solution $y_n \rightarrow y$ in $C^0([s,t])$. Hypothesis $(H3)$ guarantees that the limit function $y$ is a solution of . Because all controls $u_n$ and the limit control $u$ are admissible, $y$ is an admissible solution.
So far, the proof shows that the limit trajectory is admissible when $q$ is hold constant. Consider a sequence of admissible discrete control sequences $(w)_n$ where the number of switching orders, $0\leq k_n \leq \left\lfloor T / \delta \right\rfloor$ may depend on $n$. Since each term of this sequence has a (first) discrete component and is bounded on the (second) continuous component, then, as $n \rightarrow \infty$ there exists a subsequence $(w)_{n_l}$ and $\Lambda>0$ such that $q_{n_j}=q$ for all $l > \Lambda$. This implies $k_n \rightarrow k$. As the number of switches is constant from the $\Lambda^\text{th}$ term and $s,t$ are arbitrary, one can obtain, using the limit discrete control sequence $w$, the time intervals $[s_{i-1},s_i]$ over which $q_i$ is constant and the argument in the first paragraph of the proof. Because the trajectory is continuous and admissible on all time intervals $[s_{i-1},s_i], i=1,\cdots,K$, it is admissible on $[0,T]$.
Moreover, observe that $\forall s \in [0,t]$, $y_n(s) \in K$. Since for all $s\in [0,t]$, $y_n(s) \rightarrow y(s)$, by the compactness of $K$ we get that $\forall s\in [0,t],$ $y(s) \in K$, which completes the proof.
The arguments presented in the above proof can be slightly modified to show that the admissible trajectory set with fixed final $p$, $S^{x,q,p}_{[0,T']}$ is compact. Also in a similar way, the proof can be adapted to show that the reachable $R_{X^0}$ is closed. Indeed, by the compactness of set $X_0$, a sequence of initial conditions $(y_0)_n \in X_0$, associated with admissible trajectories $y_n \in Y^{x,q}_{[0,T']}$, converges to $y_0 \in X_0$ which is also the initial condition for the limiting trajectory $y_n \to y$.
In order to characterize the reachable set $R_{X_0}$ this paper follows the classic level-set approach [@OsherSethian88]. The idea is to describe as the negative region of a function $v$. It is well known that the function $v$ can be defined as the value function of some optimal control problem. In the case of system -, $v$ happens to be the value function of a hybrid optimal control problem.
Consider a Lipschitz continuous function $\tilde \phi : X \rightarrow {{\mathrm I\! \textsc{R} }}$ such that $$\tilde \phi(x) \leq 0 \Leftrightarrow x \in X^0.$$ Such a function always exists – for instance, the signed distance function $d_{X_0}$ from the set $X^0$. For $ L_K >0$, one can construct a bounded function $ \phi : X \rightarrow {{\mathrm I\! \textsc{R} }}$ as $$\phi(x) = \max(\min(\tilde \phi(x), L_K),- L_K).
\label{distance}$$
For a given point $s\geq 0$ and hybrid state vector $(x,q,p) \in X \times Q \times P$, define the value function to be $$v_0(x,q,p,s) = \inf_{ S_{[0,s]}^{x,q,p}} \left\{ \phi(y_{x,q,p;s}(0)) ~|~ y_{x,q,p;s}(\theta) \in K,~ \forall \theta \in [0,s] \right\}
\label{valueConst}$$ Observe that works as a level-set to the negative part of . Indeed, since contains only admissible trajectories that remain in $K$, by implies that $v_0(x,q,p,s)$ is negative if and only if $y_{x,q,p;s}(0)$ is inside $X_0$, which in turn implies that $x \in R_{X^0}(s)$.
Remark however that when defining the value function with , one includes state constraints, with the condition that $y_{x,q,p;s}(\theta) \in K$ for all times. When $K \neq {{\mathrm I\! \textsc{R} }}^d$, one cannot expect $v$ to be continuous and the HJ equation associated with may have several solutions. In order to bypass such regularity issues, this paper follows the idea of [@BFZ; @BFZ2]. Define a Lipschitz continuous function $\tilde \varphi: X \rightarrow {{\mathrm I\! \textsc{R} }}$ to be $$\tilde \varphi(x) \leq 0 \Leftrightarrow x \in K,$$ and $$\varphi(x) = \max(\min(\tilde \varphi(x), L_K), - L_K).
\label{penal}$$ Then, for a given $s\geq 0$ and $(x,q,p) \in X \times Q \times P$, define a total penalization function to be $$J(x,q,p,s ; y) = \left( \phi(y_{x,q,p;s}(0)) \bigvee \max_{\theta \in [0,s]} \varphi(y_{x,q,p;s}(\theta)) \right)$$• and then, the optimal value : $$v(x,q,p,s) = \inf_{y\in S_{[0,s]}^{x,q,p}} J(x,q,p,s ; y).
\label{valuePenal}$$
Observe that and are bounded thanks to the constructions and respectively. The idea in place is that one needs only to look at the sign of $v_0$ or $v$ to obtain information about the reachable set. Therefore, the bound $L_K$ removes the necessity of dealing with an unbounded value function besides providing a convenient value for numerical computations. In order to ensure that constructions and do not interfere with the original problems formulation (in the sense that using $\phi,\varphi$ or $\tilde \phi, \tilde \varphi$ should yield the same results), given $s,X_0$ and $x$, assuming one does use signed distance functions to sets $X_0,K$, it suffices to take $L_K > \sup_{x\rq{} \in X_0} x\rq{} e^{L_f |x - x\rq{}| s}$, where $L_f$ is the Lipschitz constant of $f$.
Main Results
============
The next proposition certifies that is indeed a level-set of and .
Assume (H1)-(H3). Define Lipschitz continuous functions $\phi$ and $\varphi$ by and respectively. Define value functions $v_0$ and $v$ by and respectively. Then, for $s \geq 0$, the reachable set is given by $$R_{X^0}(s) = \{ x ~|~ \exists (q,p) \in Q\times P, ~v_0(x,q,p,s) \leq 0 \} = \{ x ~|~ \exists (q,p) \in Q\times P, ~v(x,q,p,s) \leq 0 \}$$ \[levelSet\]
The proof begins by showing that $v_0(x,q,p,s) \leq 0 \Rightarrow v(x,q,p,s) \leq 0$. Assume $v_0(x,q,p,s) \leq 0$. Then, using lemma \[lemmaINF\], there exists an admissible trajectory such that $$\phi(y_{x,q,p;s}(0)) \leq 0,~ y_{x,q,p;s}(\theta) \in K,~ \forall \theta \in [0,s].$$ Thus, $\max_{\theta \in [0,s]} \varphi(y_{x,q,p;s}(\theta)) \leq 0$ and $$v(x,q,p,s) \leq \max(\phi(y_{x,q,p;s}(0)), \max_{\theta \in [0,s]} \varphi(y_{x,q,p;s}(\theta)) ) \leq 0$$
Now, show that $v(x,q,p,s) \leq 0 \Rightarrow v_0(x,q,p,s) \leq 0$. Assume $v(x,q,p,s) \leq 0$. Then, by lemma \[lemmaINF\] there exists a trajectory that verifies $$\max(\phi(y_{x,q,p;s}(0)), \max_{\theta \in [0,s]} \varphi(y_{x,q,p;s}(\theta)) ) \leq 0.$$ By the definition of $\varphi$, $\forall \theta \in [0,s]$, $$\max (\varphi(y_{x,q,p;s}(\theta))) \leq 0 \Rightarrow y_{x,q,p;s}(\theta) \in K,$$ which implies $v_0(x,q,p,s) \leq 0$. Therefore, $u$ and $v$ have the same negative regions.
Now, assume $y_{x,q,p;s}(s) \in R_{X^0}(s) $. Then, by definition, there exists $(q,p) \in Q\times P$ and an admissible trajectory such that $y_{x,q,p;s}(\theta) \in K$ for all time and $y_{x,q,p;s}(0) \in X_0$. This implies that $\max_{ \theta \in [0,s]}( \varphi(y_{x,q,p;s}(\theta))) \leq 0$ and $\phi(y_{x,q,p;s}(0)) \leq 0$. It follows that $v(x,q,p,s) \leq J(x,q,p,s;y) \leq 0$.
Conversely, assume $v(x,q,p,s) \leq 0$. For any optimal trajectory $\hat y$ (which is admissible thanks to proposition \[lemmaINF\]) $v(x,q,p,s) = J(x,q,p,s;\hat y) \leq 0$. Since the maximum of the two quantities is non positive only if they are both non positive one can draw the desired conclusion.
Proposition sets the equivalence between and the negative regions of and . In particular, it states that it suffices to computes $v$ or $v_0$ in order to obtain information about $R_{X_0}$. In this sense, this paper focuses on , which is associated with an optimal control problem with no state constraints.
In the sequel, it is shown that is the unique (viscosity) solution of a quasi-variational inequalities system. The first step is to state a dynamic programming principle for .
First, we present some preliminary notation. Given $t >0$, set $T = (0,t]$, $\Omega = X \times Q \times P \times T$ and denote its closure by $\overline \Omega$. For a fixed $p_0 \in P$, define $\Omega |_{p_0} = X \times Q \times \{p_0\} \times T$ and denote $\overline \Omega |_{p_0}$ the closure of $\Omega |_{p_0}$. Define $$\begin{aligned}
{{\mathcal{V} }}(\overline\Omega) &:=& \{ v ~|~ v: \overline\Omega \to {{\mathrm I\! \textsc{R} }}, ~v \text{ bounded } \}, \\
{{\mathcal{V} }}(\overline\Omega|_{p_0}) &:=& \{ v ~|~ \text{for } p_0 \in P, ~v: \overline\Omega|_{p_0} \to {{\mathrm I\! \textsc{R} }}, ~v \text{ bounded } \}
\end{aligned}$$ For $v \in {{\mathcal{V} }}(\overline \Omega)$, denote its upper and lower envelope at point $(x,q,p,s) \in \overline \Omega$ respectively as $v^*$ and $v_*$: $$\begin{aligned}
v^*(x,q,p,s) &=& \limsup\limits_{\substack{x_n \to x \\ q_n \to q \\ p_n \to p \\ s_n \to s }} v(x_n,q_n,p_n,s_n) \label{limSupDef}\\
v_*(x,q,p,s) &=& \liminf\limits_{\substack{x_n \to x \\ q_n \to q \\ p_n \to p \\ s_n \to s }} v(x_n,q_n,p_n,s_n) \label{limInfDef}
\end{aligned}$$ In the case where $p_0\in P$ is fixed and $v \in {{\mathcal{V} }}(\overline \Omega |_{p_0} )$, the upper and lower envelopes of $v$ are also given by , with $p_n = p_0$ for all $n$.
Now, fix $p=0$ and define the non-local switch operators $M,M^+,M^-: {{\mathcal{V} }}(\overline \Omega |_0) \rightarrow {{\mathcal{V} }}(\overline \Omega|_0)$ to be $$\begin{aligned}
(Mv)(x,q,0,s) &=& \inf_{\substack{w\in W(q) \\ p' \geq \delta}} v(x,g(w,q),p',s) \\
(M^+v)(x,q,0,s) &=& \inf_{\substack{w\in W(q) \\ p' \geq \delta}} v^*(x,g(w,q),p',s) \\
(M^-v)(x,q,0,s) &=& \inf_{\substack{w\in W(q) \\ p' \geq \delta}} v_*(x,g(w,q),p',s)
\end{aligned}$$
The action of these operators on the value function represents a switch that respects the lag constraint. They operate whenever a switch is activated, which is equivalent to the condition $p=0$. Therefore, they are defined only for a fixed $p=0$. Let us recall here some classical properties of operators $M,M^+$ and $M^-$ (adapted from [@Zhang]):
Let $v\in B{{\mathcal{V} }}({{}\mkern3mu\overline{\mkern-3mu\, \Omega}})$. Then $M^+ v^* \in BUSC({{}\mkern3mu\overline{\mkern-3mu\, \Omega}}) $ and $M^- v_* \in BLSC({{}\mkern3mu\overline{\mkern-3mu\, \Omega}})$. Moreover $(Mv)^* \leq M^+v^*$ and $(Mv)_* \geq M^-v_*$. \[Moperator\]
Fix $q \in Q$, $ p=0$ and $\epsilon >0$. Let $w^* \in W(x,q)$ and $p^*>0$ be such that for all $x \in X$ and $s \in T$, $(M^+v^*)(x,q,p,s) \geq v^*(x,g(w^*,q),p^*,s) - \epsilon$. Consider sequences $x_n \to x$ and $s_n \to s$. Then $$\begin{aligned}
M^+v^*(x,q,0,s) &\geq& v^*(x,g(w^*,q),p^*,s) - \epsilon\\
&\geq& \limsup\limits_{\substack{x_n \to x \\ s_n \to s }} v^*(x_n,g(w^*,q),p^*,s_n) - \epsilon \\
&\geq& \limsup\limits_{\substack{x_n \to x \\ s_n \to s }} \inf_{w \in W(x_n,q) , p\rq{} \geq \delta} v^*(x_n,g(w,q),p\rq{},s_n) - \epsilon \\
&=& \limsup\limits_{\substack{x_n \to x \\ s_n \to s }} (M^+v^*)(x_n,q,0,s_n) - \epsilon.
\end{aligned}$$• Notice that $q$ and $p$ are held constant throughout the inequalities and thus, the limsup of the jump operator considering only sequences $x_n\to x$ and $s_n\to s$ corresponds to its envelope at the limit point. Then, by the arbitrariness of $\epsilon$, this proves the upper semi-continuity of $M^+ v^*$. The lower semi-continuity of $M^- v_*$ can be obtained in a similar fashion.
Now, observe that $Mv \leq M^+v^*$. Taking the upper envelope of each side, one obtains: $$(Mv)^* \leq (M^+v^*)^* = M^+v^*.$$• By the same kind of reasoning $Mv \geq M^-v_*$ and $$(Mv)_* \geq (M^-v_*)_* = M^-v_*.$$•
The next proposition is the dynamic programming principle verified by :
The value function satisfies the following dynamic programming principle:
1. For s=0, $$v(x,q,p,0) = \max(\phi(x) , \varphi(x)),~~ \forall (x,q,p) \in X \times Q \times P,
\label{initDPP}$$
2. For $p=0$, $$v(x,q,0,s) =(Mv)(x,q,0,s), ~~ (x,q,s) \in X \times Q \times T,
\label{switchPDD}$$
3. For $(x,q,p,s) \in \Omega$, define the non-intervention zone as $\Sigma = (0,p \wedge s)$. Then, for $h \in \Sigma$, $$v(x,q,p,s) = \inf_{S_{[s-h,s]}^{x,q,p}} \left\{ v(y_{x,q,p;h}(s-h),q,p-h,s-h) \bigvee \max_{\theta \in [s-h,s]} \varphi(y_{x,q,p;h}(\theta)) \right\}
\label{eqPDD}$$
\[LemmaPDD\]
The dynamic programming principle is composed of three parts.
(i): Equality is obtained directly by definition .\
(ii): $\leq$ . Let $ (x,q,s) \in X \times Q \times T$ and $p=0$. Consider a hybrid control $a = (u(\cdot),\{ w_i,s_i \}_{i=1}^N)$ and an associated trajectory $y^a$. Construct a control $\overline a = (\overline u(\cdot),\{ \overline w_i,\overline s_i \}_{i=1}^{N-1})$ with associated trajectory $y^{\overline a}$, where $\overline u =u$, $\overline w_i = w_{i}$ and $\overline s_i = s_{i}$ for $i=1,\cdots,N-1$ and $s_N = s$, $w_N = w\rq{}$. Then, one obtains, $$\begin{aligned}
v(x,q,0,s) &\leq& J(x,q,0,s;y^a) \\
&=& J(x,g(w\rq{},q),p\rq{} ,s; y^{\overline a}),
\end{aligned}$$ where the controller must respect the condition $p\rq{}\geq \delta$ for it to be admissible. Since $\overline a$ is arbitrary, one can choose it such that $$\begin{aligned}
v(x,q,0,s) &\leq& \inf_{y^{\overline a} \in S_{[0,s]}^{(x,g(w\rq{},q),p\rq{})}} J(x,g(w\rq{},q),p\rq{} ,s; y^{\overline a}) +\epsilon_1 \\
&=& v(x,g(w\rq{},q),p\rq{},s) + \epsilon_1,
\end{aligned}$$ where $\epsilon_1 >0$. Now, choose the last switch $w\rq{}$ and $p\rq{}$ such that $$\begin{aligned}
v(x,q,0,s) &\leq& \inf_{\substack{w\rq{}\in W(q) \\ p' \geq \delta}} v(x,g(w\rq{},q),p',s) +\epsilon_1 \\
&=& (Mv)(x,q,p,s) + \epsilon_1.
\end{aligned}$$
$\geq$ . For $ (x,q,p) \in X \times Q \times P$ and $p=0$, there always exists an admissible control that $a_\epsilon$, such that there exists $\epsilon_2 >0$ and, $$v(x,q,0,s) + \epsilon_2 \geq J(x,q,0,s;y^{a_\epsilon})$$ Using the same hybrid control constructions as in the $\leq$ case, one obtains $$\begin{aligned}
J(x,q,0,s;y^{a_\epsilon}) &=& J(x,g(w\rq{},q),p\rq{},s ; y^{\bar a_\epsilon})\\
&\geq& v(x,g(w\rq{},q),p\rq{},s) \\
&\geq& \inf_{\substack{w\rq{}\in W(q) \\ p' \geq \delta}} v(x,g(w\rq{},q),p\rq{},s) \\
&=& (Mv)(x,q,0,s).
\end{aligned}$$ Relation is obtained by the arbitrariness of both $\epsilon_1, \epsilon_2$.\
(iii): $\leq$ . For $(x,q,p,s) \in \Omega$ and $0<h\leq p \wedge s$, yields $$v(x,q,p,s) \leq \max \left( \left( \phi(y_{x,q,p;s}(0)) \bigvee \max_{\theta \in [0,s-h]} \varphi(y_{x,q,p;s}(\theta)) \right) , \max_{\theta \in [s-h,s]} \varphi(y_{x,q,p;s}(\theta)) \right),
\label{ineqValueDecomp}$$ for any $y \in S^{x,q,p}_{[0,s]}$. By the choice of $h$, there is no switching between times $s-h$ and $s$. Write the admissible control $a=(u,w)$ as $a_0 = (u_0,w)$ and $a_1 = (u_1,w)$ with $$\begin{aligned}
u_0(s) = u(s), ~~ s\in[0,s-h], \\
u_1(s) = u(s),~~ s\in(s-h,s].
\end{aligned}$$ Since $a$ is admissible, both controls $a_0,a_1$ are also admissible. Denote the trajectory associated with controls $a,a_0,a_1$ respectively by $y^a, y^0, y^1$. Then, $y^a \in S^{x,q,p}_{[0,s]}$ and by continuity of the trajectory we achieve the following decomposition: $$\begin{aligned}
y^1 \in S^{x,q,p}_{[s-h,s]}, ~~~~ y^0 \in S^{y^1(s-h),q,p-h}_{[0,s-h]}.
\end{aligned}$$ The above decomposition together with inequality yields $$v(x,q,p,s) \leq \max \left( \left( \phi(y^0(0)) \bigvee \max_{\theta \in [0,s-h]} \varphi(y^0(\theta)) \right) , \max_{\theta \in [s-h,s]} \varphi(y^1(\theta)) \right),$$
And one concludes after minimizing with respect to the trajectories associated with $a_0$ and $a_1$.
The $\geq$ part uses a particular $\epsilon$-optimal controller and the same decomposition, allowing to conclude by the arbitrariness of $\epsilon$. This is possible because there is no switching between $s-h$ and $s$.
A direct consequence of proposition \[LemmaPDD\] is the Lipschitz continuity of the value function, stated in the next proposition:
Assume (H1)-(H2). Define Lipschitz continuous functions $\phi$ and $\varphi$ by and , with Lipschitz constants $L_\phi$ and $L_\varphi$ respectively. Then, for $p>0$, is Lipschitz continuous.
Fix $s,s\rq{}>0$, $x,x\rq{} \in X$, $q \in Q$, $p > 0$. Then, using $\max(A,B) - \max(C,D) \leq \max(A-B,C-D) $, one obtains, $$\begin{aligned}
| v(x,q,p,s) - v(x\rq{},q,p,s) | &\leq& \max \left( \left| \phi(y_{x,q,p;s}(0)) - \phi(y_{x',q,p;s}(0)) \right|, \phantom{ \max_{\theta}} \right. \\
&& \hspace{3cm} \left. \max_{\theta \in [0,s]} \left|\varphi(y_{x,q,p;s}(\theta)) - \varphi(y_{x',q,p;s}(\theta) )\right| \right) \\
&\leq& \max \left( L_\phi \left| y_{x,q,p;s}(0) - y_{x',q,p;s}(0) \right| ,\phantom{ \max_{\theta}} \right. \\
&& \hspace{3.2cm} \left. L_\varphi \max_{\theta \in [0,s]} \left( \left| y_{x,q,p;s}(\theta) - y_{x',q,p;s}(\theta) \right| \right) \right) \\
& \leq& L_v|x-x'|,
\end{aligned}$$ where $L_v = \max(L_\phi,L_\varphi) e^{L_f s }$.
Now, take $h>0$ and observe that $v(x,q,p,s) \geq \varphi(x)$. Then, $$\begin{aligned}
| v(x,q,p+h,s+h) - v(x,q,p,s) | &\leq& \max \left( \left| v(y_{x,q,p+h;s+h}(s),q,p,s) - v(x,q,p,s) \phantom{ \max_{\theta}} \hspace{-0.6cm} \right| ,\right. \\
&& \hspace{2.1cm} \left. \left| \max_{\theta \in [s,s+h]} \varphi(y_{x,q,p;s+h}(\theta))-\varphi(x) \right| \right) \\
&\leq& \max \left( L_v \left| y_{x,q,,p+h;s+h}(s)-y_{x,q,p;s}(s) \phantom{ \max_{\theta}} \hspace{-0.6cm} \right| \right. , \\
&& \hspace{1.1cm} L_\varphi \left. \max_{\theta \in [s,s+h]} \left|y_{s,q,p+h;s+h}(\theta)-y_{x,q,p;s}(s) \right| \right) \\
&\leq& L_f \max(L_v,L_\varphi)h.
\end{aligned}$$
In order to proceed to the HJB equations, define the Hamiltionian to be $$H(s,x,q,z) = \sup_{u\in U(q)} f(s,x,u,q) \cdot z
\label{Hamilt}$$
Before stating the next result, we recall the notion of viscosity solution used throughout this paper.
A function $u_1$ (resp. $u_2$) upper semi-continuous (u.s.c.) (resp. lower semi-continuous (l.s.c) is a viscosity subsolution (resp. supersolution) if there exists a continuously differentiable function $\psi$ such that $u_1 - \psi$ has a local maximum (resp. $u_2 - \psi$ has a local minimum) at $(x,q,p,s) \in \overline \Omega$ and $$\begin{aligned}
\partial_s \psi + \partial_p \psi+ H(s,x,q,\nabla_x \psi) \bigwedge u_1-\varphi(x) \leq 0 &\text{ if }& (x,q,p,s) \in \Omega \label{sub} \\
u_1(x,q,p,s) \leq (M^+u_1)(x,q,p,s) &\text{ if }& p=0, \\
u_1(x,q,p,s) \leq \max(\phi(x),\varphi(x)) &\text{ if }& s=0,
\end{aligned}$$ (with the inequalities signs inversed and $M^-$ instead of $M^+$ for $u_2$). A bounded function $u$ is a (viscosity) solution of - if $u^*$ is a subsolution and $u_*$ is a supersolution.
The next statement shows that the value function defined in is a solution of a quasi-variational system.
Assume (H1)-(H5). Let the Lipschitz functions $\phi$ and $\varphi$ be defined by and respectively. Then, the Lipschitz, bounded value function $v$ defined in is a viscosity solution of the quasi-variational inequality
$$\begin{aligned}
\partial_s v + \partial_p v + H(s,x,q,\nabla_x v) \bigwedge v-\varphi(x) = 0 &,& \forall (x,q,p,s) \in \Omega, \label{HJ1}\\
v(x,q,0,s) = (Mv)(x,q,0,s) &,& \forall (x,q,s) \in X \times Q \times [0,\infty[ \label{HJ2} \\
v(x,q,p,0) = \max(\phi(x),\varphi(x)) &,& \forall (x,q,p) \in X \times Q \times P.
\label{HJ3}
\end{aligned}$$
\[HJprop\]
By definition, $v$ satisfies the initial condition . The boundary condition is deducted from proposition \[LemmaPDD\]. Now, we proceed to show that $(i)$ $v$ is a supersolution and $(ii)$ a subsolution of :
First, let us prove the supersolution property $(i)$. To satisfy $\min(A,B) \geq 0$ one needs to show $A\geq 0$ and $B\geq 0$. Since $v-\varphi(x)\geq 0$, it is immediate that $B\geq 0$. Now, consider $0 < h \leq p\wedge s $. Let $\psi$ be a continuously differentiable function such that $v-\psi$ attains a minimum at $(x,q,p,s)$. Then,using proposition $(iii)$ and selecting an $\epsilon$-optimal controller, dependent on $h$, with associated trajectory $y^\epsilon_{x,q,p;s}$, it follows that $$\begin{aligned}
\psi(x,q,p,s) = v(x,q,p,s) &\geq& \inf_{S_{[0,s]}^{x,q,p}} v(y_{x,q,p;s}(s-h),q,p-h,s-h) \\
&\geq& v(y^\epsilon_{x,q,p;s}(s-h),q,p-h,s-h) - h\epsilon \\
&=& \psi(y^\epsilon_{x,q,p;s}(s-h),q,p-h,s-h) - h\epsilon
\end{aligned}$$ and then, $$\psi(x,q,p,s) -\psi(y^\epsilon_{x,q,p;s}(s-h),q,p-h,s-h) \geq - h \epsilon.$$ Since the control domain is bounded and using the continuity of $f,p$ and $\psi$ we divide by $h$ and take the limit $h \rightarrow 0$ to obtain $$\partial_s \psi + \partial_p \psi + H(s,x,q,\nabla_x \psi) \geq -\epsilon$$ and conclude that $A\geq 0$ by the arbitrariness of $\epsilon$.
For $(ii)$, observe that for $\min(A,B) \leq 0$ it suffices to show that $A\leq 0$ or $B \leq 0$. If $v(x,q,p,s)=\varphi(x)$, it implies $B \leq 0$. On the contrary, if $v(x,q,p,s) > \varphi(x)$, then there exists a $\Sigma \ni h \geq 0 $ small enough so that $$v(y^u_{x,q,p;s}(s-h),q,p-h,s-h) > \max_{\theta \in [s-h,h]} \varphi(y^u_{x,q,p;s}(\theta))$$ strictly, using the Lipschitz continuity of $f,p$ and the compactness of $U$ (which ensures the trajectories will remain near each other). Thus, proposition \[LemmaPDD\]$(iii)$ yields $$v(x,q,p,s) = \inf_{y^u \in S^{x,q,p}_{[s-h,s]}}v(y^u_{x,q,p;s}(s-h),q,p-h,s-h).$$ Fix an arbitrary $u\in U$ and consider a constant control $u(s)=u$ for $ 0< s <h$. Let $\psi$ be a continuously differentiable function such that $v-\psi$ attains a maximum at $(x,q,p,s)$. Also, without loss of generality, assume that $v(x,q,p,s)=\psi(x,q,p,s)$. Hence, $$\begin{aligned}
\nonumber v(x,q,p,s) &\leq& v(y^u_{x,q,p;s}(s-h),q,p-h,s-h) \\
\nonumber &\leq& \psi(y^u_{x,q,p;s}(s-h),q,p-h,s-h)
\end{aligned}$$ and by dividing by $h$ and taking $h \rightarrow 0$ one obtains $$\partial_s \psi + \partial_p \psi + f(s,x,u) \cdot \nabla_x \psi \leq 0.$$ Since $u$ is arbitrary and admissible, we conclude that $A\leq 0$, which completes the proof.
Theorem \[HJprop\] provides a convenient way to characterize the value function whose level-set is the reachable set defined in . However, in order to be sure that the solution that stems from - corresponds to , a uniqueness result is necessary. This is achieved by a comparison principle which is stated in the next theorem. Let $BUSC(\Omega)$ and $BLSC(\Omega)$ respectively be the space of u.s.c. and l.s.c. functions defined over the set $\Omega$.
Let $u_1 \in BUSC(\Omega)$ and $u_2 \in BLSC(\Omega)$ be, respectively, sub- and supersolution of $$\begin{aligned}
\partial_s u + \partial_p u + H(s,x,q,\nabla_x u) \bigwedge u-\varphi(x) ) = 0 &,& \forall (x,q,p,s) \in \Omega, \label{HJBinsideNUM}\\
u(x,q,0,s) -(Mu)(x,q,0,s)= 0 &,& \forall (x,q,s) \in X \times Q \times [0,\infty[ \\
u(x,q,p,0) = \max(\phi(x),\varphi(x)) &,& \forall (x,q,p) \in X \times Q \times P.
\label{HJBmainNUM}
\end{aligned}$$
Then, $u_1 \leq u_2$ in $\overline \Omega$. \[compPrinc\]
The proof is inspired by earlier work on uniqueness results for hybrid control problems. The idea is to show that $u_1 \leq u_2$ in all domain $\Omega$ and then on the boundary $p=0$. The main difficulty arises when dealing with points in the boundary $p=0$ where the system has a switching condition given by a non-local switch operator. This is tackled by the utilization of friendly giant-like test functions [@BarlesSheetal], [@Ley]. Classically, these functions are used to prove uniqueness for elliptic problems with unbounded value functions where they serve to localize some arguments regardless of the functions possible growth at infinity. This feature proves itself very useful in our case because one can properly split the domain in no-switching and switching regions. In this work, the lag condition for the switch serves as an equivalent to the landing condition– which states that after an autonomous switch the system must land at some positive distance away from the autonomous switch set [@Dharmatti2], [@Zhang].
Let $\Omega$ be defined as above, $\partial \Omega|_T = X \times Q \times P \times \{0\}$ and $\partial \Omega|_P = X \times Q \times \{0\} \times T$.
First, the comparison principle is proved for $\partial \Omega|_T$ (case $1$), followed by $\Omega$ (case $2$) and finally for $\partial \Omega|_P$ (case $3$), which concludes the proof for ${{}\mkern3mu\overline{\mkern-3mu\, \Omega}}$.
Case $1$: At a point $(x,q,p,t) \in \partial \Omega|_T$, from the sub- and supersolution properties, $$\begin{aligned}
u_1(x,q,p,0) - \max(\phi(x),\varphi(x)) &\leq& 0, \\
-u_2(x,q,p,0) + \max(\phi(x),\varphi(x)) &\leq& 0,
\end{aligned}$$ which readily yields $u_1 \leq u_2$ in $ \partial \Omega|_T$.
Case $2$: Start by using to sub- and supersolution properties of $u_1,u_2$ to obtain, in $\Omega$, $$\begin{aligned}
\min(\partial_s u_1 + \partial_p u_1 + H(s,x,q,\nabla_x u_1) , u_1-\varphi(x) ) &\leq& 0, \label{subVisc}\\
\min(\partial_s u_2 + \partial_p u_2 + H(s,x,q,\nabla_x u_2) , u_2-\varphi(x) ) &\geq& 0. \label{superVisc}
\end{aligned}$$ Expression implies that both $$u_2\geq \varphi(x)
\label{secondHSide}$$ and $$\partial_s u_2 + \partial_p u_2 + H(s,x,q,\nabla_xu_2) \geq 0.
\label{firstHSide}$$ From , one has to consider two possibilities. The first one is when $u_1 \leq \varphi(x)$. If so, together with , one has immediately $u_1 \leq u_2$. Now, if $\partial_s u_1 + \partial_p u_1 + H(s,x,q,\nabla_x u_1) \leq 0$, one turns to .
Define $v = u_1 - u_2$. Notice that $v \in BUSC(\Omega)$. The next step is to show that $v$ is a subsolution of $$\partial_s v + \partial_p v + H(s,x,q,\nabla_x v) = 0
\label{psiHJB}$$ at $(\bar x, \bar q, \bar p, \bar s) $.
Let $\psi \in C^2(\Omega)$, bounded, be such that $v-\psi$ has a strict local maximum at $(\bar x, \bar q, \bar p, \bar s) \in \Omega$. Define auxiliary functions over $\Omega^i \times \Omega^i$, $i=0,1$ as $$\begin{aligned}
\Phi_\epsilon^i (x,p,s,\xi,\pi,\varsigma)& =& u_1(x,i,p,s) - u_2(\xi, i, \pi, \varsigma) - \psi(x,p,s) \label{auxEpsilon}\\
\nonumber&& ~~~~~~~~~~~~- \frac{|x - \xi|^2}{2\epsilon} - \frac{|p - \pi|^2}{2\epsilon} - \frac{|s - \varsigma|^2}{2\epsilon}.
\end{aligned}$$
Because the boundedness of $\psi$, $u_1$ and $u_2$ the suprema points are finite, for each $i=0,1$. Denote $({{x_\epsilon,p_\epsilon,s_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon}}) \in \Omega^{\bar q} \times \Omega^{\bar q}$ a point such that $$\Phi_\epsilon^{\bar q}({{x_\epsilon,p_\epsilon,s_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon}}) = \sup_{ \Omega^{\bar q} \times \Omega^{\bar q}} \Phi_\epsilon^{\bar q} (x,p,s,\xi,\pi,\varsigma).$$
The following lemma establishes some estimations needed further in the proof:
Define $\Phi_\epsilon^i$ and $({{x_\epsilon,p_\epsilon,s_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon}})$ as above. Then, as $\epsilon \rightarrow 0$, $$\begin{aligned}
\frac{|x_\epsilon - \xi_\epsilon|^2}{\epsilon} \rightarrow 0, ~~\frac{|p_\epsilon - \pi_\epsilon|^2}{\epsilon} \rightarrow 0, ~~\frac{|s_\epsilon - \varsigma_\epsilon|^2}{\epsilon} \rightarrow 0, \label{lemmaResultA}\\
|x_\epsilon - \xi_\epsilon| \rightarrow 0, ~~|p_\epsilon - \pi_\epsilon|\rightarrow 0, ~~|s_\epsilon - \varsigma_\epsilon| \rightarrow 0,
\label{lemmaResultB}
\end{aligned}$$ and $({{x_\epsilon,p_\epsilon,s_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon}}) \rightarrow (\bar x,\bar p,\bar s,\bar x,\bar p,\bar s)$ \[lemmaConver\]
Writing $$2\Phi_\epsilon^i({{x_\epsilon,p_\epsilon,s_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon}}) \geq \Phi_\epsilon^i(x_\epsilon,p_\epsilon,s_\epsilon,x_\epsilon,p_\epsilon,s_\epsilon) + \Phi_\epsilon^i(\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon),$$ for $i=0,1$, one obtains, $$\begin{aligned}
\frac{|x - \xi|^2}{\epsilon} + \frac{|p - \pi|^2}{\epsilon} + \frac{|s - \varsigma|^2}{\epsilon} &\leq& (u_1 + u_2)(x_\epsilon,i,p_\epsilon,s_\epsilon) - (u_1 + u_2)(\xi_\epsilon,i,\pi_\epsilon,\varsigma_\epsilon) + \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \psi(x_\epsilon,p_\epsilon,s_\epsilon) - \psi(\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon) ,
\end{aligned}$$• which means that, since $\psi$, $u_1$ and $u_2$ are bounded that $$\frac{|x_\epsilon - \xi_\epsilon|^2}{\epsilon} \leq C*, ~~\frac{|p_\epsilon - \pi_\epsilon|^2}{\epsilon} \leq C*, ~~\frac{|s_\epsilon - \varsigma_\epsilon|^2}{\epsilon} \leq C^*,
\label{lemmaInegEps}$$• where $C^*$ depends on the $\sup |u_1|$, $\sup |u_2|$, $\sup |\psi|$ and is independent of $\epsilon$. Expression yields $$|x_\epsilon - \xi_\epsilon| \leq \sqrt{\epsilon C^*}, ~~|p_\epsilon - \pi_\epsilon| \leq \sqrt{\epsilon C^*}, ~~|s_\epsilon - \varsigma_\epsilon| \leq \sqrt{\epsilon C^*}.$$• which implies that the doubled terms tend to zero.
Since $ (\bar x,\bar q,\bar p,\bar s)$ is a strict maximum of $v-\psi$, one gets $({{x_\epsilon,p_\epsilon,s_\epsilon,\xi_\epsilon,\pi_\epsilon,\varsigma_\epsilon}}) \rightarrow (\bar x,\bar p,\bar s,\bar x,\bar p,\bar s)$. Remark that, since $\bar p >0$, one can always choose a suitable subsequence $\epsilon_n \rightarrow 0$ such that all $p_{\epsilon_n} >0$, avoiding thus touching the switching boundary.
A straightforward calculation allows to show that there exists $a,b \in {{\mathrm I\! \textsc{R} }}$ such that $$\begin{aligned}
(a,b,D_\epsilon) &\in& D^- u_2(\xi_\epsilon,\bar q,\pi_\epsilon,\varsigma_\epsilon) \\
(a + \partial_s \psi ,b + \partial_p \psi,D_\epsilon + \nabla_x \psi) &\in& D^+ u_1(x\epsilon,\bar q,p_\epsilon,s_\epsilon) ,
\end{aligned}$$ where $D^-,D^+$ respectively denote the sub- and super differential and $D_\epsilon= 2|x_\epsilon - \xi_\epsilon| / \epsilon$, which implies $$\begin{aligned}
a+b+ H(\varsigma_\epsilon,\xi_\epsilon,\bar q,D_\epsilon) &\geq& 0 \\
a + \partial_s \psi + b + \partial_p \psi+ H(s_\epsilon,x_\epsilon,\bar q,D_\epsilon + \nabla_x \psi) &\leq& 0,
\end{aligned}$$ which in turn yields, as $\epsilon \rightarrow 0$, $$\partial_s \psi+ \partial_p \psi - L_f | \nabla_x \psi| \leq 0$$ at $(\bar x, \bar q, \bar p, \bar s) \in \Omega$. By adequately choosing the test functions $\psi$, one can repeat the arguments to show that this assertion holds for any point in $\Omega$. Thus, this establishes that $v$ is a subsolution of in $\Omega$.
Now, take $\kappa >0$ and define a non-decreasing differentiable function $\chi_\kappa : (-\infty,0) \rightarrow {{\mathrm I\! \textsc{R} }}^+$ such that $$\chi_\kappa(x) = 0 ,~ x \leq - \kappa~;~\chi_\kappa(x) \rightarrow \infty, ~ x\rightarrow 0.$$
Take $\eta >0$,and define a test function $$\nu(x,p,s) = \eta s^2 + \chi_\kappa(-p).$$ Observe that $v - \nu$ achieves a maximum at a finite point $(x_0,\bar q,p_0,s_0) \in \Omega$. Since $\kappa$ can be made arbitrarily small one can consider $p_0 > \kappa$ without loss of generality. Therefore, using the subsolution property of $v$, by a straightforward calculation one has $$2 \eta s_0 \leq 0,$$ since $\chi_\eta\rq{}(-p_0) = 0$. The above inequality implies that $s_0 = 0$. Noticing that $\nu(x_0,p_0,s_0) = v(x_0,\bar q,p_0,s_0) = 0$, it follows $$v(x,\bar q,p,s) \leq \eta s^2 + \chi_\kappa(-p)$$ for all $s\in T$, $x\in X$ and $p > \kappa$. Letting $\eta \rightarrow 0$, $\kappa \rightarrow 0$ and from the arbitrariness of $\bar q$, we conclude that $v\leq 0$ in $\Omega$.
Case $3$: In this case the switch lock variable arrives at the boundary of the domain, incurring thus a switch, as all others variables remain inside the domain. For all $(x_0,q_0,p_0,s_0) \in \partial \Omega|_P$, for any $p \geq \delta$ one has (using case $2$ and noticing that $M^+u_1 = Mu_1$ and $M^- u_2 = Mu_2$) $$(M^+ u_1)(x_0,q_0,p_0,s_0) \leq u_1(x_0,q_0,p,s_0) \leq u_2(x_0,q_0,p,s_0).$$• Taking the infimum with respect to $p$, the above expression yields $M^+u_1 \leq M^-u_2$ in $\partial \Omega|_P$. This suffices to conclude, since that by the sub- and supersolution properties $$v = u_1 - u_2 \leq M^+u_1 - M^-u_2.$$
Numerical Analysis
==================
Numerical Scheme and Convergence
--------------------------------
Equations - can be solved using a finite differences scheme. This section proposes a class of discretization schemes and shows its convergence using the Barles-Souganidis [@souganidis] framework.
Set mesh sizes $\Delta x >0$, $\Delta p >0$, $\Delta t >0$ and denote the discrete grid point by $(x_{I},p_k,s_n)$, where $x_I = I\Delta x$, $p_k = k \Delta p$ and $s_n = n\Delta t$, with $I \in {{\mathbb{Z} }}^d$ and $k,n$ integers. The approximation of the value function is denoted $$v(x_I,q,p_k,s_n) = {{\sideset{_{}^{q}}{_{Ik}^{n}}{\mathop{v}}}}$$ and the penalization functions are denoted $\phi(x_I) = \phi_I$, $\varphi(x_I) = \varphi_I$. Define the following grids: $$\begin{aligned}
G^\# &=& I\Delta x \times Q \times \Delta p\{0,1,\cdots,n_p\} \times \Delta t \{0,1,\cdots,n_s \}, \\
G^\#_H &=& \Delta t \{0,1,\cdots,n_s \} \times I\Delta x \times Q
\end{aligned}$$ and the discrete space gradient at point $x_I$ for any general function $\mu$: $$D^\pm \mu(X_I) = D^\pm \mu_I = \left( D^\pm_{x_1} \mu_I ,\cdots, D^\pm_{x_d} \mu_I \right),$$• where $$D^\pm_{x_j} \mu_I = \pm \frac{\mu_{I^{j,\pm}} - \mu_I }{\Delta x},$$• with $$I^{j,\pm} = (i_1,\cdots,i_{j-1},i_j \pm 1,\cdots,i_d).$$•
Define a numerical Hamiltonian ${{\mathcal{H} }}: G_H^\# \times {{\mathrm I\! \textsc{R} }}^d \times {{\mathrm I\! \textsc{R} }}^d \rightarrow {{\mathrm I\! \textsc{R} }}$ destined to be an approximation of $H$. We assume that ${{\mathcal{H} }}$ verifies the following hypothesis:
(H6)
: There exists $L_{H_1},L_{H_2} > 0$ such that, for all $s,x,q \in G_H^\#$ and $A^+,A^-,B^+,B^- \in {{\mathrm I\! \textsc{R} }}^d$, $$\begin{aligned}
\nonumber | {{\mathcal{H} }}(s,x,q, A^+,A^-) - {{\mathcal{H} }}(s,x,q, B^+,B^-) | &\leq& L_{H_1}(||A^+ - B^+|| + ||A^- - B^- || \\
||{{\mathcal{H} }}(s,x,q, A^+,A^-)|| &\leq& L_{H_2}(||A^+ + A^-||).
\end{aligned}$$
(H7)
: The Hamiltonian satisfies the monotonicity condition for all $s,x,q \in G_H^\#$ and almost every $A^+,A^-\in {{\mathrm I\! \textsc{R} }}^d$: $$\partial_{A_i^+}{{\mathcal{H} }}(s,x,q, A^+,A^-) \leq 0, \text{ and } \partial_{A_i^-}{{\mathcal{H} }}(s,x,q, A^+,A^-) \geq 0.$$•
(H8)
: There exists $\L_{H_3} >0$ such that for all $s,x,q \in G_H^\#$, $s\rq{},x\rq{},q\rq{} \in T \times X \times Q$ and $A \in {{\mathrm I\! \textsc{R} }}^d$, $$|{{\mathcal{H} }}(s,x,q,A,A) - H(s\rq{},x\rq{},q\rq{},A)| \leq L_{H_3}(|s-s\rq{} | + ||x-x\rq{} || + |q-q\rq{}|).$$•
Let $\Phi : \Omega \rightarrow {{\mathrm I\! \textsc{R} }}$, $h = (\Delta x, \Delta p, \Delta t)$ and set $$\begin{aligned}
\nonumber S^\Omega _h(x,q,p,s,\lambda;\Phi)& =& \min \left( \lambda - \varphi_{I} , {{\mathcal{H} }}(s,x,q, D^+ \Phi(x,q,p,s),D^- \Phi(x,q,p,s))+ \right. \\
\nonumber && ~~~~~~~~~~~~~~~~~~~~~~\left. \frac{\lambda - \Phi(x,q,p,s)}{\Delta t} + \frac{ \lambda - \Phi(x,q,p-\Delta p,s+\Delta t) }{\Delta p} \right).
\end{aligned}$$•
Now, consider the following scheme $$S _h(x,q,p,s,\lambda;\Phi) = \left\{ \begin{array}{lll}
S^\Omega _h(x,q,p,s,\lambda;\Phi) &\text{if}& (x,q,p,s) \in \Omega \\
\lambda - \min_{w\in W(x,q),p\rq{}\geq \delta} \Phi(x,g(w,q),p\rq{},s) &\text{if}& p=0,
\end{array} \right.
\label{scheme}$$ along with the following operator $${{\mathcal{F} }}(x,q,p,s,u,\nabla u) = \left\{ \begin{array}{lll}
u-\varphi(x) \bigwedge \partial_s u + \partial_p u + H(s,x,q,\nabla_x u) &\text{if}& (x,q,p,s) \in \Omega \\
u(x,q,p,s) -(Mu)(x,q,p,s) &\text{if}& p=0. \\
\end{array} \right.
\label{contscheme}$$
Let $\Phi \in C^\infty_b(\Omega)$. Under hypothesis $(H6-H8)$ and the CFL condition $$\Delta t \left( \frac{1}{\Delta p} + \frac{1}{\Delta x} \sum_{i=1}^d \partial_{A_i^+}{{\mathcal{H} }}+\partial_{A_i^-}{{\mathcal{H} }}\right) \leq 1
\label{CFLcond}$$• the discretization scheme of is stable, monotone and consistent.
Moreover, a solution $u_h$ of converges towards the solution $u$ of as $h \rightarrow 0$.
The proof follows the lines used in the framework of Barles-Souganidis [@souganidis]. The goal is to show that the numerical scheme solutions envelopes $$\begin{aligned}
\underline u(x\rq{},q\rq{},p\rq{},s\rq{}) = \liminf\limits_{\substack{(x,q,p,s) \to (x\rq{},q\rq{},p\rq{},s\rq{})\\ h \to 0}} u_h(x,q,p,s) \\
\overline u(x\rq{},s\rq{},p\rq{},s\rq{}) = \limsup\limits_{\substack{(x,q,p,s) \to (x\rq{},q\rq{},p\rq{},s\rq{})\\ h \to 0}} u_h(x,q,p,s),
\end{aligned}$$• are respectively supersolution and subsolution of . Then, using the comparison principle in theorem \[compPrinc\], one obtains $\overline u \leq \underline u$. However, since the inverse inequality is immediate (using the definition of limsup and liminf, one gets $u \equiv \underline u = \overline u$ achieving thus the convergence.
Only the subsolution property of $\overline u$ is presented next, the proof of the supersolution property of $\underline u$ being very alike.
Firstly, the proofs shows that is stable, monotone and consistent. Observe that $S$ is proportional to $-\Phi$ in the terms outside the Hamiltonian. Since fluxes ${{\mathcal{H} }}$ are monotone by hypothesis $(H7)$ (see [@shuOsher] for details in monotone Hamiltonian fluxes) whenever the CFL condition is satisfied, the monotonicity of $S$ follows. The stability is ensured by the boundedness of $\Phi$ and hypothesis $(H6)$. Finally, hypothesis $(H8)$ and lemma \[Moperator\] are used in a straightforward fashion to obtain the consistency properties below: $$\begin{aligned}
\limsup\limits_{\substack{(x\rq{},q\rq{},p\rq{},s\rq{}) \to (x,q,p,s)\\ h \to 0}} S _h(x\rq{},q\rq{},p\rq{},s\rq{},\lambda;\Phi) \leq {{\mathcal{F} }}^*(x,q,p,s,\Phi,\nabla \Phi) \\
\liminf\limits_{\substack{(x\rq{},q\rq{},p\rq{},s\rq{}) \to (x,q,p,s)\\ h \to 0}} S _h(x\rq{},q\rq{},p\rq{},s\rq{},\lambda;\Phi) \geq {{\mathcal{F} }}_*(x,q,p,s,\Phi,\nabla \Phi)
\end{aligned}$$•
Now, choose $\Phi \in C^\infty_b(\Omega)$ such that $\overline u - \Phi$ has a strict local maximum at $(x_0,q_0,p_0,s_0) \in {{}\mkern3mu\overline{\mkern-3mu\, \Omega}}$ (without loss of generality assume $(\overline u - \Phi)(x_0,q_0,p_0,s_0) = 0$).
First, suppose $p_0>0$. Then there exists a ball centered in $(x_0,q_0,p_0,s_0)$ of radius $r >0$ such that $\overline u(x,q,p,s) \leq \Phi(x,q,p,s),~\forall (x,q,p,s) \in B((x_0,q_0,p_0,s_0),r) \subset \Omega$. Construct sequences $(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon) \to (x_0,q_0,p_0,s_0) $ and $h_\epsilon \to 0$ as $\epsilon \to 0$ such that $u_{h_\epsilon}(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon) \to \overline u(x_0,q_0,p_0,s_0)$ and $(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon) $ is a maximum of $u_{h_\epsilon} - \Phi$ in $B((x_0,q_0,p_0,s_0),r)$. Denote $\zeta_\epsilon = (u_{h_\epsilon} - \Phi)(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon)$. (Remark that $\zeta_\epsilon \to 0$ as $\epsilon \to 0$).
Then, $u_{h_\epsilon} \leq \Phi + \zeta_\epsilon$ inside the ball and since $S(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon,u_{h_\epsilon}(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon);u_{h_\epsilon})=0$, by the monotonicity property one obtains $$S(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon,\Phi(x_\epsilon,q_\epsilon,p_\epsilon,s_\epsilon) + \zeta_\epsilon;\Phi + \zeta_\epsilon) \leq 0.$$• Taking the limit (inf) $\epsilon \to 0$ together with the consistency of the scheme, one obtains the desired inequality $${{\mathcal{F} }}_*(x_0,q_0,p_0,s_0,\Phi, \nabla \Phi) \leq 0.$$•
Suppose now that $p_0=0$. Construct sequences $(x_\epsilon,q_\epsilon,p_0,s_\epsilon) \to (x_0,q_0,p_0,s_0) $ and $h_\epsilon \to 0$ as $\epsilon \to 0$ such that $u_{h_\epsilon}(x_\epsilon,q_\epsilon,p_0,s_\epsilon) \to \overline u(x_0,q_0,p_0,s_0)$. Then, $$\begin{aligned}
\liminf\limits_{\substack{x_\epsilon \to x_0 \\ q_\epsilon \to q_0 \\ s_\epsilon \to s_0 \\ h_\epsilon \to 0}} S_h(x_\epsilon,q_\epsilon,p_0,s_\epsilon,\Phi(x_\epsilon,q_\epsilon,p_0,s_\epsilon),\Phi) &=& \liminf\limits_{\substack{x_\epsilon \to x_0 \\ q_\epsilon \to q_0 \\ s_\epsilon \to s_0 \\ h_\epsilon \to 0}} (\Phi - M \Phi)(x_\epsilon,q_\epsilon,p_0,s_\epsilon) \\
&=& (\Phi - (M\Phi)_*)(x_0,q_0,p_0,s_0).
\end{aligned}$$• Since each $u_{h_\epsilon}$ is a solution of , using lemma \[Moperator\] the above expression yields at the point $(x_0,q_0,p_0,s_0)$: $$\begin{aligned}
0 &=& (\Phi - (M\Phi)_*)(x_0,q_0,p_0,s_0) \\
&\geq& (\Phi - (M\Phi)^*)(x_0,q_0,p_0,s_0) \\
&\geq& (\Phi - (M^+\Phi))(x_0,q_0,p_0,s_0) \\
&=&{{\mathcal{F} }}_*(x_0,q_0,p_0,s_0,\Phi, \nabla \Phi)
\end{aligned}$$• achieving the desired inequality.
Numerical Simulations
---------------------
For the numerical simulations, the numerical Hamiltonian ${{\mathcal{H} }}$ is discretized using a monotone Local Lax-Friedrichs scheme [@shuOsher] (where the two components of the gradient are explicit): $$\begin{aligned}
{{\mathcal{H} }}\left(t,x,q;a^+,a^-,b^+,b^-\right) &=& H \left(t,x,q;\frac{a^++a^-}{2},\frac{b^++b^-}{2}\right) - \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~ c_a \left(\frac{a^+-a^-}{2}\right) - c_b \left(\frac{b^+-b^-}{2}\right)
\end{aligned}$$• where $a^\pm = D_i^\pm v$, $b^\pm = D_j^\pm v$ and the constants $c_a,c_b$ are defined as $$\begin{aligned}
c_a = \sup_{t,x,q,r} | \partial_{r_a} H(t,x,q,r) | \\
c_b = \sup_{t,x,q,r} | \partial_{r_b} H(t,x,q,r) |.
\end{aligned}$$•
Setting $u_h^\# = [{{\sideset{_{}^{q}}{_{Ik}^{n}}{\mathop{v}}}}, {{\sideset{_{}^{q}}{_{I^{1^\pm} k}^{n}}{\mathop{v}}}}, \cdots, {{\sideset{_{}^{q}}{_{I^{d^\pm} k}^{n}}{\mathop{v}}}}, {{\sideset{_{}^{q}}{_{Ik - 1}^{n}}{\mathop{v}}}}]$, the equation $$S _h( x_{I},q,p_k,s_{n+1},{{\sideset{_{}^{q}}{_{Ik}^{n+1}}{\mathop{v}}}};u_h^\#) =0$$ allows an explicit expression of ${{\sideset{_{}^{}}{_{}^{n+1}}{\mathop{v}}}}$ as a function of past values ${{\sideset{_{}^{}}{_{}^{n}}{\mathop{v}}}}$:
$${{\sideset{_{}^{q}}{_{Ik}^{n+1}}{\mathop{v}}}} = \left\{ \begin{array}{lll}
\varphi_{I} \bigvee {{\sideset{_{}^{q}}{_{Ik}^{n}}{\mathop{v}}}} - \Delta t \left( \frac{ {{\sideset{_{}^{q}}{_{Ik}^{n}}{\mathop{v}}}} - {{\sideset{_{}^{q}}{_{Ik-1}^{n}}{\mathop{v}}}} }{\Delta p} + {{\mathcal{H} }}(t_n,x_{I},q,D^- {{\sideset{_{}^{q}}{_{Ik}^{n}}{\mathop{v}}}}, D^+ {{\sideset{_{}^{q}}{_{Ik}^{n}}{\mathop{v}}}}) \right) \\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{ if } (I,q,k,n) \in \Omega^\# \\
\min_{w\in W(x_I,q), k' \geq \delta/\Delta p} {{\sideset{_{}^{g(w,q)}}{_{Ik'}^{n}}{\mathop{v}}}}~~~~~~~~~~~~~~~~~~~~~\,~~~~~~ \text{ if } k=0.
\end{array} \right.
\label{schemeSimulation}$$
In order to illustrate the work presented, a simple vehicle model is used in the simulations. This model allows an analytic evaluation of its autonomy and is suitable for an a posteriori verification of the results.
The switch dynamics is given by $$g(w,q) = |q-w|.$$ The energetic dynamical model is given by $ f(u,q) = ( -a_x + qu , -q(a_y+u))$, where $a_x,a_y >0$ are constant depletion rates of the batterys electric energy and the reservoirs fuel (whenever the RE is on), respectively. The control domain is taken $U=[0,u_{\text{max}}]$.
Considering this dynamics, an exact autonomy of the system can be evaluated analytically. Given initial conditions $(x_0,y_0)$ the shortest time to empty the fuel reservoir is given by $t^* = y_0 / (a_y+u_{\text{max}})$. The SOC evaluated at this instant is given by $x(t^*) = x(0) -t^*(a_x - u_{\text{max}})$. If $x(t^*)\leq 0$, it means the fuel cannot be consumed fast enough before the battery is depleted. This condition can be expressed in terms of the parameters of the model as $x_0(a_y+u_{\text{max}}) \leq y_0 (a_x -u_{\text{max}})$. In this case, the autonomy is given by $T^0 = x_0/(a_x-u_{\text{max}}) $. If not, the autonomy is given by $T^1 = (x_0 + u_{\text{max}}t^*) / a_x$. Simulations are running using $\Delta x = \Delta y = 0.025$, $\Delta p = 0.05$ and $\Delta t$ is calculated using . Several instances of $x_0,y_0,a_x,a_y,u_{\text{max}}$ are tested, all with a lag of $\delta = 2$, and the theoretical and evaluated autonomy are compared. For numerical purposes, the set of initial energies is a ball of radius $2\Delta x$ around $(x_0,y_0)$ and the admissible region is set as $K=[0,1]^2$. Remark that , give a natural value $\tilde L_K$ for the numerical boundary outside set $K$.
The simulated instances use $a_x = 0.1$, $a_y=0.15$, $u_{\text{max}}=0.07$. Tests are made using three initial conditions. Table \[tab:convSimulations\] groups the error $\varepsilon =|T^* - s^*|$ between exact autonomy and the autonomy evaluated using the scheme and the algorithm running times.
Figures \[results1\], \[results2\] and \[results3\] show the minimum of the value function for all values of $p \in P^\#$ for $q=0,1$ and the corresponding reachable set of the instance $(x_0,y_0)=(0.3,0.8)$ at times $s=2.65$, $s=3.00$, $s=3.75$ respectively.
$(x_0,y_0)$ $\Delta x $ $\varepsilon$ CPUrunning time(s)
------------- ------------- --------------- --------------------
$0.05$ $0.877$ $34$
$0.04$ $0.613$ $57$
$0.03$ $0.860$ $111$
$0.02$ $0.326$ $326$
$0.05$ $1.923$ $28$
$0.04$ $1.179$ $46$
$0.03$ $0.578$ $89 $
$0.02$ $0.038$ $253$
: Convergence results and running times.
\[tab:convSimulations\]
![Reachable set and value functions at $s=2.65$. []{data-label="results1"}](ThreePlotT2p625dx0.jpg){width="1\columnwidth"}
![Reachable set and value functions at $s=3.00$. []{data-label="results2"}](ThreePlotT3p00dx0.jpg){width="1\columnwidth"}
![Reachable set and value functions at $s=3.75$. []{data-label="results3"}](ThreePlotT3p75dx0.jpg){width="1\columnwidth"}
[^1]: Renault SAS, Advanced Electronics Division, TCR RUC T 65, 78286 Guyancourt Cedex, France ([[email protected]]{}). École Nationale Supérieure de Techniques Avancées, Unité de Mathématiques Appliquées, 32, boulevard Victor, 75015, Parix Cedex 15, France ([[email protected]]{})
[^2]: École Nationale Supérieure de Techniques Avancées, Unité de Mathématiques Appliquées, 32, boulevard Victor, 75015, Parix Cedex 15, France ([[email protected]]{})
[^3]: This work was co-funded by Renault SAS under grant ANRT CIFRE n° 928/2009.
|
---
abstract: 'Embeddings of medical concepts such as medication, procedure and diagnosis codes in Electronic Medical Records (EMRs) are central to healthcare analytics. Previous work on medical concept embedding takes medical concepts and EMRs as words and documents respectively. Nevertheless, such models miss out the temporal nature of EMR data. On the one hand, two consecutive medical concepts do not indicate they are temporally close, but the correlations between them can be revealed by the time gap. On the other hand, the temporal scopes of medical concepts often vary greatly (e.g., *common cold* and *diabetes*). In this paper, we propose to incorporate the temporal information to embed medical codes. Based on the Continuous Bag-of-Words model, we employ the attention mechanism to learn a “soft” time-aware context window for each medical concept. Experiments on public and proprietary datasets through clustering and nearest neighbour search tasks demonstrate the effectiveness of our model, showing that it outperforms five state-of-the-art baselines.'
author:
-
- |
Xiangrui Cai$^1$[^1], Jinyang Gao$^2$, Kee Yuan Ngiam$^3$, Beng Chin Ooi$^2$, Ying Zhang$^1$, Xiaojie Yuan$^1$\
$^1$ Nankai University, China\
$^2$ National University of Singapore, Singapore\
$^3$ National University Health System, Singapore\
{caixiangrui, zhangying, yuanxiaojie}@dbis.nankai.edu.cn,\
{jinyang.gao, ooibc}@comp.nus.edu.sg, kee\_yuan\[email protected]\
bibliography:
- 'refs.bib'
title: 'Medical Concept Embedding with Time-Aware Attention'
---
Conclusions
===========
We introduced a model that learned the embedding and a “soft” temporal scope for each medical concept simultaneously. Based on the CBOW model, our model takes advantage of attention mechanisms to learn such temporal scopes. The experimental results on two datasets and two tasks demonstrate the effectiveness of our models compared to state-of-the-art models. Our next plan is to utilize both medical concept embeddings and the “soft” context scopes for healthcare tasks such as missing value imputation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported by National Natural Science Foundation of China (No. 61772289) and National Research Foundation, Prime Ministers Office, Singapore under CRP Award (No. NRF-CRP8-2011-08). We thank the four anonymous reviewers for their valuable comments on our manuscript. We also thank Sihan Xu for her suggestions on the organization of this paper.
[^1]: The majority of this work was completed while the 1st author was visiting Naitional University of Singapore.
|
---
abstract: 'The quarkonium plus photon production in coherent hadron-hadron interactions at the LHC is studied using the non-relativistic QCD factorization formalism. We investigate a set of kinematic distributions and compute the total cross sections for $J/\Psi + \gamma $ production. Our results demonstrate the feasibility of such process in the LHC kinematic regime and explore the possibilities for the Future Circular Collider, where higher event yields can be achieved.'
author:
- 'M.M. Machado'
- 'G. Gil da Silveira'
title: |
Study on the PDF comparisons for quarkonium + $\gamma$ production\
at the LHC and FCC energies
---
Introduction
============
Over the last years, the quarkonium production becomes the subject of intense theoretical and experimental investigations. This interest has been motivated by the observation of large discrepancies between experimental measurements of $J/\Psi$ production at the Collider Detector (CDF) at Fermilab [@CDF] and theoretical calculations based on the color-singlet model (CSM) [@csm]. Attempts to understand this discrepancy focus on new production mechanisms that allow the $c\bar{c}$ bound system to be produced in a color-octet state and evolve non-perturbatively into a charmonium, where its large mass provides a natural hard scale that allows the application of perturbative Quantum Chromodynamics (QCD). There are several mechanisms proposed for the quarkonium production at hadron colliders [@Lansberg], as the CSM, the color-octet model (COM) [@com], and the color evaporation model (CEM) [@cem]. Also, the diffractive quarkonium production is sensitive to the gluon content of the Pomeron at small-$x$ [@forward] and may be particularly useful in studying the different mechanisms for quarkonium production. Hence, considering the experimental point-of-view, the heavy-quarkonium photoproduction have an extremely clean signature through their leptonic decay modes [@forward].
Studies of $\gamma$-proton interactions at the Large Hadron Collider (LHC) provides valuable information on the QCD dynamics at high energies. The photon–hadron interactions can be divided into [*[exclusive]{}*]{} and [*[inclusive]{}*]{} reactions. In the first case, a certain particle is produced, while the target remains in the ground state (or is internally excited only). On the other hand, in inclusive interactions the particle produced is accompanied by one or more particles from the breakup of the target.
In this work, we are focused in the process $$\label{process}
p + p \rightarrow p \oplus J/\Psi + \gamma + X.$$ i.e., the inclusive quarkonium + photon production in $pp$ collisions at the LHC and Future Circular Collider (FCC) energies, and predict their cross sections as a function of the quarkonium rapidity ($y$) and transverse momentum ($p_{\perp}$). The $\oplus$ represent the presence of rapidity gaps between the colliding proton and the produced meson. Such processes are relatively easy to be detected through their leptonic decay modes, and their transverse momenta are balanced by the associated high-energy photon. While the LHC is colliding protons at staggering 13 TeV, the FCC is planned to collide protons at 100 TeV [@Mangano], and will be an important step for the future development of high energies physics. Our predictions are the first one for the energy scale of this collider. In next section we present a brief review about the quarkonium$+\gamma$ photoproduction in the nonrelativistic QCD (NRQCD) formalism. Next, we present our predictions for the rapidity and transverse momentum distributions as well as the total cross sections for $J/\Psi + \gamma$ and $\Upsilon + \gamma$ production at LHC and FCC energies. Finally, the last section summarizes our main conclusions.
The $J/\psi$ + photon photoproduction
=====================================
The total cross section for the quarkonium + photon process, $p + p \to p + M + \gamma + X $, is given by [@mehen] $$\begin{aligned}
\sigma (W^{2}_{\gamma h}) = 2 \int d\omega \frac{dN(\omega)}{d\omega}\sigma_{\gamma p\rightarrow M+\gamma +X}(W^{2}_{\gamma h}),
\label{convol}\end{aligned}$$ where $\omega$ is the photon energy, $dN(\omega)/d\omega$ is the equivalent photon spectrum, and $W^{2}_{\gamma h} = 2\omega \sqrt{S_{NN}}$ is the center-of-mass energy in the photon-hadron system, with $\sqrt{S_{NN}}$ for the hadron-hadron system. In this work we consider the LHC energy of 13 TeV and the FCC at 100 TeV.
In the NRQCD formalism, the cross section for the production of a heavy quarkonium state $M$ factorizes into $$\begin{aligned}
\sigma (ab \rightarrow M + X) = \Sigma_{n} \sigma(ab \rightarrow Q\bar{Q} [n] + X)\langle O^{M}[n]\rangle,\end{aligned}$$ where the coefficients $\sigma(ab \rightarrow Q\bar{Q}[n] + X)$ are the short-distance cross sections for the production of the heavy-quark pair $Q\bar{Q}$ in an intermediate Fock state $n$, which does not have to be color neutral. The $\langle O^{M}[n]\rangle$ are nonperturbative long-distance matrix elements, which describe the transition of the intermediate $Q\bar{Q}$ state into the physical state $M$ via soft-gluon radiation. Currently, these elements have to be extracted from global fits to quarkonium data as performed in Ref. [@review_nrqcd]. It is important to emphasize that the underlying mechanism governing the heavy quarkonium production is still subject of intense debate (for a recent review see, e.g., Refs. [@magno; @mmmmvt; @vicmairon; @vicmairon2].
![The $p_{T}$ distribution of the $J/\psi + \gamma$ production at 13 TeV for different matrix elements: Mehen, Cacciari & Kramer (CK), Butenschoen & Kniehl (BK), and Chao [*et al.*]{}[]{data-label="fig:1"}](dsdpt_13tev_MESONpars_small.eps)
In the particular case of $M + \gamma$ photoproduction, the total cross section can be expressed as follows [@mehen] $$\begin{aligned}
\sigma_{\gamma p} = \int dz dp_{T}^2 \frac{xg_p(x,Q^2)}{z(1-z)} \frac{d\sigma}{dt}(\gamma + g \rightarrow M + \gamma ),
\label{sigmagamp}\end{aligned}$$ where $$\begin{aligned}
z = \frac{p_V\cdot p}{p_{\gamma} \cdot p},\end{aligned}$$ with $p_V$, $p$, and $p_{\gamma}$ being the 4-momentum of the quarkonium, hadron, and photon, respectively. In the hadron rest frame, $z$ can be interpreted as the fraction of the photon energy carried away by the quarkonium. The partonic differential cross section in Eq. \[sigmagamp\] is given by [@ko] $$\begin{aligned}
\nonumber
\frac{d\sigma}{dt} &=& \frac{64 \pi^2}{3}\frac{e_Q^4 \alpha^2\alpha_s m_Q}{s^2} \\
&\times& \left(\frac{s^2s_1^2+t^2t_1^2+u^2u_1^2}{s_1^2t_1^2u_1^2}\right) \langle O^{V}(^3S_1^{[1]})\rangle,
\label{dsdt}\end{aligned}$$ where $e_Q$ and $m_Q$ are, respectively, the charge and mass of heavy quark constituent of the quarkonium. The Mandelstam variables can be expressed in terms of $z$ and $p_{T}$ as follows
$$\begin{aligned}
s & = & \frac{p_{T}^2+(2m_Q)^2(1-z)}{z(1-z)}, \\
t & = & - \frac{p_{T}^2+(2m_Q)^2(1-z)}{z}, \\
u & = & - \frac{p_{T}^2}{1-z},\end{aligned}$$
and they are combine into
$$\begin{aligned}
s_1 = s - 4 m_Q^2, \\
t_1 = t - 4 m_Q^2, \\
u_1 = u - 4 m_Q^2,\end{aligned}$$
with the Bjorken variable $x$ expressed like $$\begin{aligned}
x = \frac{p_{T}^2+(2m_Q)^2(1-z)}{W_{\gamma h}^2z(1-z)}.\end{aligned}$$ The long-distance matrix elements, $\langle O^{V}(^3S_1^{[1]})\rangle$, can be determined from quarkonium electromagnetic decay rates. In our calculations we use the values as given in Refs. [@mehen; @bk; @cacciari; @chao] for the $J/\Psi$ production (see Tab. \[tab1\]). In what follows we consider different parametrizations for the parton distribution functions (PDF), in particular, we employ CT10 [@ct10], CT14 [@ct14], NNPDF v. 2.1 [@nnpdf21] and 3.1 [@nnpdf31], MMHT2014 [@mmht], and MSTW2008 [@mstw].
$\langle O^{V}(^3S_1^{[1]})\rangle$ $J/\Psi$ ($10^{-3}$ GeV$^{3}$)
------------------------------------- -------------------------------- -- --
Mehen [@mehen] $ 6.60 $
BK [@bk] $ 2.24 $
CK [@cacciari] $ 1 600 $
Chao [@chao] $ 3.00 $
: Matrix elements employed to compute the partonic differential cross section for the quarkonium photoproduction.[]{data-label="tab1"}
Results
=======
Focusing in the case of $J/\psi + \gamma$ case, we compute the $p_{T}$ distribution in order to compare the different predictions among the possible matrix elements, as shown in Figs. \[fig:1\] and \[fig:2\]. We also explore recent PDF parametrizations in order the evaluate the distinct contributions and possible kinematics regions to discriminate among them.
 
In Fig. \[fig:1\] we present the comparison of the different values of the NRQCD matrix elements at 13 TeV in the LHC. Although all predicions present the same slope, one clearly sees that the CK matrix element predicts a larger event rate. The CK matrix elements are the oldest estimations which are possibly superseded by the latest ones, i.e., Chao and BK.
In Fig. \[fig:2\], we provide the predictions for the inclusive, diffractive, and photon-proton cross sections for different PDFs, now at 13 TeV (top panel) and 100 TeV (bottom panel). For these results we have employed the matrix element estimated by Chao [*et al.*]{} As a result, one can see the contribution from photoproduction is two-order of magnitude smaller than the inclusive one. Moreover, the predictions with different PDF parametrizations show a rather good agreement at the LHC energy and a slightly separation at the higher energy of FCC. In both cases, the CT10 contribution is the smallest one.
![Comparison of the vector meson rapidity distribution for various PDF parametrizations at 13 TeV. []{data-label="fig:3"}](dsdy_pdfs.eps)
We also look into the rapidity distribution in order to discriminate the different contributions. In Fig. \[fig:3\], the different parametrizations are compared in the rapidity distribution of the vector meson. Although similar results are obtained, the rapidity distributions show distinct rapidity regions where each PDF have more contribution. This is an important discriminator to compare prediction with data obtained in the LHC experiments.
Another relevant result is the comparison of the chosen charm mass, as showed in Fig. \[fig:4\], where the comparisons of different values of charm mass are presented. We consider two different values of charm mass: $m_{c}$ = 1.28 GeV/$c$ as found in the PDG [@pdg] and the value of $m_{c}$ = 1.50 GeV/$c$ employed in several works in the literature. These predicitons are compared using two different PDF parametrizations (CT10 and CT14). As one can see in Fig. \[fig:4\], there are different contributions along the central detector and more data will allow us to obtain more stringent constraints on the parametrizations.
![Rapidity distributions for the chosen charm mass values in the literatue obtained with the CT10 and CT14 PDF parametrizations. []{data-label="fig:4"}](dsdy_MC_comparison.eps)
Conclusions
===========
In summary, we have computed the cross sections for the photoproduction of quarkonium$+ \gamma$ in coherent $pp$ collisions at LHC and FCC energies using the NRQCD formalism and considering different sets of values for the matrix elements. Such processes are interesting since the final state is characterized by a low multiplicity and one rapidity gap. For the particular case o f $J/\psi + \gamma$ production, our results demonstrate that the $p_{T}$ and $y$ distributions are strongly dependent on the NRQCD matrix elements and the PDFs parametrization. A detailed data analysis could provide sensible results to constraint the proton PDF parametrizations and NRQCD models. An extension of this work will include a detail study on the $\Upsilon + \gamma$ production for the LHC and FCC energies.
This work was supported by CNPq, CAPES, and FAPERGS, Brazil.
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|
---
author:
- |
Alexander Kurz\
Institut f[ü]{}r Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT),\
76128 Karlsruhe, Germany\
E-mail:
- |
Tao Liu\
Department of Physics, University of Alberta, Edmonton AB T6G 2J1, Canada\
E-mail:
- |
Peter Marquard\
Deutsches Elektronen Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany\
E-mail:
- |
Alexander V. Smirnov\
Scientific Research Computing Center, Moscow State University, 119991, Moscow, Russia\
E-mail:
- |
Vladimir A. Smirnov\
Skobeltsyn Institute of Nuclear Physics of Moscow State University, 119991, Moscow, Russia\
E-mail:
- |
\
Institut f[ü]{}r Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT),\
76128 Karlsruhe, Germany\
E-mail:
title: 'Electron contribution to $(g-2)_\mu$ at four loops'
---
Introduction
============
The anomalous magnetic moment of the muon is among the most precisely known quantities in particle physics. At the same time there is a long-standing deviation between the experimental measurement and the theory prediction which amounts to about three standard deviations. On the experimental side there are upcoming new experiments which either use the same method as in the E821 experiment at BNL [@Bennett:2006fi; @Roberts:2010cj] but reduce the uncertainties by about a factor four, or even use a completely different technique which would eliminate doubts on possible systematic effects (for details see, e.g., Ref. [@Hertzog:2015jru]).
On the theory side it is certainly necessary to improve on the hadronic contributions, both from the vacuum polarization and from light-by-light-type diagrams. Furthermore, it is mandatory to cross check the four-loop QED contribution since the full result has only been obtained by one group [@Kinoshita:2004wi; @Aoyama:2007mn; @Aoyama:2012wk]. In a series of works [@Kurz:2013exa; @Kurz:2015bia; @Kurz:2016bau] the fermionic pieces have been confirmed. The cross check of the purely photonic part is still missing. In this contribution we report on the calculation and results of the diagrams involving closed electron loops [@Kurz:2016bau].
The method
==========
It is convenient to sub-divide the contributing four-loop diagrams in twelve classes [@Aoyama:2012wk] which are introduced in Fig. \[fig::classes\] with the help of sample Feyman diagrams.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4Ia.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4Ib.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4Ic.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4Id.eps "fig:")
I(a) I(b) I(c) I(d)
![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IIa.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IIb.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IIc.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4III.eps "fig:")
II(a) II(b) II(c) III
![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IVd.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IVa.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IVb.eps "fig:") ![ Four-loop diagram classes for $a_\mu$ containing at least one closed electron loop. The external solid lines represent muons, the solid loops denote electrons, muons or taus, and the wavy lines represent photons.[]{data-label="fig::classes"}](dia4IVc.eps "fig:")
IV(d) IV(a) IV(b) IV(c)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The approach used for the computation of the four-loop diagrams differs from the one applied in Ref. [@Aoyama:2012wk] in many ways. In [@Kurz:2016bau] we generate in a first step amplitudes for each individual vertex diagram which contain both the electron ($m_e$) and the muon mass ($m_\mu$). At this point we exploit the fact that $m_e\ll m_\mu$ and apply an asymptotic expansion which expresses each amplitude into a sum of so-called sub-diagrams. Each sub-diagram is written as a product of one-scale integrals which are much easier to compute.
The various sub-diagrams involve different types of integrals which have to be treated separately. Most of them are well studied in the literature and analytic results can be obtained. However, there are two types where this is not the case: four-loop on-shell integrals and four-loop integrals involving propagators of the form $1/(2 \ell \cdot q)$ where $q$ is the external momentum with $q^2=m_\mu^2$, in the following called “linear integrals”.
At this point we apply an appropriate projector to the magnetic form factor and expand afterwards in the photon momentum to obtain the static limit. Then we perform the traces and decompose each amplitude into a sum of scalar integrals. For simple integral types (like two-loop vacuum integrals) we can directly insert the analytic results for the integrals. The more complicated ones are reduced to so-called master integrals using the program packages [ FIRE]{} [@Smirnov:2014hma] and [crusher]{} [@crusher]. In this way we obtain an analytic result for the muon anomalous magnetic moment in term of a relatively small number \[${\cal O}(100)$\] of master integrals. This is the case for all coefficients of $(m_e/m_\mu)^n$ (we expanded up to $n=3$). Note that as far as the four-loop master integrals are concerned the odd powers of $m_e/m_\mu$ only involve linear integrals while the even powers get contributions from on-shell and linear integrals.
It is only at this point when we pass on to numerical methods since to date not all master integrals are available in analytic form. This is the origin of the uncertainty in our final results, see below.
The numerical evaluation of the four-loop on-shell master integrals is described in detail in Ref.[@Marquard:2016dcn]. A similar approach has also been used for the linear integrals, see also Ref. [@Kurz:2016bau].
Results for $(g-2)_\mu$
=======================
In this section we present results for the anomalous magnetic moment of the muon. We cast the perturbative expansion in the form $$\begin{aligned}
\frac{(g-2)_\mu}{2} \,\,=\,\, a_\mu &=&
\sum_{n=1}^\infty a_\mu^{(2n)} \left( \frac{\alpha}{\pi} \right)^n
\,,
\label{eq::amu}\end{aligned}$$ where $n$ counts the number of loops. $a_\mu^{(2n)}$ is conveniently split into several pieces according to the particles present in the loop. In particular, we have for the four-loop term $$\begin{aligned}
a_\mu^{(8)} &=& A_1^{(8)} + A_2^{(8)}(m_\mu / m_e) + A_2^{(8)}(m_\mu / m_\tau)
+ A_3^{(8)}(m_\mu / m_e, m_\mu / m_\tau)
\,,
\label{eq::Amu}\end{aligned}$$ where $A_1^{(8)}$ denotes the universal part which includes the pure photonic corrections and closed muon loops. $A_2^{(8)}(m_\mu / m_e)$ ($A_2^{(8)}(m_\mu
/ m_\tau)$) contains in addition at least one closed electron (tau) loop and $A_3^{(8)}(m_\mu / m_e, m_\mu / m_\tau)$ contains at least one electron and one tau loop.
In Table \[tab::res\] the results from the individual diagram classes contributing to $A_2^{(8)}(m_\mu / m_e)$ are shown. For practical reasons only the sum is presented for the classes I(b)+I(c) and II(b)+II(c) and a further splitting is carried out in case more than one electron loop is present (see Ref. [@Kurz:2016bau] for a detailed discussion.)
$A_2^{(8)}(m_\mu/m_e)$ [@Kurz:2016bau; @Kurz:2015bia] literature
------------------------ ----------------------------------------------------------- ----------------------------------------------------------- ---------------------
I(a0) $\hphantom{-00}7.223076$ $\hphantom{-00}7.223077 \pm 0.000029$ [@Kinoshita:2004wi]
$\hphantom{-00}7.223076$ [@Laporta:1993ds]
I(a1) $\hphantom{-00}0.494072$ $\hphantom{-00}0.494075 \pm 0.000006$ [@Kinoshita:2004wi]
$\hphantom{-00}0.494072$ [@Laporta:1993ds]
I(a2) $\hphantom{-00}0.027988$ $\hphantom{-00}0.027988 \pm 0.000001$ [@Kinoshita:2004wi]
$\hphantom{-00}0.027988$ [@Laporta:1993ds]
I(a) $\hphantom{-00}7.745136$ $\hphantom{-00}7.74547 \pm 0.00042$ [@Aoyama:2012wk]
I(bc0) $\hphantom{-00}8.56876 \pm 0.00001$ $\hphantom{-00}8.56874 \pm 0.00005$ [@Kinoshita:2004wi]
I(bc1) $\hphantom{-00}0.1411 \pm 0.0060$ $\hphantom{-00}0.141184 \pm 0.000003$ [@Kinoshita:2004wi]
I(bc2) $\hphantom{-00}0.4956 \pm 0.0004$ $\hphantom{-00}0.49565 \pm 0.00001$ [@Kinoshita:2004wi]
I(bc) $\hphantom{-00}9.2054 \pm 0.0060$ $\hphantom{-00}9.20632 \pm 0.00071$ [@Aoyama:2012wk]
I(d) $\hphantom{0}\text{$-$}\hphantom{0}0.2303 \pm 0.0024$ $\hphantom{0}\text{$-$}\hphantom{0}0.22982 \pm 0.00037$ [@Aoyama:2012wk]
$\hphantom{0}\text{$-$}\hphantom{0}0.230362 \pm 0.000005$ [@Baikov:1995ui]
II(a) $\hphantom{0}\text{$-$}\hphantom{0}2.77885$ $\hphantom{0}\text{$-$}\hphantom{0}2.77888 \pm 0.00038$ [@Aoyama:2012wk]
$\hphantom{0}\text{$-$}\hphantom{0}2.77885$ [@Laporta:1993ds]
II(bc0) $\hphantom{0}\text{$-$}12.212631$ $\hphantom{0}\text{$-$}12.21247 \pm 0.00045$ [@Kinoshita:2004wi]
II(bc1) $\hphantom{0}\text{$-$}\hphantom{0}1.683165 \pm 0.000013$ $\hphantom{0}\text{$-$}\hphantom{0}1.68319 \pm 0.00014$ [@Kinoshita:2004wi]
II(bc) $\hphantom{0}\text{$-$}13.895796 \pm 0.000013$ $\hphantom{0}\text{$-$}13.89457 \pm 0.00088$ [@Aoyama:2012wk]
III $\hphantom{-0}10.800 \pm 0.022$ $\hphantom{-0}10.7934 \pm 0.0027$ [@Aoyama:2012wk]
IV(a0) $\hphantom{-}116.76 \pm 0.02$ $\hphantom{-}116.759183 \pm 0.000292$ [@Kinoshita:2004wi]
$\hphantom{-}111.1 \pm 8.1$ [@Calmet:1975tw]
$\hphantom{-}117.4 \pm 0.5$ [@Chlouber:1977dr]
IV(a1) $\hphantom{-00}2.69 \pm 0.14$ $\hphantom{-00}2.697443 \pm 0.000142$ [@Kinoshita:2004wi]
IV(a2) $\hphantom{-00}4.33 \pm 0.17$ $\hphantom{-00}4.328885 \pm 0.000293$ [@Kinoshita:2004wi]
IV(a) $\hphantom{-}123.78\pm 0.22$ $\hphantom{-}123.78551 \pm 0.00044$ [@Aoyama:2012wk]
IV(b) $\hphantom{0}\text{$-$}\hphantom{0}0.38 \pm 0.08$ $\hphantom{0}\text{$-$}\hphantom{0}0.4170 \pm 0.0037$ [@Aoyama:2012wk]
IV(c) $\hphantom{-00}2.94 \pm 0.30$ $\hphantom{-00}2.9072 \pm 0.0044$ [@Aoyama:2012wk]
IV(d) $\hphantom{0}\text{$-$}\hphantom{0}4.32 \pm 0.30$ $\hphantom{0}\text{$-$}\hphantom{0}4.43243 \pm 0.00058$ [@Aoyama:2012wk]
: Final results for the different classes and comparison with the literature.[]{data-label="tab::res"}
It is interesting to note that in some cases our coefficients have smaller uncertainties (e.g. II(bc)) whereas for others we have obtained an uncertainty which is much worse than the one of [@Aoyama:2012wk] (e.g. IV(c) or IV(d)). This can be traced back to complicated master integrals which at the moment can only be evaluated with a few-digit precision. Let us stress that, if necessary, the precision of our result can be improved systematically.
Our final result for $A_2^{(8)}(m_\mu/m_e)$ is given by $$\begin{aligned}
A_2^{(8)} &=& 126.34(38) + 6.53(30) = 132.86(48)\,,\end{aligned}$$ where the first number after the first equality sign originates from the light-by-light-type diagrams IV(a), IV(b) and IV(c). Our final numerical uncertainty amounts to approximately ${0.5} \times
(\alpha/\pi)^4 \approx {1.5} \times 10^{-11}$. It is larger than the uncertainty in Ref. [@Aoyama:2012wk]. Nevertheless it is sufficiently accurate as can be seen by the comparison to the difference between the experimental result and theory prediction which is given by [@Aoyama:2012wk] $$\begin{aligned}
a_\mu({\rm exp}) - a_\mu({\rm SM}) &\approx& 249(87) \times 10^{-11}
\,.
\label{eq::amu_diff}\end{aligned}$$ Note that the uncertainty in Eq. (\[eq::amu\_diff\]) receives approximately the same amount from experiment and theory (i.e. essentially from the hadronic contribution). Even after a projected reduction of the uncertainty by a factor four both in $a_\mu({\rm exp})$ and $a_\mu({\rm SM})$ our numerical precision is a factor ten below the uncertainty of the difference.
Conclusions
===========
In this contribution we reported on the calculation of the four-loop QED corrections to $a_\mu$ which involve closed electron loops [@Kurz:2016bau; @Kurz:2015bia] \[see $ A_2^{(8)}(m_\mu / m_e)$ in Eq. (\[eq::Amu\])\]. In Ref. [@Kurz:2016bau] also the contribution $A_3^{(8)}(m_\mu / m_e, m_\mu / m_\tau) $ with at least one electron and one tau loop have been computed and the results for $A_2^{(8)}(m_\mu / m_\tau)$ can be found in Ref. [@Kurz:2013exa]. For all contributions perfect agreement with the results of Ref. [@Aoyama:2012wk] have been obtained. The only missing four-loop contribution which still has to be cross-checked is the universal part $A_1^{(8)}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
P.M. was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704. The work of V.S. was supported by the Alexander von Humboldt Foundation (Humboldt Forschungspreis).
[99]{}
|
---
author:
- |
$\,^1$, A.Arbet-Engels$\,^2$, D.Baack$\,^3$, M.Balbo$\,^1$, M.Beck$\,^{2,a}$, N.Biederbeck$\,^3$, A.Biland$\,^2$, M.Blank$\,^4$, T.Bretz$\,^{2,a}$, K.Bruegge$\,^3$, M.Bulinski$\,^3$, J.Buss$\,^3$, M.Doerr$\,^4$, D.Dorner$\,^4$, D.Elsaesser$\,^3$, D.Hildebrand$\,^2$, R.Iotov$\,^4$, M.Klinger$\,^{2,a}$, K.Mannheim$\,^4$, S.A.Mueller$\,^2$, D.Neise$\,^2$, M.Noethe$\,^3$, A.Paravac$\,^4$, W.Rhode$\,^3$, B.Schleicher$\,^4$, K.Sedlaczek$\,^3$, A.Shukla$\,^4$, L.Tani$\,^2$, F.Theissen$\,^{2,a}$ and R.Walter$\,^1$ (the FACT Collaboration)\
[$^1$]{}University of Geneva, Department of Astronomy, Chemin d’Ecogia 16, 1290 Versoix, Switzerland\
[$^2$]{}ETH Zurich, Institute for Particle Physics and Astrophysics, Otto-Stern-Weg 5, 8093 Zurich, Switzerland\
[$^3$]{}TU Dortmund, Experimental Physics 5, Otto-Hahn-Str. 4, 44221 Dortmund, Germany\
[$^4$]{}Universität Würzburg, Institute for Theoretical Physics and Astrophysics, Emil-Fischer-Str. 31, 97074 Würzburg, Germany\
[$^a$]{}also at RWTH Aachen University, Physics Institute III A, 52074 Aachen, Germany\
E-mails: ,\
title: '5.5 years multi-wavelength variability of Mrk 421: evidences of leptonic emission from the radio to TeV'
---
firstaubox
Introduction\[sec:introduction\]
================================
Mrk421 is a bright nearby high-energy-peaked blazar ($z = 0.031$), perfectly suitable for the study of the broadband emission from a relativistic jet. Mrk421 features frequent and bright GeV and TeV flares. Its spectral energy distribution (SED) has two humps peaking in the X-rays and in GeV energies. The precise model of the blazar emission is still under discussions. It is generally agreed that the low energy hump is created by relativistic emitting electrons, though the high energy one has been explained by hadronic or leptonic emission, or by a mixture of the two. A one-zone leptonic synchrotron self-Compton model (SSC) [@abdo_2011ApJ...736..131A] can be used to explain the complete SED of Mrk421. Hadronic emission has been proposed as a result of proton synchrotron [@2015MNRAS.448..910C], hadrons can also interact with the leptonic synchrotron photons creating cascades of pions and muons, which then decay and emit $\gamma$-rays and neutrinos [@2001APh....15..121M].
The X-ray and TeV variability of Mrk421 suggests either that relativistic shocks move within the jet or that the emission region is much smaller than the gravitational radius , possibly driven by the interactions between stars/clouds and the jet, or by magnetic reconnections.
Mrk421 has been the target of numerous multi-wavelength (MWL) campaigns aiming to investigate the emission processes within its jet . Most studies reported strong correlation between the X-ray and TeV emission, with a maximum lag of $\sim$5 days . The highest variability was found in X-rays, where also a harder-when-brighter behaviour was discovered.
Here, we report results of variability and correlation studies, based on about 5.5 years long multi-wavelength monitoring campaign, which includes dense TeV observations with the First G-APD Cherenkov Telescope (FACT) [@2013arXiv1311.0478D]. Unlike other TeV observational campaigns, FACT was not triggered in case of flare detected elsewhere, but observed densely and regularly over the years. At lower energies, we used continuous radio, optical, ultraviolet, X-ray, and GeV data, obtained quasi-simultaneously with FACT observations.
Multi-wavelength data\[sec:data\]
=================================
Nine different instruments contributed data to the MWL dataset used in this study. The light curves span from December 14, 2012 to April 18, 2018. During this time period, Mrk421 was observed in various flux states. Flares and increased flux were observed in all considered bands. Some flares, observed in the X-ray and TeV band, lasted for a few days, so could be individually identified (see table \[tab:flares\]).
FACT is a 3.8m imaging air Cherenkov telescope located in La Palma, Spain [@2013JInst...8P6008A], operating since late 2011 [@2013arXiv1311.0478D], and remotely controlled since July 2012 (fully robotic observations were achieved in late 2017). Its SiPM camera and the state-of-the-art feedback system[@2014JInst...9P0012B], allows to observe even in bright ambient light conditions [@2013arXiv1307.6116K]. Analysis techniques and quality checks, which were applied to the Mrk421 data, are reported in [@2017ICRC...35..779H; @2017ICRC...35..612M; @2019arXiv190203875B; @2005ICRC....5..215R]. Using simulated data, the energy threshold of the telescope for this source is estimated as 850GeV.
We used data from the Fermi Large Area Telescope (LAT) [@2009ApJ...697.1071A] to cover the GeV $\gamma$-rays band with energy 100 MeV $<$ E $<$ 300GeV. The PASS8 pipeline, Fermi Science Tool v10r0p5 package, and background sources from the LAT 4-year Point Source Catalogue were used for the modelling.
X-ray data originates from multiple space-based telescopes instruments: Swift/BAT, Swift/XRT and MAXI. Swift/BAT covers 15-50keV band, data reduction pipeline is based on the BAT analysis software `HEASOFT` version 6.13 [@2013ApJS..207...19B]. Using Swift/XRT [@2005SSRv..120..165B] data, we obtained the light curves for the 0.3-2keV and 2-10keV bands directly from the on-line Swift-XRT products generation tool [^1] which adopts the `HEASOFT` software version 6.22. MAXI [@2009PASJ...61..999M] is sensitive from 2keV to 20keV, and the light curve for Mrk421 is publicly available [^2].
Optical observations were performed by multiple ground- and space-based telescopes. In the V-band, as a part of a blazar monitoring campaign [@2009arXiv0912.3621S], Mrk421 was observed by the 1.54m Kuiper and Bok telescopes. Data from Cycle 5 to 10 are available online[^3] and was used in this study. In the UV, the source is observed by Swift/UVOT [@roming_2005SSRv..120...95R]. The on-, off-method using the HEASOFT package version 6.24 along with UVOT CALDB version 20170922 was adopted for the data reduction.
Regular observations of Mrk421 at 15GHz were performed by the the OVRO 40 meter radio telescope. The light curve is available through the public archive[^4].
Timing and correlation analysis
===============================
We performed cross-correlation, auto-correlation, Bayesian Block and fractional variability analysis of the considered MWL light curves to investigate the physical processes responsible for the emission in all these bands. We also estimated the best-fit parameters of the GeV to radio response profile to generate a synthetic radio light curve from the GeV one.
Fractional variability
----------------------
Following the prescription in [@Vaughan_2003MNRAS.345.1271V] and uncertainties estimation suggested in [@Poutanen_2008MNRAS.389.1427P] the fractional variability analysis was performed for all Mrk421 light curves. The highest variability ($F_{var}=1.33$) was found in the hard X-rays (Swift/BAT) and the lowest in the radio ($F_{var} = 0.15$). Beyond the X-ray band, the variability drops at GeV energies ($F_{var}=0.34$) and increases again at TeV energies ($F_{var}=0.92$). The two distinctive humps in the variability spectrum match the SED, as already previously reported . The two emission humps are more variable towards higher energies. As simultaneous (within a day) X-ray and TeV fluxes are correlated, a single parameter (the cutoff energy) is indeed driving the main variations observed in both components.
Light curves correlations
-------------------------
Long term unevenly sampled light curves allow to estimate correlations for sub-sampling time lags. We performed such an analysis using the discrete cross-correlation function (DCF) as proposed in [@Peterson_1998PASP..110..660P]. Originally underestimated uncertainties of the DCF were recalculated using flux randomisation and random subset selection process for MC simulations. The final lag between the light curves is estimated using the centroid method [@Peterson_1998PASP..110..660P; @2003ApJ...584L..53U] at 80% maximum. The variability time scale observed in the TeV and X-ray bands are short, $\sim$3 days, which is consistent with the models where the emission in the jet is dominated by relativistic electron cooling.
A strong cross-correlation was found between the TeV and X-ray light curves. The combined (all X-ray bands) X-rays lag is ($0.02\pm0.41$) days ($1\sigma$). The combined cross-correlation between X-ray and TeV bands and respective lag distribution are shown in Fig. \[fig:dcf\_all\]. This result is more constraining than previously reported lags on more sparse and shorter data sets . The X-ray and TeV light curves are not correlated with the GeV, optical or radio light curves. The low energy bands are correlated with the GeV band although with wide response and lags of 40 to 70 days.
![Combined DCF cross-correlations of TeV (FACT) and X-rays (MAXI, Swift/BAT, Swift/XRT) variability. Left: DCF values as a function of lag. Gray lines denote the 1$\sigma$ uncertainties. Right: lag distribution corresponding to the maximum DCF value.[]{data-label="fig:dcf_all"}](images/cc_tev_xrays_all.eps){width="0.83\columnwidth"}
A Bayesian block algorithm was applied to the X-ray and TeV light curves to identify individual flares assuming simultaneous flares in these two bands. A flare was defined as an increased flux for at least two days with amplitude at least twice the amplitude of the previous Bayesian block. The identified flares are listed in table \[tab:flares\]. 29 of the TeV flares were coincident with ones in X-rays. This indicates that particle populations with different energy distributions are necessary to explain all the observations.
Bands Number Time ranges, MJD
----------------- -------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
TeV only: 2 56689-56692, 57006-57015
TeV and X-rays: 29 56317-56330, 56369-56383, 56389-56400, 56441-56449, 56650-56670, 56696-56700, 56751-56755, 56976-57005, 57039-57053, 57065-57070, 57091-57099, 57110-57121, 57188-57192, 57368-57379, 57385-57390, 57422-57431, 57432-57449, 57504-57511, 57531-57538, 57545-57550, 57728-57753, 57756-57769, 57770-57775, 57787-57793, 57850-57866, 58103-58113, 58129-58142, 58162-58167, 58185-58196
: List of TeV and X-ray flares sorted by the spectral bands in which they were detected.[]{data-label="tab:flares"}
GeV to radio response
---------------------
Since there is a wide correlation between the GeV and radio light curves, we attempted to reproduce the radio light curve by convolving the GeV light curve with a specially constructed response profile . The response profile, original and synthetic radio light curves are shown in Fig. \[fig:fermi\_radio\_conv\]. We found that the response has $t_{rise}=3$ days, $t_{fall}=7.7$ days, for a delay of $\Delta t = 43$ days. Need for such a delay was also reported previously in 3C273 . Adopting such an approach, we were able to well reproduce the radio light curve except for one very fast radio flare (MJD 56897).
![Synthetic radio light curve (top) derived from the *Fermi*-LAT light curve and OVRO 15GHz radio light curve (bottom). The GeV to radio response profile is depicted in the top right corner. The $y$ axis is in arbitrary units.[]{data-label="fig:fermi_radio_conv"}](images/convolution_radio.eps){width="0.83\columnwidth"}
Results and conclusions\[sec:conclusions\]
==========================================
We performed correlations, variability and timing analysis of 5.5 years of dense multi-wavelength data of Mrk421 and found that:
1. The highest variability is exhibited in the X-rays and at TeV energies. 95% of the short X-ray and TeV flares are coincident. The lag between the TeV and X-ray light curves is not significant and is shorter than ($0.02\pm0.41$) days.
2. The radio light curve can be reproduced by convolving the GeV light curve with a delayed fast rising and slowly decaying response profile.
The fractional variability of Mrk421 and the correlated TeV and X-ray emission indicate that the main source of variability is dominated by a synchronous change of the cutoff energies of the low and high-energy components. The zero lag correlation between the TeV and X-ray light curves of Mrk421 indicates that these two emissions are driven by the same physical parameter and are consistent with a leptonic emission scenario. The observed X-ray and TeV variability time scales do not match the prediction for proton cooling for shock acceleration. The proton acceleration times for lepto-hadronic models are also much longer.
The GeV to radio response is a strong indication that synchrotron processes dominate the low energy emission component. A fast rise of the response profile after a delay of $\approx$43 days and a slow decay can be interpreted as an emitting region moving outwards in a conical jet: first becoming transparent to gamma rays, later to the radio. The fast rise after a delay may indicate a discontinuity in the jet properties.
*Acknowledgements:* The important contributions from ETH Zurich grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the Swiss SNF and the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) and HAP (Helmoltz Alliance for Astroparticle Physics) are gratefully acknowledged. Part of this work is supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876 “Providing Information by Resource-Constrained Analysis”, project C3. We are thankful for the very valuable contributions from E. Lorenz, D. Renker and G. Viertel during the early phase of the project. We thank the Instituto de Astrofísica de Canarias for allowing us to operate the telescope at the Observatorio del Roque de los Muchachos in La Palma, the Max-Planck-Institut für Physik for providing us with the mount of the former HEGRA CT3 telescope, and the MAGIC collaboration for their support. This research has made use of public data from the *OVRO* 40-m telescope [@Richards_2011ApJS..194...29R], the Bok Telescope on Kitt Peak and the 1.54 m Kuiper Telescope on Mt. Bigelow [@2009arXiv0912.3621S], MAXI [@2009PASJ...61..999M], *Fermi*-LAT [@2009arXiv0912.3621S] and *Swift* [@2004NewAR..48..431G].
[^1]: http://www.swift.ac.uk/user\_objects/
[^2]: http://maxi.riken.jp/star\_data/J1104+382/J1104+382.html
[^3]: http://james.as.arizona.edu/$\sim$psmith/Fermi/DATA/photdata.html
[^4]: http://www.astro.caltech.edu/ovroblazars/
|
---
abstract: 'Recent work applying multidimentional coherent electronic spectroscopy at dilute samples in the gas phase is reviewed. The development of refined phase cycling approaches with improved sensitivity has opened up new opportunities to probe even dilute gas-phase samples. In this context, first results of two-dimensional spectroscopy performed at doped helium droplets reveal the femtosecond dynamics upon electronic excitation of cold, weakly-bound molecules, and even the induced dynamics from the interaction with the helium environment. Such experiments, offering well-defined conditions at low temperatures, are potentially enabling the isolation of fundamental processes in the excitation and charge transfer dynamics of molecular structures which so far have been masked in complex bulk environments.'
address:
- '$^1$Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany'
- '$^2$Freiburg Institute of Advanced Studies (FRIAS), University of Freiburg, Albertstr. 19, D-79194 Freiburg, Germany'
author:
- 'Lukas Bruder$^1$, Ulrich Bangert$^1$, Marcel Binz$^1$, Daniel Uhl$^1$ and Frank Stienkemeier$^{1,2}$'
bibliography:
- 'CMDSgasphase\_1.bib'
- 'LB\_added\_citations.bib'
title: Coherent multidimensional spectroscopy in the gas phase
---
April 2019
[*Keywords*]{}: multidimensional spectroscopy, nonlinear optics, ultrafast spectroscopy, molecular beams, cluster beams, helium nanodroplets
Introduction
============
The development of coherent multidimensional spectroscopy (CMDS) in the optical regime has greatly improved the toolkit of ultrafast spectroscopy[@hochstrasser_two-dimensional_2007; @cho_coherent_2008; @nuernberger_multidimensional_2015]. The method may be regarded as an extension of pump-probe spectroscopy, where pump and probe steps are both spectrally resolved, while maintaining high temporal resolution in the sub 50fs regime[@jonas_two-dimensional_2003]. By spreading the nonlinear response onto multidimensional frequency-correlation maps, improved spectral decongestion is achieved and the analysis of couplings within the system or to the environment is greatly simplified.
The concept of CMDS was originally developed in NMR spectroscopy[@aue_twodimensional_1976] and was first implemented at optical frequencies about 25 years ago[@hamm_structure_1998; @hybl_two-dimensional_1998]. Nowadays, routinely used methods in the optical regime comprise two-dimensional infrared (2DIR) spectroscopy and its counter part in the visible spectral range, 2D electronic spectroscopy (2DES)[@hochstrasser_two-dimensional_2007]. 2DIR yields insights into structure and dynamics of molecular networks with high chemical sensitivity, while 2DES accesses in addition electronic degrees of freedom (DOFs) and provides information about the coupled electronic-nuclear dynamics albeit with reduced chemical selectivity. Notably, also combinations of both methods have been reported[@oliver_correlating_2014; @vanwilderen_mixed_2014; @courtney_measuring_2015] and extensions to the THz spectral regime exist[@woerner_ultrafast_2013].
In recent years, 2DIR and 2DES have provided decisive information about many ultrafast phenomena including energy and charge transfer in biological systems[@brixner_two-dimensional_2005; @read_cross-peak-specific_2007; @lewis_probing_2012; @romero_quantum_2014; @fuller_vibronic_2014; @dostal_situ_2016; @scholes_using_2017; @thyrhaug_identification_2018], the real-time analysis of solvent dynamics[@asbury_water_2004; @cowan_ultrafast_2005; @moca_two-dimensional_2015] or the mapping of reaction pathways in chemical reactions[@nuernberger_multidimensional_2015], to name a few examples. Furthermore, multiple-quantum coherence CMDS studies have proven to be sensitive probes for many-body effects[@stone_two-quantum_2009; @turner_coherent_2010; @cundiff_optical_2012; @dai_two-dimensional_2012].
So far, CMDS has been mostly performed in the condensed phase, where the wide range of accessible systems has lead to a rich variety of studies, ranging from molecular aggregates[@milota_two-dimensional_2009; @ginsberg_two-dimensional_2009; @nemeth_tracing_2009; @dostal_direct_2018] to multichromophoric photosynthic systems[@brixner_two-dimensional_2005; @read_cross-peak-specific_2007; @myers_two-dimensional_2010; @collini_coherently_2010; @dostal_situ_2016], thin films[@provencher_direct_2014; @oudenhoven_dye_2015; @de_sio_tracking_2016; @ostrander_energy_2017] and bulk crystal structures[@huxter_vibrational_2013; @bakulin_real-time_2015; @batignani_probing_2018] as well as different types of semiconductor nanomaterials[@moody_exciton-exciton_2011; @bylsma_quantum_2012; @cundiff_optical_2012; @graham_two-dimensional_2012; @mehlenbacher_energy_2015].
Yet, most of these systems exhibit a large number of degrees of freedom (DOFs) with commonly complex interrelations, resulting often in highly congested data and significant spectral broadening. As such, detailed theoretical calculations are essential for the interpretation of the experiments. However, the large parameter spaces of condensed phase systems force to strong approximations and simplifications in theoretical models, even with nowadays numerical capacities. This makes the precise analysis and interpretation of the experiments, despite the advanced spectroscopic methods, up to date a challenging task.
CMDS studies of isolated gas-phase systems resolve these issues and thus provide an invaluable complementary view on condensed phase experiments. In the gas phase, isolated model systems of confined complexity can be synthesized with a high degree of control[@hertel_ultrafast_2006]. Systems may be prepared in well-defined initial states[@scoles_atomic_1988; @stienkemeier_use_1995; @toennies_superfluid_2004], and high resolution data can be acquired[@smalley_molecular_1977; @levy_laser_1980; @wewer_laser-induced_2004] while perturbations by the environment are eliminated. This also improves the situation for theorists and will assist the development of more accurate models, ultimately leading to a better understanding of primary ultrafast processes.
In addition, the information content deducible from these experiments is increased by a palette of highly selective detection methods exclusively accessible in the gas phase. These include ion-mass[@wiley_timeflight_1955; @schultz_efficient_2004; @lippert_femtosecond_2004; @bruder_phase-modulated_2015] and photoelectron kinetic energy spectrometry[@kruit_magnetic_1983; @stolow_femtosecond_2004], velocity map imaging (VMI)[@eppink_velocity_1997; @vallance_molecular_2004; @r.ashfold_imaging_2006; @greaves_velocity_2010] and even electron-ion coincidence detection[@helm_images_1993; @radloff_internal_1997; @ergler_time-resolved_2005]. The additional information gained from these detection types can be extremely helpful in disentangling complex dynamics and sensing the system’s reaction energy landscape including dark states[@stolow_femtosecond_2004], non-radiative internal conversion pathways[@wolf_probing_2017] and reaction intermediates [@dantus_realtime_1987]. As such, the gas-phase approach opens new possibilities for high precision multidimensional spectroscopy studies unveiling an unprecedented amount of details.
However, this development has been so far precluded by insufficient sensitivity to probe highly dilute gas-phase samples. Only few examples of CMDS in the gas phase exist to date.
![Landscape showing typical temperatures and densities of gas phase samples in comparison to the condensed phase. So far, CMDS has been exclusively performed in the condensed phase and gas cells which cover only a small fraction of the available parameter space covered by nowadays available targets. The work of our group extends this range to CMDS experiments on cluster-isolation and helium droplet targets. MOT: atoms in magneto-optical traps; BEC: Bose-Einstein condensate of atoms.[]{data-label="fig:overview"}](overview_1.pdf){width="0.8\linewidth"}
Some groups have reported 2DES studies of alkali atom vapors at particle densities of $\geq 10^{10}$cm$^{-3}$[@tian_femtosecond_2003; @tekavec_fluorescence-detected_2007; @dai_two-dimensional_2010] and very recently down to $10^8$cm$^{-3}$[@Yu_2019]. These simple target systems, however, do not imply a generalization of the method’s applicability to more advanced systems. For rubidium vapors, also a high-resolution 2D spectroscopy scheme based on frequency combs has been recently demonstrated capable of even resolving the atomic hyperfine levels[@lomsadze_frequency_2017; @lomsadze_tri-comb_2018]. Furthermore, high-resolution multidimensional spectroscopy in the frequency domain using nanosecond lasers has been performed on several molecular vapors[@Chen_review_2016]. Yet, considering the wide range of unique target systems available in the gas phase and the large parameter space they cover, CMDS has been so far restricted to a small portion of targets. This is illustrated in Fig. \[fig:overview\] where the landscape of gas-phase samples provided by different experimental techniques is plotted with respect to sample density and internal temperature.
Only very recently, the Brixner group and our group demonstrated the first 2DES studies of gas-phase molecules and incorporated some of the afore mentioned new photoionization detection schemes[@roeding_coherent_2018; @bruder_coherent_2018]. Brixner and coworkers probed a thermal gas of NO$_2$ molecules combined with selective mass spectrometry[@roeding_coherent_2018]. Our group studied cold ($T =380$mK) Rb$_2$ and Rb$_3$ molecules prepared with matrix isolation in a cluster beam apparatus, detected with photoelectron and ion-mass spectrometry[@bruder_coherent_2018]. These experiments, in principle, continue the pioneering early work of Zewail and coworkers[@zewail_femtochemistry:_2000], advancing the field of Femtochemistry to a new direction and extending the range of target systems towards more fundamental quantum systems which may now come within reach (Fig.\[fig:overview\]).
Besides the extreme demands on sensitivity, another circumstance has prolonged the development of gas-phase multidimensional spectroscopy. The vacuum technologies required for advanced sample preparation in the gas phase are not common to the CMDS community. On the other side, the molecular beam and related communities are mostly not aware of CMDS techniques. Hence, this combination of techniques requires expertise from rather disjunct communities and calls for a novel fusion of disciplines and their specialized technologies.
In this context, we review in the present paper the recent extension of CMDS to the gas phase in conjunction with the 2DES experiments developed and conducted in our group. Along this, we will provide a brief introduction to both methodologies, CMDS on the one hand, gas-phase sample preparation and detection on the other hand. We examine the technical challenges and solutions of gas-phase 2DES, and conclude with a discussion of future perspectives.
Principle of 2DES
=================
Principle and advantages of 2DES
--------------------------------
![Principle of 2D spectroscopy. (a) Simplified four-level system. (b) Pulse sequence used in 2D spectroscopy. The sample is excited with 3 or 4 pulses (indicated by dashed pulse envelope). The blue trace indicates the signal, which represents an oscillating dipole during the coherence times $\tau, t$. In between pulse 2 and 3 (time interval $T$), the system’s time evolution (dashed trace) is probed. All three time variables $\tau, t, T$ are systematically scanned in the experiment. (c) 2D frequency-correlation map obtained by a 2D Fourier transform of the data set with respect to the coherence times $\tau, t$ directly correlating excitation ($\omega_\tau$) and detection ($\omega_t$) frequencies. Peaks A, B on the diagonal represent the $\ket{g} \leftrightarrow \ket{a}, \ket{b}$ resonances, with inhomogeneous and homogeneous lineshapes along the diagonal and antidiagonal, respectively. Peak C denotes an excited state absorption from $\ket{a}$ to the higher lying state $\ket{c}$ (typically appearing with negative amplitude). AB and BA denote cross peaks, indicating couplings between states $\ket{a}$ and $\ket{b}$. The time evolution of all features is tracked as a function of $T$.(d) Linear absorption spectrum of the same system. Most spectral features overlap and are difficult to infer from the data. Likewise, a characterization of the system’s inhomogeneity becomes difficult. []{data-label="fig:2D_scheme"}](2D_scheme.pdf){width="0.6\linewidth"}
The principle of CMDS is described in detail in several review articles and books[@jonas_two-dimensional_2003; @cho_coherent_2008; @ogilvie_chapter_2009; @schlau-cohen_two-dimensional_2011; @cundiff_optical_2012; @branczyk_crossing_2014; @nuernberger_multidimensional_2015; @fuller_experimental_2015; @moody_advances_2017; @oliver_recent_2018; @maiuri_electronic_2018; @hamm_concepts_2011; @fayer_watching_2011]. Here, we provide only a brief introduction to the basic concept of 2DES and highlight its most important features.
In 2D spectroscopy, the sample (here approximated by a four-level model system, Fig.\[fig:2D\_scheme\]a) is excited with a sequence of three to four optical pulses (Fig.\[fig:2D\_scheme\]b) and the third-order nonlinear response of the system is probed as a function of the pulse delays $\tau,\, T$ and $t$. The time intervals $\tau$ and $t$ (termed coherence times), track the evolution of induced electronic coherences. A Fourier transform with respect to these time variables yields the 2D frequency-correlation maps (Fig.\[fig:2D\_scheme\]c) as parametric function of the third time variable $T$ (termed evolution time). Consequently, pump and probe steps are both frequency-resolved with $\omega_\tau$ representing the pump/excitation frequency and $\omega_t$ the probe/detection frequency axis, respectively.
The detected signals are categorized in stimulated emission (SE), ground state bleach (GSB) and exited state absorption (ESA) each occurring as rephasing (RP) (photo echo) and non-rephasing (NRP) signals. GSB probes the time evolution on the system’s ground state manifold whereas SE and ESA probe the dynamics of the excited state. Thereby ESA involves the excitation to a higher-lying state (Fig.\[fig:2D\_scheme\]). In most cases, GSB and SE pathways appear as positive and ESA as negative signals in the 2D maps which simplifies their identification and separation.
Furthermore, peaks on the diagonal reflect the linear absorption/emission spectrum of the sample, however, with the additional information of 2D lineshapes readily dissecting homogeneous (along antidiagonal) from inhomogeneous (along diagonal) broadening[@lazonder_easy_2006]. This provides decisive information about static and dynamic inhomogeneities in the probed ensemble[@maiuri_electronic_2018]. Off-diagonal features directly disclose couplings among excited states of the system from which different types of interaction and relaxation dynamics can be inferred, e.g. coherent excitonic interactions or spontaneous decay pathways[@lewis_probing_2012]. Excited state absorption (ESA) to higher lying states may be also induced. These contributions typically appear with inverted (negative) amplitude which simplifies their identification and separation from other contributions. All this information is in most cases hard to retrieve from one-dimensional spectra (Fig.\[fig:2D\_scheme\]d), indicating the great advantage of 2D spectroscopy.
Furthermore, due to the Fourier transform-concept of CMDS, the time-frequency resolution automatically adapts to the system’s time scales and spectral line widths[@jonas_two-dimensional_2003]. As such, broad bandwidth, ultrashort femtosecond pulses can be used to yield simultaneously high temporal and frequency resolution down to the Fourier limit, while probing transitions and correlations in a large spectral range[@ma_broadband_2016; @kearns_broadband_2017].
Technical challenges
--------------------
2DES faces two major technical challenges. First, tracking the femto- to sub-femtosecond beats of electronic coherences (during intervals $\tau,\, t$) requires interferometric measurements with high phase/timing stability among the optical pulses (typically $\leq \lambda / 50$[@hamm_concepts_2011]). This demand is slightly relaxed in 2DIR spectroscopy, since vibrational coherences evolve on roughly an order of magnitude lower frequencies. Second, the third-order 2D signals, subject to three to four light-matter interactions, are often weak and are covered by dominating background contributions, e.g. the linear system response or scattered light. This calls for highly sensitive detection with large dynamic range.
In the past 25 years, both issues have been experimentally solved. A number of active and passive phase stabilization concepts have been developed to meet the demands of interferometric stability[@tian_femtosecond_2003; @brixner_phase-stabilized_2004; @grumstrup_facile_2007; @vaughan_coherently_2007; @tekavec_fluorescence-detected_2007; @selig_inherently_2008; @bristow_versatile_2009; @turner_invited_2011; @rehault_two-dimensional_2014; @draeger_rapid-scan_2017]. These are combined with phase matching[@hybl_two-dimensional_1998] or phase cycling[@tian_femtosecond_2003; @tan_theory_2008] schemes or combinations of both[@fuller_pulse_2014; @dostal_direct_2018] to select the desired nonlinear signal contributions and provide highly sensitive background-free detection. An overview of the different experimental techniques has been recently published[@fuller_experimental_2015].
![Phase matching and phase cycling in 2D spectroscopy. (a) Phase matching of three incident pulses with different $\vec{k}_i$-vectors induce a nonlinear polarization in the sample which radiates off in $\vec{k}_s$-direction where it is heterodyned with the local oscillator (LO) and isolated with a mask. (b) Phase cycling with four collinear pulses exciting a sample. The phase $\phi_i$ of each pulse is modified throughout the experiment. (c), (d) Example Feynman diagrams showing signal contributions selected with phase matching/cycling, respectively. []{data-label="fig:MatchingCycling"}](phase_matching_cycling_2.pdf){width="0.7\linewidth"}
Phase matching (Fig.\[fig:MatchingCycling\]a) relies on coherent four-wave-mixing (FWM), where the sample is excited with three laser pulses in the so-called boxcar geometry[@jonas_two-dimensional_2003]. The third light-matter interaction stimulates the coherent emission of the signal wave in phase matching direction, where it is background-free detected and frequency resolved with an optical spectrometer. Thereby, amplitude and phase of the signal are determined by heterodyned detection with a fourth optical field (termed local oscillator)[@hamm_concepts_2011].
In phase cycling (Fig.\[fig:MatchingCycling\]b), collinear pulse trains are used to induce four light-matter interactions, leaving the sample in a population state after the fourth pulse. The final population is detected with incoherent observables yielding the nonlinear response of the system. At the same time, specific phase patterns are imprinted on the pulse trains by manipulating the carrier envelope phase (CEP) of each pulse, which results in a distinct phase signature of the detected signal[@tan_theory_2008]. By applying a unique set of phase combinations (typically 16 or 27), the desired nonlinear signal is identified and isolated in the post processing while other contributions destructively cancel. Here, the signal’s amplitude and phase are deduced by adequate combination of extracted signal contributions.
The development of these concepts has solved some important technical issues of 2D spectroscopy experiments, having in recent years paved the way for widespread implementation in the condensed phase. Yet, other experimental issues exist that are less discussed in literature. These include timing uncertainties due to chirped optical pulses[@tekavec_effects_2010], pulse overlap effects due to finite pulse durations[@perlik_finite_2017], incomplete spectral overlap with the sample[@de_a._camargo_resolving_2017], laser intensities beyond the weak perturbation regime[@chen_nonperturbative_2017], pulse propagation effects in the studied medium itself[@li_pulse_2013; @spencer_pulse_2015], photo bleaching of samples and scattering light contributions[@augulis_two-dimensional_2011]. These points make 2DES still a sophisticated experimental task requiring specialized expertise in ultrafast nonlinear optics and related fields.
Experimental implementation of gas-phase 2DES
=============================================
Action-based 2D spectroscopy
----------------------------
The idea of gas-phase 2DES is to study isolated model systems, which implies very low ensemble concentrations (typically particle densities $\leq 10^{11}$cm$^{-3}$[@hertel_ultrafast_2006]). This requires orders of magnitude higher detection sensitivity than in condensed phase experiments (Fig.\[fig:overview\]) and thus poses a severe technical challenge. The phase matching approach is ruled out by this criteria, as it relies on the coherent emission from a macroscopic polarization induced in the sample. Therefore, the method cannot be scaled down to low target densities and, to the best of our knowledge, phase-matching 2DES experiments have not been demonstrated for particle densities $\leq 10^{12}$cm$^{-3}$[@dai_two-dimensional_2012].
In contrast, the phase cycling concept relies on the detection of a specific phase signature encoded in a nonlinear population state excited in individual particles. The respective nonlinear signal is deduced by mapping the population state with action-based detection of incoherent observables, e.g. spontaneous emission, depletion or photoionization. This approach does not rely on a macroscopic ensemble effect and, in principle, may be scaled down to the single-molecule level[@brinks_visualizing_2010]. As such, phase cycling has facilitated the development of action-based 2D spectroscopy, which opened a plethora of possibilities to incorporate new detection types. The combination with fluorescence[@tekavec_fluorescence-detected_2007; @de_two-dimensional_2014; @draeger_rapid-scan_2017], photocurrent[@nardin_multidimensional_2013; @karki_coherent_2014; @vella_ultrafast_2016], ion-mass[@roeding_coherent_2018; @bruder_coherent_2018] as well as with optical microscopy[@goetz_coherent_2018; @tiwari_spatially-resolved_2018] and even with high resolution photoemission electron microscopy[@aeschlimann_coherent_2011; @aeschlimann_perfect_2015] has been demonstrated.
Pulse shaping versus continuous phase modulation
------------------------------------------------
Experimentally, phase cycling is implemented by pulse shaping based on spatial light modulators (SLMs)[@weiner_femtosecond_2000; @vaughan_coherently_2007; @turner_invited_2011] or acousto-optical modulators (AOMs)[@tian_femtosecond_2003; @draeger_rapid-scan_2017; @seiler_coherent_2017]. Alternatively, a phase modulation (PM) technique based on continuous phase modulation with acousto-optical frequency shifters (AOFSs) is used[@tekavec_fluorescence-detected_2007; @nardin_multidimensional_2013; @karki_coherent_2014; @bruder_phase-modulated_2015; @vella_ultrafast_2016; @bruder_coherent_2018; @Yu_2019]. The latter may be regarded as shot-to-shot quasi-continuous phase cycling[@nardin_multidimensional_2013].
Both approaches have their strengths and weaknesses depending on the application. Pulse shaping can drastically simplify the optical setups for 2DES experiments[@draeger_rapid-scan_2017; @seiler_coherent_2017] and provide highest experimental flexibility. Amplitude, phase and polarization shaping permit the generation of arbitrary pulse sequences to perform a vast array of nonlinear spectroscopy experiments with a single apparatus[@brixner_femtosecond_2001]. Another advantage of pulse shapers is their ability for inherent pulse compression to yield transform-limited pulses in the 10-fs-regime[@goetz_coherent_2018].
On the contrary, the PM approach requires larger assemblies of optics and is more restricted in the manipulation of pulse properties. Yet, flexible signal selection protocols have been also implemented with the PM technique[@bruder_efficient_2015; @Yu_2019] and pulse durations $<20$fs have been reported[@vella_ultrafast_2016]. The continuously operated AOFSs in the PM technique have the advantage of imprinting particularly clean, high purity phase manipulation with very low distortion (reported artifacts $\leq 50$dB[@lai_nonlinear_2014; @bruder_phase-modulated_2017]), whereas pulse shapers require careful calibration and may produce artifacts due to space-time couplings[@frei_space-time_2009], thermal phase instabilities[@bruhl_minimization_2017] or if operated at high update rates.
In view of gas-phase experiments, the signal-to-noise performance and detection efficiency are particularly important factors. In 2DES, it is recommended to use moderate laser intensities to avoid the contribution of higher-order (larger than third order) signals to the data. Therefore, large statistics is best reached with low laser intensities and high laser repetition rates. Here, the PM technique has the clear advantage of providing shot-to-shot phase manipulation up to laser repetition rates in the MHz-regime[@tekavec_fluorescence-detected_2007] which is combined with highly sensitive lock-in detection[@tekavec_fluorescence-detected_2007; @bruder_efficient_2015; @bruder_delocalized_2019]. Most pulse shapers are restricted to update rates of $\leq 1$kHz[@draeger_rapid-scan_2017]. However, with the recent development of pulse shapers extending update rates to 100kHz [@kearns_broadband_2017], the gap to the PM technique may be closed.
Furthermore, gas-phase experiments may require much larger scanning ranges of pulse delays than typical in the condensed phase, where perturbations by the environment induce rapid dephasing of electronic coherences often within $\leq 100$fs. In the gas phase, broadening effects are considerably smaller and electronic coherences can be detected over hundreds of picoseconds[@bruder_phase-modulated_2015; @lomsadze_frequency_2017], enabling high resolution experiments. To exploit this feature in 2DES, coherence times have to be scanned over large time intervals, accordingly, which is possible in the PM approach[@bruder_phase-modulated_2015] but not with most pulse shapers where pulse delays are constrained to $\leq 1$ps[@draeger_rapid-scan_2017].
In our group, we favored the PM concept for the realization of gas-phase 2DES and we will in the following describe its experimental scheme in more detail. We note, that Brixner et al. realized gas-phase 2DES based on pulse shaping technology and we refer to their work for more information[@roeding_coherent_2018].
Phase modulation 2DES combined with photoionization
---------------------------------------------------
Phase modulation 2DES combined with fluorescence detection is described in detail in the original publication from the Marcus group[@tekavec_fluorescence-detected_2007]. Here, we provide only a brief description of the technique with the focus on the photoionization gas-phase experiments performed in our laboratory.
{width="0.7\linewidth"}
The experimental scheme and a sketch of the setup is shown in Fig.\[fig:PMsetup\]. A collinear pulse train of four phase-modulated laser pulses prepares a nonlinear population state in the sample, which is probed upon photoionization. The ionization is either performed with a separate fifth pulse or by absorbing additional photons from pulse 4. Pulse 1-4 are generated in a nested three-fold optical interferometer fed by the output of a noncollinear optical parametric amplifier (NOPA) (640-900nm tuning range). Pulse 5 is produced from a second NOPA to enable independent wavelength tuning (540-900 nm) or from fourth harmonic generation (FHG) of the amplified oscillator pulses to yield deep ultraviolet (UV) pulses (260nm). The relative pulse delays are controlled by motorized translation stages.
The multipulse excitation sequence generally induces a large number of signals. The desired third-order rephasing (RP) and non-rephasing (NRP) signal contributions are selected from the total signal by phase modulation of the excitation pulses combined with lock-in detection. To this end, pulse1-4 are passed through individual AOFSs (AOFS 1-4, Fig.\[fig:PMsetup\]b) which are phase-locked driven at distinct radio frequencies $\Omega_i$. AOFS 1-4 shift the frequency of transmitted pulses by the value $\Omega_1 = 109.995$MHz, $\Omega_2 = 110.000$MHz, $\Omega_3 = 110.001$MHz and $\Omega_4 = 110.009$MHz, respectively. This is equivalent to a shot-to-shot modulation of the CEP $\phi_i$ of each pulse[@nardin_multidimensional_2013] in increments of $\Delta \phi_i = \Omega_i / \nu_\mathrm{rep}$ between consecutive laser shots ($\nu_\mathrm{rep}=200\,$kHz denotes the laser repetition rate).
The nonlinear mixing of the modulated electric fields in the sample leads to characteristic beat notes in the photoionization yield. According to the phase cycling conditions for RP and NRP signals ($S_\mathrm{RP}$ and $S_\mathrm{NRP}$), the modulation frequencies are: $$\begin{aligned}
S_\mathrm{RP} : \phi_\mathrm{RP} (t) &=& -\phi_1 + \phi_2 + \phi_3 - \phi_4 = 3\,\mathrm{kHz}\\
S_\mathrm{NRP} : \phi_\mathrm{NRP} (t) &=& -\phi_1 + \phi_2 - \phi_3 + \phi_4 = 13\,\mathrm{kHz} \, .\end{aligned}$$
The signals are extracted from the photoelectron/-ion count rates with lock-in detection. For the lock-in amplification, an external reference signal is used constructed from the optical interference of pulse 1-4. For this purpose, pulse pairs 1,2 and 3,4 are split off at BS4 and 5, respectively and are subsequently stretched in time with a monochromator (Fig.\[fig:PMsetup\]b). The pulse stretching ensures a non-vanishing interference signal over sufficiently long scanning ranges of $\tau$ and $t$. The acquired beat signals of both pulse pairs ($\Omega_{21} = 5$kHz and $\Omega_{43} = 8$kHz) are electronically mixed to yield sum- and difference frequency-sidebands at 3 and 13kHz, respectively.
In the post processing, the sum of demodulated RP and NRP signals is Fourier transformed with respect to the time delays $\tau$ and $t$ to yield the complex-valued 2D frequency-correlation spectrum $\tilde{S}(\omega_\tau, T, \omega_t)$ as a parametric function of $T$. Its real part represents the 2D absorption spectrum which is analyzed in the experiments.
The here employed lock-in detection scheme has several advantages. RP and NRP signals are retrieved simultaneously in a single 2D scan of positive coherence times $\tau$ and $t$. Amplitude and phase of the signal are recovered through phase-synchronous lock-in detection. Heterodyning with the external reference leads to rotating frame sampling which reduces the required delay sampling points by several orders of magnitude. Phase/timing jitter introduced in the optical interferometers appears as correlated noise in the signal and reference and thus cancels out in the lock-in demodulation, resulting in a highly efficient passive phase stabilization of the setup. As such, a phase stabilization better than $\lambda /200$ has been achieved in a deep-UV interferometer ($\lambda = 266$nm)[@wituschek_stable_2019]. Eventually, the lock-in amplification considerably improves the general sensitivity of the setup. The signal-to-noise (SN) advantage of the PM technique is clearly demonstrated in an electronic quantum interference measurement combined with photoionization which served as a precurser experiment to our gas-phase 2DES experiments (Fig.\[fig:PMvsClassic\])[@bruder_phase-modulated_2015].
![Performance advantage of the PM technique demonstrated in a quantum beat experiment. (a), (b): Time domain signal of electronic coherences excited in gaseous Rb atoms, with (a) and without (b) using the PM technique. In (a), rotating frame sampling leads to a downshift of the quantum beat frequencies. (c), (d): Respective Fourier transform spectra showing a clear SN advantage for the PM case. Adapted from Ref.[@bruder_phase-modulated_2015] - Published by the PCCP Owner Societies. []{data-label="fig:PMvsClassic"}](PMvsClassic_v3.pdf){width="0.8\linewidth"}
Pathways in photoionization-2DES
--------------------------------
{width="0.9\linewidth"}
There is a distinct difference between 2D spectroscopy experiments using phase matching and phase cycling. In case of phase matching, for each signal type (SE, GSB, ESA) exists one RP and one NRP pathway (and their complex conjugate). With phase-cycling, for each contribution exists an additional pathway whose signal is phase shifted by $\pi$. Example RP pathways, as detected in our photoionization experiments, are shown in Fig.\[fig:pathways\]. Here, the relative amplitude with which the pathways contribute to the signal strongly depend on the ionization probability. While SE1- ESA1 pathways are probed by two-photon ionization, the ESA2 process requires only one photon to the continuum and therefore dominates the ESA signal in the photoionization 2D spectra. On the contrary, SE2 and GSB2 require three photons for ionization and are usually negligible. As such, the net SE and GSB signals contribute with positive amplitude, whereas the net ESA signal strictly appears with negative amplitude in the 2D absorption spectra. This is in analogy to phase matching based 2D spectroscopy where the ESA amplitudes are also of opposite sign to SE/GSB signals.
The negative sign of ESA features in contrast to the other signal contributions, simplifies their identification, which is of advantage, in particular in congested spectra. Note that with other detection types the situation can differ. In fluorescence detection, the sign of the ESA peaks depends on the degree of quenching of fluorescence from the $\ket{f}$ state[@perdomo-ortiz_conformation_2012].
Phasing of 2D spectra
---------------------
Related to the phase shift among signal contributions is the general *phasing* issue in 2D spectra[@gallagher_faeder_two-dimensional_1999; @turner_comparison_2011]. The correct phase information can only be retrieved if the total phase of the complex-valued 2D response function $S(\tau,T,t)$ is correctly determined. Otherwise the absorptive and dispersive line shapes are not correctly separated in the 2D absorption spectrum leading to distorted or even inverted peak shapes which might be interpreted incorrectly.
While phasing of the 2D signals is intricate in FWM-based 2D spectroscopy, it is much simpler in collinear 2D spectroscopy experiments. Pulse shaper setups are intrinsically phased through the calibration of the device. In the phase modulation approach, phasing is done by calibrating the phase offset between the signal and reference in the lock-in detection. To this end, at coherence times set to zero ($\tau = t = 0\,$fs), the phase of the demodulated RP/NRP signal is adjusted to zero through adjusting a global phase factor applied in the lock-in electronics or in the post processing[@tekavec_fluorescence-detected_2007]. With this procedure, phase shifts between the signal and the reference accumulated in the different electronic circuits of the setup are compensated.
This procedure is required for the initial calibration of any PM-2DES setup, or whenever electronics are changed. Any reference sample may be used for the calibration. In our experiments, we phased the setup with photoionization signals of gaseous Rb atoms which provide a simple, well-defined 2D spectrum with isolated sharp peaks that allows for direct examination of any phase offset (Fig.\[fig:2Deffusiv\]).
![Reference measurement to phase the setup. (a) Photoelectron-detected absorptive 2D-spectrum of gaseous Rb. Used to phase the setup according to the expected peak shape and sign: positive, absorptive GSB/SE features and negative, absorptive ESA features. (b) Different color-coding to amplify weak contributions. Homogeneous/inhomogeneous broadenings are beyond the spectral resolution of the measurement, explaining the absence of peak elongations along the diagonal. (c) Relevant energy levels of atomic Rb for the reference measurement. Probed are the D line transitions (via GSB/SE) as well as the $5 {}^2P_{3/2} \rightarrow 5 {}^2D_{3/2, 5/2}$ transition (via ESA).[]{data-label="fig:2Deffusiv"}](2D_effusiv_with_unsat_plot.pdf){width="0.9\linewidth"}
Preparation of gas-phase samples
================================
Thermal vapors and molecular beams
----------------------------------
A great variety of experiments on atoms, molecules and molecular complexes are performed on gas phase samples, driven in particular by two main characteristics of such targets: (a) probing systems without the interaction between individual constituents or/and without the interaction with an environment; (b) establishing low temperature conditions and corresponding quantum state selectivity.
With respect to (a), already gas cells containing a vapor pressure of the sample may evolve only weak perturbations in spectroscopic measurements. Albeit, the coherent excitation of molecular vapors (molecular densities $\sim 10^{18}$cm$^{-3}$) may lead to cascading effects which compromise the nonlinear response of the sample[@grimberg_ultrafast_2002]. Likewise, propagation effects may occur in gas cells[@li_pulse_2013; @spencer_pulse_2015]. Experiments on particle beams prepared in high or ultra-high vacuum (UHV) environments circumvent these issues and in addition provide conditions (pressures below $\approx 10^{-5}$mbar) where the mean free path for extracting ions and electrons is suitable for an unperturbed detection. Furthermore, detection methods employing electron multipliers and corresponding high voltages cannot be operated at higher vacuum pressures.
With respect to (b), in the gas phase a variety of cooling and trapping methods are at hand to reach temperatures even down to nanokelvin temperatures (cf. Fig. \[fig:overview\])[@krems2009cold]. A central technique is based on the cooling by means of a supersonic expansion in molecular beams [@scoles_atomic_1988], reaching temperatures in the low Kelvin range. Ultracold temperatures (below mK) mostly involve laser cooling methods, as well as evaporative cooling in shallow traps [@krems2009cold]. Low temperatures are for many experiments instrumental for guaranteeing quantum state selectivity, preferable in all degrees of freedom, as well as providing well-defined structural properties. Finally, in comparison with ordinary gas targets, molecular beam as well as trapping methods are in many cases a prerequisite for providing an interaction volume having a distinct higher target density in comparison to the background gas inside the vacuum apparatus. Furthermore, Doppler broadening is minimized even in fast molecular beams when intersected perpendicularly by the laser beams.
It is intriguing that independent of the very different experimental techniques providing gas-phase targets, like e.g. size-selected molecular or cluster ion beams, decelerated molecular beams[@van_de_meerakker_taming_2008], helium droplet isolation, or ultracold atoms in magnetooptical traps (cf. Fig. \[fig:overview\]), the target density typically is in the range of about $10^8$cm$^{-3}$. Of course, such densities are many orders of magnitudes below corresponding bulk target densities. However, the sensitivity and selectivity of signals detecting angular resolved and energy resolved single electrons or mass-selected ions even in sophisticated coincidence methods, in combination with generally fast regenerating targets offer unique options of experimental techniques not being available on bulk liquid or solid systems.
![Gas-phase sample preparation. (a) Skimmed seeded supersonic beam generation. (b) Cluster isolation technique.[]{data-label="fig:Beams1"}](Mol_cluster_beam_1.pdf){width="0.7\linewidth"}
{width="0.9\linewidth"}
The most commmon molecular beam technique is the generation of a skimmed seeded supersonic beam, (Fig. \[fig:Beams1\]a). In an adiabatic expansion of high-pressure rare gases (He, Ar, Kr, Xe) into vacuum an internally cold beam traveling at supersonic speed is formed [@scoles_atomic_1988; @Demtroeder_1988], seeded with target molecules at much lower pressure, e.g. from a heated reservoir. In this way the molecules adapt in many collisions during the expansion process to the narrow speed distribution and the low temperature of the seed gas. In this way, both the directionality and density in the target volume is much higher, and the internal temperature is much lower in comparison with e.g. an effusive gas beam (molecules exiting a reservoir though a pin hole without collisions).
In our first studies on 2DES in molecular beams we used the helium nanodroplet isolation (HENDI) technique, detailed out in the next section, because of its prospects and options for generating specific larger molecular structures at millikelvin temperatures.
Cluster isolation technique
---------------------------
Rare gas (Rg) clusters of variable sizes (Rg$_N, 1<N<10^{12}$) can be readily condensed in supersonic expansions at appropriate conditions. Depending on the rare gas, typically high stagnation pressures, $5-100$bar, and low temperatures, down to 4K in case of He have to be applied[@Hagena1972; @Hagena1987; @Haberland1994]. Because of the low binding energies of rare gas atoms to the clusters and the high surface-to-volume ratio, the clusters very efficiently evaporatively cool to specific low temperatures. In helium, the terminal temperature is 380mK[@Hartmann:1995] which is well below the transition temperature to superfluidity. The liquid state and the superfluidity provides peculiar properties, in particular frictionless flow and efficient cooling which has been confirmed in many helium cluster studies, [@Grebenev:1998; @Choi:2006] and explain why such clusters are appropriately called droplets. All rare gas clusters can be loaded with atoms and molecules by the pickup technique [@Lewerenz:1995; @Toennies:1998], where during inelastic collisions, e.g. in a cell containing a low vapor pressure of the dopant atoms or molecules, these are attached to the clusters. In comparison with seeded beams the needed partial pressure for doping a large cluster with unit probability is on the order of $10^{-5} - 10^{-4}$mbar, significantly extending the range of molecules suitable for establishing such low densities without fragmentation. One can dope large clusters even with thousands of atoms or molecules [@Tiggesbaumker:2007]. A variety of doping techniques has been developed, including laser ablation[@claas_characterization_2003; @mudrich_kilohertz_2007; @katzy_doping_2016] and dopants from electrospray (ESI) sources[@bierau_catching_2010; @filsinger_photoexcitation_2012; @florez_ir_2015]. In this way, also charged particles have been doped. In combination with ion traps, cluster-isolated spectroscopy of large bio-molecules up to 12000Dalton has been performed[@bierau_catching_2010].
In helium, generally, dopants aggregate inside the liquid droplet and in this way one can specifically synthesizes even larger atomic or molecular structures (Fig. \[fig:Beams2\]b) and/or model solvation effects by adding specific solvent molecules. On the other hand, the larger clusters of heavier rare gas atoms (Ne, Ar, Kr, Xe) all form solid clusters. For such solid clusters, it has been shown that larger molecules upon doping do not submerge and are immobile[@dvorak_spectroscopy_2012; @dvorak_spectroscopy_2012-1]. In this way, multiple doping leads to a variable surface coverage of the doped molecules (cf. Fig. \[fig:Beams2\]b)[@muller_cooperative_2015; @izadnia_singlet_2017].
![Comparison of linear absorption spectra of PTCDA (3, 4, 9, 10-perylenetetracarboxylicdianhydride) in different environments. Purple: gas-phase absorption in a heated vapor cell,[@Stienkemeier:2001]. Red: measurement in a room temperature solvent (dimethyl slufoxide)[@Bulovic:1996_2]. Black: helium droplet isolated monomer spectrum[@Wewer:2003].[]{data-label="fig:PTCDA"}](PTCDA.pdf){width="0.6\linewidth"}
All kinetic energy from the doping process as well as internal energy of the formed aggregates are dissipated via the evaporated cooling of the cluster. In this way, low temperature targets are formed, e.g., for helium droplets at millikelvin temperatures. Since the rare gas clusters are transparent at all wavelength down to the VUV, the dopants are selectively probed in laser experiments operating at IR, VIS or UV wavelengths[@Toennies:1998; @Stienkemeier:2006].
Fig. \[fig:PTCDA\] demonstrates the advantage of helium droplet isolation in the comparison of linear absorption spectra of PTCDA molecules at different conditions. The spectrum in a room temperature solvent shows the typical broad absorption bands of the S$_1 \leftarrow$ S$_0$ first singlet-to-singlet transition (red curve in Fig. \[fig:PTCDA\]). Even the gas-phase absorption in a heated vapor cell does not lead to better-resolved details (purple curve in Fig. \[fig:PTCDA\]) because of the large number of thermally populated states. The helium droplet isolated spectrum, however, clearly resolves in detail the vibrational structure of the molecule.
With the latter technique, the broadening of lines in vibronic spectra typically is about 1 cm$^{-1}$ [@Stienkemeier:2001]. The main source of broadening often is the Pauli repulsion of the electron density with the surrounding helium. For atoms having low ionization potentials and corresponding extended electron density distributions, large blue-shifts and repulsive interactions may appear upon excitation of electronic states. The repulsive nature of helium with respect to electrons can even lead to the formation of so called “bubbles” [@Dalfovo:1994], i.e. a helium void around e.g. atomic dopants. For the same reasons alkai atoms, dimers and trimers do not submerged in helium but are located at dimple-like structures on the surface of helium droplets (cf. Fig. \[fig:Beams2\]b) having binding energies only on the order of 10cm$^{-1}$ [@Ancilotto1995; @Stienkemeier:1996].
The just introduced peculiar binding properties of alkali-doped helium droplets preferably leads to the formation of high-spin states upon the formation of alkali molecules or clusters (Fig. \[fig:Mol\_formation\])[@stienkemeier_use_1995; @Higgins:1998; @Buenermann:2007]. Dissipation of binding energy upon the formation of molecules leads to high desorption rates of strongly bound entities during the doping process. In this way, in particular weakly bound molecules can be studied, which might be very difficult to form by other techniques. Alkali molecules in weakly-bound high-spin states have been probed in the first 2DES studies on helium droplets.
During the last 20 years, helium droplet isolation has been applied to a large variety of spectroscopic techniques. The results have been reviewed in various publications and we refer to these for further information[@Callegari:2001; @Stienkemeier:2001; @toennies_superfluid_2004; @Stienkemeier:2006; @Choi:2006; @Barranco:2006; @Tiggesbaumker:2007; @CallegariErnst:2011].
![He droplet assisted formation mechanism of the investigated Rb molecules. By picking up several atoms, molecules are formed on the cluster surface. Evaporation of He atoms efficiently dissipates the released binding energy and cools the formed molecule to its vibrational ground state. Due to this mechanism, the formation of the Rb molecules in their lowest weakly-bound high-spin state is preferred. The higher binding energy of the low-spin electronic ground state molecules leads to desorption or droplet destruction, due to which these molecular configuaritons are normally not detected in the experiments. Further downstream, the prepared doped droplets are probed via 2DES. Graphic taken from Ref.[@bruder_coherent_2018], licensed under the [Creative Commons Attribution 4.0 International License.](https://creativecommons.org/licenses/by/4.0/) []{data-label="fig:Mol_formation"}](Rb3_formation.pdf){width="0.9\linewidth"}
Helium nanodroplet beam apparatus
---------------------------------
A typical helium nanodroplet apparatus is depicted in Fig. \[fig:Beams2\]a. Helium droplets (He$_N$) with an average size of $N\approx 10000$ helium atoms per droplet form in a supersonic expansion at $P_0 = 50$bar stagnation pressure and about $T_0 = 15$K nozzle temperature. The molecular beam machine consists of a differentially pumped linear chain of HV/UHV vacuum chambers guiding the initially formed helium droplet beam via the doping unit to different detection chambers. Laser pulses can be introduced alternativly into a fluorescence detector, a magnetic bottle-type electron time-of-flight (TOF) spectrometer or a ion-TOF mass spectrometer, respectively. The mildly focussed laser and the droplet beam intersect perpendicularly. Since the droplet beam is travelling at about 400m/s and the repetition rate of the laser is 200kHz, each set of 2DES laser pulses acts on a fresh section of the target beam. Typical signal rates are one ion/electron per laser shot at target densities of about $10^{8}$ droplets per cm$^{-3}$. The magnetic bottle spectrometer for photoelectron spectroscopy has a resolution $\Delta E/E\approx 2$% and includes retarder electrodes for shifting electron flight times. Further details of the machine used for 2DES can be found in other publications[@bruder_phase-modulated_2015].
Gas-phase 2DES of isolated, cold molecules
==========================================
![PECs of Rb$_2$ triplet (a) and Rb$_3$ quartet (b) manifolds. Arrows indicate the probed transitions. In (a), the perturbation of the $0_\mathrm{g}^+$ state by the helium environment is schematically shown as dashed curve. The Rb$_2$ PEC graphic is adapted from Ref.[@bruder_coherent_2018], licensed under the [Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/). The Rb$_3$ PEC graphic is adapted by permission from Springer Nature: Ref.[@hauser_advances_2009], License Number: 4606941432839.[]{data-label="fig:PECs"}](Rb2_Rb3_PECs.pdf){width="0.95\linewidth"}
2DES of weakly-bound rubidium molecules
---------------------------------------
{width="0.9\linewidth"}
Recently, we have combined PM-2DES with HENDI and studied Rb$_2$ and Rb$_3$ molecules prepared in their weakly-bound high-spin states. These experiments constitute the first 2DES study of isolated, cold molecules prepared at sub-Kelvin temperatures.
Fig.\[fig:PECs\] shows the potential energy curves (PECs) of the molecules. Both molecules have been previously studied with HENDI using high resolution steady-state laser spectroscopy[@allard_investigation_2006; @nagl_heteronuclear_2008; @nagl_high-spin_2008] and femtosecond quantum beat spectroscopy[@mudrich_spectroscopy_2009; @gruner_vibrational_2011; @giese_homo-_2011]. The steady-state laser absorption and emission spectra are shown in Fig.\[fig:Rb2Rb32DES\]a, b. The Rb$_2$ molecule shows a pronounced absorption at the $ 1^3\Pi_\mathrm{g}\leftarrow a^3\Sigma_\mathrm{u}^+ $ excitation with resolved spin-orbit (SO) couplings of the excited state. Note, that the $1_\mathrm{g}$ absorption does not appear in HENDI experiments. In the Rb$_3$ molecule, the $1^4 A_{1,2}^{''} \leftarrow 1^4 A_2^{'}$ absorption peak is observed. Emission spectra have been only reported for the Rb$_2$ molecule (Fig.\[fig:Rb2Rb32DES\]b). 2D spectra taken of the same molecules attached to helium droplets are shown in Fig.\[fig:Rb2Rb32DES\]c for photoion detection and in (d) for photoelectron detection. Both are taken under different ionization conditions with the purpose to selectively amplifying certain features (see discussion below).
The 2D frequency-correlation maps exhibit high quality, in particular if considering the challenging experimental conditions. They show sharp, well-separated spectral features, which is not common in condensed phase studies, indicating the resolution advantage of the gas-phase approach. Remarkably, these spectra were taken for very small number densities of doped droplets being only $n \approx 10^7$cm$^{-3}$ which corresponds to roughly 300 absorbers inside the laser interaction volume. For these conditions, the integral optical density (OD) of the sample estimates to $\mathrm{OD}=-\log_{10}(I/I_0)\sim 10^{-11}$[@bruder_coherent_2018]. This is several orders of magnitude lower than in previous 2D spectroscopy studies, where the OD typically ranges between 0.1 and 1. Our experiments thus indicate a drastic improvement in sensitivity and open up new possibilities for an expansion to other fields, e.g. ultra cold atom clouds[@bloch_many-body_2008] and ion crystals[@gessner_nonlinear_2014] or towards single-molecule studies[@brinks_visualizing_2010].
In comparison with the previous 1D spectroscopy measurements of the molecules, the advantages and additional information gained by 2D spectroscopy become apparent. While the 2D spectra show the same absorption lines as in the 1D steady-state spectroscopy, correlated to the absorption bands, additional ESA pathways (negative peaks) and cross-peaks (red shifted positive peaks) are revealed. The ESA features expose the different ionization pathways as a function of the molecular excitation and show the position of respective Frank-Condon (FC) windows to higher-lying states[@bruder_coherent_2018]. This information was not available in previous photoionization studies of these molecules[@mudrich_spectroscopy_2009], but, in principle, may be gained by narrow-band two-color pump-probe ionization experiments. The advantage of the 2DES approach is, that high spectral resolution is gained even when using broadband femtosecond pulses (see pulse spectra in Fig.\[fig:Rb2Rb32DES\]a,b), that cover all transitions simultaneously and also permit femtosecond temporal resolution.
Furthermore, in Fig.\[fig:Rb2Rb32DES\]a, we show a direct comparison of line shapes obtained from steady-state spectroscopy and 2DES. To this end, the 2D spectra were integrated along certain horizontal/vertical spectral intervals[@bruder_coherent_2018]. We find a remarkably good match of absorption and emission profiles and equal spectral resolution for both methods, confirming that the Fourier transform concept of 2D spectroscopy indeed achieves optimum spectral resolution. Note that the relative amplitude missmatch in the absorptive profiles of Rb$_2$ corresponds to different ionization probabilities of the respective states.
For the Rb$_2$ emission spectrum, the situation is slightly different. The steady-state spectroscopy captures mainly the emitted fluorescence of free gas-phase molecules which tend to desorb from the droplet surface after their excitation[@vangerow_dynamics_2015; @sieg_desorption_2016]. This explains the absence of the ESA resonance (negative peak at 13250cm$^{-1}$) and the well resolved vibronic features around 13300cm$^{-1}$.
In contrast, the 2DES measurements provide spectral information with femtosecond time resolution (as a function of the evolution time $T$) and reproduce the spectrally broadened response of the Rb$_2$ molecules while being still attached to the droplet surface. As such, an almost identical pump and probe profile of the $a^3\Sigma_\mathrm{u}^+ \rightarrow 1^3\Pi_\mathrm{g}$ transition is observed. The absence of a Stokes shift is due to a very narrow FC window between the shallow ground state potential and the $1^3\Pi_\mathrm{g}$ state[@bruder_coherent_2018].
The femtosecond time resolution of the 2DES study, furthermore, has revealed the coherent WP dynamics of the Rb$_2$ molecule (not shown), which permitted a refined interpretation of the Stokes peak appearing at 12900cm$^{-1}$. While this feature was previously interpreted as the emission from vibrationally relaxed free gas-phase Rb$_2$ molecules[@allard_investigation_2006], the 2DES experiments point to an ultrafast intramolecular relaxation into the outer potential well of the $1^3\Pi_g$ state, catalyzed by the helium perturbation. This population transfer shows a remarkable efficiency, taking place within $< 100$fs[@bruder_coherent_2018].
System-bath couplings in a superfluid environment
-------------------------------------------------
{width="0.9\linewidth"}
Using the helium nanodroplets as a matrix to isolate and cool down molecules, has the advantage of forming enclosed nanometer-sized model systems in an ultra high vacuum environment. This allows us to expand our studies beyond pure intramolecular dynamics towards intermolecular effects and the exploration of system-bath couplings induced by well-controlled environmental parameters.
For instance, different types of system-bath interactions can be modeled by co-doping of the droplets with rare gas atoms or solvent molecules (Fig.\[fig:Beams2\]b)[@muller_cold_2009; @dvorak_size_2014]. Alkali-metals show here an exceptional behavior and induce already a significant interaction with the pure helium droplets. This is explained by the strong Pauli-repulsion between the loosely-bound valence electrons of the alkali-metal atoms with the closed-shell $1s^2$ configuration of the helium atoms[@Buenermann:2007]. Alkali atoms and molecules thus serve as ideal probes to sense the interaction potentials and dynamical behavior of the superfluid droplets and allow us to explore the properties of the quantum fluid itself.
An example of static guest-host interaction has been already discussed above for the perturbation of the Rb$_2$ $(1)^3\Pi_g,\,0_\mathrm{g}^+$ potential, opening up an ultrafast molecular relaxation channel which is not observed in the gas phase. The Rb$_3$ excitation reveals, on the contrary, an example of the ultrafast dynamic droplet response when impulsively pumping energy into the system through an impurity (Fig.\[fig:Rb3Stokes\]).
Upon excitation of the Rb$_3$ molecule, its electron density distribution expands, causing a repulsion of the surrounding helium atoms on a few-picosecond time scale, while the heavy Rb$_3$ molecule effectively remains in position and slowly desorbs from the droplet surface on a much longer time scale (estimated to be $>10$ps).
The 2DES experiment allows us to directly follow the initial fast repulsion of the quantum liquid. Here, we observe a dynamic Stokes shift along the probe-frequency axis of the 2D spectra (Fig.\[fig:Rb3Stokes\]a), which reflects the system’s relaxation on the excited state of the Rb$_3$-He$_N$ interaction potential (Fig.\[fig:Rb3Stokes\]b). Our time-resolved study reveals a rearrangement of the helium density towards the Rb$_3^*$He$_N$ equilibrium state within 2.5ps. Note, that the Rb$_3$ peak on the diagonal position reflects the dynamics on the system’s ground state where the Rb$_3$-He$_N$ interaction is of static character.
The here discussed example of system-bath dynamics represents a unique case where a single, isolated molecule interacts with a homogeneous environment. It allows us to directly probe the system-bath interaction potential without inhomogeneous broadening. This is in contrast to condensed phase studies, where a statistical ensemble of molecules is probed in an inhomogeneous environment. There, local bath fluctuations typically lead to a diffusion of the lineshape over time[@asbury_water_2004; @moca_two-dimensional_2015] rather than resolving a dynamic Stokes shift. As such, our experimental approach provides an interesting alternative route to elucidate the influence of environmental parameters on molecular processes.
Photoionization as a selective and versatile probe
==================================================
One benefit of the photoionization is the vast array of highly developed electron/ion detectors enabling for instance energy- or mass-resolved spectroscopy. Depending on the detection method, one hereby is able to add extra dimensions to the 2D spectroscopy measurements which permits further disentanglement of excitation and reaction pathways.
Coherent two-dimensional electronic mass spectrometry
-----------------------------------------------------
![Ion-mass detected 2DES of gaseous NO$_2$ molecules. Absorptive 2D correlation spectra extracted from the NO$_2^+$ parent ion in a) and the ion fragment NO$^+$ in b). Adapted from Ref.[@roeding_coherent_2018], licensed under the [Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/).[]{data-label="fig:Brixner"}](Brixner.pdf){width="0.8\linewidth"}
A first demonstration of 2D spectroscopy combined with ion-mass detection has been recently reported by Brixner et al.[@roeding_coherent_2018]. Here, a warm effusive beam of gaseous NO$_2$ molecules was studied. Multiphoton excitation and ionization of the sample was induced by four collinear VIS pulses produced by an acousto-optical pulse shaper. Rapid shot-to-shot phase cycling was incorporated to improve signal to noise performance[@draeger_rapid-scan_2017] and to isolate the nonlinear response from NO$_2^+$ and NO$^+$ ion yields (Fig.\[fig:Brixner\]).
Both ion signals reveal clear differences in their line shape and sign of amplitudes, which points to different excitation pathways leading to the ionic products. In accordance with their hypothesis, a laser intensity analysis shows different high-order multiphoton processes for the detected cationic signals (8’th order for NO$_2^+$ and 10’th order for NO$^+$). The high-order nonlinearity reveals the challenging experimental conditions in this study, however, also leads to ambiguities due to many overlapping high-order pathways that cannot be discriminated. This might be resolved in the future by incorporating additional phase cycling steps or extended pulse sequences.
In general, the multiphoton ion-mass-detected 2D spectroscopy approach shows the potential to study ionization pathways and ultrafast autoionization processes in highly excited Rydberg manifolds of gaseous molecules, where 2D spectroscopy may provide additional information about transient intermediate states and improves the analysis of complex high-order multiphoton processes.
In the same fashion to this study, we have combined ion-mass detection with the phase modulation approach. In an early demonstration, we combined phase-modulated quantum interference measurements with a quadrupole ion mass filter for sensitive detection of RbHe excimer formations[@bruder_phase-modulated_2015]. For our 2DES experiments, we used instead an ion-TOF spectrometry arrangement (Fig.\[fig:IonTOF\]). The 2D spectra of specific masses are recorded by means of TOF-gating using boxcar integrators. To this end, boxcar windows are placed on the respective mass peaks in the ion-TOF transients and the boxcar output is fed into the lock-in detection and processed as discussed in section 3.
![Mass-resolved 2DES. (a) Ion time-of-flight (TOF) trace of photoionized rubidium molecules desorbed from helium nanodroplets (recorded with additional, delayed ionization laser). (b) 2D spectrum recorded without mass selection. (c-d) 2D spectrum recorded at masses Rb$^{+}$ and Rb$_2^{+}$, respectively. []{data-label="fig:IonTOF"}](Ion_TOF.pdf){width="0.8\linewidth"}
For the above discussed study of Rb molecules, the ion-TOF distribution (Fig.\[fig:IonTOF\]a) shows three significant peaks, which correspond to the Rb$^{+}$, Rb$_2^{+}$ and Rb$_3^{+}$ species. Examples of obtained 2D spectra (for population time $T=700$fs) are shown in (Fig.\[fig:IonTOF\]c,d) along with a 2D spectrum recorded without mass-gating as a reference (Fig.\[fig:IonTOF\]b). From these measurements, we can learn more details about the photo-induced dissociation dynamics and can identify specific dissociation channels.
For the used laser frequencies, Rb atoms may only be ionized via off-resonant three-photon excitation, which is a negligible process for the applied low laser intensities. Thus, the Rb$^+$ cations reflect the dissociation products of Rb$_2$ and Rb$_3$ molecules and carry the nonlinear response of both parent molecules, respectively, as reflected in the 2D spectrum of Fig.\[fig:IonTOF\]c. Here, we directly see, that the ESA pathway in the Rb$_3$ molecule leads to a dominant production of Rb$^+$ ions, whereas neither dimer (Fig.\[fig:IonTOF\]d) nor trimer ions (not shown) are detectable for this excitation/ionization channel. This stands in contrast to the photodynamics in the Rb$_2$ molecule, where excitation to the $(1)^3\Pi_g$ manifold and subsequent ionization leads only to a small production of Rb$^+$ fragments and most signal is detected in the Rb$_2^+$ 2D spectrum.
Interestingly, the Rb$_2$ ESA$_1$ excitation pathway is absent in all ion-detected measurements but can be clearly observed in photoelectron measurements (Fig.\[fig:Rb2Rb32DES\]d). This may be explained by a direct transition into the ion continuum or an ultrafast autoionization induced via the ESA pathway. Both processes would lead to immediate ionization by pulses 3, 4 before desorption of the excited molecule from the droplet surface has taken place. This is followed by the solvation of the cation into the helium droplet accompanied by the formation of a surrounding high density shell of He atoms (*snowball* formation), which can be detected at large masses ($> 500$amu)[@vangerow_dynamics_2015]. In contrast, the lower-lying ESA$_2$ transition can be observed in ion-detected spectra at certain population times (Fig.\[fig:Rb2Rb32DES\]c), which indicates the absence of a coupling to the ion continuum for slightly lower-lying states.
Eventually, the sum of the 2D spectra recorded for individual mass-species (Fig.\[fig:IonTOF\]c,d) matches with the spectrum obtained from integral ion yields (Fig.\[fig:IonTOF\]b), confirming that no signals are omitted in the mass-selective detection.
Both examples, from the Brixner group and our group, show the added information, one may gain by combining coherent 2D spectroscopy with mass spectrometry. As an advantage over conventional pump-probe mass spectrometry, 2DES provides spectral information for pump and probe steps and is able to track the coherent molecular dynamics. This information may help to decipher complex ultrafast photoreactions including involved dissociative dynamics.
State-selectivity and modulation contrast
-----------------------------------------
![Comparison of photoionization schemes and modulation contrast. a) Oscillation of the population probability in a two-level system as a function of the relative delay and phase of excitation pulses. b,c) Mapping of populations to the ion continuum by photoionization. A nonlinear population state is induced by the 4-pulse sequence used in action-detected 2D spectroscopy (red arrow), which is detected via multiphoton ionization (a) or one-photon ionization (b). The ionization of ground ($\ket{g}$) and excited states ($\ket{e}$) is shown by green/blue arrows. Dashed lines indicate resonant or virtual intermediate states in the multiphoton ionization. []{data-label="fig:Ionization"}](Modulation_contrast.pdf){width="0.8\linewidth"}
A crucial point in all action-detected 2DES is the modulation contrast. The measurements rely on the detection of small modulations of a nonlinear population, induced by the coherent interactions of four optical pulses. Systematic modulation of the pulses’ phase induces an alternation of the probability to reach the final population state. By applying well-defined phase patterns on the excitation pulses, the desired third-order nonlinear response of the system can be isolated from population signals modulated at different patterns and non-modulated background contributions.
The sensitivity of this detection concept critically depends on the detection contrast between the final excited population state and the complementary state (e.g. the ground state). This becomes clear when considering the simple case of a two-level system, where a coherence induced between both states leads to a complementary (antiphase) oscillation of the excited and ground state population as a function of the relative pulse delay/phase (Fig.\[fig:Ionization\]a). If excited and ground state populations are detected with equal probability, the modulation contrast is lost and a constant signal is measured. Hence, a high contrast between the detection efficiency of both states is important.
An alternative explanation is given by the Feynman diagrams in Fig\[fig:pathways\]b. For each process, always two complementary pathways exist (denoted as 1 and 2), which differ in the final population state (being $\ket{e}$ and$\ket{g}$ or $\ket{f}$ and$\ket{e}$). Both pathways are identical except for the 4’th interaction, leading to an antiphase modulation of the signal yield. If both contributions (i.e. final population states) are detected with the same efficiency, the modulations cancel each other leading to a depletion of the signal.
In case of multiphoton ionization of the sample, a reasonably strong modulation contrast is naturally given, since complementary states require different numbers of photons for the ionization (Fig.\[fig:Ionization\]b). Here, care must be taken to avoid saturation and hence loss of modulation contrast due to an intense ionization laser which is close to resonance with the probed transition ($\ket{g}\rightarrow \ket{e}$). The least photons are required for the ionization of ESA pathways ending in a high-lying population state (ESA2 in Fig\[fig:pathways\]b), due to which these signals are generally amplified in mulitphoton detection. Likewise, any state may be selectively amplified by suitable choice of ionization laser wavelength and consideration of resonant intermediate levels. This property might be exploited to amplify and discriminate certain features in the 2D spectra in order to further disentangle overlapping spectral features from different species (Fig.\[fig:Rb2Rb32DES\]).
While multiphoton ionization may add complexity to the measurements due to the influence of intermediate levels, direct one-photon ionization has the advantage of mapping all populations directly to the ion continuum (Fig.\[fig:Ionization\]c). Yet, one-photon ionization with UV pulses tends to generate high background signals by ionizing constituents of background gas and ground state molecules. Moreover, the ionization probability of a one-photon process solely depends on the bound-free wavefunction overlap which generally shows only small variations between different bound states and thus causes a loss of modulation contrast in the detection. This problem can be solved with photoelectron spectrometry, where modulation contrast is readily recovered by selecting the photoelectrons only from specific populations (and excluding their complementary population states). We have recently implemented such a scheme based on photoelectron TOF measurements in a magnetic bottle spectrometer. Our first results show an increased detection efficiency as compared to other methods[@bruder_delocalized_2019].
Outlook
=======
In conclusion, action-detected 2DES has already demonstrated a high degree of sensitivity in measurements including both high spectral resolution and full femtosecond dynamics, thereby facilitating the combination with extremely dilute samples like molecular beams at target densities down to $10^7$cm$^{-3}$ or even below. The combination with HENDI yields studies at millikelvin temperatures and unprecedented high resolution enabling a new level in the interpretation of dynamics in comparison with theory. The HENDI technique provides a plethora of tailored model systems ranging from weakly-bound van der Waals molecules, microsolvated systems up to specifically designed large organic complexes. Other prospective targets like size-selected free ions or charged clusters may open alternative avenues where 2DES could provide new insight, thus, being the ideal playground to study photoinduced ultrafast processes.
Furthermore, with the capabilities demonstrated already, coherent spectroscpoy may find its way into new disciplines. E.g., the field of Quantum Optics could pivotally gain from corresponding new approaches. Ultracold ensembles in optical lattices or cold Rydberg gases might be studied with respect to the full dynamics of all coupled states, or Markovian vs. non-Markovian dissipation when probing interactions with external modes. On the other hand, high sensitive detection methods, as introduced above, are prerequisite for experiments on non-trivial quantum effects exploiting e.g. correlation experiments with single photon light sources. With the ongoing development of high repetition rate and high photon flux sources, exciting experiments have come into reach.
New dimensions accessible in the gas phase
------------------------------------------
The photoionization detection proved to be a valuable extension in 2DES. The variety of detection schemes available at UHV conditions grants a high control of the detection process, thereby simplifying 2D spectra through precise selectivity, and adding a considerable amount of complementary information. The incorporation of further detection schemes will open extra dimensions in multidimensional spectroscopy schemes. Velocity-map-imaging combined with sophisticated online data processing offers not only selected photoelectron energies but also electron emission angles as extra dimension. VMI ion images can provide directionality in dissociation processes. Alignment of molecules in strong laser fields or by means of well-established molecular beam methods gives access to the stereodynamics of chemical reactivity. Finally, detecting multiple electrons and ions in coincidence and correlation methods would even further enhance dimensions and selectivity in new types of studies.
However, most of the advanced techniques are limited to low data acquisition rates, which are still in the kHz range for digitizing time-of-flight traces or even below 50Hz for CCD-cameras. This conflicts with an ideal high SN ratio at high repetition rates and shot-to-shot modulation. As a first step into this direction we have already shown that undersampling schemes allow to combine high modulation frequencies with low sampling rates[@Bruder:18]. Another issue is to properly process the full amount of information (e.g. multiple peaks in TOF) without an excessive number of lock-in hardware. In this direction software based lock-in algorithms[@Karki_lockin_2013] as well as advanced post-processing methods are on the way to readily scale the number of demodulators and to enable detailed shot-to-shot data processing.
Coherent spectroscopy methods in the extreme-UV spectral range
--------------------------------------------------------------
The introduced phase-cycling methods inherently enable the detection of higher harmonic processes, e.g. in high-harmonic demodulation at phase-modulation experiments[@bruder_efficient_2015]. Apart from studying multiple quantum coherences (MQC)[@bruder_efficient_2015; @bruder_delocalized_2019; @Yu_2019], it has been demonstrated, that this can be employed for phase-modulation experiments using light generated in higher harmonic generation (HHG) processes[@bruder_phase-modulated_2017]. In a recent approach, we successfully performed an extension of this work at the seeded Free-Electron Laser (FEL) FERMI, where by means of phase modulation of the UV seed laser, attosecond wave packet interferometry at XUV photon energies (28eV) was done[@wituschek_tracking_2019]. Control of femtosecond pulse timing and CEP at higher harmonics in the XUV has been demonstrated, only acting on the fundamental UV laser pulses. In view of the rich options based on HHG XUV light sources covering pulse durations down to the attosecond range, the prospective extension of multidimensional coherent methods at high photon energies would open a new field including inner shell processes, site specifity in molecular complexes, and attosecond time resolution.
Of course, many specific aspects realizing these kind of new experiments are still to be worked out and will remain challenging. However, the recent steps in 2DES are encouraging for many more exciting experiments that are underway.
Funding by the European Research Council within the Advanced Grant “COCONIS” (694965), by the Bundesministerium f[ü]{}r Bildung und Forschung (05K16VFB) and by the Deutsche Forschungsgemeinschaft (IRTG 2079) are acknowledged.
References {#references .unnumbered}
==========
|
---
abstract: 'The $K$-user multiple-input and multiple-output (MIMO) broadcast channel (BC) with no channel state information at the transmitter (CSIT) is considered, where each receiver is assumed to be equipped with reconfigurable antennas capable of choosing a subset of receiving modes from several preset modes. Under general antenna configurations, the sum linear degrees of freedom (LDoF) of the $K$-user MIMO BC with reconfigurable antennas is completely characterized, which corresponds to the maximum sum DoF achievable by linear coding strategies. The LDoF region is further characterized for a class of antenna configurations. Similar analysis is extended to the $K$-user MIMO interference channels with reconfigurable antennas and the sum LDoF is characterized for a class of antenna configurations.'
author:
- 'Minho Yang, Sang-Woon Jeon, and Dong Ku Kim, [^1] [^2] [^3]'
title: Linear Degrees of Freedom of MIMO Broadcast Channels with Reconfigurable Antennas in the Absence of CSIT
---
Blind interference alignment, broadcast channels, degrees of freedom (DoF), multiple-input and multiple-output (MIMO), reconfigurable antennas.
Introduction
============
Recently, there have been considerable researches on characterizing the *degrees of freedom* (DoF) of wireless networks. As current wireless networks become very complicated, exact capacity characterization is so difficult that many researchers have actively studied approximate capacity characterizations in the shape of DoF. The DoF is the prelog factor of capacity, providing an intuitive metric for the number of interference-free communication channels that wireless networks can attain at the high signal-to-noise ratio (SNR) regime. Hence, it is regarded as a primary performance metric for multiantenna and/or multiuser communication systems. Cadambe and Jafar recently made a remarkable progress on understanding DoF of multiuser wireless networks showing that the sum DoF of the $K$-user interference channel (IC) is given by $K/2$ [@Cadambe:08]. An innovative methodology called *interference alignment* (IA) has been proposed to obtain $K/2$ DoF, which aligns multiple interfering signals into the same signal space at each receiver. The concept of such signal space alignment has been successfully adapted to various network environments, e.g., see [@Viveck2:09; @Viveck1:09; @Tiangao:10; @Suh:11; @Tiangao:12; @Jeon4:12; @Jeon:14] and the references therein. More recently, different strategies of IA were further developed in terms of ergodic IA [@Nazer11:09; @Jeon5:13; @Jeon2:11; @Jeon2:14] and real IA [@Motahari:09; @Motahari2:09].
Note that most of the previous researches including the aforementioned IA techniques have focused on DoF of wireless networks under the assumption that each transmitter perfectly knows global channel state information (CSI). However, for many practical communication systems, acquiring the exact CSI value at transmitters is very challenging due to channel feedback delay, system overhead, and so on. Motivated by these practical restrictions, implementing IA under a more relaxed CSI condition has been actively studied in the literature. Maddah-Ali and Tse made a breakthrough in [@Maddah-Ali:12] demonstrating that completely outdated CSI is still useful to improve DoF of the $K$-user multiple-input and single-output (MISO) broadcast channel (BC). Preceded by [@Maddah-Ali:12], there have been a series of researches for studying IA techniques exploiting outdated or delayed CSI at transmitters [@Vaze:11; @Abdoli:11; @Vaze:12_dCSIT; @Abdoli:11_IC; @Abdoli:13]. In [@Vaze:11; @Abdoli:11; @Vaze:12_dCSIT], similar DoF gains were shown in MIMO BC under delayed CSIT and, in particular, the DoF region of the two-user MIMO BC with delayed CSIT was completely characterized in [@Vaze:12_dCSIT]. In the context of IC, it has been first shown in [@Maleki:12] that IA can achieve more than one DoF in the three-user SISO IC under delayed CSIT, which is then extended to the $K$-user case in [@Abdoli:11_IC; @Abdoli:13].
Although there is still a practical demand for further relaxing CSI requirements at the transmitter side, it has been proved in [@Jafar:05] that the DoF of the $K$-user MISO BC collapses to one for isotropic fading if the transmitter cannot acquire any information about CSI. In terms of isotropic fading and no CSIT, similar DoF degradation was further shown in MIMO BC and IC [@Vaze:12; @Huang:12; @Zhu:12; @Vaze:12_IC]. On the other hand, IA without CSIT, called *blind IA*, has been recently proposed in [@Jafar:12] for a class of heterogeneous block fading models[^4] achieving larger DoF than that achievable for the isotropic fading model. In addition, it was shown that blind IA obtains similar DoF gain for a class of homogeneous block fading models[^5] [@Zhou:12; @Zhou:12_dio; @Zhou:12_Imp].
In [@Gou:10; @Gou:11], Gou, Wang, and Jafar have first proposed a blind IA technique exploiting *reconfigurable antennas*. As shown in Fig. \[reconfigurableantenna\], reconfigurable antennas are capable of dynamically adjusting their radiation patterns in a controlled and reversible manner through various technologies such as solid state switches or microelectromechanical switches (MEMS), which can be conceptually modeled as antenna selection that each RF-chain of reconfigurable antennas chooses one of receiving mode among several preset modes at each time instant, see also [@Gou:11 Section I] for the concept of reconfigurable antennas. Based on a remarkable observation that even for time-invariant channels, reconfigurable antenna can artificially create channel matrices correlated across time in some specific structure, the authors in [@Gou:11] show that the optimal sum DoF of the $K$-user $M \times 1$ MISO BC is given by $\frac{MK}{M+K-1}$ when each user is equipped with a reconfigurable antenna whose RF-chain can choose one receiving mode from $M$ preset modes. Subsequently, in [@Wang:10], the achievability result in [@Gou:11] is generalized to the $K$-user $M \times N$ MIMO BC where each user is equipped with a set of reconfigurable antennas whose RF-chains are able to choose $N$ receiving modes from $M$ preset modes, showing that the sum DoF of $\frac{MNK}{M+NK-N}$ is achievable. The idea of blind IA using reconfigurable antennas is further extended to ICs consisting of receivers with reconfigurable antennas [@Wang:11; @Wang:11_2; @Lu:13; @Wang:14; @Lu:14].
In this paper, we consider the *$K$-user MIMO BC assuming a general reconfigurable antenna environment*. In particular, the transmitter is equipped with $M$ antennas and user $k$, $k=1,\cdots,K$, is equipped with a set of reconfigurable antennas whose RF-chains can choose $L_{k}$ receiving modes from $N_{k}$ preset modes ($N_{k}\geq L_k$), which includes the conventional non-reconfigurable antenna model ($N_k=L_k$ for this case). We focus on the *linear DoF (LDoF) with no CSIT*, i.e., the maximum DoF achievable by linear coding strategies with no CSIT, see also [@Lashgari:13; @Lashgari:14; @Kao:14] for the definition of LDoF. For general antenna configurations, we completely characterize the sum LDoF of the $K$-user MIMO BC with reconfigurable antennas in the absence of CSIT. We further characterize the LDoF region for a specific class of antenna configurations. Therefore, the main contributions of this paper are two-folds: 1) we generalize the previous achievability results in [@Gou:11; @Wang:10] assuming a certain class of antenna configurations to general antenna configurations, 2) we show the converse of our achievable DoF in the LDoF sense, which implies that the achievability result in [@Wang:10] is also optimal in the LDoF sense. Our analysis is further applied to a class of $K$-user MIMO IC with reconfigurable antennas and the sum LDoF is characterized for a class of antenna configurations, which generalizes the achievable sum DoF result in [@Lu:13].
The rest of this paper is organized as follows. In Section \[sec:system\_model\], we introduce the $K$-user MIMO BC with reconfigurable antennas. In Section \[sec:main\_results\], we first define the LDoF and state the main result of this paper, the sum LDoF and LDoF region of the $K$-user MIMO BC with reconfigurable antennas. We present the converse and achievability of the main results in Section \[sec:converse\] and \[sec:achievability\], respectively and finally conclude in Section \[sec:conclusion\].
System Model {#sec:system_model}
============
![$K$-user MIMO BC with reconfigurable antennas.[]{data-label="system"}](reconfigurableantenna.png){width="3.5in"}
![$K$-user MIMO BC with reconfigurable antennas.[]{data-label="system"}](system.png){width="3in"}
Notation
--------
For integer values $a$ and $b$, $a\setminus b$ and $a|b$ denote the quotient and the remainder respectively when dividing $a$ by $b$. For a set $\mathcal{A}$, $|\mathcal{A}|$ is the cardinality of $\mathcal{A}$. For a vector space $\mathcal{V}$, $\operatorname{dim}(\mathcal{V})$ is the dimension of $\mathcal{V}$. For a matrix $\mathbf{A}$, $\mathbf{A}^{T}$, $|\mathbf{A}|$, $\operatorname{rank}(\mathbf{A})$, and $\mathcal{R}(\mathbf{A})$ are the transpose, determinant, rank, and column space of $\mathbf{A}$ respectively. For matrices $\mathbf{A}$ and $\mathbf{B}$, $\mathbf{A}\otimes \mathbf{B}$ is the Kronecker product of $\mathbf{A}$ and $\mathbf{B}$. For a set of matrices $\{\mathbf{A}_{i}\}_{i=1,\cdots,n}$, $\operatorname{diag}(\mathbf{A}_{1},\cdots,\mathbf{A}_{n})$ denotes the block-diagonal matrix consisting of $\{\mathbf{A}_{i}\}$. Also $\mathbf{I}_{a}$, $\mathbf{1}_{a\times b}$, and $\mathbf{0}_{a\times b}$ denote the $a \times a$ identity matrix, the $a \times b$ all-one matrix, and the $a \times b$ all-zero matrix respectively and let $\mathbf{0}_{a} = \mathbf{0}_{a \times a}$.
$K$-user MIMO BC with Reconfigurable Antennas
---------------------------------------------
Consider the $K$-user MIMO BC depicted in Fig. \[system\] in which the transmitter is equipped with $M$ antennas and user $k \in \mathcal{K} = \{1,\cdots,K \}$ is equipped with a set of reconfigurable antennas whose RF-chains are able to choose $L_{k}$ receiving modes from $N_{k}$ preset modes at every time instant, where $N_{k} \geq L_{k}$. Note that, if $N_{k} = L_{k}$, then user $k$ is equivalent to be equipped with $L_{k}$ conventional (non-reconfigurable) antennas.
The received signal vector of user $k$ at time $t$ is given by $$\begin{aligned}
\label{eq:in_out}
\mathbf{y}_{k}(t) = \boldsymbol{\Gamma}_{k}(t) \mathbf{H}_{k}(t) \mathbf{x}(t) + \mathbf{z}_{k}(t) \end{aligned}$$ where $\mathbf{H}_{k}(t) \in \mathbb{C}^{N_{k}\times M}$ is the channel matrix from the transmitter to $N_k$ preset modes of user $k$ at time $t$, $\mathbf{x}(t) \in \mathbb{C}^{M} $ is the transmit signal vector at time $t$, $\mathbf{z}_{k}(t) \in \mathbb{C}^{L_{k}}$ is the additive noise vector of user $k$ at time $t$, and $\boldsymbol{\Gamma}_{k}(t) \in \{0,1\}^{L_{k} \times N_{k}}$ is the selection matrix of user $k$ at time $t$. In particular, each row vector of $\boldsymbol{\Gamma}_{k}(t)$ consists of zero values except for a single element of one value and is different from each other. That is, $\mathbf{\Gamma}_k(t)$ extracts $L_{k}$ elements out of the $N_{k}$ elements in $\mathbf{H}_{k}(t)\mathbf{x}(t)$ and if user $k$ is equipped with conventional antennas, i.e., $L_{k} = N_{k}$, then $\boldsymbol{\Gamma}_{k}(t) = \mathbf{I}_{N_{k}}$ so that $\boldsymbol{\Gamma}_{k}(t)$ can be omitted in . The transmitter should satisfy the average power constraint $P$, i.e., $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{t=1}^{n} \|\mathbf{x}(t)\|^{2}\leq P$, where $\|\cdot\|$ denote the norm of a vector. The elements of $\mathbf{z}_{k}(t)$ are independent and identically distributed (i.i.d.) drawn from $\mathcal{CN}(0,1)$.
We assume that channel coefficients are i.i.d. drawn from a continuous distribution and remain constant across time, i.e., $\mathbf{H}_{k}(t) = \mathbf{H}_{k}$ for all $t \in \mathbb{N}$. Global channel state information (CSI) is assumed to be available only at the users, but not at the transmitter. i.e., no CSIT. Furthermore, we assume that each user selects its receiving modes in a predetermined pattern independent of channel realization, which are revealed to the transmitter. That is, $\boldsymbol{\Gamma}_{k}(t)$ is not a function of $\{\mathbf{H}_j\}_{j\in \mathcal{K}}$ for all $k\in\mathcal{K}$ and $t\in\mathbb{N}$.
For notational convenience, from , we define the $n$ time-extended input–output relation as $$\begin{aligned}
\label{eq:in_out_extended}
\mathbf{y}^n_{k} = \boldsymbol{\Gamma}^n_{k} \mathbf{H}^n_{k} \mathbf{x}^n + \mathbf{z}^n_{k} \end{aligned}$$ where $$\begin{aligned}
\mathbf{\Gamma}^n_k&=\operatorname{diag}\left(\mathbf{\Gamma}_k(1),\cdots,\mathbf{\Gamma}_k(n)\right),\\
\mathbf{H}^n_k&= \mathbf{I}_{n} \otimes \mathbf{H}_k, \\ %\operatorname{diag}\left(\mathbf{H}_k;n\right),\\
\mathbf{y}^n_{k}&=\left[\mathbf{y}^T_{k}(1)\cdots\mathbf{y}^T_{k}(n)\right]^T,\\
\mathbf{x}^n_{k}&=\left[\mathbf{x}^T_{k}(1)\cdots\mathbf{x}^T_{k}(n)\right]^T,\\
\mathbf{z}^n_{k}&=\left[\mathbf{z}^T_{k}(1)\cdots\mathbf{z}^T_{k}(n)\right]^T.\end{aligned}$$
Linear Degrees of Freedom and Main Results {#sec:main_results}
==========================================
Linear Degrees of Freedom
-------------------------
In this paper, we confine the transmitter to use linear precoding techniques, in which DoF represents the dimension of the linear subspace of transmitted signals [@Lashgari:14]. Consider a linear precoding scheme with block length $n$, in which the transmitter sends the information symbols of user $k$, denoted by $\mathbf{s}_{k} \in \mathbb{C}^{m_{k}(n)}$, through the $n$ time-extended beamforming matrix $\mathbf{V}^n_{k} \in \mathbb{C}^{n M \times m_{k}(n)}$. Hence, the $n$ time-extended transmit signal vector is given by $$\mathbf{x}^{n} = \sum_{j=1}^K\mathbf{V}_{j}^{n}\mathbf{s}_{j}$$ and, from , the $n$ time-extended received signal vector of user $k$ is given by $$\begin{aligned}
\mathbf{y}_{k}^{n} = \sum\limits_{j=1}^{K} \boldsymbol{\Gamma}_{k}^{n}\mathbf{H}_{k}^{n}\mathbf{V}_{j}^{n}\mathbf{s}_{j} + \mathbf{z}_{k}^{n}. \notag\end{aligned}$$
Based on such a linear precoding scheme, we define the linear degrees of freedom as the follow, see also [@Lashgari:14] for more details.
\[def:LDoF\] The linear degrees of freedom (LDoF) of $K$-tuple ($d_{1}, \cdots, d_{K}$) is said to be achievable if there exist a set of beamforming matrices $\mathbf{V}_{j}^{n}$ and selection matrices $\boldsymbol{\Gamma}_{j}^{n}$ for $j=1,\cdots,K$ almost surely satisfying $$\begin{gathered}
\dim\left ( \operatorname{Proj}_{\mathcal{I}_{j}^{c}}\mathcal{R}(\boldsymbol{\Gamma}_{j}^{n}\mathbf{H}_{j}^{n}\mathbf{V}_{j}^{n}) \right ) = m_{j}(n), \notag \\
d_{j} = \underset{n \rightarrow \infty}{\lim} \frac{m_{j}(n)}{n} \notag\end{gathered}$$ where $\mathcal{I}_{j} = \mathcal{R}( \boldsymbol{\Gamma}_{j}^{n}\mathbf{H}_{j}^{n}[\mathbf{V}_{1}^{n}\cdots\mathbf{V}_{j-1}^{n}\mathbf{V}_{j+1}^{n}\cdots\mathbf{V}_{K}^{n}] )$ and $\operatorname{Proj}_{\mathcal{A}^{c}}\mathcal{B}$ denotes the vector space induced by projecting the vector space $\mathcal{B}$ onto the orthogonal complement of the vector space $\mathcal{A}$.
The LDoF region $\mathcal{D}$ is the closure of the set of all achievable LDoF tuples satisfying Definition \[def:LDoF\] and the sum LDoF is then given by $$\begin{aligned}
d_{\Sigma} = \max_{(d_{1},\cdots,d_{K}) \in \mathcal{D}} \left\{\sum_{k=1}^{K}d_{k} \right\} \notag .\end{aligned}$$
Main Results
------------
For convenience of representation, the following parameters are defined. $$\begin{aligned}
L_{\max} &= \max_{k\in\mathcal{K}}\{L_{k}\}, \notag \\
T_{k} & = \min(M,N_{k}) \mbox{ for } k \in \mathcal{K}, \notag \\
\Lambda &= \{ k \in \mathcal{K} : T_{k} > L_{\max} \}, \notag \\
\eta & =\frac{\sum\limits_{i \in \Lambda}\frac{T_{i}L_{i}}{T_{i}-L_{i}}}{1+\sum\limits_{i\in\Lambda}\frac{L_{i}}{T_{i}-L_{i}}}. \label{Lambda}\end{aligned}$$
In the following, we completely characterize the sum LDoF of the $K$-user MIMO BC with reconfigurable antennas.
\[main\_thm\] For the K-user MIMO BC with reconfigurable antennas defined in Section \[sec:system\_model\], the sum LDoF is given by $$\begin{aligned}
d_{\Sigma} = \min ( M,\max(L_{\max}, \eta ) ). \label{eq:sumLDoF}\end{aligned}$$
We refer to Section \[subsec:converse1\] for the converse proof and Section \[achievability\_thm1\] for the achievability proof.
From Theorem \[main\_thm\], $N_k$ greater than $M$ cannot further increase $d_{\Sigma}$. Therefore, the number of preset modes $N_k$ for maximizing $d_{\Sigma}$ is enough to set $N_k=M$ for $k\in\mathcal{K}$. Note that this remark is valid only in MIMO BC with reconfigurable antennas and it is shown in [@Wang:14] that the number of preset modes greater than that of transmit antennas can increase sum DoF in MIMO IC with reconfigurable antennas.
![Sum LDoF $d_{\Sigma}$ with respect to $K$ when $M=4$ and $L=1$.[]{data-label="ex1"}](ex1.png){width="3.5in"}
Consider the symmetric $K$-user MIMO BC with reconfigurable antennas in Section \[sec:system\_model\] in which $N_k=N$ and $L_k=L$ for all $k\in\mathcal{K}$. For this case, $$\begin{aligned}
d_{\Sigma}=\min\left(M,\max\left(L,\frac{KL\min(M,N)}{KL+\min(M,N)-L}\right)\right)\end{aligned}$$ from Theorem \[main\_thm\]. To figure out the impact of reconfigurable antennas, let us focus on the limiting case where $K$ tends to infinity. Then $$\begin{aligned}
\lim_{K\to\infty} d_{\Sigma}=\min(M,\max(L, \min(M,N)))=\min(M,N)\end{aligned}$$ regardless of $L$. Note that $d_{\Sigma}=\min(M,L)$ for the symmetric $K$-user MIMO BC without reconfigurable antennas, which corresponds to the case where $N=L$. Therefore, reconfigurable antennas can significantly improve the sum LDoF as both $M$ and $N$ increase. Figure \[ex1\] plots $d_{\Sigma}$ with respect to $K$ when $M=4$ and $L=1$. As the number of preset modes $N$ increases, the DoF gain from reconfigurable antennas increases compared to the conventional (nonreconfigurable) antenna model, i.e., $N=L$.
We further derive the LDoF region $\mathcal{D}$ for a class of antenna configurations in the following theorem.
\[main\_thm2\] Consider the $K$-user MIMO BC with reconfigurable antennas defined in Section \[sec:system\_model\]. If $\ M > L_{\max}$ and $N_{k} > L_{\max}$ for all $ k \in \mathcal{K}$, then the LDoF region $\mathcal{D}$ consists of all $K$-tuples $(d_{1},\cdots, d_{K})$ satisfying $$\begin{aligned}
\label{eq:ineq1}
\frac{d_{k}}{L_{k}} + \sum\limits_{j =1, j\neq k}^{K} \frac{d_{j}}{T_{j}} \leq 1\end{aligned}$$ for all $k \in \mathcal{K}$.
We refer to Section \[subsec:converse2\] for the converse proof and Section \[achievability\_thm2\] for the achievability proof.
![LDoF region $\mathcal{D}$ for the $2$-user MIMO BC with reconfigurable antennas, where $M,N_1,N_2>\max(L_1,L_2)$.[]{data-label="dof_region"}](dof_region.png){width="3in"}
![LDoF region $\mathcal{D}$ when $K=2$, $M=4$, $L_1=2$, and $L_2=1$.[]{data-label="ex2"}](ex2.png){width="4in"}
Consider the $2$-user MIMO BC with reconfigurable antennas in Section \[sec:system\_model\] in which $M,N_1,N_2>\max(L_1,L_2)$. From Theorem \[main\_thm2\], the LDoF region $\mathcal{D}$ is then given as in Fig. \[dof\_region\]. For the conventional (nonreconfigurable) antenna model, where $N_1=L_1$ and $N_2=L_2$, $\mathcal{D}$ is given by the time-sharing region between $(L_1,0)$ and $(0,L_2)$. Hence $\mathcal{D}$ enlarges as $N_1$ and $N_2$ increase, which demonstrate the benefit of reconfigurable antennas. Figure \[ex2\] plots $\mathcal{D}$ when $K=2$, $M=4$, $L_1=2$, and $L_2=1$.
From Theorem \[main\_thm\], the sum LDoF is derived for a class of the $K$-user MIMO IC with reconfigurable antennas in the following. We omit the formal definition of LDoF for the $K$-user MIMO IC with reconfigurable antennas, which can be straightforwardly defined in the same manner as in Definition \[def:LDoF\].
![$K$-user MIMO IC with reconfigurable antennas.[]{data-label="system_IC"}](system_IC.png){width="3in"}
\[main\_thm3\] Consider the $K$-user MIMO IC with reconfigurable antennas depicted in Fig. \[system\_IC\] in which transmitter $k \in \mathcal{K}$ is equipped with $M_{k}$ antennas and user $k$ is equipped with a set of reconfigurable antennas whose RF-chains are able to choose $L_{k}$ receiving mode from $N_{k}$ preset modes, where $N_{k} \geq L_{k}$. If $M_{k} \geq N_{k}$ for all $k \in \mathcal{K}$, then the sum LDoF is given by $$\begin{aligned}
\label{eq:dof_ic}
d_{\Sigma,\text {IC}} = \max(L_{\max}, \eta_{\text {IC}})\end{aligned}$$ where $\eta_{\text {IC}}$ is defined as $\eta$ with $\Lambda = \{ k \in \mathcal{K} : N_{k} > L_{\max} \}$ and $T_{k} = N_{k}$ for all $k \in \Lambda$
Obviously, the achievable LDoF of the $K$-user MIMO IC with reconfigurable antennas defined in Corollary \[main\_thm3\] is upper bounded by $d_{\Sigma}$ of the $K$-user MIMO BC with reconfigurable antennas where the transmitter is equipped with $\sum_{k=1}^K M_k$ antennas and user $k \in \mathcal{K}$ is equipped with a set of reconfigurable antennas whose RF-chains are able to choose $L_{k}$ receiving modes from $N_{k}$ preset modes. Hence, from Theorem \[main\_thm\], the LDoF of the considered $K$-user MIMO IC is upper bounded by , which completes the converse proof of Corollary \[main\_thm3\]. We refer to Section \[achievability\_thm3\] for the achievability proof.
Consider the symmetric MIMO IC with reconfigurable antennas in Fig. \[system\_IC\] in which $N_k=N$ and $L_k=L$ for all $k\in\mathcal{K}$ ,where $N\geq L$. If $M\geq N$, then from Corollary \[main\_thm3\], $$\begin{aligned}
d_{\Sigma, \text{IC}}=\max\left(L,\frac{KLN}{KL+N-L}\right),\end{aligned}$$ which attains $\lim_{K\to \infty}d_{\Sigma, \text{IC}}=N$. Note that the symmetric $K$-user MIMO IC without reconfigurable antennas is given by $d_{\Sigma, \text{IC}}=L$, which corresponds to the case where $M\geq N=L$. Therefore, similar to the symmetric MIMO BC case, reconfigurable antenna can significantly improve the sum LDoF as both $M$ and $N$ increase with $M\geq N$.
The following two remarks summarize the contributions of Theorem \[main\_thm\] and Corollary \[main\_thm3\], compared with the previous results in [@Wang:10; @Lu:13].
Consider the $K$-user MIMO BC with reconfigurable antennas defined in Section \[sec:system\_model\]. If $M = N_{k}$ and $L_{k} = L$ for all $k \in \mathcal{K}$ where $M > L$, then $$\begin{aligned}
d_{\Sigma}=\frac{MLK}{M+LK-L}\notag\end{aligned}$$ from Theorem \[main\_thm\], which coincides with the previous achievability result in [@Wang:10]. Hence, Theorem \[main\_thm\] not only generalizes the result in [@Wang:10] but it also shows the converse in the LDoF sense for general $M$, $\{N_k\}_{k\in\mathcal{K}}$, and $\{L_k\}_{k\in\mathcal{K}}$.
Consider the $K$-user MIMO IC with reconfigurable antennas defined in Corollary \[main\_thm3\]. If $M_k=N_k > 1$ and $L_k = 1$ for all $k\in\mathcal{K}$, then $$\begin{aligned}
d_{\Sigma,\text{IC}} =\frac{\sum\limits_{k= 1}^K\frac{N_{k}}{N_{k}-1}}{1+\sum\limits_{k=1}^K\frac{1}{N_{k}-1}} \notag\end{aligned}$$ from Corollary \[main\_thm3\], which coincides with the previous achievability result in [@Lu:13]. Hence, Corollary \[main\_thm3\] not only generalizes the result in [@Lu:13] but it also shows the converse in the LDoF sense for a broader class of antenna configurations.
Converse {#sec:converse}
========
In this section, we prove the converse of Theorem \[main\_thm\], \[main\_thm2\].
Converse of Theorem \[main\_thm\] {#subsec:converse1}
---------------------------------
First divide the entire parameter space into three cases as follows:
- Case 1: $M \leq L_{\max}$.
- Case 2: $M > L_{\max}$ and $N_{k} \leq L_{\max}$ for all $ k \in \mathcal{K}$.
- Case 3: $M > L_{\max}$ and $N_{k} > L_{\max}$ for some $k \in \mathcal{K}$.
Then the right hand side of is given by $$\begin{aligned}
\min ( M,\max(L_{\max}, \eta ) )=
\begin{cases}
M & \mbox{for Case 1}, \\
L_{\max} & \mbox{for Case 2}, \\
\max ( L_{\max} , \eta ) & \mbox{for Case 3}. \\
\end{cases}
\label{eq:converse_total}\end{aligned}$$
For Case 1, an achievable sum LDoF is trivially upper bounded by the number of transmit antennas. Consequently, we have $$\begin{aligned}
d_{\Sigma} \leq M & \ \mbox{ for Case 1}. \label{eq:converse_case1}\end{aligned}$$
For Case 2, consider the *extended $K$-user MIMO BC* by substituting $N_{k} = L_{\max}$ and $L_{k}=L_{\max}$ for all $k \in \mathcal{K}$ from the original $K$-user MIMO BC with reconfigurable antennas. That is, for the extended $K$-user MIMO BC, all users are equipped with $L_{\max}$ conventional antennas. Obviously, the sum DoF of the extended $K$-user MIMO BC provides an upper bound on $d_{\Sigma}$. From the fact that the received signals of all the user are statistically equivalent in the extended $K$-user MIMO BC so that any receiver can decode all messages from the transmitter, $d_{\Sigma}$ is further bounded by the sum DoF of point-to-point MIMO BC where transmitter and receiver are equipped with $M$ and $L_{\max}$ conventional antennas respectively, given by $\min(M, L_{\max}) = L_{\max}$. Therefore, we have $$\begin{aligned}
d_{\Sigma} \leq L_{\max} & \ \mbox{ for Case 2}. \label{eq:converse_case2}\end{aligned}$$ Hence, for the rest of this subsection, we prove that $$\begin{aligned}
d_{\Sigma} & \leq \max (L_{\max}, \eta) \notag\end{aligned}$$ by assuming that $M > L_{\max}$ and $N_{k} > L_{\max}$ for some $k \in \mathcal{K}$, which is Case 3. Suppose that user $k$ satisfies the condition $T_k>L_{\max}$ (equivalently $k\in\Lambda$). Then, consider the *extended $K$-user MIMO BC with reconfigurable antennas at user $k$* by substituting $N_{i}=L_{\max}$ and $L_{i}=L_{\max}$ for all $i \in \mathcal{K} \setminus \Lambda$ and $L_{i} = N_{i}$ for all $i \in \Lambda\setminus\{k\}$ from the original $K$-user MIMO BC with reconfigurable antennas. Hence, users in $\mathcal{K} \setminus \Lambda$ have $L_{\max}$ conventional antennas and user $i \in \Lambda\setminus\{k\}$ has $N_{i}$ conventional antennas. Only user $k$ is equipped with reconfigurable antennas in this extended model. Again, the sum DoF of this model provides an upper bound on $d_{\Sigma}$. Then, the received signal vector of user $i$ is given by $$\begin{aligned}
\mathbf{y}_{i}(t) =\begin{cases}
\boldsymbol{\Gamma}_{k}(t)\mathbf{H}_{k} \mathbf{x}(t) + \mathbf{z}_{k}(t) & \mbox{if } i = k, \\
\mathbf{G}_{i}\mathbf{x}(t) + \mathbf{z}_{i}(t) & \textrm{otherwise}
\end{cases}\end{aligned}$$ where $\mathbf{G}_{i} \in \mathbb{C}^{\max(N_{i},L_{\max}) \times M}$ for $i \in \mathcal{K}\setminus \{k\}$ satisfies the channel assumption in Section \[sec:system\_model\].
![Extended $K$-user MIMO BC with reconfigurable antennas based on the rearranged user index, where we assume that if $|\Lambda|=1$, then the set of user $K-|\Lambda|+2$ through user $K$ is empty and if $|\Lambda|=K$, then the set of user 2 through $K-|\Lambda|+1$ is empty.[]{data-label="system_extended"}](system_extended.png){width="3in"}
For convenience, we rearrange the users in ascending order of $L_{i}$ and denote the new index of user $i$ as $\sigma(i)$. We assume that $\sigma(k)=1$ without loss of generality. From now on, we denote the index $i$ as the rearranged user index. Fig. \[system\_extended\] illustrates the extended model based on the rearranged user index. Hence, the $n$ time-extended received signal vector of user $i$ with linear precoding is given by $$\begin{aligned}
\label{eq:ch_extended}
\mathbf{y}_{i}^{n} = \begin{cases}
\sum\limits_{j=1}^{K} \boldsymbol{\Gamma}_{i}^{n}\mathbf{H}_{i}^{n}\mathbf{V}_{j}^{n}\mathbf{s}_{j} + \mathbf{z}_{i}^{n} &\mbox{if } i = 1, \\
\sum\limits_{j=1}^{K} \mathbf{G}_{i}^{n}\mathbf{V}_{j}^{n}\mathbf{s}_{j} + \mathbf{z}_{i}^{n} & \textrm{otherwise} \\
\end{cases}\end{aligned}$$ where $\mathbf{G}_{i}^{n}= \mathbf{I}_{n} \otimes \mathbf{G}_{i}$ for $i \in \mathcal{K}\setminus \{1\}$. Also, we define an increasing sequence $\Delta_{i}$ for $i\in\mathcal{K}$ as $$\begin{aligned}
\Delta_{i} = \begin{cases}
L_{k} & \mbox{if } \sigma^{-1}(i) = k,\\
L_{\max} & \mbox{if } \sigma^{-1}(i) \in \mathcal{K}\setminus \Lambda, \\
T_{\sigma^{-1}(i)} & \textrm{otherwise}.
\end{cases} \notag\end{aligned}$$ Note that $\Delta_{i}$ is the rank of $\mathbf{\Gamma}_1(t)\mathbf{H}_1$ for $i=1$ and the rank of $\mathbf{G}_{i}$ for $i = 2,\cdots, K$, almost surely.
In the following, we introduce three key lemmas used for proving the converse of Theorem \[main\_thm\]. The first lemma provides an equivalent condition for decodability of messages [@Lashgari:14].
\[lemma\_fano\] For two matrices $\mathbf{A}$, $\mathbf{B}$ with the same row size, $$\begin{aligned}
\operatorname{dim}(\operatorname{Proj}_{\mathcal{R}(\mathbf{A})^{c}}\mathcal{R}(\mathbf{B})) = \operatorname{rank} \left ([ \mathbf{A} \ \mathbf{B} \right ]) - \operatorname{rank} \left(\mathbf{A} \right). \notag\end{aligned}$$
We refer to [@Lashgari:14 Lemma 1] for the proof.
The second lemma states that mode switching does not decrease the dimension of the interference space of user 1 almost surely.
\[lemma\_sel\] Consider the extended $K$-user MIMO BC with reconfigurable antennas at user 1 depicted in Fig. \[system\_extended\]. Let $\mathbf{G}_{1}\in\mathbb{C}^{L_1\times M}$ denote the matrix consisting of the first through the $L_{1}$th row vectors of $\mathbf{H}_{1}$ and $\mathbf{G}_{1}^{n}= \mathbf{I}_{n} \otimes \mathbf{G}_1$. For any mode switching pattern $\boldsymbol{\Gamma}_{1}^{n}$, the following relation holds almost surely: $$\begin{aligned}
\operatorname{rank} \left (\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \geq \operatorname{rank} \left (\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] \right ). \label{eq_sel}\end{aligned}$$
We refer to Appendix \[proof\_sel\] for the proof.
Although the definition of $\mathbf{G}^n_{1}$ in Lemma \[lemma\_sel\] is not consistent with those of $\mathbf{G}^n_{i}$ for $i = 2,\cdots, K$ in , we adopt this notation for easy presentation of the converse proof. The third lemma shows the relation of the dimensions of the interference space between user $i$ and user $i-1$.
\[lemma\_equi\] Consider the extended $K$-user MIMO BC with reconfigurable antennas at user 1 depicted in Fig. \[system\_extended\]. The following relations hold almost surely: $$\begin{aligned}
\label{eq: relation_1}
\frac{1}{\Delta_{i-1}}\operatorname{rank} \left (\mathbf{G}_{i-1}^{n}[\mathbf{V}_{i}^{n} \cdots \mathbf{V}_{K}^{n}] \right )
&\geq \frac{1}{\Delta_{i}}\operatorname{rank} \left (\mathbf{G}_{i}^{n}[\mathbf{V}_{i+1}^{n} \cdots \mathbf{V}_{K}^{n}] \right )
+ \frac{1}{\Delta_{i}} \operatorname{dim}(\operatorname{Proj}_{\mathcal{I}_{i}^{c}}\mathcal{R}(\mathbf{G}_{i}^{n}\mathbf{V}_{i}^{n})),\end{aligned}$$ for $i = 2,\cdots, K-1$ and $$\begin{aligned}
\label{eq: relation_2}
\frac{1}{\Delta_{K-1}}\operatorname{rank} \left (\mathbf{G}_{K-1}^{n}\mathbf{V}_{K}^{n} \right )
&\geq \frac{1}{\Delta_{K}} \operatorname{dim}(\operatorname{Proj}_{\mathcal{I}_{K}^{c}}\mathcal{R}(\mathbf{G}_{K}^{n}\mathbf{V}_{K}^{n})). \end{aligned}$$
We refer to Appendix \[proof\_equi\] for the proof.
We are now ready to prove the converse of Theorem \[main\_thm\]. From the definition of $m_{1}(n) $, we have
$$\begin{aligned}
m_{1}(n)
&= \operatorname{dim}(\operatorname{Proj}_{\mathcal{I}_{1}^{c}}\mathcal{R}(\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}\mathbf{V}_{1}^{n})) \notag \\
&= \operatorname{rank} \left (\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{1}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) - \operatorname{rank}\left (\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \label{eq_dof11}\\
&\leq n \Delta_{1} - \operatorname{rank}\left (\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \notag \\
&\overset{a.s.}{\leq} n \Delta_{1} - \operatorname{rank} \left (\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \label{eq_dof12}\\
&\overset{a.s.}{\leq} n \Delta_{1} - \sum\limits_{i=2}^{K} \frac{\Delta_{1}}{\Delta_{i}} \operatorname{dim}(\operatorname{Proj}_{\mathcal{I}_{i}^{c}}\mathcal{R}(\mathbf{G}_{i}^{n}\mathbf{V}_{i}^{n})) \label{eq_dof14} \\
&= n \Delta_{1} - \sum\limits_{i=2}^{K} \frac{\Delta_{1}}{\Delta_{i}} m_{i}(n) \label{eq_dof15}\end{aligned}$$
where , , , and follow from Lemma \[lemma\_fano\], \[lemma\_sel\], \[lemma\_equi\], and Definition \[def:LDoF\] respectively. Then, by dividing both sides by $n$ and letting $n$ to infinity, we have $$\begin{aligned}
\label{dof_result21}
\sum\limits_{i=1}^{K} \frac{1}{\Delta_{i}} d_{i} \leq 1.\end{aligned}$$
Rearranging with respect to the original index, i.e., $\sigma^{-1}(i)$ provides $$\begin{aligned}
\label{dof_result2}
\frac{1}{L_{k}} d_{k} + \sum\limits_{i \in \Lambda\setminus \{ k\} } \frac{1}{T_{i}} d_{i}+\sum\limits_{i \notin \Lambda } \frac{1}{L_{\max}} d_{i} \leq 1.\end{aligned}$$
Since holds for all $k \in \Lambda$, we have total $|\Lambda|$ inequalities composing the outer region of $\mathcal{D}$. Then, we obtain an upper bound on $d_{\Sigma}$ by solving the linear programming in the following lemma.
\[lemma:linear\_program\] \[lemma\_upper\] Consider the following optimization problem assuming that $M > L_{\max}$ and $N_{k} > L_{\max}$ for some $k \in \mathcal{K}$: $$\begin{aligned}
\operatorname{maximize} \sum\limits_{i=1}^{K} d_{i} \notag \end{aligned}$$ $$\begin{aligned}
\textrm{subject to } & \frac{1}{L_{k}} d_{k} + \sum\limits_{i \in \Lambda\setminus \{ k\} } \frac{1}{T_{i}} d_{i}+\sum\limits_{i \notin \Lambda } \frac{1}{L_{\max}} d_{i} \leq 1, \ \ \forall k\in \Lambda,
\notag \\
& d_i \geq 0, \ \ \forall i\in\mathcal{K}. \notag\end{aligned}$$ Then $$\begin{aligned}
\sum\limits_{i=1}^{K} d_{i} \leq
\max \left ( \eta, L_{\max} \right ) \notag.\end{aligned}$$
We refer to Appendix \[proof\_upper\] for the proof.
Therefore, combining , , and the result in Lemma \[lemma:linear\_program\] provides $$\begin{aligned}
d_{\Sigma} \leq \min( M , \max ( L_{\max}, \eta ) )\notag,\end{aligned}$$ which completes the converse proof of Theorem \[main\_thm\].
Converse of Theorem \[main\_thm2\] {#subsec:converse2}
----------------------------------
Notice that the condition $M > L_{\max}$ and $N_{k} > L_{\max}$ in Theorem \[main\_thm2\] is a special class of Case 3 defined in Section \[subsec:converse1\] satisfying that $\Lambda=\mathcal{K}$. Therefore, in the same manner in Section \[subsec:converse1\], we have with $\Lambda=\mathcal{K}$. Therefore, $$\begin{aligned}
\frac{d_{k}}{L_{k}} + \sum\limits_{i =1, i\neq k}^{K} \frac{d_{i}}{T_{i}} \leq 1, & \ \ \forall k \in\mathcal{K}, \notag\end{aligned}$$ which completes the converse proof of Theorem \[main\_thm2\].
Achievability {#sec:achievability}
=============
In this section, we prove achievability of Theorems \[main\_thm\], \[main\_thm2\], and Corollary \[main\_thm3\]. The proposed blind IA scheme generalizes those in [@Gou:10; @Gou:11; @Wang:10], but it cannot be straightforwardly obtained from [@Gou:10; @Gou:11; @Wang:10] due to general antenna configurations of $M$, $\{N_{k}\}_{k\in\mathcal{K}}$, and $\{L_{k}\}_{k\in\mathcal{K}}$ considered in this paper. For better understanding, we also provide an example for the proposed blind IA scheme based on the two-user case in Appendix \[appendix:example\].
Achievability of Theorem \[main\_thm\] {#achievability_thm1}
--------------------------------------
First divide the entire parameter space into four cases as follows:
- Case 1: $M \leq L_{\max}$.
- Case 2: $M > L_{\max}$ and $N_{k} \leq L_{\max}$ for all $ k \in \mathcal{K}$.
- Case 3-1: $M > L_{\max}$, $N_{k} > L_{\max}$ for some $k \in \mathcal{K}$, and $\eta \leq L_{\max}$,
- Case 3-2: $M > L_{\max}$, $N_{k} > L_{\max}$ for some $k \in \mathcal{K}$, and $\eta > L_{\max}$,
where Cases 1 and 2 are identical to those in Section \[subsec:converse1\] and Case 3-1 and Case 3-2 are two partitions of Case 3 in Section \[subsec:converse1\]. The right side of is then given by $$\begin{aligned}
\label{eq:dof_total}
\min( M , \max ( L_{\max} , \eta ) ) =
\begin{cases}
M & \mbox{for Case 1}, \\
L_{\max} & \mbox{for Case 2 or Case 3-1}, \\
\eta & \mbox{for Case 3-2}. \\
\end{cases} \end{aligned}$$
For Cases 1, 2, and 3-1, the sum DoF is trivially achievable by only supporting the user having the maximum number of RF-chains. Hence, $$\begin{aligned}
\label{eq:dof1_2}
d_{\Sigma} =
\begin{cases}
M & \mbox{ for Case 1}, \\
L_{\max} & \mbox{ for Case 2 or Case 3-1} \\
\end{cases}\end{aligned}$$ is achievable. For the rest of this subsection, we prove that $$\begin{aligned}
d_{\Sigma} = \eta = \frac{\sum\limits_{k \in \Lambda}\frac{T_{k}L_{k}}{T_{k}-L_{k}}}{1+\sum\limits_{k \in \Lambda}\frac{L_{k}}{T_{k}-L_{k}}} \end{aligned}$$ is achievable by assuming that $M > L_{\max}$, $N_{i} > L_{\max}$ for some $i \in \mathcal{K}$, and $\eta > L_{\max}$, which is Case 3-2. For this case, only the users in $\Lambda$ are supported, i.e., $d_{i} = 0$ for all $i \notin \Lambda$. Suppose that $\Lambda=\{1,2,\cdots,|\Lambda|\}$ without loss of generality. For easy representation, let us define $S_{i}$, $U_{i}$, and $W_{i}$ for $i \in \Lambda$ and define $U$ and $W$ as $$\begin{aligned}
S_{i} & = \begin{cases}
\frac{T_{i}}{L_{i}}-1 & \mbox{if } T_{i}|L_{i} = 0, \\
T_{i} \setminus L_{i} & \textrm{otherwise},
\end{cases} \notag
\\
U_{i} & = S_{i}\prod_{p \in \Lambda\setminus\{i\}} (T_{p} - L_{p}), \notag
\\
W_{i} & = \prod_{p \in \Lambda\setminus\{i\}} S_{p}, \notag
\\
U & = \prod_{p \in \Lambda} (T_{p} - L_{p}), \notag
\\
W & =\prod_{p\in\Lambda}S_{p} \label{def:ach_parameter} .\end{aligned}$$
### Transmit beamforming design {#subsubsec:transmit_signal}
To construct transmit beamforming, we adopt a bottom-up approach as in the following steps. [**Step 1**]{} (Alignment block): As the first step, we construct alignment blocks, which will be used for building alignment units in the next step. The basic concept of alignment block in this paper is similar to those in [@Gou:11; @Wang:10]. Define $\mathbf{I}_{M,T_{i}} = [\mathbf{I}_{T_{i}} \ \mathbf{0}_{T_{i},M-T_{i}}]^{T}$ and the $j$th information vectors of user $i$ as $\mathbf{s}_{j}^{[i]}\in \mathbb{C}^{L_{i} T_{i}}$, which consists of $L_{i} T_{i}$ independent information symbols, where $j = 1, \cdots, U_{i}W_{i}$. Then the $j$th alignment block of user $i$, denoted by $\mathbf{v}_{j}^{[i]} \in \mathbb{C}^{MT_{i}}$, is defined as $$\begin{aligned}
\mathbf{v}_{j}^{[i]} =\left [(\mathbf{v}_{j,1}^{[i]})^{T} (\mathbf{v}_{j,2}^{[i]})^{T} \cdots (\mathbf{v}_{j,S_{i}+1}^{[i]})^{T}\right]^{T} \notag\end{aligned}$$ where for $k=1,\cdots,S_{i}+1$, $$\begin{aligned}
\label{eq:align_block}
\mathbf{v}_{j,k}^{[i]} =\begin{cases}
(\boldsymbol{\Phi}\otimes \mathbf{I}_{M,T_{i}})\mathbf{s}_{j}^{[i]}
\in \mathbb{C}^{(T_{i}|L_{i}) M} &\mbox{if } T_{i}|L_{i} \neq 0 \mbox{ and } k=1,\\
(\mathbf{I}_{L_{i}} \otimes \mathbf{I}_{M,T_{i}}) \mathbf{s}_{j}^{[i]}
\in \mathbb{C}^{L_{i} M}&\mbox{otherwise}.
\end{cases}\end{aligned}$$ Here $\boldsymbol{\Phi} \in \mathbb{C}^{T_{i}|L_{i} \times L_{i}}$ is a random matrix whose entries are i.i.d. drawn from a continuous distribution. From , the following relation holds: $$\begin{aligned}
\label{eq_blockre}
\mathbf{v}_{j,k}^{[i]} = \begin{cases}
(\boldsymbol{\Phi}\otimes \mathbf{I}_{M})\mathbf{v}_{j,S_{i}+1}^{[i]} & \mbox{if } T_{i}|L_{i} \neq 0 \mbox{ and } k = 1, \\
\mathbf{v}_{j,S_{i}+1}^{[i]} & \textrm{otherwise}.
\end{cases}\end{aligned}$$
[**Step 2**]{} (Alignment unit): Next, we build an alignment unit using $U_{i}$ alignment blocks. Specifically, $\mathbf{v}_{1+(j-1)U_{i}}^{[i]}$ through $\mathbf{v}_{jU_{i}}^{[i]}$ are used for building the $j$th alignment unit of user $i$, denoted by $\mathbf{u}_{j}^{[i]} \in \mathbb{C}^{MT_{i}U_{i}}$ where $j = 1,\cdots, W_{i}$, which is given as $$\begin{aligned}
\mathbf{u}_{j}^{[i]} =\left [(\mathbf{u}_{j,1}^{[i]})^{T} (\mathbf{u}_{j,2}^{[i]})^{T} \cdots (\mathbf{u}_{j,S_{i}+1}^{[i]})^{T}\right]^{T} \label{eq:align_unit}\end{aligned}$$ where $$\begin{aligned}
\mathbf{u}_{j,k}^{[i]} =
\left [
\begin{array}{c}
\mathbf{v}_{1+(j-1)U_{i},(1-k)|S_{i}+1}^{[i]} \\
\mathbf{v}_{2+(j-1)U_{i},(2-k)|S_{i}+1}^{[i]} \\
\vdots \\
\mathbf{v}_{jU_{i},(U_{i}-k)|S_{i}+1}^{[i]}
\end{array}
\right] \in \mathbb{C}^{MU} \notag %\prod_{p \in \Lambda} (T_{p} - L_{p})\end{aligned}$$ for $k = 1,\cdots,S_{i}$ and $$\begin{aligned}
\mathbf{u}_{j,S_{i}+1}^{[i]} =
\left [
\begin{array}{c}
\mathbf{v}_{1+(j-1)U_{i},S_{i}+1}^{[i]} \\
\mathbf{v}_{2+(j-1)U_{i},S_{i}+1}^{[i]} \\
\vdots \\
\mathbf{v}_{jU_{i},S_{i}+1}^{[i]}
\end{array}
\right] \in \mathbb{C}^{L_{i}MU_{i}}.
\notag\end{aligned}$$ From , the following relations hold for $k = 1,\cdots,S_{i}$: $$\begin{aligned}
\label{eq_blockre2}
\mathbf{u}_{j,k}^{[i]} = \begin{cases}
\mathbf{u}_{j,S_{i}+1}^{[i]} & \mbox{if } T_{i}|L_{i} = 0,\\
(\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{k}^{[i]} \otimes \mathbf{I}_{M})\mathbf{u}_{j,S_{i}+1}^{[i]} & \textrm{otherwise}\\
\end{cases}\end{aligned}$$ where $\mathbf{Q}_{k}^{[i]}\in \mathbb{C}^{(T_{i}-L_{i}) \times L_{i}S_{i}}$ is the block-diagonal matrix consisting of $S_{i}$ blocks whose blocks are all $\mathbf{I}_{L_{i}}$ except that the $k$th block is $\boldsymbol{\Phi}$. For convenience, let us call $\mathbf{u}_{j,k}^{[i]}$ as the $k$th sub-unit of $\mathbf{u}_{j}^{[i]}$.
[**Step 3**]{} (Transmit signal vector for user $i$): We then construct the transmit signal vector for user $i$ using $\mathbf{u}_{1}^{[i]}$ through $\mathbf{u}_{W_{i}}^{[i]}$. The transmit signal for user $i$, denoted by $\mathbf{x}_{i}\in\mathbb{C}^{MUW + M\sum_{i \in \Lambda} L_{i}U_{i}W_{i}}$, is defined as $$\begin{aligned}
\label{eq_x1x2}
\mathbf{x}_{i} = \left[ \mathbf{x}_{i,1}^{T} \ \mathbf{0}_{C_{1,i-1}\times 1}^{T} \ \mathbf{x}_{i,2}^{T} \ \mathbf{0}_{C_{i+1,|\Lambda|}\times 1}^{T} \right ]^{T}\end{aligned}$$ where $C_{l,m} = \sum_{p=l}^{m}L_{p}MU_{p}W_{p}$ and $\mathbf{x}_{i,1}$ consists of $\{ \mathbf{u}_{j,k}^{[i]} \}_{k=1,\cdots,S_{i}}^{j=1,\cdots,W_{i}}$, total $W$ sub-units, and $\mathbf{x}_{i,2}$ consists of $\{ \mathbf{u}_{j,k}^{[i]} \}_{k=S_{i}+1}^{j=1,\cdots,W_{i}}$, total $W_{i}$ sub-units, defined as in the followings. $$\begin{aligned}
\label{eq_x1x3}
\mathbf{x}_{i,1} = \left [
\begin{array}{c}
\mathbf{u}_{f^{[i]}(1)}^{[i]}
\\
\mathbf{u}_{f^{[i]}(2)}^{[i]}
\\
\vdots
\\
\mathbf{u}_{f^{[i]}(W)}^{[i]}
\end{array}
\right ] \in \mathbb{C}^{M U W}, %\prod_{p \in \Lambda}(T_{p} - L_{p})
\
\mathbf{x}_{i,2} = \left [
\begin{array}{c}
\mathbf{u}_{1,S_{i}+1}^{[i]}
\\
\mathbf{u}_{2,S_{i}+1}^{[i]}
\\
\vdots
\\
\mathbf{u}_{W_{i},S_{i}+1}^{[i]}
\end{array}
\right ] \in \mathbb{C}^{ L_{i} M U_{i} W_{i}}\end{aligned}$$ Here, $f^{[i]}$ for ${i\in\Lambda}$ is a function on $\{l \in \mathbb{N} :1 \leq l \leq W \}$ to $\{(j,k) \in \mathbb{N}^{2} : 1 \leq j \leq W_{i}, 1 \leq k \leq S_{i} \}$, defined by $f^{[i]}(l) = (f_{1}^{[i]}(l), f_{2}^{[i]}(l))$ such that $$\begin{aligned}
\label{eq:g}
&f_{1}^{[i]}(l) = ((l-1)\setminus\prod_{p=1}^{i}S_{p}) \prod_{p=1}^{i-1}S_{p} + 1 + (l-1)|\prod_{p=1}^{i-1}S_{p}, \notag
\\
&f_{2}^{[i]}(l) = ((l-1)|\prod_{p=1}^{i}S_{p}) \setminus \prod_{p=1}^{i-1}S_{p} +1. \end{aligned}$$ The following lemma shows that every element of $\{ \mathbf{u}_{j,k}^{[i]} \}_{k=1,\cdots,S_{i}}^{j=1,\cdots,W_{i}}$ appears once in $\mathbf{x}_{i,1}$.
\[lemma:inverse\] Let $\mathcal{A} = \{l \in \mathbb{N} : 1 \leq l \leq W \}$, $\mathcal{B} = \{(j,k) \in \mathbb{N}^{2} : 1 \leq j \leq W_{i}, 1 \leq k \leq S_{i} \}$. Let $f^{[i]}$ for $i\in\Lambda$ be a function on $\mathcal{A}$ to $\mathcal{B}$ defined in and let $g^{[i]}$ for $i\in\Lambda$ be a function on $\mathcal{B}$ to $\mathcal{A}$ defined by $$\begin{aligned}
g^{[i]}(j,k) = 1 + ((j-1)\setminus\prod_{p=1}^{i-1}S_{p}) \prod_{p=1}^{i}S_{p} + (j-1)|\prod\limits_{p = 1}^{i-1}S_{p} + (k-1)\prod\limits_{p=1}^{i-1}S_{p}. \label{eq:f}\end{aligned}$$ Then $g^{[i]}$ is the inverse function of $f^{[i]}$.
We refer to Appendix \[properties1\] for the proof.
[**Step 4**]{} (Transmit signal vector): Finally, transmit signal vector $\mathbf{x}^{n}$ is the sum of the transmit signal vector for each user as $$\begin{aligned}
\label{eq_total}
\mathbf{x}^{n} = \sum\limits_{i\in\Lambda}\mathbf{x}_{i}\end{aligned}$$ where $$\begin{aligned}
\label{eq_duration}
n = UW + \sum_{i \in \Lambda} L_{i}U_{i}W_{i}\end{aligned}$$ because $\mathbf{x}_{i}\in\mathbb{C}^{MUW + M\sum_{i \in \Lambda} L_{i}U_{i}W_{i}}$. That is, at time $t=1 ,\cdots, UW + \sum_{i \in \Lambda} L_{i}U_{i}W_{i}$, the transmitter sends from the $((t-1)M+1)$th to the $(tM)$th elements of $ \sum_{i\in\Lambda}\mathbf{x}_{i}$ through $M$ antennas.
### Mode switching patterns at receivers {#subsubsec:mode_switching}
Based on the proposed transmit beamforming stated above, we design the mode switching patterns at receivers, which is fixed regardless of channel realizations. From and , we have $$\begin{aligned}
\mathbf{x}^{n} =
[\underbrace{(\sum_{i\in\Lambda}\mathbf{x}_{i,1})^{T}}_{\mbox{\footnotesize block}\ 1}\ \underbrace{\mathbf{x}_{1,2}^{T} \ \mathbf{x}_{2,2}^{T} \ \cdots \ \mathbf{x}_{|\Lambda|,2}^{T}}_{\mbox{\footnotesize block}\ 2} ]^{T}. \label{eq:total1}\end{aligned}$$ Let us denote $\sum_{i\in\Lambda}\mathbf{x}_{i,1}$ and $[\mathbf{x}_{1,2}^{T} \ \mathbf{x}_{2,2}^{T} \ \cdots \ \mathbf{x}_{|\Lambda|,2}^{T}]^{T}$ in $\mathbf{x}^{n}$ as block 1 and block 2 respectively. Subsequently, the received signal vector of user $i$ is divided as $$\begin{aligned}
\mathbf{y}_{i}^{n} = \left [ \mathbf{y}_{i,0}^{T} \ \mathbf{y}_{i,1}^{T} \cdots \mathbf{y}_{i,|\Lambda|}^{T} \right ]^{T}
\notag\end{aligned}$$ where $\mathbf{y}_{i,0}$ and $\mathbf{y}_{i,j}$ for $j = 1, \cdots, |\Lambda|$ are the received signal vectors induced by $\sum_{i\in\Lambda}\mathbf{x}_{i,1}$ and $\mathbf{x}_{j,2}$ respectively. Now, we design each user’s mode switching pattern during blocks 1 and 2 in the following. For convenience, we simply call a selection pattern (of user $i$ at time $t$) to denote a specific selection matrix $\mathbf{\Gamma}_i(t)$. We omit rigorous description of selection patterns, nonetheless one can infer them from associated channel matrices induced by selection matrices, i.e., $\mathbf{\Gamma}_i(t)\mathbf{H}_{i}$.
[**Mode switching pattern during block 1**]{}: From , block 1 is divided as $$\begin{aligned}
\sum_{i\in\Lambda}\mathbf{x}_{i,1} = \left [
\begin{array}{c}
\sum\limits_{i \in \Lambda}\mathbf{u}_{f^{[i]}(1)}^{[i]} \\
\vdots \\
\sum\limits_{i \in \Lambda}\mathbf{u}_{f^{[i]}(W)}^{[i]}
\end{array}
\right ]. \label{eq:block1}\end{aligned}$$
Note that the time interval for transmitting block 1 is $$\begin{aligned}
\label{eq:time_block1}
1\leq t\leq UW.\end{aligned}$$ During block 1, user $i$ exploits a set of $S_{i}$ selection patterns repeatedly over the entire time interval in . The channel matrix associated with the $j$th selection pattern, denoted by $\mathbf{H}_{i,j} \in \mathbb{C}^{L_{i} \times M}$, is given by $$\begin{aligned}
\begin{array}{cc}
\mathbf{H}_{i,j} = \left [
\begin{array}{c}
\mathbf{h}_{i,1+(j-1)L_{i}} \\
\mathbf{h}_{i,2+(j-1)L_{i}} \\
\vdots \\
\mathbf{h}_{i,jL_{i}}
\end{array}
\right ] & j = 1,\cdots, S_{i} \\
\end{array} \notag\end{aligned}$$ where $\mathbf{h}_{k,l}\in \mathbb{C}^{1\times M}$ is the $l$th row vector of $\mathbf{H}_{k}$ for $k\in\Lambda$ and $l = 1,\cdots, N_{k}$. For this case, at each time instant, each user chooses the selection pattern of which index is same as that of the currently transmitted sub-unit of his transmit signal vector. One can see from that the sub-unit of user $i$ transmitted at time $t=1,\cdots,UW$ is given by $\mathbf{u}_{f^{[i]}(l(t))}^{[i]}$ where $l(t) = 1 + (t-1)\setminus U$. Then, at time $t=1,\cdots,UW$, the user $i$ receives the transmit signal vector using the $f_{2}^{[i]}(l(t))$th selection pattern, associated with $\mathbf{H}_{i,f_{2}^{[i]}(l(t))}$. As a result, the received signal vector of user $i$ during $t=1,\cdots,UW$ is given by $$\begin{aligned}
\label{eq:y_i0}
\mathbf{y}_{i,0}=
\left [
\begin{array}{c}
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(1)} )\sum\limits_{j \in \Lambda}\mathbf{u}_{f^{[j]}(1)}^{[j]}
\\
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(2)} )\sum\limits_{j \in \Lambda}\mathbf{u}_{f^{[j]}(2)}^{[j]}
\\
\vdots
\\
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(W)} ) \sum\limits_{j \in \Lambda}\mathbf{u}_{f^{[j]}(W)}^{[j]}
\end{array} \right ]. \end{aligned}$$
[**Mode switching pattern during block 2**]{}: We divide block 2 into desired signal and interference signal parts of user $i$, in which desired signal part is $\mathbf{x}_{i,2}$ and interference signal part is the rest of block 2 except $\mathbf{x}_{i,2}$. Note that the time interval for transmitting $\mathbf{x}_{i,2}$ is $$\begin{aligned}
\label{eq:time_block2_desired}
a_{i}+1 \leq t \leq a_{i+1}\end{aligned}$$ where $a_{i} = UW + \sum_{p=1}^{i-1}L_{p}U_{p}W_{p}$. Let us define $\mathbf{H}_{i,S_{i}+1} \in \mathbb{C}^{(T_{i} - L_{i}S_{i}) \times M}$ and $\mathbf{H}_{i,j,k} \in \mathbb{C}^{(L_{i}-T_{i}|L_{i}) \times M}$ as $$\begin{aligned}
\label{eq:H_i_S}
\mathbf{H}_{i,S_{i}+1} = \left [
\begin{array}{c}
\mathbf{h}_{i,L_{i}S_{i}+1} \\
\mathbf{h}_{i,L_{i}S_{i}+2} \\
\vdots \\
\mathbf{h}_{i,T_{i}}
\end{array}
\right ] \end{aligned}$$ and $$\begin{aligned}
\label{eq:H_ijk}
\mathbf{H}_{i,j,k} =
\left [
\begin{array}{c}
\mathbf{h}_{i, (j-1) L_{i} + (k-1)|L_{i}+1} \\
\mathbf{h}_{i, (j-1) L_{i} + k|L_{i}+1} \\
\vdots \\
\mathbf{h}_{i, (j-1) L_{i} + (k-2 + L_{i}-(T_{i}|L_{i}))|L_{i}+1}
\end{array}
\right ] \ \end{aligned}$$ for $j = 1,\cdots, S_{i}$ and $k = 1,\cdots, L_{i}$.
First consider the desired signal part of user $i$ during block 2. For the transmission of $\mathbf{x}_{i,2}$, the mode switching pattern of user $i$ differs according to the value of $T_{i}|L_{i}$. If $T_{i}|L_{i}=0$, then user $i$ exploits a single selection pattern repeatedly over the entire time interval in . The channel matrix associated with the selection pattern is given by , where $\mathbf{H}_{i,S_{i}+1} \in \mathbb{C}^{L_{i} \times M}$ in this case. If $T_{i}|L_{i}\neq 0$, then user $i$ exploits a set of $L_{i}S_{i}$ selection patterns repeatedly over the entire time interval in , i.e., the number of $L_{i}U_{i}W_{i}/(L_{i}S_{i}) = \prod_{p \in \Lambda \setminus \{i\}} S_{p} (T_{p}-L_{p})$ repetitions. The channel matrix associated with the $j$th selection pattern, denoted by $\mathbf{H}_{i,S_{i}+1,j} \in \mathbb{C}^{L_{i} \times M}$ for $j = 1 , \cdots , L_{i}S_{i} $ which can be constructed from and , is given by $$\begin{aligned}
\label{eq_chan}
\mathbf{H}_{i,S_{i} + 1,j} =
\left [
\begin{array}{c}
\mathbf{H}_{i,S_{i}+1}
\\
\mathbf{H}_{i,(j-1)\setminus L_{i}+1, (j-1)|L_{i} + 1}
\end{array}
\right ].\end{aligned}$$ Then, at time $t=a_i+1,\cdots,a_{i+1}$, user $i$ receives the transmit signal vector using the selection pattern corresponding to $\mathbf{H}_{i,S_{i}+1,l_{i}(t)}$ where $l_{i}(t) =1 + (t-a_{i}-1)|(L_{i}S_{i})$. As a result, the received signal vector of user $i$ during $t=a_i+1,\cdots,a_{i+1}$ is given by $$\begin{aligned}
\mathbf{y}_{i,i}=
\left [
\begin{array}{c}
( \mathbf{I}_{U_{i}/S_{i}} \otimes \mathbf{H}_{i,S_{i}+1}')\mathbf{u}_{1,S_{i}+1}^{[i]}
\\
( \mathbf{I}_{U_{i}/S_{i}} \otimes \mathbf{H}_{i,S_{i}+1}')\mathbf{u}_{2,S_{i}+1}^{[i]}
\\
\vdots
\\
( \mathbf{I}_{U_{i}/S_{i}} \otimes \mathbf{H}_{i,S_{i}+1}')\mathbf{u}_{W_{i},S_{i}+1}^{[i]}
\end{array}
\right ] \label{eq_rs3_desired}\end{aligned}$$ where $$\begin{aligned}
\mathbf{H}_{i,S_{i}+1}' = \begin{cases}
\mathbf{I}_{L_{i}S_{i}} \otimes \mathbf{H}_{i,S_{i}+1} & \mbox{if } T_{i}|L_{i}=0, \\
\operatorname{diag}\left ( \mathbf{H}_{i,S_{i}+1,1},\mathbf{H}_{i,S_{i}+1,2}, \cdots, \mathbf{H}_{i,S_{i}+1,L_{i}S_{i}} \right ) & \textrm{otherwise}.
\end{cases}
\notag\end{aligned}$$
Now consider the interference signal part of user $i$ during block 2. From , the time interval for transmitting $\mathbf{x}_{i',2}$, where $i' \in \Lambda\setminus \{i\}$, is given by $$\begin{aligned}
\label{eq:time_block2_intf}
a_{i'}+1 \leq t \leq a_{i'+1}.\end{aligned}$$ For the transmission of $\mathbf{x}_{i',2}$, user $i$ exploits the same set of $S_{i}$ selection patterns used for block 1 again over the entire time interval in . For this case, at each time instant, each user chooses the selection pattern of which index is the same as that used to receive the first sub-unit of the alignment unit to which the currently transmitted sub-unit belongs. Specifically, from , the sub-unit of user $i'$ transmitted at time $t= a_{i'}+1 ,\cdots, a_{i'+1}$ is given by $\mathbf{u}_{l(t),S_{i'}+1}^{[i']}$ where $l(t) = 1+ (t-1-a_{i'}) \setminus (L_{i'}U_{i'})$. Since $\mathbf{u}_{l(t),1}^{[i']} = \mathbf{u}_{f^{[i']}(g^{[i']}(l(t),1))}^{[i']}$ from Lemma \[lemma:inverse\], $\mathbf{u}_{l(t),1}^{[i']}$ is a summand of $\sum_{j \in \Lambda}\mathbf{u}_{f^{[j]}(g^{[i']}(l(t),1))}^{[j]}$, which means from that $\mathbf{u}_{l(t),1}^{[i']}$ is transmitted simultaneously with $\mathbf{u}_{f^{[i]}(g^{[i']}(l(t),1))}^{[i]}$ in block 1. That is, user $i$ exploits the $f_{2}^{[i]}(g^{[i']}(l(t),1))$th selection pattern to receive $\mathbf{u}_{f^{[i]}(g^{[i']}(l(t),1))}^{[i]}$ so that, at time $t= a_{i'}+1 ,\cdots, a_{i'+1}$, user $i$ receives the transmit signal vector using the $f_{2}^{[i]}(g^{[i']}(l(t),1))$th selection pattern, associated with $\mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(l(t),1))}$. As a result, the received signal vector of user $i$ induced by $\mathbf{x}_{i',2}$ is $$\begin{aligned}
\label{eq_rs3}
\mathbf{y}_{i,i'} =
\left [
\begin{array}{c}
(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(1,1))})\mathbf{u}_{1,S_{i'}+1}^{[i']}
\\
(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(2,1))})\mathbf{u}_{2,S_{i'}+1}^{[i']}
\\
\vdots
\\
(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(W_{i'},1))} ) \mathbf{u}_{W_{i'},S_{i'}+1}^{[i']}
\end{array} \right ] .\end{aligned}$$
### Interference cancellation at receivers
In the following, we show that user $i$ can eliminate all interference signals contained in $\mathbf{y}_{i,0}$ in using the received interference signal parts during block 2, i.e., $\mathbf{y}_{i,i'}$ in for all $i'\in\Lambda\setminus\{i\}$. First, we introduce the following lemma, which plays a key role to verify such interference cancellation.
\[lemma:const\] Let $\mathcal{A} = \{l \in \mathbb{N} : 1 \leq l \leq W \}$, $\mathcal{B} = \{(j,k) \in \mathbb{N}^{2} : 1 \leq j \leq W_{i}, 1 \leq k \leq S_{i} \}$. Let $f^{[i]}$ for $i\in\Lambda$ be a function on $\mathcal{A}$ to $\mathcal{B}$ defined in and let $g^{[i]}$ for $i\in\Lambda$ be a function on $\mathcal{B}$ to $\mathcal{A}$ defined in . For $i,i' \in \Lambda$ where $i\neq i'$ and $(j,k), (j,k') \in \mathcal{B}$, the following relation holds: $$\begin{aligned}
f_{2}^{[i]}(g^{[i']}(j,k)) = f_{2}^{[i]}(g^{[i']}(j,k')) \notag\end{aligned}$$
We refer to Appendix \[properties2\] for the proof.
Consider $(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(l)} )\mathbf{u}_{f^{[i']}(l)}^{[i']}$ for $i'\in\Lambda\setminus\{i\}$ and $1\leq l \leq W$, which is an interference vector in $\mathbf{y}_{i,0}$. We have $$\begin{aligned}
\label{eq_rsequ}
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(l)} )\mathbf{u}_{f^{[i']}(l)}^{[i']}
&= (\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f^{[i']}(l)))})\mathbf{u}_{f^{[i']}(l)}^{[i']}\notag \\
& = (\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))})\mathbf{u}_{f^{[i']}(l)}^{[i']} \end{aligned}$$ where the first and second equalities follow from Lemma \[lemma:inverse\] and Lemma \[lemma:const\] respectively. If $T_{i'}|L_{i'} = 0$, then, substituting into , we have $$\begin{aligned}
\label{eq_inf1}
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(l)} )\mathbf{u}_{f^{[i']}(l)}^{[i']}
=(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))})\mathbf{u}_{f_{1}^{[i']}(l),S_{i'}+1}^{[i']}\end{aligned}$$ where $U = L_{i}U_{i}$ for this case. If $T_{i'}|L_{i'} \neq 0$, then, substituting into , we have $$\begin{aligned}
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(l)} )\mathbf{u}_{f^{[i']}(l)}^{[i']}
&=(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))}) (\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{f_{2}^{[i']}(l)}^{[i']} \otimes \mathbf{I}_{M})\mathbf{u}_{f_{1}^{[i']}(l),S_{i'}+1}^{[i']} \notag \\
& = (\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{f_{2}^{[i']}(l)}^{[i']}\otimes \mathbf{I}_{L_{i}})(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))})\mathbf{u}_{f_{1}^{[i']}(l),S_{i'}+1}^{[i']} .
\label{eq_inf2}\end{aligned}$$ Here comes from the following relation: $$\begin{aligned}
\label{eq_inter}
& (\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))})(\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{f_{2}^{[i']}(l)}^{[i']} \otimes \mathbf{I}_{M}) \notag\\
& = (\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{f_{2}^{[i']}(l)}^{[i']} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))}) \notag\\
& = ((\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{f_{2}^{[i']}(l)}^{[i']}) \mathbf{I}_{L_{i}U_{i}})
\otimes ( \mathbf{I}_{L_{i}} \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))}) \notag\\
& = (\mathbf{I}_{{U_{i}}/{S_{i}}} \otimes \mathbf{Q}_{f_{2}^{[i']}(l)}^{[i']}\otimes \mathbf{I}_{L_{i}})(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))})\end{aligned}$$ where follows from the mixed-product property that for matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, and $\mathbf{D}$ in which the matrix products $\mathbf{AC}$ and $\mathbf{BD}$ can be defined, $(\mathbf{A} \otimes \mathbf{B}) (\mathbf{C} \otimes \mathbf{D}) = \mathbf{AC} \otimes \mathbf{BD}$, see [@horn1991topics Lemma 4.2.10].
From , user $i$ is able to extract the following vector from $\mathbf{y}_{i,i'}$: $$\begin{aligned}
\label{eq_inf4}
(\mathbf{I}_{L_{i}U_{i}} \otimes \mathbf{H}_{i,f_{2}^{[i]}(g^{[i']}(f_{1}^{[i']}(l),1))})\mathbf{u}_{f_{1}^{[i']}(l),S_{i'}+1}^{[i']}.\end{aligned}$$ Then, user $i$ constructs $(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(l)} )\mathbf{u}_{f^{[i']}(l)}^{[i']}$ using from the relations in and and subtracts it from $\mathbf{y}_{i,0}$. In the same manner, user $i$ can remove all interference vectors in $\mathbf{y}_{i,0}$.
### Achievable LDoF
Let us denote the remaining signal vector after cancelling all interference vectors in $\mathbf{y}_{i,0}$ as $\mathbf{y}_{i,0}'$. Combining $\mathbf{y}_{i,0}'$ with $\mathbf{y}_{i,i}$, user $i$ has $$\begin{aligned}
\label{eq_result}
\left [
\begin{array}{c}
\mathbf{y}_{i,0}' \\
\mathbf{y}_{i,i} \\
\end{array}
\right ]
=
\left [
\begin{array}{c}
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(1)} )\mathbf{u}_{f^{[i]}(1)}^{[i]}
\\
\vdots
\\
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,f_{2}^{[i]}(\prod_{p \in \Lambda} S_{p})} ) \mathbf{u}_{f^{[i]}(\prod_{p \in \Lambda} S_{p})}^{[i]}
\\
( \mathbf{I}_{U_{i}/S_{i}} \otimes \mathbf{H}_{i,S_{i}+1}')\mathbf{u}_{1,S_{i}+1}^{[i]}
\\
\vdots
\\
( \mathbf{I}_{U_{i}/S_{i}} \otimes \mathbf{H}_{i,S_{i}+1}')\mathbf{u}_{W_{i},S_{i}+1}^{[i]}
\end{array} \right ].\end{aligned}$$ By classifying by alignment units, is decomposed into $W_{i}$ segments as follows: $$\begin{aligned}
\label{eq_result2}
\left [
\begin{array}{c}
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,1} )\mathbf{u}_{j,1}^{[i]}
\\
\vdots
\\
(\mathbf{I}_{U} \otimes \mathbf{H}_{i,S_{i}} )\mathbf{u}_{j,S_{i}}^{[i]}
\\
( \mathbf{I}_{U_{i}/S_{i}} \otimes \mathbf{H}_{i,S_{i}+1}')\mathbf{u}_{j,S_{i}+1}^{[i]}
\end{array} \right ] \ \ j = 1, \cdots , W_{i}.\end{aligned}$$ By classifying by alignment blocks, is further decomposed into $U_{i}W_{i}$ segments as follows: $$\begin{aligned}
\label{eq_result3}
\left [
\begin{array}{c}
(\mathbf{I}_{N_{i}} \otimes \mathbf{H}_{i,(j-1)|S_{i}+1} )\mathbf{v}_{j,1}^{[i]}
\\
(\mathbf{I}_{N_{i}} \otimes \mathbf{H}_{i,(j-2)|S_{i}+1} )\mathbf{v}_{j,2}^{[i]}
\\
\vdots
\\
(\mathbf{I}_{N_{i}} \otimes \mathbf{H}_{i,(j-S_{i})|S_{i}+1} )\mathbf{v}_{j,S_{i}}^{[i]}
\\
\mathbf{H}_{i,S_{i}+1,j}'\mathbf{v}_{j,S_{i}+1}^{[i]}
\end{array} \right ] \ \ j = 1, \cdots , U_{i}W_{i}\end{aligned}$$ where $$\begin{aligned}
\mathbf{H}_{i,S_{i}+1,j}' =
\begin{cases}
\mathbf{I}_{L_{i}} \otimes \mathbf{H}_{i,S_{i}+1} & \mbox{if } T_{i}|L_{i} = 0, \\
\operatorname{diag}\left( \mathbf{H}_{i,S_{i+1},((j-1)|S_{i})L_{i}+1}, \mathbf{H}_{i,S_{i+1},((j-1)|S_{i})L_{i}+2}, \cdots, \mathbf{H}_{i,S_{i+1},((j-1)|S_{i}+1)L_{i}} \right) & \textrm{otherwise}.
\end{cases} \notag\end{aligned}$$
If $T_{i}|L_{i}=0$, then, substituting into and switching the rows, we have $$\begin{aligned}
\label{eq_result4}
& (\mathbf{I}_{L_{i}} \otimes \mathbf{H}_{i} ) (\mathbf{I}_{L_{i}} \otimes \mathbf{I}_{M,T_{i}}) \mathbf{s}_{j}^{[i]} = (\mathbf{I}_{L_{i}} \otimes [\mathbf{H}_{i}]_{T_{i}} )\mathbf{s}_{j}^{[i]} \end{aligned}$$ where $[\mathbf{H}_{i}]_{T_{i}}$ is the leading principal minor of $\mathbf{H}_{i}$ of order $T_{i}$. Since $(\mathbf{I}_{L_{i}} \otimes [\mathbf{H}_{i}]_{T_{i}} )$ is non-singular almost surely, user $i$ can obtain $\mathbf{s}_{j}^{[i]}$ from almost surely.
If $T_{i}|L_{i} \neq 0$, then, substituting into , we have $$\begin{aligned}
\label{eq_result5}
\underbrace{
\left [
\begin{array}{c}
\boldsymbol{\Phi} \otimes \mathbf{H}_{i,(j-1)|S_{i}+1}
\\
\mathbf{I}_{N_{i}} \otimes \mathbf{H}_{i,(j-2)|S_{i}+1}
\\
\vdots
\\
\mathbf{I}_{N_{i}} \otimes \mathbf{H}_{i,(j-S_{i})|S_{i}+1}
\\
\mathbf{H}_{i,S_{i}+1,j}'
\end{array} \right ]}_{\mathbf{B}_{j}^{[i]}} (\mathbf{I}_{L_{i}} \otimes \mathbf{I}_{M,T_{i}}) \mathbf{s}_{j}^{[i]} .\end{aligned}$$ It can be easily verified that $\mathbf{B}_{j}^{[i]} = \mathbf{C}_{j}^{[i]} (\mathbf{I}_{L_{i}} \otimes [\mathbf{H}_{i}]_{T_{i}} )$ where $\mathbf{C}_{j}^{[i]} \in \mathbb{C}^{L_{i}T_{i} \times L_{i}T_{i}}$ for $j = 1, \cdots, U_{i}W_{i}$ is a non-singular matrix almost surely so that user $i$ can obtain $\mathbf{s}_{j}^{[i]}$ from almost surely. Consequently, user $i$ is able to $\mathbf{s}_{1}^{[i]}$ through $\mathbf{s}_{U_{i}W_{i}}^{[i]}$ almost surely.
Since total $L_{i}T_{i}U_{i}W_{i}$ independent information symbols are delivered almost surely to user $i$ during the period given in , the achievable LDoF of user $i$ is given by $$\begin{aligned}
d_{i}= & \ \frac{\frac{T_{i}L_{i}}{T_{i}-L_{i}}\prod\limits_{p \in \Lambda} S_{p}(T_{p} - L_{p})}
{\prod\limits_{p \in \Lambda} S_{p}(T_{p} - L_{p}) + \sum\limits_{p \in \Lambda} \frac{L_{p}}{T_{p}-L_{p}}\prod\limits_{q \in \Lambda} S_{q}(T_{q} - L_{q}) } \\
=& \
\frac{\frac{T_{i}L_{i}}{T_{i}-L_{i}}}
{1 + \sum\limits_{p \in \Lambda} \frac{L_{p}}{T_{p}-L_{p}}} .\end{aligned}$$ Therefore, $$\begin{aligned}
\label{eq:dof3_2}
d_{\Sigma}=\sum\limits_{i=1}^{K}d_{i} = \frac{\sum\limits_{i\in\Lambda}\frac{T_{i}L_{i}}{T_{i}-L_{i}}}
{1 + \sum\limits_{i \in \Lambda} \frac{L_{i}}{T_{i}-L_{i}}} \end{aligned}$$ is achievable for Case 3-2. In conclusion, from , , and , $d_{\Sigma}=\min( M , \max ( L_{\max} , \eta ) )$ is achievable, which completes the achievability proof of Theorem \[main\_thm\].
Achievability of Theorem \[main\_thm2\] {#achievability_thm2}
---------------------------------------
For notational convenience, we define the inequality in as $I_{1k}$ and the inequality $d_{k} \geq 0$ as $I_{2k}$ in the rest of this subsection. Since the LDoF region $\mathcal{D}$ in Theorem \[main\_thm2\] is a polyhedron, it suffices to show that all vertices of $\mathcal{D}$ are achievable. Hence, our achievability proof begins with characterizing vertices of $\mathcal{D}$. The following lemma establishes a condition for a $K$-tuple in $\mathbb{R}^{K}$ to be a vertex of $\mathcal{D}$.
\[active\] Consider the LDoF region $\mathcal{D}$ in Theorem \[main\_thm2\]. If $\mathbf{d} \in \mathbb{R}^{K}$ is a vertex of $\mathcal{D}$, then only $K$ inequalities among $\{I_{1k}, I_{2k} \}_{k \in\mathcal{K}}$ should be active [^6] at $\mathbf{d}$ while $I_{1k}$ and $I_{2k}$ for $k \in \mathcal{K}$ cannot be active simultaneously at $\mathbf{d}$.
Assume that $I_{11}$ and $I_{21}$ are active at $\mathbf{d} = (d_{1},\cdots,d_{K}) \in \mathcal{D}$. Combining $I_{11}$ and $I_{21}$, it is followed by $$\begin{aligned}
\sum\limits_{i=1}^{K} \frac{d_{i}}{T_{i}}=1 \notag .\end{aligned}$$ Hence, $\mathbf{d}$ must not be a zero vector and we can find an index $k^{*}\in\mathcal{K}$ such that $I_{2k^{*}}$ is not active at $\mathbf{d}$, i.e., $d_{k^{*}} > 0$. Then, we have $$\begin{aligned}
\frac{d_{k^{*}}}{L_{k^{*}}} + \sum\limits_{i =1, i\neq k^{*}}^{K} \frac{d_{i}}{T_{i}} = 1 + (\frac{1}{L_{k^{*}}} -\frac{1}{T_{k^{*}}}) d_{k^{*}} > 1 \notag,\end{aligned}$$ which means that $\mathbf{d}$ does not satisfy $I_{1k^{*}}$ so that $\mathbf{d} \notin \mathcal{D}$. Contradicting the assumption, $I_{1k}$ and $I_{2k}$ for $k \in \mathcal{K}$ cannot be active simultaneously at $\mathbf{d}$ and, as a result, for $\mathbf{d} \in \mathcal{D}$, at most $K$ inequalities among $\{I_{1k}, I_{2k} \}_{k \in\mathcal{K}}$ are active at $\mathbf{d}$. Furthermore, if $\mathbf{d}$ is a vertex of $\mathcal{D}$, then at least $K$ inequalities among $\{I_{1k}, I_{2k} \}_{k \in\mathcal{K}}$ should be active on $\mathbf{d}$ because a vertex of a polyhedron is expressed as an intersection of at least $K$ faces of the polyhedron. Therefore, only $K$ inequalities among $\{I_{1k}, I_{2k} \}_{k \in\mathcal{K}}$ should be active at $\mathbf{d}$, which completes the proof of Lemma \[active\].
Consider a $K$-tuple $\mathbf{d} = (d_{1},\cdots,d_{K}) \in \mathbb{R}^{K}$ such that $K$ inequalities among $\{I_{1k}, I_{2k} \}_{k \in\mathcal{K}}$ are active at $\mathbf{d}$ while $I_{1k}$ and $I_{2k}$ for $k \in \mathcal{K}$ are not active simultaneously at $\mathbf{d}$ and let $\Lambda_{i}=\{k \in \mathcal{K} : I_{ik} \mbox{ is active at } \mathbf{d} \}$ for $i = 1, 2$. Note that, from Lemma \[active\], $\{ \Lambda_{1}$, $\Lambda_{2}\}$ is a partition of $\mathcal{K}$. Assume $\Lambda_{1}=\{1,\cdots,J\}$ and $\Lambda_{2}=\{J+1,\cdots,K\}$ without loss of generality. Composing the $K$ inequalities active at $\mathbf{d}$, we have $$\begin{aligned}
\mathbf{A}_{1} [d_{1} \cdots d_{J}]^{T} + \left [ \frac{1}{T_{J+1}} \mathbf{1}_{J\times 1} \ \cdots \ \frac{1}{T_{K}} \mathbf{1}_{J\times 1} \right ][d_{J+1} \cdots d_{K}]^{T} & = \mathbf{1}_{J\times1} \notag \\
[d_{J+1} \cdots d_{K}]^{T} & = \mathbf{0}_{(K-J)\times 1} \label{eq:inequalities_active}\end{aligned}$$ where $$\begin{aligned}
\begin{array}{l}
\mathbf{A}_{1} =
\left[
\begin{array}{c}
\begin{array}{ccccc}
\frac{1}{N_{1}} & \frac{1}{T_{2}} & \frac{1}{T_{3}} & \cdots & \frac{1}{T_{J}} \\
\frac{1}{T_{1}} & \frac{1}{N_{2}} & \frac{1}{T_{3}} & \cdots & \frac{1}{T_{J}} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\frac{1}{T_{1}} & \frac{1}{T_{2}} & \frac{1}{T_{3}} & \cdots & \frac{1}{N_{J}} \\
\end{array}
\end{array}
\right]
\end{array} \notag .\end{aligned}$$ Since $\mathbf{A}_{1}$ is non-singular from Lemma \[det\] in Appendix \[proof\_upper\], from , we have $$\begin{aligned}
\mathbf{d} = [(\mathbf{A}_{1}^{-1} \mathbf{1}_{J \times 1})^{T} \ \mathbf{0}_{(K-J) \times 1}^{T}]^{T} \notag\end{aligned}$$ From , $\mathbf{A}_{1}^{-1} \mathbf{1}_{J \times 1}$ can be calculated easily, which results that $$\begin{aligned}
\label{eq_dof}
d_{i} =
\begin{cases}
\frac{\frac{T_{i}N_{i}}{T_{i}-N_{i}}}{1+\sum\limits_{k\in\Lambda_{1}}\frac{N_{k}}{T_{k}-N_{k}}} & \mbox{if } i \in \Lambda_{1}, \\
0 & \textrm{otherwise}.
\end{cases}\end{aligned}$$ Since is achievable by supporting users in $\Lambda_{1}$ with the scheme proposed in Case 3-2 of Section \[achievability\_thm1\], all the vertices of $\mathcal{D}$ are achievable, which completes the achievability proof of Theorem \[main\_thm2\].
Achievability of Corollary \[main\_thm3\] {#achievability_thm3}
-----------------------------------------
If $L_{\max} \geq \eta_{\text IC}$, then it is achievable by supporting an user with the maximum number of RF-chains. For the rest of this section we prove that $d_{\Sigma,IC} = \eta_{\text IC}$ assuming $L_{\max} < \eta_{\text IC}$. It can be shown that is achievable by modifying the achievable scheme derived for Case 3-2 in Section \[achievability\_thm1\] as follows. Transmitter $k\in \{ k \in \mathcal{K} : N_{k} > L_{\max} \}$ constructs transmit signal vector for user $k$ in accordance with Step 1 through Step 3 in Section \[subsubsec:transmit\_signal\] by setting $M = M_{k}$ and $T_{k} = N_{k}$ and sends it as in Step 4 in Section \[subsubsec:transmit\_signal\]. Note that since the beamforming strategy in Section \[subsubsec:transmit\_signal\] does not require cooperation among transmitters, it can be directly applied to interference channels. Then each user receives and decodes the transmitted signal in accordance with the procedure in Section \[subsubsec:mode\_switching\]. Note that it can be easily shown that is achievable, which completes the achievability proof of Corollary \[main\_thm3\].
Concluding Remarks {#sec:conclusion}
==================
In this paper, the DoF of the $K$-user MIMO BC with reconfigurable antennas under no CSIT has been studied. We completely characterized the sum LDoF of the $K$-user MIMO BC with reconfigurable antennas under general antenna configurations and further characterized the LDoF region for a class of antenna configurations. Our results provide a comprehensive understanding of reconfigurable antennas on the LDoF of the $K$-user MIMO BC, which demonstrates that reconfigurable antennas are beneficial for a broad class of antenna configurations. In particular, the DoF gain from reconfigurable antennas enlarges as both the number of transmit antennas and the number of preset modes increase. Our analysis has been further extended to characterizing the sum LDoF of the $K$-user MIMO IC with reconfigurable antennas for a class of antenna configuration, which leads to similar argument for the $K$-user MIMO BC with reconfigurable antennas.
Proof of Technical Lemmas
=========================
Proof of Lemma \[lemma\_sel\] {#proof_sel}
-----------------------------
Let us define $\mathbf{G}_{1}^{c}\in \mathbb{C}^{(N_1-L_1)\times M}$ as the submatrix consisting of the ($L_{1}+1$)th through the $N_1$th rows of $\mathbf{H}_{1}$. We will prove Lemma \[lemma\_sel\] for given realization of $\mathbf{G}_{1}$ and $[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$. That is, ‘almost sure’ in the rest of the proof is due to the randomness of $\mathbf{G}_1^c$. Since Lemma \[lemma\_sel\] trivially holds if $\mathbf{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n} = \mathbf{G}_{1}^{n}$ or $\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] = \mathbf{0}$ , we assume $\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n} \neq \mathbf{G}_{1}^{n}$ and $\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}] \neq \mathbf{0}$ from now on.
For convenience, denote $\operatorname{rank}(\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}])=r\geq 1$. Define the set of column indices consisting of $r$ linearly independent columns of $\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$ as $\mathcal{I}$. Then construct $\mathbf{A}_1\in\mathbb{C}^{nL_1\times r}$ by choosing $r$ column vectors of $\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$ whose indexes are in $\mathcal{I}$ and construct $\mathbf{A}_2\in\mathbb{C}^{nL_1\times r}$ by choosing $r$ column vectors of $\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$ whose indexes are in $\mathcal{I}$. Clearly, $\mathbf{A}_1$ is of full-rank. There exist $\binom{nL_{1}}{r}$ choices of constructing $r\times r$ submatrices from $\mathbf{A}_1$ (or $\mathbf{A}_2$) and the determinant of each of these submatrices can be expressed as a polynomial with respect to the entries of $\mathbf{G}_{1}^{c}$. Suppose that all $r \times r$ submatrix of $\mathbf{A}_{2}$ of which determinant is a zero polynomial with respect to the entries of $\mathbf{G}_{1}^{c}$, i.e., a constant polynomial whose coefficients are all equal to zero. In this case, $\mathbf{A}_{2}$ is not of full-rank regardless of the entries of $\mathbf{G}_{1}^{c}$. Hence, any matrix constructed from $\mathbf{A}_{2}$ by substituting the entries of $\mathbf{G}_{1}^{c}$ with arbitrary values is not of full-rank either. Let us now define $\mathbf{A}_{3}$ constructed from $\mathbf{A}_{2}$ by substituting $\mathbf{G}_{1}^{c}$ with $\mathbf{P} \mathbf{G}_{1}$, where $\mathbf{P} \in \mathbb{C}^{(N_{1}-L_{1})\times L_{1}}$, which is not of full-rank from the above argument. Then, we can represent $\mathbf{A}_{3}$ as $\mathbf{A}_{3} = \mathbf{Q} \mathbf{A}_{1}$ for some matrix $\mathbf{Q} \in \mathbb{C}^{n L_{i} \times n L_{i}}$. If all square submatrices of $\mathbf{P}$ are non-singular, $\mathbf{Q}$ becomes invertible so that $\mathbf{A}_{1}$ and $\mathbf{A}_{3}$ have the same rank. We can easily find such $\mathbf{P}$, for example, Vandermonde matrix or Cauchy matrix [@bader2007petascale]. However, from the fact that $\mathbf{A}_{3}$ is not of full-rank, the result that $\mathbf{A}_3$ and $\mathbf{A_1}$ have the same rank contradicts the assumption that $\mathbf{A}_{1}$ is of full-rank. Consequently, there exists at least one $r \times r$ submatrix of $\mathbf{A}_{2}$ of which determinant is not a zero polynomial with respect to the entries of $\mathbf{G}_{1}^{c}$. Then now consider some $r \times r$ submatrix of $\mathbf{A}_{2}$ of which determinant is not a zero polynomial with respect to the entries of $\mathbf{G}_{1}^{c}$. Since the entries of $\mathbf{G}_{1}^{c}$ are i.i.d drawn from a continuous distribution, for given $\mathbf{G}_{1}$ and $[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$, the determinant of the considered submatrix of $\mathbf{A}_{2}$ is non-zero almost surely. Hence, $\mathbf{A}_{2}$ is of full-rank almost surely. Since $\mathbf{A}_{2}$ is a submatrix of $\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$, the rank of $\boldsymbol{\Gamma}_{1}^{n}\mathbf{H}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$ is grater than or equal to that of $\mathbf{G}_{1}^{n}[\mathbf{V}_{2}^{n} \cdots \mathbf{V}_{K}^{n}]$ almost surely, which complete the proof of Lemma \[lemma\_sel\].
Proof of Lemma \[lemma\_equi\] {#proof_equi}
------------------------------
In order to prove Lemma \[lemma\_equi\], we need the following lemmas. The first lemma comes from the submodularity property for rank of matrices [@Lashgari:14; @Lovasz1983].
\[lemma\_submod\] For matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ with the same number of rows, $$\begin{aligned}
\operatorname{rank}[\mathbf{A} \mathbf{C}] - \operatorname{rank}[\mathbf{C}] \geq \operatorname{rank}[\mathbf{A}\mathbf{B} \mathbf{C}] - \operatorname{rank}[\mathbf{B}\mathbf{C}] \notag .\end{aligned}$$
We refer to [@Lovasz1983] for the proof.
The second lemma is one of the key properties for multantenna systems without CSIT, which means that there is no spatial preference in the received signal space without CSIT.
\[lemma\_stat\] Let $\mathbf{A}_{i,m}$ be the submatrix consisting of arbitrary $m$ row vectors of $\mathbf{G}_{i}$ and $\mathbf{B}_{j,m}$ be the submatrix consisting of arbitrary $m$ row vectors of $\mathbf{G}_{j}$. Then, for all $i,j,k \in \mathcal{K}$, the following property holds almost surely: $$\begin{aligned}
\operatorname{rank}(\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]) = \operatorname{rank}(\mathbf{B}^n_{j,m}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]) \notag\end{aligned}$$ where $\mathbf{A}_{i,m}^{n}= \mathbf{I}_{n} \otimes \mathbf{A}_{i,m} $ and $\mathbf{B}_{j,m}^{n}=\mathbf{I}_{n} \otimes \mathbf{B}_{j,m} $.[^7]
For $i = j$, it can be straightforwardly derived from the proof in Lemma \[lemma\_sel\]. Hence, we assume $i \neq j$ in the rest of the proof. We first prove that $$\begin{aligned}
\label{eq:rank_inequal}
\operatorname{rank}(\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}])
\overset{a.s.}{\leq}
\operatorname{rank}(\mathbf{B}_{j,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}])\end{aligned}$$ for given realizations of $\mathbf{A}_{i,m}$ and $[\mathbf{V}_{k}^{n} \cdots \mathbf{V}_{K}^{n}]$. Note that, for given $\mathbf{A}_{i,m}$ and $[\mathbf{V}_{k}^{n} \cdots \mathbf{V}_{K}^{n}]$, $\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]$ is deterministic but $\mathbf{B}_{j,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]$ is random induced by $\mathbf{B}_{j,m}$.
If $\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]=\mathbf{0}$, then trivially holds. Then now consider the case where $\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]\neq \mathbf{0}$. For convenience, denote $\operatorname{rank}(\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]) = r \geq 1$. Define the set of column indices consisting of $r$ linearly independent columns of $\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]$ as $\mathcal{I}$. Then construct $\mathbf{C}_1\in\mathbb{C}^{nm\times r}$ by choosing $r$ column vectors of $\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]$ whose indexes are in $\mathcal{I}$ and $\mathbf{C}_2\in\mathbb{C}^{nm\times r}$ by choosing $r$ column vectors of $\mathbf{B}_{j,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}]$ whose indexes are in $\mathcal{I}$. Clearly, $\mathbf{C}_1$ is of full-rank.
There exist $\binom{nm}{r}$ $r \times r$ choices of constructing $r\times r$ submatrices from $\mathbf{C}_2$ and the determinant of each of these submatrices can be expressed as a polynomial with respect to the entries of $\mathbf{B}_{j,m}$. Then from the same argument in the proof of Lemma \[lemma\_sel\], we can show that there exists at least one $r \times r$ submatrix of $\mathbf{C}_{2}$ of which determinant is not a zero polynomial with respect to the entries of $\mathbf{B}_{j,m}$. Now consider one of such $r\times r$ submatrices of $\mathbf{C}_2$. Since the entries of $\mathbf{B}_{j,m}$ are i.i.d drawn from a continuous distribution, for given $\mathbf{A}_{i,m}$ and $[\mathbf{V}_{k}^{n} \cdots \mathbf{V}_{K}^{n}]$, its determinant is non-zero almost surely. Hence, $\mathbf{C}_{2}$ is of full-rank almost surely and, as a result, holds. Similarly, we can also prove $\operatorname{rank}(\mathbf{A}_{i,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}])
\overset{a.s.}{\geq}
\operatorname{rank}(\mathbf{B}_{j,m}^{n}[\mathbf{V}_{k}^{n}\cdots\mathbf{V}_{K}^{n}])$. In conclusion, Lemma \[lemma\_stat\] holds.
We are now ready to prove Lemma \[lemma\_equi\]. Let us define $\mathbf{Z}^{i,j} = (\mathbf{G}_{i}^{n}[\mathbf{V}_{j}^{n}\cdots\mathbf{V}_{K}^{n}])^{T}$ and $\mathbf{Z}_{k}^{i,j} = (\mathbf{g}_{i,k}^{n}[\mathbf{V}_{j}^{n}\cdots\mathbf{V}_{K}^{n}])^{T}$, where $\mathbf{g}_{i,k}$ is the $k$th row vector of $\mathbf{G}_{i}$ and $\mathbf{g}_{i,k}^{n}= \mathbf{I}_{n} \otimes \mathbf{g}_{i,k}$. Then, for $i=2,\cdots,K$,
\[eq\_equi4\] $$\begin{aligned}
\operatorname{rank}\left( \mathbf{Z}^{i-1,i} \right) & \overset{a.s.}{=} \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]\label{eq_app0} \\
& = \operatorname{rank} \left ( \mathbf{Z}_{1}^{i-1,i} \right)
+ \sum\limits_{k=2}^{\Delta_{i-1}}
\left(
\operatorname{rank} \left [ \mathbf{Z}_{k}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{k-1}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]
\right) \nonumber
\\
%
& \overset{a.s.}{=} \operatorname{rank} \left ( \mathbf{Z}_{\Delta_{i}-1}^{i-1,i} \right)
+ \sum\limits_{k=2}^{\Delta_{i-1}}
\left(
\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \mathbf{Z}_{k-1}^{i-1,i}\cdots \mathbf{Z}_{1}^{i-1,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{k-1}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]
\right)
\label{eq_app1} \\
%
& \geq \sum\limits_{k=1}^{\Delta_{i-1}}
\left(
\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}-1}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]
\right)
\label{eq_app2} \\
%
& \overset{a.s.}{=} \Delta_{i-1} \left(\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}-1}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
\right)
\label{eq_app3} \\
& \overset{a.s.}{=} \frac{\Delta_{i-1}}{(\Delta_{i} - \Delta_{i-1})}
\sum\limits_{k=1}^{\Delta_{i} - \Delta_{i-1}}
\left (
\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}+k}^{i,i} \mathbf{Z}_{\Delta_{i-1}-1}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}-1}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
\right )
\label{eq_app5}\\
%
& \geq \frac{\Delta_{i-1}}{(\Delta_{i} - \Delta_{i-1})} \sum\limits_{k=1}^{\Delta_{i} - \Delta_{i-1}}
\left(
\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}+k}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}+k-1}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
\right)
\label{eq_app4} \\
%
& = \frac{\Delta_{i-1}}{(\Delta_{i} - \Delta_{i-1})}
\left (
\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i}}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
\right)
\notag \\
& \overset{a.s.}{=} \frac{\Delta_{i-1}}{(\Delta_{i} - \Delta_{i-1})}
\left(
\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i}}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]
- \operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]
\right)
\label{eq_app6} \\
& \overset{a.s.}{=} \frac{\Delta_{i-1}}{(\Delta_{i} - \Delta_{i-1})} \left(\operatorname{rank} \left ( \mathbf{Z}^{i,i} \right ) - \operatorname{rank} \left ( \mathbf{Z}^{i-1,i} \right) \right). \label{eq_app7} \end{aligned}$$
Here holds since $[ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i}]$ is a submatrix of $\mathbf{Z}^{i-1,i}$ and $\mathbf{Z}^{i-1,i} = [ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i}] \mathbf{F} $ almost surely for a matrix $\mathbf{F}$ such that $$\begin{aligned}
\mathbf{F} =
\left \{
\begin{array}{cc}
\mathbf{I}_{n} \otimes
\left [
\mathbf{I}_{\Delta_{i-1}}^{T} \ \left ((\left[\mathbf{G}_{i-1}\right]_{\Delta_{i-1}}^{T})^{-1} (\left[\mathbf{G}_{i-1}\right]_{\Delta_{i-1}}^{c})^{T} \right )^{T} \right ]^{T} & \textrm{if } \mathbf{G}_{i-1} \textrm{ is a tall matrix}, \\
\mathbf{I}_{n\Delta_{i-1}} & \textrm{otherwise} \\
\end{array}
\right.\end{aligned}$$ where $[\mathbf{G}_{i-1}]_{\Delta_{i-1}}$ is the leading principal minor of $\mathbf{G}_{i-1}$ of order $\Delta_{i-1}$, $[\mathbf{G}_{i-1}]_{\Delta_{i-1}}^{c}$ is the remainder part of $\mathbf{G}_{i-1}$ except $[\mathbf{G}_{i-1}]_{\Delta_{i-1}}$. Lemma \[lemma\_stat\] is used for , , , and and Lemma \[lemma\_submod\] is used for and . Also, follows since $\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i}}^{i,i} \cdots \mathbf{Z}_{1}^{i,i} \right]\overset{a.s.}{=}\operatorname{rank} \left ( \mathbf{Z}^{i,i} \right )$ and $\operatorname{rank} \left [ \mathbf{Z}_{\Delta_{i-1}}^{i-1,i} \cdots \mathbf{Z}_{1}^{i-1,i} \right]\overset{a.s.}{=}\operatorname{rank} \left ( \mathbf{Z}^{i-1,i} \right )$. From the fact that $\operatorname{rank} \mathbf{A} = \operatorname{rank} \mathbf{A}^{T}$ for a matrix $\mathbf{A}$ whose elements are complex numbers [@horn2012matrix], becomes $$\begin{aligned}
\frac{1}{\Delta_{i-1}}\operatorname{rank} \left (\mathbf{G}_{i-1}^{n}[\mathbf{V}_{i}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \overset{a.s.}{\geq} \frac{1}{\Delta_{i}}\operatorname{rank} \left (\mathbf{G}_{i}^{n}[\mathbf{V}_{i}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \notag .\end{aligned}$$ Then, for $i=2,\cdots,K$, we have
$$\begin{aligned}
\frac{1}{\Delta_{i-1}}\operatorname{rank} \left (\mathbf{G}_{i-1}^{n}[\mathbf{V}_{i}^{n} \cdots \mathbf{V}_{K}^{n}] \right )
& \overset{a.s.}{\geq} \frac{1}{\Delta_{i}}\operatorname{rank} \left (\mathbf{G}_{i}^{n}[\mathbf{V}_{i}^{n} \cdots \mathbf{V}_{K}^{n}] \right ) \notag \\
&= \frac{1}{\Delta_{i}} \left( \operatorname{rank} (\mathbf{G}_{i}^{n}[\mathbf{V}_{i+1}^{n} \cdots \mathbf{V}_{K}^{n}] ) + \operatorname{dim}(\operatorname{Proj}_{(\mathcal{I}_{i}')^{c}}\mathcal{R}(\mathbf{G}_{i}^{n}\mathbf{V}_{i}^{n})) \right)\label{eq_equi1} \\
&\geq \frac{1}{\Delta_{i}} \left( \operatorname{rank} \left (\mathbf{G}_{i}^{n}[\mathbf{V}_{i+1}^{n} \cdots \mathbf{V}_{K}^{n}]\right ) + \operatorname{dim}(\operatorname{Proj}_{\mathcal{I}_{i}^{c}}\mathcal{R}(\mathbf{G}_{i}^{n}\mathbf{V}_{i}^{n})) \right)\label{eq_equi2}\end{aligned}$$
where $\mathcal{I'}_{i} = \mathcal{R}(\mathbf{G}_{i}^{n}[\mathbf{V}_{i+1}^{n} \cdots \mathbf{V}_{K}^{n}])$. Here follows from Lemma \[lemma\_fano\] and follows since $\mathcal{I}_{i}' \subseteq \mathcal{I}_{i}$, which is given by $\mathcal{I}_{i} = \mathcal{R}( \mathbf{G}_{i}^{n}[\mathbf{V}_{1}^{n} \cdots \mathbf{V}_{i-1}^{n},\mathbf{V}_{i+1}^{n},\cdots,\mathbf{V}_{K}^{n}] )$ from Definition \[def:LDoF\]. Therefore holds. In the same manner, we can proof , which completes the proof of Lemma \[lemma\_equi\].
Proof of Lemma \[lemma\_upper\] {#proof_upper}
-------------------------------
Let us assume that $M > L_{\max}$ and $N_{k} > L_{\max}$ for some $k \in \mathcal{K}$, i.e., $\Lambda \neq \emptyset$. Let $\Lambda = \{1,\cdots,|\Lambda| \}$ without loss of generality and $\mathbf{A} = [\mathbf{A}_{1} \mathbf{A}_{2}]$ such that $$\begin{aligned}
\label{eq_A1}
\begin{array}{l}
\mathbf{A}_{1} =
\left[
\begin{array}{c}
\begin{array}{ccccc}
\frac{1}{L_{1}} & \frac{1}{T_{2}} & \frac{1}{T_{3}} & \cdots & \frac{1}{T_{|\Lambda|}} \\
\frac{1}{T_{1}} & \frac{1}{L_{2}} & \frac{1}{T_{3}} & \cdots & \frac{1}{T_{|\Lambda|}} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\frac{1}{T_{1}} & \frac{1}{T_{2}} & \frac{1}{T_{3}} & \cdots & \frac{1}{L_{|\Lambda|}} \\
\end{array}
\end{array}
\right]
\end{array}\end{aligned}$$ and $\mathbf{A}_{2} = \frac{1}{L_{\max}} \mathbf{1}_{|\Lambda|\times (K-|\Lambda|)}$. Then the optimization problem in Lemma \[lemma:linear\_program\] is rewritten as $$\begin{gathered}
\textrm{maximize} \sum\limits_{i=1}^{K} d_{i} \notag \\
\textrm{subject to} \ \mathbf{Ad} + \mathbf{x} = \mathbf{1}_{|\Lambda| \times 1} \label{eq:opt_constraint}, \\
\ \ \ \ \ \ \mathbf{d} \geq \mathbf{0}, \mathbf{x} \geq \mathbf{0} \notag\end{gathered}$$ where $\mathbf{d} = [d_{1} \cdots d_{K}]^{T}$, $\mathbf{x} = [x_{1} \cdots x_{|\Lambda|}]^{T}$, see also [@chong2013introduction].
The following lemma provides non-singularity of $\mathbf{A}_{1}$.
\[det\] For the matrix $\mathbf{A}_{1}$ defined in , the determinant of $\mathbf{A}_{1}$ is given by $$\begin{aligned}
|\mathbf{A}_{1}| = \prod\limits_{k\in\Lambda}\frac{T_{k}-L_{k}}{T_{k}L_{k}}\left ( 1 + \sum\limits_{k\in\Lambda} \frac{L_{j}}{T_{j}-L_{j}} \right ) .\end{aligned}$$ Consequently, since $T_{i} > L_{i}$ for $i \in \Lambda$, $\mathbf{A}_{1}$ is non-singular.
It can be easily verified by mathematical induction.
Notice the the optimal $\mathbf{d}$ should satisfy . Then, subtracting $\mathbf{x}$ from both sides of and multiplying them by $\mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}$, which is possible from Lemma \[det\], we have $$\begin{aligned}
\sum\limits_{i=1}^{|\Lambda|} d_{i} + \frac{\mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{1}_{|\Lambda|\times 1}}{L_{\max}}\sum\limits_{i=|\Lambda|+1}^{K} d_{i} = \mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{1}_{|\Lambda|\times 1} - \mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{x}.
\label{eq_upper2}\end{aligned}$$ Note that $$\begin{aligned}
\mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{1}_{|\Lambda|\times 1}
&=\sum_{i\in \Lambda} [\mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}]_{i}\nonumber\\
&=\sum_{i\in \Lambda} \frac{|\mathbf{A}_{1}|_{T_{i}=L_{i}=1}}{|\mathbf{A}_{1}|}\nonumber\\
&=\sum_{i\in \Lambda}\frac{\prod\limits_{k\in\Lambda, k\neq i}\frac{T_{k}-L_{k}}{T_{k}L_{k}}}
{\prod\limits_{k\in\Lambda}\frac{T_{k}-L_{k}}{T_{k}L_{k}}\left ( 1 + \sum\limits_{j\in\Lambda} \frac{L_{j}}{T_{j}-L_{j}} \right )}\nonumber\\
&=\frac{\sum\limits_{i \in \Lambda}\frac{T_{i}L_{i}}{T_{i}-L_{i}}}{1+\sum\limits_{i\in\Lambda}\frac{L_{i}}{T_{i}-L_{i}}} = \eta. \label{eq:eta}\end{aligned}$$ Here the third and fourth equalities follow from Cramer’s rule [@gockenbach2011finite Lemma 176] and Lemma \[det\] respectively. Substituting into , we have $$\begin{aligned}
\sum\limits_{i=1}^{|\Lambda|} d_{i} + \frac{\eta}{L_{\max}}\sum\limits_{i=|\Lambda|+1}^{K} d_{i} = \eta - \mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{x}.
\label{eq_upper5}\end{aligned}$$ Therefore, $$\begin{aligned}
\sum\limits_{i=1}^{K} d_{i} & \leq \max(\eta,L_{\max}) \left ( \frac{1}{\eta}\sum\limits_{i=1}^{|\Lambda|} d_{i} + \frac{1}{L_{\max}}\sum\limits_{i=|\Lambda|+1}^{K} d_{i} \right ) \nonumber\\
& = \max(\eta,L_{\max}) \frac{1}{\eta} \left ( \eta - \mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{x} \right ) \label{eq_upper3}\\
&\leq \max(\eta,L_{\max}) \label{eq_upper4} \end{aligned}$$ where follows from and follows since $\mathbf{1}_{1\times |\Lambda|}\mathbf{A}_{1}^{-1}\mathbf{x} \geq 0$. In conclusion, Lemma \[lemma\_upper\] holds.
Proof of Lemma \[lemma:inverse\] {#properties1}
--------------------------------
We have $$\begin{aligned}
g^{[i]}(f^{[i]}(l)) & = ((l-1) \setminus \prod\limits_{p=1}^{i}S_{p})\prod\limits_{p=1}^{i}S_{p} + (l-1) | \prod\limits_{p=1}^{i-1}S_{p} +
(((l-1) | \prod\limits_{p=1}^{i}S_{p})\setminus \prod\limits_{p=1}^{i-1}S_{p})\prod\limits_{p=1}^{i-1}S_{p} + 1 \notag \\
& = ((l-1) \setminus \prod\limits_{p=1}^{i}S_{p})\prod\limits_{p=1}^{i}S_{p} + (l-1) | \prod\limits_{p=1}^{i}S_{p} + 1 \label{eq_inv} \\
& = l \notag .\end{aligned}$$ Here follows that $(l-1) | \prod_{p=1}^{i-1}S_{p} = ((l-1) | \prod_{p=1}^{i}S_{p})|\prod_{p=1}^{i-1}S_{p}$. In conclusion, Lemma \[lemma:inverse\] holds.
Proof of Lemma \[lemma:const\] {#properties2}
------------------------------
We now prove that, for $i, i' \in \mathcal{K}$ where $i \neq i'$, $$\begin{aligned}
f_{2}^{[i]}(g^{[i']}(j,k)) =
\begin{cases}
((j-1)|\prod\limits_{p=1}^{i'-1}S_{p} ) \setminus \prod\limits_{p=1}^{i-1}S_{p} + 1 & \mbox{if } i < i', \\
((j-1)\setminus\prod\limits_{p=1}^{i'-1}S_{p})|\prod\limits_{p=i'+1}^{i}S_{p})\setminus \prod\limits_{p=i'+1}^{i-1}S_{p} + 1 & \mbox{if } i > i'.
\end{cases}
\label{eq:const_proof}\end{aligned}$$ From the definition of $f^{[i]}$ and $g^{[i']}$, holds trivially for $i < i'$. Hence, assume $i > i'$ in the rest of this section.
For easy representation of the proof, for $i > i'$, denote, $$\begin{aligned}
& a_0 = (j-1)\setminus\prod_{p=1}^{i'-1}S_{p} \notag \\
& a_1 = a_0 \setminus \prod_{p=i'+1}^{i}S_p, \ \ b_1 = a_0 | \prod_{p=i'+1}^{i}S_p \notag \\
& a_2 = b_1 \setminus \prod_{p=i'+1}^{i-1}S_p, \ \ b_2 = b_1 | \prod_{p=i'+1}^{i-1}S_p \notag\end{aligned}$$ From the definition of $g^{[i']}$, the following relation holds for $j \in \mathcal{A}$, $k \in \mathcal{B}$, and $i>i'$: $$\begin{aligned}
g^{[i']}(j,k) - 1 & = a_0 \prod_{p=1}^{i'}S_{p} + c \notag \\
& = a_1 \prod_{p=1}^{i}S_{p} + b_1\prod_{p=1}^{i'}S_{p} + c \label{eq:const_proof2}\end{aligned}$$ where $c = (j-1)|\prod_{p = 1}^{i'-1}S_{p} + (k-1)\prod_{p=1}^{i'-1}S_{p}$ and the second equality follows since $a_0 = a_1 \prod_{p=i'+1}^{i}S_p + b_1$. From the fact that $b_1 \leq \prod_{p = i' +1}^{i}S_p - 1$ and $c < \prod_{p = 1}^{i'}S_p$, one can see that $b_1\prod_{p=1}^{i'}S_{p} + c < \prod_{p=1}^{i}S_{p}$, which results from that $$\begin{aligned}
(g^{[i']}(j,k) - 1)|\prod_{p = 1}^{i}S_p & = b_1\prod_{p=1}^{i'}S_{p} + c \notag \\
& = a_2\prod_{p=1}^{i-1}S_{p} + b_2\prod_{p=1}^{i'}S_{p} + c \label{eq:const_proof3}\end{aligned}$$ where the second equality follows that $b_1 = a_2 \prod_{p=i'+1}^{i-1}S_p + b_2$. Since $b_2 \leq \prod_{p = i' +1}^{i-1}S_p - 1$ and $c < \prod_{p = 1}^{i'}S_p$, one can see that $b_2\prod_{p=1}^{i'}S_{p} + c < \prod_{p=1}^{i-1}S_{p}$, which results from that $$\begin{aligned}
f_{2}^{[i]}(g^{[i']}(j,k)) & = ((g^{[i']}(j,k) - 1)|\prod_{p = 1}^{i}S_p) \setminus \prod_{p=1}^{i-1}S_p + 1 \notag \\
& = a_2 + 1, \notag \end{aligned}$$ which completes the proof of Lemma \[lemma:const\].
Blind IA for a Two-User Example {#appendix:example}
===============================
For better understanding of the proposed blind IA stated in Section \[achievability\_thm1\], we provide a two-user example here. Consider the two-user MIMO BC with reconfigurable antennas defined in Section \[sec:system\_model\] where $M = N_{1} = N_{2} = 3$, $L_{1} = 1$, and $L_{2} = 2$. From , $T_{1}=T_{2}=3$, $S_{2}=W_{1}=1$, and $S_{2}=U_{1}=U_{2}=U=W_{2}=W = 2$.
### Transmit beamforming design {#transmit-beamforming-design}
In Step 1, user 1 needs two information vectors ($U_{1}W_{1}=2$) of which size is three ($T_{1}L_{1} = 3$) and user 2 needs four information vectors ($U_{2}W_{2}=4$) of which size is six ($T_{2}L_{2} = 6$). Let us denote the information vectors of user 1 as $\mathbf{s}_{1}^{[1]}$ and $\mathbf{s}_{2}^{[1]} \in \mathbb{C}^{3}$ and denote the information vectors of user 2 as $\mathbf{s}_{1}^{[2]}$, $\mathbf{s}_{2}^{[2]}$, $\mathbf{s}_{3}^{[2]}$, and $\mathbf{s}_{4}^{[2]} \in \mathbb{C}^{6}$. Then, from , the alignment block of user 1 $\mathbf{v}_{j}^{[1]}$ for $j=1,2$ is given by $$\begin{aligned}
\mathbf{v}_{j}^{[1]}
=
\left[
\begin{array}{c}
\mathbf{v}_{j,1}^{[1]} \\ \hline
\mathbf{v}_{j,2}^{[1]} \\ \hline
\mathbf{v}_{j,3}^{[1]}
\end{array}
\right]
=
\left[
\begin{array}{c}
(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{j}^{[1]} \\ \hline
(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{j}^{[1]} \\ \hline
(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{j}^{[1]}
\end{array}
\right]
=
\left[
\begin{array}{c}
\mathbf{I}_{3} \\ \hline
\mathbf{I}_{3} \\ \hline
\mathbf{I}_{3}
\end{array}
\right] \mathbf{s}_{j}^{[1]} \in \mathbb{C}^{9} \notag, \ j = 1, 2\end{aligned}$$ and from , the alignment block of user 2 $\mathbf{v}_{j}^{[2]}$ for $j=1,2,3,4$ are given by $$\begin{aligned}
&
\mathbf{v}_{j}^{[2]}
=
\left[
\begin{array}{c}
\mathbf{v}_{j,1}^{[1]} \\ \hline
\mathbf{v}_{j,2}^{[1]} \\
\end{array}
\right]
=
\left[
\begin{array}{c}
(\boldsymbol{\Phi} \otimes \mathbf{I}_{3}) \mathbf{s}_{j}^{[2]} \\ \hline
(\mathbf{I}_{2} \otimes \mathbf{I}_{3}) \mathbf{s}_{j}^{[2]} \\
\end{array}
\right]
=
\left[
\begin{array}{c}
\boldsymbol{\Phi} \otimes \mathbf{I}_{3} \\ \hline %\phi_{11}\mathbf{I}_{3} \ \phi_{12}\mathbf{I}_{3}
\mathbf{I}_{6} \\
\end{array}
\right] \mathbf{s}_{j}^{[2]} \in \mathbb{C}^{9}, \notag \ j = 1, 2, 3, 4\end{aligned}$$ where $\boldsymbol{\Phi} = [\phi_{11} \ \phi_{12}] \in \mathbb{C}^{1 \times 2}$ is a random matrix of which entries are i.i.d. continuous random variables.
In Step 2, we construct one $(W_{1} = 1)$ alignment unit of user 1 and two $(W_{2} = 2)$ alignment units of user 2. Then, from , alignment unit of user 1 $\mathbf{u}_{1}^{[1]}$ is given by $$\begin{aligned}
\mathbf{u}_{1}^{[1]}
= \left[
\begin{array}{c}
\mathbf{u}_{1,1}^{[1]} \\ \hline
\mathbf{u}_{1,2}^{[1]} \\ \hline
\mathbf{u}_{1,3}^{[1]} \\
\end{array}
\right ]
= \left[
\begin{array}{c}
\mathbf{v}_{1,1}^{[1]} \\
\mathbf{v}_{2,2}^{[1]} \\ \hline
\mathbf{v}_{1,2}^{[1]} \\
\mathbf{v}_{2,1}^{[1]} \\ \hline
\mathbf{v}_{1,3}^{[1]} \\
\mathbf{v}_{2,3}^{[1]} \\
\end{array}
\right ]
=
%\left[
%\begin{array}{c}
%(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{1}^{[1]} \\
%(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{2}^{[1]} \\ \hline
%(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{1}^{[1]} \\
%(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{2}^{[1]} \\ \hline
%(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{1}^{[1]} \\
%(\mathbf{I}_{1} \otimes \mathbf{I}_{3} ) \mathbf{s}_{2}^{[1]} \\
%\end{array}
%\right ]
%=
\left[
\begin{array}{c}
\mathbf{I}_{6} \\ \hline
\mathbf{I}_{6} \\ \hline
\mathbf{I}_{6} \\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ] \in \mathbb{C}^{18} \notag\end{aligned}$$ and the alignment unit of user 2 $\mathbf{u}_{j}^{[2]}$ for $j=1,2$ is given by $$\begin{aligned}
\mathbf{u}_{j}^{[2]}
=
\left[
\begin{array}{c}
\mathbf{u}_{j,1}^{[2]} \\ \hline
\mathbf{u}_{j,2}^{[2]} \\
\end{array}
\right ]
=
\left[
\begin{array}{c}
\mathbf{v}_{2j-1,1}^{[2]} \\
\mathbf{v}_{2j,1}^{[2]} \\ \hline
\mathbf{v}_{2j-1,2}^{[2]} \\
\mathbf{v}_{2j,2}^{[2]} \\
\end{array}
\right ]
%=
%\left[
%\begin{array}{c}
%(\boldsymbol{\Phi} \otimes \mathbf{I}_{3}) \mathbf{s}_{2j-1}^{[2]} \\
%(\boldsymbol{\Phi} \otimes \mathbf{I}_{3}) \mathbf{s}_{2j}^{[2]} \\ \hline
%(\mathbf{I}_{2} \otimes \mathbf{I}_{3}) \mathbf{s}_{2j-1}^{[2]} \\
%(\mathbf{I}_{2} \otimes \mathbf{I}_{3}) \mathbf{s}_{2j}^{[2]} \\
%\end{array}
%\right ]
= \left[
\begin{array}{cc}
\boldsymbol{\Phi} \otimes \mathbf{I}_{3} & \mathbf{0}_{3 \times 6} \\
\mathbf{0}_{3 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{I}_{3} \\ \hline
\mathbf{I}_{6} & \mathbf{0}_{6}\\
\mathbf{0}_{6} & \mathbf{I}_{6}\\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{2j-1}^{[2]} \\
\mathbf{s}_{2j}^{[2]}
\end{array}
\right ] \in \mathbb{C}^{18}. \notag\end{aligned}$$
In Step 3, we construct the transmit signal vector for each user. For this case, $f^{[i]}$ for $i=1,2$ is given by $$\begin{aligned}
\left [ f^{[1]}(1) \ f^{[1]}(2) \right ] = \left [ (1,1) \ (1,2) \right ], \
\left [ f^{[2]}(1) \ f^{[2]}(2) \right ] = \left [ (1,1) \ (2,1) \right ] \notag\end{aligned}$$ Then, from , we have $$\begin{aligned}
& \mathbf{x}_{1,1} =
\left [ (\mathbf{u}_{1,1}^{[1]})^{T} \ (\mathbf{u}_{1,2}^{[1]})^{T} \right ]^{T}
, \
\mathbf{x}_{1,2} = \mathbf{u}_{1,3}^{[1]},
\notag
\\
& \mathbf{x}_{2,1}
=
\left [ (\mathbf{u}_{1,1}^{[2]})^{T} \ (\mathbf{u}_{2,1}^{[2]})^{T} \right ]^{T}
, \
\mathbf{x}_{2,2} =
\left [ (\mathbf{u}_{1,2}^{[2]})^{T} \ (\mathbf{u}_{2,2}^{[2]})^{T} \right ]^{T}.
\notag\end{aligned}$$ Subsequently, from , the transmit signal vector for user 1 is given by $$\begin{aligned}
\mathbf{x}_{1}
=\left [
\begin{array}{c}
\mathbf{x}_{1,1} \\ \hline
\mathbf{x}_{1,2} \\ \hline
\mathbf{0}_{24 \times 1} \\
\end{array}
\right]
=
\left [
\begin{array}{c}
\mathbf{u}_{1,1}^{[1]} \\
\mathbf{u}_{1,2}^{[1]} \\ \hline
\mathbf{u}_{1,3}^{[1]} \\ \hline
\mathbf{0}_{24 \times 6} \\
\end{array}
\right ]
=\left [
\begin{array}{c}
\mathbf{I}_{6} \\
\mathbf{I}_{6} \\ \hline
\mathbf{I}_{6} \\ \hline
\mathbf{0}_{24 \times 6} \\
\end{array}
\right]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ] \in \mathbb{C}^{42} \notag\end{aligned}$$ and the transmit signal vector for user 2 is given by $$\begin{aligned}
\mathbf{x}_{2}
=\left [
\begin{array}{c}
\mathbf{x}_{2,1} \\ \hline
\mathbf{0}_{6 \times 1} \\ \hline
\mathbf{x}_{2,2} \\
\end{array}
\right]
=
\left [
\begin{array}{c}
\mathbf{u}_{1,1}^{[2]} \\
\mathbf{u}_{2,1}^{[2]} \\ \hline
\mathbf{0}_{6 \times 1} \\ \hline
\mathbf{u}_{1,2}^{[2]} \\
\mathbf{u}_{2,2}^{[2]} \\
\end{array}
\right ]
\notag =
\left [
\begin{array}{cccc}
\boldsymbol{\Phi} \otimes \mathbf{I}_{3} & \mathbf{0}_{3 \times 6} & \mathbf{0}_{3 \times 6} & \mathbf{0}_{3 \times 6}\\
\mathbf{0}_{3 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{I}_{3} & \mathbf{0}_{3 \times 6} & \mathbf{0}_{3 \times 6}\\
\mathbf{0}_{3 \times 6} & \mathbf{0}_{3 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{I}_{3} & \mathbf{0}_{3 \times 6} \\
\mathbf{0}_{3 \times 6} & \mathbf{0}_{3 \times 6} & \mathbf{0}_{3 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{I}_{3} \\ \hline
\mathbf{0}_{6} & \mathbf{0}_{6} & \mathbf{0}_{6} & \mathbf{0}_{6}
\\ \hline
\mathbf{I}_{6} & \mathbf{0}_{6} & \mathbf{0}_{6} & \mathbf{0}_{6}\\
\mathbf{0}_{6} & \mathbf{I}_{6} & \mathbf{0}_{6} & \mathbf{0}_{6}\\
\mathbf{0}_{6} & \mathbf{0}_{6} & \mathbf{I}_{6} & \mathbf{0}_{6}\\
\mathbf{0}_{6} & \mathbf{0}_{6} & \mathbf{0}_{6} & \mathbf{I}_{6}
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[2]} \\
\mathbf{s}_{2}^{[2]} \\
\mathbf{s}_{3}^{[2]} \\
\mathbf{s}_{4}^{[2]}
\end{array}
\right ] \in \mathbb{C}^{42}. \notag\end{aligned}$$ In Step 4, the overall transmit signal vector is given by $$\begin{aligned}
\mathbf{x}^{n} = \mathbf{x}_{1} + \mathbf{x}_{2} \notag\end{aligned}$$ where $n = 14$.
### Mode switching patterns at receivers {#mode-switching-patterns-at-receivers}
From , the time interval for transmitting block 1 is given by $1 \leq t \leq 4$. For this case, user 1 has two selection patterns and user 2 has one selection pattern, in which the associated channel matrices are given by $$\begin{aligned}
& \mathbf{H}_{1,j} = \mathbf{h}_{1,j} \in \mathbb{C}^{1 \times 3}, \ j = 1, 2, \label{eq:example_pattern} \\
& \mathbf{H}_{2,1} =
\left [\mathbf{h}_{2,1}^{T} \ \mathbf{h}_{2,2}^{T} \right ]^{T} \in \mathbb{C}^{2 \times 3} \notag\end{aligned}$$ respectively. As explained in Section \[achievability\_thm1\], when block 1 is transmitted, each user chooses the selection pattern of which index is the same as that of the currently transmitted sub-unit of his transmit signal vector. Since the indexes of the transmitted sub-unit of users 1 and 2 are 1, 1, 2, 2 and 1, 1, 1, 1 respectively for $1 \leq t \leq 4$, from , the received signal vectors of users 1 and 2 during $1 \leq t \leq 4$ are given by $$\begin{aligned}
& \mathbf{y}_{1,0} =
\left [
\begin{array}{c}
\begin{array}{cc}
\mathbf{H}_{1,1} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{H}_{1,1} \\
\mathbf{H}_{1,2} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{H}_{1,2} \\
\end{array}
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ]
+
\left [
\begin{array}{cccc}
\boldsymbol{\Phi} \otimes \mathbf{H}_{1,1} & \mathbf{0}_{1 \times 6} & \mathbf{0}_{1 \times 6} & \mathbf{0}_{1 \times 6} \\
\mathbf{0}_{1 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{1,1} & \mathbf{0}_{1 \times 6} & \mathbf{0}_{1 \times 6} \\
\mathbf{0}_{1 \times 6} & \mathbf{0}_{1 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{1,2} & \mathbf{0}_{1 \times 6} \\
\mathbf{0}_{1 \times 6} & \mathbf{0}_{1 \times 6} & \mathbf{0}_{1 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{1,2} \\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[2]} \\
\mathbf{s}_{2}^{[2]} \\
\mathbf{s}_{3}^{[2]} \\
\mathbf{s}_{4}^{[2]}
\end{array}
\right ] , \label{eq:example_y10}
\\ &
\mathbf{y}_{2,0} =
\left [
\begin{array}{c}
\begin{array}{cc}
\mathbf{H}_{2,1} & \mathbf{0}_{2 \times 3} \\
\mathbf{0}_{2 \times 3} & \mathbf{H}_{2,1} \\
\mathbf{H}_{2,1} & \mathbf{0}_{2 \times 3} \\
\mathbf{0}_{2 \times 3} & \mathbf{H}_{2,1} \\
\end{array}
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ]
+
\left [
\begin{array}{cccc}
\boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} \\
\mathbf{0}_{2 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} \\
\mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} & \mathbf{0}_{2 \times 6} \\
\mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} \\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[2]} \\
\mathbf{s}_{2}^{[2]} \\
\mathbf{s}_{3}^{[2]} \\
\mathbf{s}_{4}^{[2]}
\end{array}
\right ] \label{eq:example_y20}\end{aligned}$$ respectively.
The time interval for transmitting block 2 is given by $5 \leq t \leq 14$. First, we consider the time interval for transmitting $\mathbf{x}_{1,2}$, given by $5 \leq t \leq 6$. Note that this part corresponds to the desired signal part of block 2 for user 1 and the interference signal part of block 2 for user 2. Since $T_{1}|L_{1} = 0$, user 1 exploits one selection pattern repeatedly over the time interval $5 \leq t \leq 6$, in which the associated channel matrix is given by $$\begin{aligned}
\mathbf{H}_{1,3} = \mathbf{h}_{1,3} \in \mathbb{C}^{1 \times 3} \notag .\end{aligned}$$ On the other hands, user 2 exploits the same selection pattern used for receiving during block 1 over the time interval $5 \leq t \leq 6$, in which there is one selection pattern associated with $\mathbf{H}_{2,1}$ for this case. Then, user 2 receives the transmit signal for $5 \leq t \leq 6$ using the selection pattern associated with $\mathbf{H}_{2,1}$. As a result, from and , the received signal vectors of users 1 and 2 during $5 \leq t \leq 6$ are given by $$\begin{aligned}
\mathbf{y}_{1,1} =
\left [
\begin{array}{c}
\begin{array}{cc}
\mathbf{H}_{1,3} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{H}_{1,3} \\
\end{array}
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ],
\
\mathbf{y}_{2,1} =
\left [
\begin{array}{c}
\begin{array}{cc}
\mathbf{H}_{2,1} & \mathbf{0}_{2 \times 3} \\
\mathbf{0}_{2 \times 3} & \mathbf{H}_{2,1} \\
\end{array}
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ] \label{eq:example_x11}\end{aligned}$$ respectively. Next, we consider the time interval for transmitting $\mathbf{x}_{2,2}$, given by $7 \leq t \leq 14$. This part corresponds to the desired signal part of block 2 for user 2 and the interference signal part of block 2 for user 1. Since $T_{2}|L_{2} \neq 0$, user 2 exploits two ($L_{2}S_{2} = 2)$ selection patterns, which repeat four ($S_{1}(T_{1}-L_{1}) = 4$) times periodically over the time interval $7 \leq t \leq 14$. The associated channel matrices are given by $$\begin{aligned}
\mathbf{H}_{1,2,1} = \left [ \mathbf{h}_{2,3}^{T} \ \mathbf{h}_{2,1}^{T} \right ]^{T} \in \mathbb{C}^{2 \times 3}, \
\mathbf{H}_{1,2,2} = \left [ \mathbf{h}_{2,3}^{T} \ \mathbf{h}_{2,2}^{T} \right ]^{T} \in \mathbb{C}^{2 \times 3}. \notag\end{aligned}$$ On the other hands, user 1 exploits the same selection pattern used for receiving block 1 over the time interval $7 \leq t \leq 14$, in which the associated channel matrices are given in . When the interference signal part of block 2 is transmitted, user 2 chooses the selection pattern of which index is the same as that used to receive the first sub-unit of the alignment unit to which the currently transmitted sub-unit belongs. One can see that the sub-units transmitted for $7 \leq t \leq 10$ and $11 \leq t \leq 14$ is $\mathbf{u}_{1,3}^{[2]}$ and $\mathbf{u}_{2,3}^{[2]}$ respectively and user 1 exploits the selection pattern associated with $\mathbf{H}_{1,1}$ to receive $\mathbf{u}_{1,1}^{[2]}$ and the selection pattern associated with $\mathbf{H}_{1,2}$ to receive $\mathbf{u}_{2,1}^{[2]}$ in block 1. Hence, user 1 exploits the selection pattern associated with $\mathbf{H}_{1,1}$ for $7 \leq t \leq 10$ and the selection pattern associated with $\mathbf{H}_{1,2}$ for $11 \leq t \leq 14$. As a result, from and , the received signal vectors of user 1 and 2 during $7 \leq t \leq 14$ are given by $$\begin{aligned}
&
\mathbf{y}_{1,2} =
\left [
\begin{array}{cccccccc}
\mathbf{I}_{2} \otimes \mathbf{H}_{1,1} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6}\\
\mathbf{0}_{2 \times 6} & \mathbf{I}_{2} \otimes \mathbf{H}_{1,1} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6}\\
\mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{I}_{2} \otimes \mathbf{H}_{1,2} & \mathbf{0}_{2 \times 6}\\
\mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{I}_{2} \otimes \mathbf{H}_{1,2}\\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[2]} \\
\mathbf{s}_{2}^{[2]} \\
\mathbf{s}_{3}^{[2]} \\
\mathbf{s}_{4}^{[2]}
\end{array}
\right ], \label{eq:example_y12}
\\ &
\mathbf{y}_{2,2} =
\left [
\begin{array}{cccccccc}
\mathbf{H}_{2,2}' & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} \\
\mathbf{0}_{4 \times 6} & \mathbf{H}_{2,2}' & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} \\
\mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{H}_{2,2}' & \mathbf{0}_{4 \times 6} \\
\mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{H}_{2,2}' \\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[2]} \\
\mathbf{s}_{2}^{[2]} \\
\mathbf{s}_{3}^{[2]} \\
\mathbf{s}_{4}^{[2]}
\end{array}
\right ] \label{eq:example_y22}\end{aligned}$$ respectively, where $\mathbf{H}_{2,2}' = \operatorname{diag}(\mathbf{H}_{2,2,1},\mathbf{H}_{2,2,2}) \in \mathbb{C}^{4 \times 6}$.
### Interference cancellation and achievable LDoF
From , after cancelling all interference vectors in $\mathbf{y}_{1,0}$, user 1 has $$\begin{aligned}
\left [
\begin{array}{c}
\mathbf{y}_{1,0} - (\mathbf{I}_{4} \otimes \boldsymbol{\Phi}) \mathbf{y}_{1,2} \\ \hline
\mathbf{y}_{1,1} \\
\end{array}
\right ]
=
\left [
\begin{array}{cc}
\mathbf{H}_{1,1} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{H}_{1,1} \\
\mathbf{H}_{1,2} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{H}_{1,2} \\ \hline
\mathbf{H}_{1,3} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{H}_{1,3} \\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ] \label{eq_example1}\end{aligned}$$ Sorting the rows in , we have $$\begin{aligned}
\left [
\begin{array}{c}
\begin{array}{cc}
\mathbf{H}_{1} & \mathbf{0}_{ 3} \\
\mathbf{0}_{3} & \mathbf{H}_{1} \\
\end{array}
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[1]} \\
\mathbf{s}_{2}^{[1]}
\end{array}
\right ] \label{eq_example2}\end{aligned}$$ Obviously, user 1 can obtain $\mathbf{s}_{1}^{[1]}$ and $\mathbf{s}_{2}^{[1]}$ from almost surely.
Similarly, from , after cancelling all interference vectors in $\mathbf{y}_{2,0}$, user 2 has $$\begin{aligned}
\left [
\begin{array}{c}
\mathbf{y}_{2,0} - \mathbf{1}_{2 \times 1} \otimes \mathbf{y}_{2,1} \\ \hline
\mathbf{y}_{2,2} \\
\end{array}
\right ]
=
\left [
\begin{array}{cccccccc}
\boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} \\
\mathbf{0}_{2 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} \\
\mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} & \mathbf{0}_{2 \times 6} \\
\mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \mathbf{0}_{2 \times 6} & \boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} \\ \hline
%
\mathbf{H}_{2,2}' & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} \\
\mathbf{0}_{4 \times 6} & \mathbf{H}_{2,2}' & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} \\
\mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{H}_{2,2}' & \mathbf{0}_{4 \times 6} \\
\mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{0}_{4 \times 6} & \mathbf{H}_{2,2}' \\
\end{array}
\right ]
\left [
\begin{array}{c}
\mathbf{s}_{1}^{[2]} \\
\mathbf{s}_{2}^{[2]} \\
\mathbf{s}_{3}^{[2]} \\
\mathbf{s}_{4}^{[2]}
\end{array}
\right ] \label{eq_example3}\end{aligned}$$ Then, can be decomposed into four segments as in the following. $$\begin{aligned}
\left [
\begin{array}{cc}
\boldsymbol{\Phi} \otimes \mathbf{H}_{2,1} \\
\mathbf{H}_{2,2}'\\
\end{array}
\right ]
\mathbf{s}_{i}^{[2]}
=
\left [
\begin{array}{cc}
\phi_{11} \mathbf{h}_{2,1} & \phi_{12} \mathbf{h}_{2,1} \\
\phi_{11} \mathbf{h}_{2,2} & \phi_{12} \mathbf{h}_{2,2} \\
\mathbf{h}_{2,3} & \mathbf{0}_{1 \times 3} \\
\mathbf{h}_{2,1} & \mathbf{0}_{1 \times 3} \\
\mathbf{0}_{1 \times 3} & \mathbf{h}_{2,3} \\
\mathbf{0}_{1 \times 3} & \mathbf{h}_{2,2} \\
\end{array}
\right ]
\mathbf{s}_{i}^{[2]}, & \ \ \ i = 1, 2, 3, 4 \label{eq_example4}\end{aligned}$$ It can be easily shown that user 2 can obtain $\mathbf{s}_{i}^{[2]}$ for all $i$ from almost surely.
As a result, the transmitter delivers $6$ information symbols to user 1 and $24$ information symbols to user 2 during $14$ time slots. Consequently, the achievable sum LDoF is given by $\frac{15}{7}$.
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[^1]: M. Yang and D. K. Kim were funded by the Ministry of Science, ICT $\&$ Future Planning (MSIP), Korea in the ICT R$\&$D Program 2014. S.-W. Jeon was supported in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) \[NRF-2013R1A1A1064955\].
[^2]: M. Yang and D. K. Kim are with the School of Electrical and Electronic Engineering, Yonsei University, Seoul, South Korea (e-mail: {navigations, dkkim}@yonsei.ac.kr).
[^3]: S.-W. Jeon is with the Department of Information and Communication Engineering, Andong National University, Andong, South Korea (e-mail: [email protected]).
[^4]: Certain users experience smaller coherence time/bandwidth than others (See [@Jafar:12] for more details).
[^5]: All users experience independent block fading with the same coherence time, but different offsets (See [@Zhou:12] for more details).
[^6]: An inequality $f(\mathbf{x}) \leq 0 $ is said to be active at $\mathbf{x}^{*}$ if $f(\mathbf{x}^{*}) = 0$ [@chong2013introduction Definition 20.1].
[^7]: The maximum value of $m$ depends on $i$ and $j$, see the definition of $\{\mathbf{G}_l\}_{l\in\mathcal{K}}$.
|
---
author:
- |
Rutger Boels\
The Mathematical Institute, University of Oxford\
24-29 St. Giles, Oxford OX1 3LP, United Kingdom\
${[email protected]}$
bibliography:
- 'lib.bib'
title: 'A quantization of twistor Yang-Mills theory through the background field method'
---
Introduction
============
Even more than 50 years after its introduction, Yang-Mills theories in four space-time dimensions continue to be a fascinating and rich area of research. Applications range from the very physical in the standard model to the very theoretical in the study of geometric invariants of (four) manifolds. However, it is fair to say that they are, in some sense, not very well understood. Their non-perturbative behaviour for instance is not under analytic control. The only known exceptions to this involve either less space-time dimensions or supersymmetry. In other words, in situations where there is an extra underlying symmetry to exploit. In addition, also the perturbative behaviour of Yang-Mills theory for the calculation of scattering amplitudes is a very active area of research still because of its computational complexity when using standard methods even at tree level, especially when large numbers of external particles are involved. That is not to say these scattering amplitudes are not interesting: in order to discover new physics at LHC for instance one needs a good quantitative control of the ‘old’ physics contained in these amplitudes. However, for the strong force for instance the coupling constant is not parametrically small in the region of interest and one needs to calculate loop corrections which is prohibitively complex when using ordinary Feynman diagrams. Then one has to resort to other calculational methods.
However, there are several results in the literature which show that even ordinary perturbation theory based on the non-supersymmetric space-time Yang-Mills action misses part of an underlying structure of the theory. The first of these is the classic result of Parke and Taylor [@parktaylor] that a particular class of amplitudes have a simple expression at tree level. These are the amplitudes which involve two gluons of one, and arbitrarily many gluons of the opposite helicity. This result was related to a construction on twistor space by Nair [@nair]. The connection between perturbative amplitudes and twistor space was further elaborated upon by Witten [@witten], who showed that it can be extended at least to all tree level amplitudes. In addition, he also speculated that these results could be derived from an underlying topological string theory which should be equivalent to $\mathcal{N}=4$ super Yang-Mills theory. Note that Witten’s twistor string proposal can be understood as an attempt to answer the question *why* some results in perturbative Yang-Mills theory like the Parke-Taylor amplitude are so simple.
This inspired a lot of activity in the last few years, which up to now has mainly focused on obtaining new calculational techniques for scattering amplitudes. Two outstanding developments here are the recursive techniques of Britto, Cachazo, Feng and Witten [@BCFW] and the Feynman-like rules of Cachazo, Svrček and Witten [@CSW]. Both these techniques rearrange the ordinary perturbative expansion of Yang-Mills theories into something much more simple. A natural physical explanation for this possibility is that there is some underlying symmetry in the problem which is not manifest in the usual Yang-Mills Lagrangian. In previous work [@lionel1; @us; @usII] this natural conjecture was confirmed by constructing explicitly an action on twistor space with a (linear) gauge symmetry which is larger than the ordinary gauge symmetry. In [@us; @usII] it was shown that both the CSW rules and text-book perturbation theory can be obtained as Feynman rules for the twistor action[^1].
In related work, Mansfield [@mansfield] obtained an action which reproduces the CSW rules on-shell by a non-linear and non-local canonical field transformation from the light-cone formulation of four-dimensional Yang-Mills. It is expected that that field transformation is exactly the space-time equivalent of the twistor gauge transformation of [@usII]. In this article we provide some evidence for this claim; a full treatment will appear elsewhere. In more physical terms, from a space-time point of view we conjecture the twistor action provides the right auxiliary fields to linearise a specific non-linear space-time symmetry and is in that sense a ‘superfield’ formulation[^2].
A remaining open problem in all this is extending the analysis to loop level. Ordinary text-book perturbation theory is fine, but in straightforwardly applying the CSW rules to loop level [@qmul], one encounters problems as at one loop the diagrams generically only reproduce the cut-constructible pieces of amplitudes. This is a problem in particular for non-supersymmetric gauge theories, where amplitudes are known to involve more than just cut-constructible parts. Recently some interesting light was shed on this in [@BSTrecent], although a full understanding is still lacking. From the view of twistor Yang-Mills theory however, we seem to have a gauge theory which interpolates between a well-defined and a not-so-well-defined perturbation theory. In this article the twistor action will be quantised in a gauge close to, but not equivalent to space-time gauge where all the divergences are simply four-dimensional. In this gauge the regularization and renormalization properties will be shown to reduce to standard space-time problems, which can be resolved by standard techniques. This is part of the main message of this paper: standard four dimensional field theory techniques extend to Yang-Mills theories on twistor space.
This article is structured as follows: We will begin by giving a brief review and clarification of the twistor action approach to Yang-Mills theory and point out some of its more salient features. After this the background field method will be set up, with the quantum field in the background field version of the space-time gauge. This will lead to specific Feynman rules which can then be identified with the space-time Yang-Mills rules derived from the Chalmers and Siegel action. Put differently, the calculation shows that the space-time quantum effective action can be lifted to twistor space by a simple lifting prescription. In particular, the $\beta$ function calculation, performed explicitly in an appendix for the Chalmers and Siegel action, lifts directly, as well as renormalizability arguments. The next section contains some investigations into using the formalism to calculate scattering amplitudes. In an appendix an alternative gauge for obtaining CSW rules is constructed.
In this article dotted and undotted Greek letters from the beginning of the alphabet indicate spinor indices. Our spinor conventions are $\omega \cdot \lambda = \omega^\alpha \lambda_{\alpha} = \omega^\alpha \lambda^\beta \epsilon_{\beta \alpha}$, $\pi \cdot \mu = \pi_{{\dot{\alpha}}} \mu^{{\dot{\alpha}}} = \pi_{{\dot{\alpha}}} \mu_{{\dot{\beta}}} \epsilon^{{\dot{\beta}}{\dot{\alpha}}}$. We normalise the isomorphism between the cotangent bundle and the spin-bundles such that $g_{\mu \nu} = \frac{1}{2} \epsilon_{\alpha \beta} \epsilon_{{\dot{\alpha}}{\dot{\beta}}}$, where (symbolically) $\mu = \alpha{\dot{\alpha}}$, $\nu = \beta {\dot{\beta}}$. Furthermore, fields on twistor space are denoted by Roman symbols, while space-time fields are bold. Quantum and background fields will be denoted by lower and upper case letters respectively. Finally, we normalise the natural volume-form on a ${\mathbb{CP}}^1$ such that it includes a factor of $\frac{1}{2 \pi i}$.
Twistor Yang-Mills theory
=========================
In this section the twistor formulation of Yang-Mills theory will be reviewed and clarified. Although this is not immediately obvious from the exposition here, what is discussed in this section is a Euclidean, *off-shell* version of the Penrose-Ward correspondence and the interested reader is referred to [@woodhouse] for an introduction to twistor space useful from the point of view of this paper.
Some twistor geometry
---------------------
As usual, four-dimensional space-time arises in the twistor program as the space of holomorphic lines embedded in ${\mathbb{CP}}^3$’. The prime indicates the removal of a ${\mathbb{CP}}^1$, which is necessary mathematically to obtain interesting cohomology as ${\mathbb{CP}}^3$ is compact, and physically to obtain a notion of a point at ‘infinity’ which is needed to define scattering amplitudes. Consider homogeneous coordinates $(\omega^{\alpha}, \pi_{{\dot{\alpha}}})$ for a point in ${\mathbb{CP}}^3$’ where $\alpha$ and ${\dot{\alpha}}$ run from $1$ to $2$. Then a holomorphic line corresponds to an embedding equation $$\omega^{\alpha} = x^{\alpha {\dot{\alpha}}} \pi_{{\dot{\alpha}}}.$$ Note that this equation makes sense since the symmetry group of twistor space contains naturally two $SL(2,\mathbb{C})$ subgroups which can and will be identified with the chiral components of the complexified Lorentz group. In order for this equation to be solved for $x$ a reality condition is needed. In this paper we will be interested in Euclidean signature for which Euclidean spinor conjugation is needed, $$\widehat{\left(\begin{array}{c} \pi_1 \\ \pi_2 \end{array} \right)} = \left(\begin{array}{c} - \bar{\pi}_2 \\ \bar{\pi}_1 \end{array} \right).$$ The main difference to the Lorentzian conjugation is the absence of the application of the parity operator which interchanges spin bundles. In Euclidean signature there is a *unique* point $x$ associated to every pair $\omega, \pi$, $$x^{\alpha {\dot{\alpha}}} = \left(\frac{\omega^{\alpha} \hat{\pi}^{{\dot{\alpha}}} - \hat{\omega}^{\alpha} \pi^{{\dot{\alpha}}} }{\pi \hat{\pi}} \right)$$ this equation exposes twistor space for Euclidean signature space-time as a $CP^1$ fibre-bundle over space with the above equation furnishing the needed projection, which will be denoted by $p$.
In the following explicit coordinates $x^{\alpha {\dot{\alpha}}}, \pi_{{\dot{\alpha}}}, \hat{\pi}_{{\dot{\alpha}}}$ will be used to parametrize the twistor space. This choice leads to a basis of anti-holomorphic one-forms, $$\bar{e}_0 = \frac{\hat{\pi}^{{\dot{\alpha}}} d \hat{\pi}_{{\dot{\alpha}}}}{(\pi \hat{\pi})^2} \quad \bar{e}^\alpha = \frac{ dx^{\alpha {\dot{\alpha}}} \hat{\pi}_{{\dot{\alpha}}}}{(\pi \hat{\pi})}$$ which is naturally dual to a set of $(0,1)$ vectors, $${\bar\partial}_{0} \equiv (\pi \hat{\pi}) \pi_{{\dot{\alpha}}} \frac{\partial}{\partial \hat{\pi}_{{\dot{\alpha}}}} \quad {\bar\partial}_{\alpha} \equiv \pi^{{\dot{\alpha}}} \frac{\partial}{\partial x^{\alpha {\dot{\alpha}}}}.$$ With the above basis of one-forms, one-form fields $A$ can be expanded as $$A = \bar{e}^\alpha A_\alpha+ \bar{e}^0 A_0$$ Note that $A_{\alpha}$ and $A_0$ have holomorphic weight $+1$ and $+2$ respectively compared to the original weight of $A$. In the following the word ‘weight’ will always refer to holomorphic weight.
Lifting fields to twistor space
-------------------------------
One of the successes of the twistor program has always been the fact that on-shell fields, including self-dual fields, correspond to certain cohomology classes on twistor space. However, here we will be interested in lifting off-shell fields from space-time to twistor space, clarifying the procedure first employed in [@lionel1]. The objective is to lift Yang-Mills theory, such as captured in the Chalmers and Siegel action $$S_{\textrm{CS}} = {\, \mathrm{tr}\;}\int d^4\!x\, \frac{1}{2} {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}} F^{{\dot{\alpha}}{\dot{\beta}}}[{\mathbf{A}}] - \frac{1}{4} {\, \mathrm{tr}\;}\int d^4x\, {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}} {\mathbf{B}}^{{\dot{\alpha}}{\dot{\beta}}}$$ from Euclidean space-time to twistor space. First of all, the self-dual two form ${\mathbf{B}}$ lifts as $${\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}} = \int_{{\mathbb{CP}}^1}{\mathrm{d} k}\, H^{-1} B_0 H \pi_{{\dot{\alpha}}} \pi_{{\dot{\beta}}}$$ Here ${\mathrm{d} k}$ is the natural weightless volume-form on ${\mathbb{CP}}^1$ and $B_0$ is the zeroth component of a weight minus 4, anti-holomorphic form. Anti-holomorphic $n$-forms will be denoted by $(0,n)$. $H$ is a holomorphic frame of the gauge bundle over $p^{-1}(x)$ such that the covariant derivative of it vanishes on the sphere, ${\bar\partial}_A H|_{p^{-1}(x)}=0$. This covariant derivative involves a connection of the Riemann sphere which is always trivial in perturbation theory[^3]. Denote this connection (or more precisely, it’s $(0,1)$ part) as $A_0$. With this input the Chalmers and Siegel action becomes, $$\begin{aligned}
\label{eq:twacthalfway}
S_{CS} = {\, \mathrm{tr}\;}\int d^4\!x {\mathrm{d} k}\, & \frac{1}{2} B_0 H F_{{\dot{\alpha}}{\dot{\beta}}}(x)[{\mathbf{A}}] H^{-1} \pi^{{\dot{\alpha}}} \pi^{{\dot{\beta}}} - \nonumber \\
& \frac{1}{4} {\, \mathrm{tr}\;}\int d^4x {\mathrm{d} k}_1 {\mathrm{d} k}_2\, H_1^{-1} B^{(1)}_0 H_1 H_2^{-1} B^{(2)}_0 H_2 (\pi_{1} \pi_2)^2\end{aligned}$$ Now we would like to lift the gauge field ${\mathbf{A}}_{\mu}$ to twistor space. We already know from basic twistor theory (see [@woodhouse] for instance) that it is represented by a weight $0$ $(0,1)$ form, say $A$, which is the pull-back of the space-time connection to the twistor space. If this form is ${\bar\partial}$ closed, then it corresponds to an anti-self-dual connection on space-time. The curvature of this form naturally splits into a curvature tensor involving only space-directions, say $F_{\alpha \beta} = [{\bar\partial}_\alpha + A_\alpha, {\bar\partial}_\beta + A_\beta]$ and two curvature tensors with one leg along the fibre $F_{0 \alpha} = [{\bar\partial}_0 + A_0, {\bar\partial}_\alpha + A_\alpha] $. Since there is no physical interpretation of the latter curvature and since we want $A$ to be just the pull-back of the physical space-time connection, it will be set to zero: we will study gauge connections on twistor space such that $F_{0 \alpha}$ vanishes. As this is a weight $3$ operator, the condition it vanishes can be added to the action with a Lagrange multiplier of weight $-3$, $$\label{eq:accons}
S \,+\!= \frac{1}{2} \int d^4\!x {\mathrm{d} k}\, B^{\alpha} F_{0 \alpha}$$ This constraint also gives a neat way of lifting the vector potential $A$: For calculational ease, let $F_{0\alpha}$ act on a field in the fundamental, $$F_{0\alpha} \phi = [({\bar\partial}_A)_{0}, ({\bar\partial}_A)_{\alpha}] \phi$$ Since $A_0$ is pure gauge, it can be always be gauged away. This is implemented through the use of the holomorphic frames $H$. We arrive at $$F_{0\alpha} \phi = {\bar\partial}_{0}\left(H^{-1} ({\bar\partial}_\alpha + A_{\alpha}) H \right) H^{-1} \phi$$ The constraint sets this quantity to be zero. The quantity in brackets is then a holomorphic function of weight one and therefore $$\label{eq:vectcohom}
H^{-1}\left( {\bar\partial}_\alpha + A_{\alpha} \right)H = {\mathbf{A}}_{\alpha {\dot{\alpha}}}(x) \pi^{{\dot{\alpha}}}$$ for some vector field ${\mathbf{A}}_{\alpha {\dot{\alpha}}}$ which is only a function of $x$. This can easily be inverted to give $$\label{eq:eqforpentranA}
{\mathbf{A}}_{\alpha {\dot{\alpha}}}(x) = \int {\mathrm{d} k}H^{-1} \left({\bar\partial}_\alpha + A_{\alpha} \right)H \frac{\hat{\pi}_{{\dot{\alpha}}} }{\pi \hat{\pi}}$$ In the same way as above, we easily derive $$\epsilon^{\alpha \beta} F_{\alpha \beta}(x, \pi) = H F[{\mathbf{A}}]_{{\dot{\alpha}}{\dot{\beta}}}(x) H^{-1} \pi^{{\dot{\alpha}}} \pi^{{\dot{\beta}}}$$ With this expression taken into account and the constraint in (\[eq:accons\]) added, the action (\[eq:twacthalfway\]) becomes $$\begin{aligned}
\label{eq:twymactionXXX}
S = \frac{1}{2} & \int d^4\!x {\mathrm{d} k}B_0 \left({\bar\partial}^{\alpha} A_{\alpha} + g A^{\alpha} A_{\alpha} \right) + B^{\alpha} \left({\bar\partial}_{\beta} A_0 - {\bar\partial}_0 A_\beta + g[A_\beta,A_0]\right) \nonumber\\
& - \frac{1}{4} \int d^4\!x {\mathrm{d} k}_1 {\mathrm{d} k}_2 H^{-1}_1 B^0(\pi_1) H_1 H^{-1}_2 B^0(\pi_2) H_2 (\pi_1 \pi_2)^2 (\pi_{1} \pi_2)^2\end{aligned}$$
This action has a clear geometrical expression on twistor space, as the fields $A_0$, $A_\alpha$, $B_0$ and $B_\alpha$ can naturally be combined into anti-holomorphic one forms $A$ and $B$ of weight $0$ and $-4$ respectively, $$\begin{aligned}
\label{eq:twymaction}
S[A,B] = \frac{1}{2} &\int_{{{\mathbb{PT}}}} D^3Z \wedge B \wedge \left({\bar\partial}A + A \wedge A \right) -\nonumber \\ & \frac{1}{4} \int_{{{\mathbb{PT}}}\times_\mathbb{M} {{\mathbb{PT}}}}
{\, \mathrm{tr}\;}H_1^{-1}B_1H_1\wedge H_2^{-1}B_2H_2\wedge D^3Z_1\wedge D^3Z_2\end{aligned}$$ where ${{\mathbb{PT}}}$ is projective twistor space ${\mathbb{CP}}^3$’ with a space-time point removed, ${{\mathbb{PT}}}\times_\mathbb{M} {{\mathbb{PT}}}=\{(Z_1,Z_2)\in{{\mathbb{PT}}}\times{{\mathbb{PT}}}|p(Z_1)=p(Z_2)\}$ with $p$ the projection map and subscript 1 or 2 denotes dependence on $Z_1$ or $Z_2$. This form of the action is the restriction to the $\mathcal{N}=0$ fields of the $\mathcal{N} =4$ form of the twistor Yang-Mills action as studied in [@us; @usII] and this action appeared in this form first in [@lionel1], although there only the twistor lift of $B$ was studied.
Gauge invariances
-----------------
The twistor action is invariant under $$\label{eq:gaugeinv_reg}
A \rightarrow A+ {\bar\partial}_A \chi \quad B \rightarrow B+ g [B,\chi] + {\bar\partial}_A \omega$$ which can either be verified explicitly, or inferred directly from the $\mathcal{N}=4$ formulation. Here $\chi$ and $\omega$ are functions on projective twistor space of weight $0$ and $-4$ respectively. This apparently innocuous remark is actually very important: $\chi$ is a function of $6$ real variables, instead of the usual $4$. Hence this formulation of Yang-Mills theory has *more* gauge symmetry than the usual one. It is the existence of this symmetry which is the underlying physical reason for the existence of MHV methods in general. For instance, the Parke-Taylor formula (and its supersymmetric analogs) can be viewed as a consequence of this symmetry, as shown in [@usII].
Note that in the ‘lifting picture’ the gauge symmetries have quite dissimilar origins: the gauge symmetry in $A$ is in effect space-time gauge symmetry + gauge symmetry on the $P_1$ for $A_0$ and $A_\alpha$ separately, but since $F_{0\alpha}$ vanishes, this can be enlarged to the symmetry group above. The gauge invariance in $B_0$ is a consequence of some leeway in the ‘lifting’ formula. The full gauge symmetry in $B$ is actually the most interesting since it is a direct consequence of ‘quantising with constraints’: when one quantises a theory with constraints, in a real sense also the momentum conjugate to the constraint must be eliminated. This is familiar from the usual setup of gauge theories as explained in, for instance, [@itzyksonzuber] as this leads to the usual gauge fixing procedure. The gauge symmetry associated to $B$ in the above action is generated by $F_{0 \alpha}$ and gauge fixing this in the usual manner is therefore the right procedure by the same calculation as for the ordinary gauge symmetry.
The same procedure as employed here for the gluon can be extended readily to spin-0 and spin-$\frac{1}{2}$ fields in an obvious fashion. The lifting of these fields will entail separate new gauge invariances. One wants to require full gauge invariance with respect to these as this is necessary to allow an invertible field transformation. Requiring this leads to additional towers of $B^2$ like terms in the action through the Noether procedure. These will generate, in gauge which will be discussed in the next section, additional towers of MHV-like vertices and amplitudes as was shown in [@usII].
Quantization generalities
-------------------------
Standard path integral quantization of the twistor version of Yang-Mills theory will involve a gauge choice as this is needed to invert the kinetic operator. In this section some some general aspects of this will be explored. Before the consequences of gauge invariance will be explored it is perhaps useful to point out some features which are apparent in this form of the action. First of all the mass-dimension of the fields as counted by space-time derivatives are rather odd, as they are $$[A_0] = 0 \quad [A_\alpha] = 1 \quad [B_0] = 2 \quad [B_A] = 2$$ The vanishing dimension of the field $A_0$ is worrying as one expects this will lead to infinite series of possible counterterms. Very naively, this action is not power-counting renormalizable! However, it is also clear the quadratic part of this action is definitely not canonical and involves the fibre coordinates, which makes power counting non-standard. This is related to a second comment: although an action on a six-dimensional space is studied for a four-dimensional theory, one does not expect Kaluza-Klein modes, since that argument requires a *canonical* six-dimensional kinetic term. Thirdly, this action is non-local. However, the non-locality is restricted to the ${\mathbb{CP}}^1$ fibre, not on space-time, so this is not a problem provided this remains true in perturbation theory.
### Space-time gauge
The gauge invariance of the action can be exploited in multiple ways. The same techniques as in [@us; @usII] also apply here. In particular, when restricted to a gauge in which $$\label{eq:spacetimegauge_reg}
{\bar\partial}_0^\dagger a_0 =0 \quad \quad {\bar\partial}_0^\dagger b_0 =0$$ the above action reduces down to the Chalmers and Siegel form of the non-supersymmetric Yang-Mills action, which is perturbatively equivalent to the usual one. This gauge will be referred to as ‘space-time gauge’. Note that it does not fix the complete gauge symmetry contained in (\[eq:gaugeinv\_reg\]). The residual gauge symmetry is exactly the usual space-time one, which of course has to be fixed further in order to quantise the theory. Any ordinary gauge choice will do for this. The calculation in [@us] will not be reproduced here, as it is an obvious specialisation of the argument in section \[sec:bgtwym\]. In effect this article extends the above observation to a general class of gauges.
As the action reduces to the usual space-time action in a particular partial gauge (and the associated ghosts can be shown to decouple), it should be obvious that path integral quantization of this theory is perfectly fine in this gauge: here it is just the usual quantization and *every* perturbative calculation can therefore be reproduced using the twistor action to all orders in perturbation theory. The real question is if the same holds true in other gauges. Put differently, the question is whether the extra symmetry found in our formulation of Yang-Mills at tree level is in some way anomalous. More precisely, the question is whether physical quantities like scattering amplitudes are invariant under the extra gauge symmetry.
### On quantization in CSW gauge
In addition, as the Queen Mary group has shown [@qmul], the most straightforward application of the Cachazo-Svrcek-Witten [@CSW] rules already does calculate some loop level effects: these rules can be used at the one-loop level to calculate the cut-constructible parts of scattering amplitudes. As shown explicitly in [@usII] these CSW rules can be derived directly from the twistor action by changing gauge to the axial (space-cone) like gauge $$\label{eq:CSWgauge}
\eta^{\alpha} A_{\alpha} =0 \quad \eta^{\alpha} B_{\alpha} =0$$ which we will call CSW gauge. Here $\eta$ is an arbitrary spinor, normalised such that $\eta \hat{\eta} =1$. That the CSW rules can be derived in this way is a possibility which is clear from the original article[@CSW]: the twistor action in the $\mathcal{N}=4$ case can also be obtained as the reduction to single trace-terms of the conjectured effective action of Witten’s twistor-string theory. Hence the original derivation of the rules applies here and this is fleshed out in [@usII]. Therefore there are already two gauges in which our twistor formulation makes some sense at loop level: we have full consistency for the space-time gauge and partial consistency (at least self-consistency) for the CSW gauge.
However, the CSW rules do not calculate full amplitudes in non-supersymmetric Yang-Mills theories. A particular example of this drawback are the amplitudes with all helicities equal. These are zero at tree level for all pure Yang-Mills theories and vanish at loop level for supersymmetric ones. If one picks a preferred helicity for the MHV amplitudes used in the CSW rules, say the set which is ‘mostly minus’, then it is obvious there is no one-loop diagram which has only minus on the external legs, although there are diagrams for ‘only’ plus.[^4] In the approach of Mansfield these missing diagrams are thought to be generated by a Jacobian factor which appears if the canonical transformation is regulated properly. This scenario was supported by the calculations in [@BSTrecent] where a modified, non-canonical transformation was employed. This however invalidates the equivalence theorem and calculating amplitudes should then be done by combining contributions from many different sources.
In the twistor action approach a non-trivial Jacobian might translate to a partly anomalous gauge symmetry: the lifting formulae only guarantee space-time gauge symmetry, but nothing beyond that. A more mundane explanation would be the fact that there are regularization issues when one tries to quantise the twistor action at loop level in the CSW gauge. The most natural regularization treats the six dimensional twistor space as $(4-2\epsilon) + 2$: one performs a half-Fourier transform w.r.t. the space-time directions and employs dimensional regularization there. However, this already encounters several problems. It is well-known in four dimensions for instance that axial gauges need careful regulating [@leibbrandt] since Fourier transforms of $$\sim \frac{1}{\eta^{\alpha} \pi^{{\dot{\alpha}}} p_{\alpha {\dot{\alpha}}} }$$ are ill-defined and a pole prescription is necessary. This problem afflicts the twistor action, since in CSW gauge the propagators can be shown to behave like $:A_0 B_0: = \frac{\delta(\eta \pi p)}{p^2}$, and this delta function is obtained as $$\delta(\eta \pi p) \sim {\bar\partial}_0 \frac{1}{\eta \pi p }$$ A drawback of implementing the Mandelstam-Leibbrandt regulating prescription is that it leads to very large algebraic complexity. In addition, there is the usual problem using chiral indices in any dimensional regularization scheme. In a supersymmetric theory one would rely on dimensional reduction to keep all the spinor algebra in four dimensions. This is however known to be in grave danger of being inconsistent (non-unitary) in non-supersymmetric theories which leads to the inclusion of $\epsilon$-scalars. Furthermore, as we are regulating a gauge theory, Pauli-Villars is flawed as this would break gauge invariance, apart from the non-generic form mass-terms on twistor space take.
In appendix \[app:difgaugcond\] a gauge condition equivalent to the CSW gauge at tree level is constructed using ’t Hooft’s trick which has slightly better loop behaviour, but which remains problematic for basically the same reasons indicated above.
Background field method for twistor Yang-Mills {#sec:bgtwym}
==============================================
The existence of the CSW rules and their loop-level application suggests a trick: If one is given a gauge theory with two gauges, one of which is well-defined at loop level and one which makes results most transparent, the obvious game to play is the quantization of this theory using the background field method. Within this method the quantum effective action is calculated by integrating over the quantum field, and when interpreted as the generating functional of 1PI diagrams, this can be used to calculate S-matrix elements for the background fields. As pointed out in [@abbottgrisaru] in the context of four-dimensional field theory, one can put the background field into a different gauge than the quantum field. As our action is very close to the four-dimensional one, it is reasonable to expect the same trick can be performed here. Hence in this paper the twistor Yang-Mills action will be studied with the quantum field in the background field version of the space-time gauge. Then some properties of the S-matrix will be studied if the background field is put into the CSW gauge. A very important product of the analysis is the study of renormalization properties of our action. Actually, this was the original reason to study background field formulations, both in the literature as for us. In particular a background field calculation will also easily yield the $\beta$ function of the theory, and it is this purpose for which the background field method is usually employed.
Since the twistor action reduces to the Chalmers and Siegel formulation of Yang-Mills theory in ordinary space-time gauge, as a suitable warm-up for the calculation about to be presented one should therefore first treat that action in the background field formalism. We refer the reader to appendix \[app:modelcalc\] for the details of that calculation. Below the same procedure is performed for twistor Yang-Mills, which is shown to reduce the calculation down to the space-time one worked out in the appendix.
Begin by splitting the fields $\tilde{A}$ and $\tilde{B}$ into a background and quantum part $$\label{eq:splitfieldsTWYM}
\tilde{A} = A + a \quad
\tilde{B} = B + b$$ indicated by capital and lower case letters respectively. The action will be expanded in quantum fields, ignoring the terms linear in the quantum field as they will not contribute to the quantum effective action which is the generating functional of 1PI diagrams. Subsequently integrating out the quantum field requires a gauge choice. As the action is invariant under $$\begin{aligned}
\tilde{A} &\rightarrow& \tilde{A} + {\bar\partial}_{\tilde{A}} \chi \\
\tilde{B} &\rightarrow& B + [\tilde{B}, \chi] + {\bar\partial}_A \omega\end{aligned}$$ there are two obvious choices one can make for the symmetry transformations of quantum and background field: $$\begin{aligned}
\label{eq:gaugesym1}
A &\rightarrow A + {\bar\partial}_{A} \chi & A &\rightarrow A\nonumber \\
B &\rightarrow B + [B, \chi] + {\bar\partial}_{ A} \omega & B &\rightarrow B \nonumber \\
a &\rightarrow a + [a, \chi] & a &\rightarrow a + {\bar\partial}_{A+a} \nonumber \chi\\
b &\rightarrow b + [b, \chi] + [a,\omega]& b &\rightarrow b + [B+b, \chi]+ {\bar\partial}_{A+a} \omega\end{aligned}$$ The objective is to completely fix the second symmetry, while keeping the first (referred to as background gauge symmetry). As the quantum fields transform in the adjoint under the background symmetry, writing elliptic gauge fixing conditions such as Lorenz gauge and the background field version of space-time gauge (\[eq:spacetimegauge\_reg\]) requires one to lift the derivatives to covariant derivatives. In addition, although its almost immaterial to our calculation, one should promote the Lagrange multiplier to transform in the adjoint of the background symmetry. Hence the ghost and anti-ghost will also transform in that adjoint by the usual BRST symmetry.
Gauge fixing the background field
---------------------------------
The background version of space-time gauge (\[eq:spacetimegauge\_reg\]) reads: $$\label{eq:spacetimegauge_BG}
{\bar\partial}_0^\dagger(H[A_0] a_0 H[A_0]^{-1}) =0 \quad \quad {\bar\partial}_0^\dagger(H[A_0] b_0 H[A_0]^{-1}) =0$$ Note that this gauge condition involves a choice of metric on a ${\mathbb{CP}}^1$. These conditions are solvable since Yang-Mills connections on a ${\mathbb{CP}}^1$ are trivial. We can therefore transform to the frame where $A_0$ is zero, solve the equations and the transform back. Since both $a_0$ and $b_0$ are part of a one form on ${\mathbb{CP}}^1$ they are automatically ${\bar\partial}$ closed and by the gauge condition co-closed in the frame where $A_0$ is zero. They are therefore harmonic. As $a_0$ has weight zero and $b_0$ has weight $-4$, there are no non-trivial harmonic forms $a_0$ and there is a two dimensional space of harmonic forms $b_0$ by a standard cohomology calculation. Hence we obtain $$a_0 = 0 \quad b_0 = \frac{3 H {{\mathbf b}}_{{\dot{\alpha}}{\dot{\beta}}}(x) H^{-1} \hat{\pi}^{{\dot{\alpha}}} \hat{\pi}^{{\dot{\beta}}} }{(\pi \hat{\pi})^2}$$ where $H$ are the holomorphic frames encountered before and we apologise for the appearance of a normalisation factor proportional to $(2 \pi i)$. Note that these frames are functionals of the background field $A_0(x,\pi)$. This solution can be put back into the action. The field $b_\alpha$ is now a Lagrange multiplier for a very simple condition, $$({\bar\partial}_0 +A_0) a_\alpha = 0.$$ This can be solved in the same way as before, which yields $$a_\alpha = H {{\mathbf a}}_{\alpha {\dot{\alpha}}}(x) H^{-1} \pi^{{\dot{\alpha}}}.$$ In the original derivation [@us] the ghosts which came from the space-time gauge fixing decoupled. In the present context, due to the presence of a coupling to the background field in both the gauge fixing condition \[eq:spacetimegauge\_BG\] and the symmetry transformations (of the symmetry we are trying to fix!), the argument is slightly more convoluted. First note that we can still argue that the coupling of the *quantum* fields to ghosts is off-diagonal, so that these quantum fields can safely be ignored. That leaves a possible one-loop contribution to the effective action, since the diagonal part of the ghost action takes the form $$\label{eq:ghostdecop}
\sim \bar{c}(x,\pi) ({\bar\partial}_0 +A_0)^{\dagger}({\bar\partial}_0 +A_0)c(x,\pi)$$ The ghosts in this action however do not have a space-time kinetic term. This leads to a contribution to the effective action which seems to diverge wildly since it is proportional to $$\int d^4p = \delta^{(4)}(0).$$ However, it is well known[^5] that contributions like this vanish in dimensional regularization, just like tad-poles. Hence the ghosts which came from fixing space-time gauge can safely be ignored in perturbation theory, as long as dimensional regularization is employed. See also the discussion in section \[sec:towardssmatrix\] for a second reason why this factor can safely be ignored.
Reduction to space-time fields
------------------------------
At this point it is clear that the quantum fields only live on space-time and the calculation starts to become equivalent to the calculation in \[app:modelcalc\]. In particular the fields in a loop are now a standard four dimensional vector field and self-dual tensor field. This argument therefore neatly avoids any regularization problems special to the twistor space formulation. The quantum self-dual tensor field ${{\mathbf b}}_{{\dot{\alpha}}{\dot{\beta}}}$ can be integrated out to yield $$\begin{aligned}
\label{eq:YMbackgroundaction}
S[A,B, a,b] & = S[A,B] + \frac{1}{4} \int d^4x F_{{\dot{\alpha}}{\dot{\beta}}}[{{\mathbf a}}] F^{{\dot{\alpha}}{\dot{\beta}}}[{{\mathbf a}}]+ \nonumber \\
& \frac{3 g}{4 } \int d^4x F_{{\dot{\alpha}}{\dot{\beta}}}[{{\mathbf a}}] \int dk\left[ \right(H^{-1} ({\bar\partial}^\alpha + A^{\alpha}) H \left), {{\mathbf a}}_{\alpha {\dot{\gamma}}} \right] \frac{\pi^{{\dot{\gamma}}} \hat{\pi}^{{\dot{\alpha}}} \hat{\pi}^{{\dot{\beta}}}}{(\pi \hat{\pi})^2} + \nonumber \\
&\frac{9 g^2}{16} \int d^4x \left[\int dk\left[ \left(H^{-1} ({\bar\partial}^\alpha + A^{\alpha}) H \right), {{\mathbf a}}_{\alpha {\dot{\gamma}}} \right] \frac{\pi^{{\dot{\gamma}}} \hat{\pi}^{{\dot{\alpha}}} \hat{\pi}^{{\dot{\beta}}}}{(\pi \hat{\pi})^2} \right]^2 \nonumber + \\
& \frac{3 g}{4} \int d^4x {{\mathbf a}}^\alpha_{{\dot{\alpha}}} {{\mathbf a}}_{\alpha {\dot{\beta}}} \int dk H^{-1} B_0 H \pi^{{\dot{\alpha}}} \pi^{{\dot{\alpha}}}\end{aligned}$$ As a final step one can now use (\[eq:vectcohom\]) to argue that the coupling of quantum to background fields is the same as in the Chalmers and Siegel Lagrangian (\[eq:backgroundChalSieg\]), since with this argument in hand the $\pi$ integrals can simply be performed. Actually, this is the calculation which lead to the derivation of (\[eq:vectcohom\]) in the first place.
The kinetic term for the quantum fields is just the ordinary Yang-Mills one up to a non-perturbative term. Therefore, dimensional regularization can be employed as usual. We will employ the original ’t Hooft-Veltman [@thooftveltman] scheme which keeps the fields outside the loop in 4 dimensions. A second remark is on the structure of the vertices: every background field vertex involves an infinite amount of $A_0$ fields through the holomorphic frames. Note that this is permitted since the mass dimension of $A_0$ is zero. As a final remark note that this theory is expected to diverge no worse than Yang-Mills theory as the quantum field is simply a gluon: in this particular background gauge twistor Yang-Mills theory is therefore power counting renormalizable. This in contrast to naive power counting based on the mass dimension of the $a_0$ field.
Another way of summarising the above observation is that in this particular background gauge the following diagram commutes, $$\label{dia:diagram}
\xymatrix{\textrm{twistor YM} \ar[r]^{\textrm{quant.}}& \textrm{twistor YM} \ar@{<->}[d]^{Penrose }\\
\textrm{YM on }R^4 \ar@{<->}[u]^{Penrose } & \textrm{YM on }R^4 \ar@{<-}[l]^{\textrm{quant.}}
}$$ In a very real sense this result is also expected, since the quantum effective action can always be calculated on space-time for the Chalmers and Siegel action as a functional of the space-time fields ${\mathbf{A}}(x)$ and ${\mathbf{B}}(x)$. The method outlined in the previous section allows one in principle to lift any functional, so the part which is added in this section is the top arrow.
Just as in the case described in [@us] there is a residual gauge symmetry. When the background field is also in the space-time gauge these are those transformations for which the transformation parameter $\chi$ is a function of $x$ only. Putting the frames back in we arrive at $$\label{eq:resgaugeBG}
\chi(x,\pi) = H \chi(x) H^{-1}$$ Of course this can be checked by direct calculation, as these are the transformations for which the gauge-covariant Laplacian vanishes, $$({\bar\partial}_0 +A_0)^{\dagger}({\bar\partial}_0 +A_0) \chi = 0.$$ What needs checking is whether or not the additional gauge fixing conditions break the carefully preserved background gauge invariance. Below this is verified explicitly by imposing the background version of Lorenz gauge on the background field.
### Residual gauge fixing in Lorenz gauge {#residual-gauge-fixing-in-lorenz-gauge .unnumbered}
The gauge one would like to impose on the quantum field ${{\mathbf a}}$ is the usual background field version of the Lorenz gauge, $$(\partial_\mu + [{\mathbf{A}}_\mu,) {{\mathbf a}}^{\mu}=0.$$ However, it is not immediately obvious this is actually invariant under the background symmetry, the right hand side of (\[eq:gaugesym1\]). The problem is that the quantum field ${{\mathbf a}}_{\alpha {\dot{\alpha}}}(x)$ does not transform nicely under this symmetry, whereas the field $a_\alpha(x,\pi)$ does. In addition, the weights of the fields are odd: since we want to write down a Lagrange multiplier on space-time, it must be weightless from the point of view of twistor space and this is difficult to achieve with two weight $1$ fields. We must write down a term which is gauge covariant under the background symmetry, a Lorentz scalar, weightless and reduces to the Lorenz gauge if the background field obeys the background gauge.
From the form of the effective vertices in the Yang-Mills action (\[eq:YMbackgroundaction\]) it follows that the thing to look for consists of integrations over multiple spheres. This solves the weightless condition. The natural building block is of course the background covariant derivative, $$\label{eq:backgroundgaugederiv}
{\bar\partial}_\alpha + [A_{\alpha}(x,\pi_1),$$ which transforms in the adjoint at $\pi_1$ if and only if it acts on something which transforms in the adjoint at $\pi_1$. Since this must be true locally, a functional of $a_\alpha(x,\pi_2)$ needs to be constructed which transforms as an adjoint field at $\pi_1$. This is naturally constructed by using link operators in terms of holomorphic frames $(\sim ({\bar\partial}+ A_0)^{-1})$, see [@lionel1]) $$\sim \int_{{\mathbb{CP}}^1} dk_2 H_1 H_2^{-1} a^\alpha(x,\pi_2) H_2 H_1^{-1} \frac{\hat{\pi}^{{\dot{\alpha}}}}{\pi \hat{\pi}}.$$ This construction transforms in the adjoint of the background gauge symmetry at $\pi_1$. If we therefore act on it with the background covariant derivative, (\[eq:backgroundgaugederiv\]) and integrate over $\pi_1$ we obtain the desired background gauge covariant gauge fixing term. Now we can rewrite that term in terms of the field ${{\mathbf a}}_{\alpha {\dot{\alpha}}}(x)$ $$\int \frac{\hat{\pi}_1^{{\dot{\alpha}}}}{\pi_1 \hat{\pi_1}} \left(\pi_1^{{\dot{\gamma}}} \frac{\partial}{\partial x^{\alpha {\dot{\gamma}}}} + g [A_{\alpha}(x,\pi_1)\right) \left(H_1 a^{\alpha {\dot{\alpha}}}(x) H_1^{-1}(x)\right)$$ Some further massaging gives $$\left(\partial_{\mu} + [{\mathbf{A}}_{\mu},\right) {{\mathbf a}}^{\mu}(x)$$ with ${\mathbf{A}}$ given by \[eq:eqforpentranA\]. The point of this exercise is that while the above term does not look background gauge covariant, by its construction it is and in addition it is a Lorentz scalar and weightless as required. Note that this gauge condition involves the metric on $\mathbb{R}^4$.
Renormalization and a conjecture
--------------------------------
Now we have all the ingredients to discuss the renormalization properties of the twistor action in the background space-time gauge. If we also impose the twistor version of the background Lorenz gauge constructed above to fix the residual gauge symmetry, the perturbation series is completely well-defined and by employing dimensional regularization maintains Lorentz invariance and the space-time part of the gauge invariance. A direct consequence of the diagram \[dia:diagram\] is therefore that the twistor action, in this particular gauge, is therefore as renormalizable as the Chalmers and Siegel action. Since that action reduces to Yang-Mills theory by integrating out the ${\mathbf{B}}$ field, it is expected that the Chalmers and Siegel action is renormalizable. From the point of view of the twistor action, the counterterms contain by the lifting formula an infinite sequence of terms. As noted before, this possibility is a consequence of the fact that the mass dimension of the field $A_0$ is zero. However, the calculation in the current section shows that by background gauge invariance, only 3 towers counter-terms are non-trivial, since in the space-time action only $$F[{\mathbf{A}}]^2 \quad \quad {\mathbf{B}}^2 \quad \quad {\mathbf{B}}F[{\mathbf{A}}]$$ counterterms are needed.
These terms have an intriguing structure from the twistor point of view. Lorentz and space-time gauge invariance only restrict the renormalization $Z$ factors to $$\begin{aligned}
A_0 & \rightarrow Z_{A_0} A^R_0 \nonumber\\
B_0 & \rightarrow Z_{B_0} B_0 + \left(Z_{BA} F[{\mathbf{A}}]_{{\dot{\alpha}}{\dot{\beta}}}\right) \frac{\hat{\pi}^{{\dot{\alpha}}}\hat{\pi}^{{\dot{\beta}}}}{(\pi \hat{\pi})^2} \nonumber\\
A_\alpha & \rightarrow Z_{A_\alpha} A^R_\alpha \nonumber\\
B_\alpha & \rightarrow Z_{B_\alpha} B^R_\alpha \nonumber\\
g & \rightarrow Z_g g^R\end{aligned}$$ where a possible renormalization of $B_\alpha$ by $\sim \int F_{0\alpha}$ has been discarded since that term vanishes by the constraint. It is easy to see that $Z_{A_0} = Z_{A_\alpha}$, since by gauge invariance, these should appear on an equal footing in $F_{0\alpha}$ and are equal to the usual $Z_A$ renormalization constant. In addition, it is also easy to see that the Feynman rules in the background gauge employed above do not generate contributions to $B^\alpha F_{0\alpha}$, so $$\label{eq:suspi}
Z_{B_\alpha} Z_{A}= 1$$ The usual space-time gauge symmetry argument yields $$\label{eq:renormfacscoupl}
Z_A Z_g =1$$ We are therefore left with three independent renormalization constants[^6] $$Z_A \quad Z_{AB} \quad Z_{B_0}$$ Three constants seems superfluous, since Yang-Mills theory itself only needs one. This suggests that there are more relations between the constants, which are not obvious in the chosen set of gauge conditions. Of course, the same question can be asked in the background approach to the Chalmers and Siegel action itself. On a slightly speculative note, we will conjecture one: we suspect that the $BF$ term never diverges. In other words, $$Z_{B_0} (Z_{A} + Z_{BA})= 1 \quad \quad (\textrm{all loops?})$$ This is true at the one-loop level. Furthermore, in the supersymmetric version of the twistor action, the $BF$ term is part of what seems to be an F-term. The underlying observation is that twistor space has a conformal symmetry, so all terms contributing to the local term on twistor space should be ’conformal’. Note that the natural extension of the conjecture is the expectation that *all* local terms on twistor space are in some definite sense protected from quantum corrections. However, at this point this is nothing but a conjecture, which needs further checking. It would of course already be nice to have a definite translation of the usual supersymmetric non-renormalization theorems into twistor Yang-Mills language.
Using the calculation in the second appendix, it is clear that the twistor Yang-Mills theory has a non-zero $\beta$ function: scale-invariance is broken. Up to a field redefinition (the $Z_{BA}$ term), it can be seen that the $\beta$ function arises by comparing the coefficient in front of the $BF$ and $B^2$ terms. In other words, the beta function is related to the size of the twistor ${\mathbb{CP}}^1$. This is also expected, as this can be related to the size of the excised twistor line which corresponds to $\infty$. Removing this line breaks the symmetry group of the space from the conformal down to the Poincare group.
### $\mathcal{N}=4$ {#mathcaln4 .unnumbered}
The above analysis does have a nice interpretation in $\mathcal{N}=4$ SYM where a background gauge calculation can be set up just as in this article: if the quantum effects leave supersymmetry unbroken, than the $\beta$ function vanishes. This follows from the observation that in the twistor formulation of $\mathcal{N}=4$ theory [@us] $a$ and $b$ are parts of the same super multiplet. They should therefore have the same renormalization constant $Z_A$ if $\mathcal{N}=4$ supersymmetry is unbroken by quantum effects. Therefore by (\[eq:suspi\]) $$Z_A =1 \quad \quad (\textrm{in } \mathcal{N}=4)$$ holds to all orders in perturbation theory. By (\[eq:renormfacscoupl\]) $$Z_g=1 \quad \quad (\textrm{in } \mathcal{N}=4).$$ then follows which in turn implies a perturbatively vanishing $\beta$ function. Of course, the real technical difficulty in this argument lies in proving the assumption that the quantum effects do not break $\mathcal{N}=4$. By the close relation of our techniques to space-time arguments this is fully expected (including the usual caveat about the existence of a supersymmetric regulator), but the background gauge choice employed in this article does break manifest (linear) $\mathcal{N}=4$ supersymmetry. This is probably comparable to the way a Lorenz gauge in real Chern-Simons theory introduces dependency on a metric.
Towards Yang-Mills S-matrix {#sec:towardssmatrix}
===========================
In ordinary Yang-Mills the background field method can be used to calculate S-matrix elements with the background field in a different gauge than the quantum field [@abbottgrisaru; @abbott]. This observation is based on the fact that the quantum effective action obeys $$\label{eq:abbottargument}
\Gamma[A] = \tilde{\Gamma}[\hat{A},A] \restriction_{\hat{A}=A}$$ In ordinary Yang-Mills theory, the left hand side of this equation is the quantum effective action calculated in the background field method, while the right hand side is the effective action of the theory defined by shifting the quantum field, basically undoing (\[eq:splitfieldsTWYM\]). This equation is derived from the observation that the only difference for the calculation of the effective action between the background field path integral and the usual path integral is the fact that they employ a Legendre transform with different sources: the background field integral has a source $J a$, while the usual path integral has a source $J(\tilde{a}-A)$. The background fields in the right hand side will then appear solely in the gauge fixing part of the action. As the S-matrix is independent of the gauge-fixing functional, it is independent of the background field gauge choice, which can be checked explicitly. This is of course nothing but the statement that physical states correspond to BRST cohomology. We expect that the same proof can in principle be used in the twistor action in a general background gauge.
Note that it is already clear from the formulae in the previous section that even low-point Green’s functions calculated using the background field method contain vertices with a large number of fields in any other gauge than space-time gauge. Two-point functions for instance here calculate already infinite towers of effective vertices! These towers disappear in the case where the background field obeys the space-time gauge condition. It can be taken however as a clear indication that it might be possible to calculate large classes of effective vertices with a few simple diagrams.
The background field in CSW gauge
---------------------------------
As indicated before, the background field will be put into CSW gauge, as then at tree level just the MHV formalism is obtained [@usII]. Although one could calculate in principle with a Lorenz gauge gauge quantum field, in this case it is more convenient to employ the light-cone formalism for the quantum field. The convenience stems from the fact that the space-time projection of the background field in CSW gauge obeys $$\label{eq:lightcone}
\eta^{\alpha} \xi^{\dot \alpha} {\mathbf{A}}_{\alpha \dot\alpha} =0$$ The origin of the arbitrary spinor $\xi$ is elucidated below. This equation follows from equation (\[eq:vectcohom\]), since evaluating that equation on $\pi=\xi$ gives, $$\eta^{\alpha} \xi^{\dot \alpha} {\mathbf{A}}_{\alpha \dot\alpha} = \eta^{\alpha} \xi^{\dot \alpha} \left(H^{-1}(\xi) \partial_{\alpha \dot{\alpha}} H(\xi) \right)$$ Recall that the holomorphic frames are defined to be the solution to ${\bar\partial}_A H|_{p^{-1}(x)}=0$. Solving this equation however requires a boundary condition, in this case the value of the holomorphic frame at a base-point. We pick this point to be $\xi$ and normalise $H(\xi)= 1$. From this short observation equation (\[eq:lightcone\]) follows.
The combination $\xi^{{\dot{\alpha}}} \eta^{\alpha}$ forms a null vector in four dimensional space-time, therefore the above result exhibits the close link between the twistor CSW and space-time light-cone gauge. Since the projection of the background field to space-time gauge which couples to the quantum field obeys a light-cone gauge condition, it is natural to impose light-cone gauge on the quantum field as well. In the following for calculational ease the spinor direction indicated by $\eta$ and $\xi$ will be denoted by $1$ and $\dot{1}$ respectively. So for arbitrary spinors $m^{\alpha}$,$n^{\dot\alpha}$, $$m^{\alpha} = m_1 \eta^{\alpha} +m_2 \hat{\eta}^{\alpha} \quad n^{\dot \alpha} = n_1 \xi^{\dot \alpha} + n_2 \hat{\xi}^{\dot \alpha}$$ The light-cone gauge condition on the quantum field therefore becomes $${{\mathbf a}}_{2 \dot{2}} =0.$$ In light-cone coordinates, it is natural to study the physical fields $A_{2 \dot{1}}$ and $A_{1 \dot{2}}$. By equation (\[eq:vectcohom\]), it follows that the Yang-Mills field on space-time splits into two series of fields on twistor space in the CSW gauge. Roughly we have $$\begin{aligned}
A_{2 \dot{1}} &= a_0 + (a_0 a_0) + \ldots \\
A_{1 \dot{2}} &= b_0 (a_0 + (a_0 a_0) + \ldots),\end{aligned}$$ where the second equality follows by the field equation.
Self-dual sector
----------------
In space-time Yang-Mills it is known that the truncation to just the $BF$ part of the Chalmers and Siegel action generates exactly one series of amplitudes, at one loop: the amplitudes with all helicities equal. This is precisely the series of amplitudes which appear to be projected out in the MHV formalism, and as a first consistency check one would like to know if these are non-zero in this approach. The background coupled action follows by the same method as employed in the previous section, $$\begin{aligned}
\nonumber
S[A,B, a,b] = S[A,B] + &\frac{1}{2} \int d^4x {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}}[{{\mathbf a}}] F^{{\dot{\alpha}}{\dot{\beta}}}[{{\mathbf a}}] + \frac{g}{4} \int d^4\!x f_{abc} {\mathbf{B}}^a_{{\dot{\alpha}}{\dot{\beta}}} {{\mathbf a}}_{\alpha}^{{\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\beta}},c} \nonumber\\
& + \frac{g}{4} \int d^4\!x f_{abc} {{\mathbf b}}^a_{{\dot{\alpha}}{\dot{\beta}}} {\mathbf{A}}_{\alpha}^{ \{{\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\beta}}\},c} .\end{aligned}$$ Here the fields ${\mathbf{B}}$ and ${\mathbf{A}}$ denote the space-time projection of the twistor fields $A$ and $B$. These fields are put in the CSW gauge. Hence the complete tree-level perturbation theory is automatically trivial, as there are no more vertices whatsoever. As argued above, it is convenient to impose ${{\mathbf a}}_{2 \dot{2}} =0$ on the quantum field. Writing out the components of the action yields, $$\begin{aligned}
S[A,B, a,b] = S[A,B] + & \frac{1}{2} ( \int (- {{\mathbf b}}_{\dot1\dot1} \partial_{2 \dot2} {{\mathbf a}}_{1\dot2}) + {{\mathbf b}}_{\dot2\dot2} (\partial_{1 \dot1} {{\mathbf a}}_{2\dot1} - \partial_{2 \dot1} {{\mathbf a}}_{1\dot1} + [{{\mathbf a}}_{1\dot1},{{\mathbf a}}_{2 \dot1}] ) - \nonumber \\
& {{\mathbf b}}_{\dot1\dot2} \left(\partial_{1 \dot2} {{\mathbf a}}_{2\dot1} - \partial_{2 \dot1} {{\mathbf a}}_{1\dot2} - \partial_{2 \dot2} {{\mathbf a}}_{1\dot1} + [{{\mathbf a}}_{1\dot2},{{\mathbf a}}_{2 \dot1}] \right) + \nonumber \\
& \frac{1}{2} ({\mathbf{B}}_{\dot2\dot2} [{{\mathbf a}}_{1\dot1}, {{\mathbf a}}_{2\dot1}] - {\mathbf{B}}_{\dot1\dot2} ([{{\mathbf a}}_{1\dot2}, {{\mathbf a}}_{2\dot1}]) + \nonumber \\
& \frac{1}{2} ( {{\mathbf b}}_{\dot2\dot2}\left( [{\mathbf{A}}_{1\dot1}, {{\mathbf a}}_{2\dot1}] - [{\mathbf{A}}_{2\dot1}, {{\mathbf a}}_{1\dot1}] \right) - {{\mathbf b}}_{\dot1\dot2} \left([{\mathbf{A}}_{1\dot2}, {{\mathbf a}}_{2\dot1}] - [{\mathbf{A}}_{2\dot1}, {{\mathbf a}}_{1\dot2}] \right) ).\end{aligned}$$ Now, following similar steps as in [@chalmsieg], the fields ${{\mathbf b}}_{1 \dot1}, {\mathbf{B}}_{1 \dot 1}$ can be integrated out. The last field can be integrated out because there are no quantum corrections for this field. Note that this is equivalent to studying the field equation for the twistor field $B_0$ and evaluating the resulting equation on $\pi=\hat{\xi}$. The quantum field will then decouple. This will set $\hat{\eta}^{\alpha} a_{\alpha}(\hat{\xi}) =0$, which is exactly ${\mathbf{A}}_{1 \dot2}$. The obtained solutions are $${\mathbf{A}}_{1 \dot 2}=0 \quad {{\mathbf a}}_{1 \dot 2}=0.$$ With these solutions there are no more quantum corrections for ${\mathbf{B}}_{\dot1 \dot2}$, and the only place ${{\mathbf b}}_{\dot1\dot2}$ features is now in the kinetic term. Therefore these to can be integrated out exactly, and $$\begin{aligned}
{{\mathbf a}}_{1 \dot 1} = p_{1 \dot 2} \bar{\phi} & {{\mathbf a}}_{2 \dot 1} = p_{2 \dot 2} \bar{\phi} \nonumber \\
{\mathbf{A}}_{1 \dot 1} = p_{1 \dot 2} \bar{\Phi} & {\mathbf{A}}_{2 \dot 1} = p_{2 \dot 2} \bar{\Phi}\end{aligned}$$ is obtained. Again, integrating out $B_{\dot1 \dot2}$ can also be performed by studying the field equation for the twistor field $B_0$ and evaluating this on $\pi = a \xi + b \hat{\xi}$. As the left hand side of the field equation is proportional to $\pi^{\dot \alpha} \pi{\dot \beta}$ by the constraint, all components of this equation should vanish separately.
The obtained solutions can be plugged into the remaining parts of the action and with the definition ${{\mathbf b}}_{\dot2 \dot2} = \phi $ the following result $$\label{eq:determinant}
S = S[A,B] + {\, \mathrm{tr}\;}\int \phi \Box \bar{\phi} + \frac{1}{2} ( \phi [\partial_{\alpha \dot 2} \bar{\phi},\partial^{\alpha}_{\dot 2}\bar{\phi}]) + \frac{1}{2} ( \phi [\partial_{\alpha \dot 2} \Phi,\partial^{\alpha}_{\dot 2}\bar{\phi}]) + \frac{1}{2} ( {\mathbf{B}}_{\dot2 \dot2} [\partial_{\alpha \dot 2} \bar{\phi},\partial^{\alpha}_{\dot 2}\bar{\phi}])$$ is obtained. Here the full background action is retained. In other words the solutions to the field equations are only used for the background fields coupling to the quantum field. This is possible here because the background fields $B$ whose field equations are needed only appear in the classical action. Several things follow from this action. First of all, it can be checked there are no higher than one-loop diagrams. Second, with the background field in CSW gauge the only quantum effects in this theory arise as a field determinant. Thirdly, no loops with external $B's$ will be generated as there are simply no diagrams for pure loops and there are also no tree-level vertices which could give external $B$’s by dressing. If the background field was put in “space-time+light-cone” gauge and treated in the light-cone formalism as well, this is diagrammatically simply what is obtained from using only MHV three (3) vertices as building blocks. In the present setup the calculation is slightly different, as will be illustrated below.
### The four-point all plus amplitude
It is instructive to study the four-point all plus amplitude calculated in the above frame-work. Since ${\mathbf{A}}_{2 \dot 1}$ can be expanded in terms of $a_0$ twistor fields, there are in principle 3 different contributions. These can be diagrammatically represented by diagrams with the same topology as in the ordinary light-cone case. Within dimensional regularization, the bubble contributions vanish, which leaves the box and the triangles. The box diagram can easily be seen to be equivalent to the lightcone calculation. Therefore we will only need the expansion of ${\mathbf{A}}_{2 \dot 1}$ to second order, $$\begin{aligned}
{\mathbf{A}}_{2 \dot 1} & \equiv \eta^{\alpha} \hat{\xi}^{{\dot{\alpha}}} {\mathbf{A}}_{\alpha {\dot{\alpha}}} = H(\hat{\xi}) (\eta^{\alpha} \hat{\xi}^{{\dot{\alpha}}} \partial_{\alpha{\dot{\alpha}}})H^{-1}(\hat{\xi}) \\
& = (a_0) + (a_0)^2 + \ldots\end{aligned}$$ One wasy this computation can be done is by an expansion of the frames $H$, $$H(\hat{\xi}) = 1 + \int_{{\mathbb{CP}}^1} \frac{a_0(\pi_1)}{\hat{\xi} \pi_1} \frac{\hat{\xi} \xi}{\pi_1 \xi} + \int_{({\mathbb{CP}}^1)^2} \frac{a_0(\pi_1)a_0(\pi_2)}{(\hat{\xi} \pi_1)(\pi_1 \pi_2)} \frac{\hat{\xi} \xi}{\pi_2 \xi} + \ldots$$ which leads after some algebra to $$\begin{aligned}
\label{eq:ettlemorrisresem}
{\mathbf{A}}_{2 \dot 1} & = A_z(p) = \braket{\hat{\xi} \xi} \left(\frac{a_0(q_1)}{\eta \xi q_1} + (\eta \xi p)\frac{a_0(q_1) a_0(q_2)}{(\eta \xi q1)(\eta q1 \eta q2) (\eta \xi q2)} + \ldots \right).\end{aligned}$$ Here the obvious momentum constraint has been suppressed. Indeed, inserting the external field normalizations and performing all the sphere integrals using the delta functions shows explicitly that the calculation of the triangles are also *exactly* equivalent to the light-cone calculation, diagram by diagram. Hence it follows that the correct scattering amplitudes are reproduced (see e.g. [@BSTrecent]). Moreover, expressions of the type displayed above (especially \[eq:ettlemorrisresem\]) are very closely related to the light-cone approach to MHV diagrams advocated by [@mansfield]. Actually the coefficients found by Ettle and Morris [@ettlemorris] can all be reproduced by an extension of the above argument. This will be discussed elsewhere.
### Towards the off-shell all-plus vertex
From the setup described above, all the all-plus amplitudes should follow from equation \[eq:determinant\]. Hence these amplitudes are generated by a determinant. This is in the spirit of [@BSTrecent]. One of the aims of the present work was to see if it were possible to calculate complete vertices in the twistor quantum effective action. One of the goals would be to elucidate the twistor structure of the one-loop amplitudes. In particular, as all-plus amplitudes localize on lines in twistor space [@wittencachazo], it is natural to expect in a twistor action formalism that there exists an off-shell *local* vertex in the quantum effective action which reproduces those amplitudes. Remarkably, this is very easy to write down in a CSW gauge as it can be verified that[^7] $$\Gamma^{(1)}[a_0] = \int d^4x \int_{({\mathbb{CP}}^1)^2} \partial_{\alpha {\dot{\alpha}}} K_{21} (\partial^{\alpha}_{{\dot{\beta}}} (a_0)_{(1)} \partial_{\beta}^{{\dot{\alpha}}} K_{12} \partial^{\beta {\dot{\beta}}} (a_0)_{(2)}$$ reproduces the known answer as a local vertex. Here $$K_{12} = \left({\bar\partial}_0 + a_0\right)^{-1}_{12}$$ is the full Green’s function on the ${\mathbb{CP}}^1$ sphere. Note this can never be invariant under the full twistor space gauge symmetry as this expression vanishes in space-time gauge. It is therefore perhaps best interpreted as an effective vertex in a gauge-fixed formalism. Unfortunately, apart from the indirect argument that the self-dual Yang-Mills theory generates all all plus amplitudes, we were unable to connect the above vertex to the calculation of the amplitudes directly.
Note however, that the vertex can be promoted to a quantity invariant under space-time gauge transformations. Promoting derivitives to full covariant derivatives using equation \[eq:eqforpentranA\] would lead to unwanted contributions to the scattering amplitudes, as there would be more $a_0$ fields floating around. In contrast, note that $$\tilde{A}_{\alpha {\dot{\alpha}}} = \int \frac{\hat{\pi}_{{\dot{\alpha}}} a_{\alpha}}{ \pi \hat{\pi}}$$ transforms like a space-time connection under gauge transformations on twistor space which only depend on spacetime: $$\delta\tilde{A}_{\alpha {\dot{\alpha}}} = \int \frac{\hat{\pi}_{{\dot{\alpha}}} ({\bar\partial}_\alpha f(x) + [a_{\alpha,f(x)])}}{\pi \hat{\pi}} = \partial_{\alpha {\dot{\alpha}}} f + [\tilde{A}_{\alpha {\dot{\alpha}}},f]$$ Using this new covariant derivative a vertex invariant under space-time gauge transformations can be devised. We do not know if the extra contributions generated by the ’tilde’ covariant derivatives control any scattering amplitude: they most definitely do no generate the four point $+++-$ amplitudes[@wittencachazo]. Furthermore, these amplitudes would not be generated within the self-dual theory. It is interesting to note that the structure of the all-plus amplitudes also arises in other contexts [@stieberger].
The full theory
---------------
The same method as in the self-dual sector can in principle be applied to the complete theory. A full treatment will be deferred to future work, but it is possible to predict the result: by this stage it is natural to expect that it amounts to taking light-cone Yang-Mills theory, induce a separation between background and quantum field and apply a twistor lift to the background fields only, keeping these in CSW gauge. The tree level results should be given by just the MHV rules, as we can use the solution to the background field equations only for the background fields coupling to the loops, leaving the tree level action intact. Denoting $+$ and $-$ helicity fields by ${{\mathbf{C}}}$ and ${{\bar{\mathbf{C}}}}$ for background and ${{\mathbf{c}}}$ and ${{\bar{\mathbf{c}}}}$ for quantum fields we obtain schematically (for the one loop calculation) $$\begin{aligned}
\label{eq:edguess}
S_{\textrm{C's}}[{\mathbf{A}},{\mathbf{B}}] + S_{\textrm{light-cone}}[{{\mathbf{c}}}, {{\bar{\mathbf{c}}}}] +& {{\mathbf{c}}}{{\bar{\mathbf{c}}}}{{\mathbf{C}}}+ {{\mathbf{c}}}{{\bar{\mathbf{c}}}}{{\bar{\mathbf{C}}}}+ {{\mathbf{c}}}{{\mathbf{c}}}{{\bar{\mathbf{C}}}}+ {{\bar{\mathbf{c}}}}{{\bar{\mathbf{c}}}}{{\mathbf{C}}}\nonumber \\
+& {{\bar{\mathbf{c}}}}{{\bar{\mathbf{c}}}}{{\mathbf{C}}}{{\mathbf{C}}}+ {{\mathbf{c}}}{{\mathbf{c}}}{{\bar{\mathbf{C}}}}{{\bar{\mathbf{C}}}}+ {{\mathbf{c}}}{{\bar{\mathbf{c}}}}{{\bar{\mathbf{C}}}}{{\mathbf{C}}}.\end{aligned}$$ Here ${{\mathbf{C}}}$ and ${{\bar{\mathbf{C}}}}$ are given by $${{\mathbf{C}}}= {\mathbf{A}}_{1 \dot2}(x) \quad \quad {{\bar{\mathbf{C}}}}= {\mathbf{A}}_{2 \dot1}(x).$$ which can be lifted straight to twistor space in the CSW gauge. Interestingly, this decouples the two terms of (\[eq:eqforpentranA\]). This is also expected, as the linearised $A$ on-shell shows explicitly one term is one and the other the other helicity. Now it can easily be checked that one of these contains at least one $A_{\alpha}$ which gets turned into a $b_0$ by the field equation, while the other only contains $A_0$ fields. As the action for background fields only generates MHV diagrams at tree level, it is easy to see that the Feynman rules derived for the action above will then generate loop diagrams for both the all $+$ and all $-$ amplitudes at the same time: one follows from combining what are simply the MHV $3$-vertices (the one with the external $A_\alpha$) knotted into a loop, dressed with MHV-trees. The other one then follows as a straight field determinant, without any forestry. Note that this is a neat realisation of both separate scenarios sketched in [@BSTrecent] within one framework. Actually it is easy to see that applying Mansfield’s canonical field transformation to only the background fields in equation (\[eq:edguess\]) will realise this scenario. We conjecture this constitutes a full quantum completion of the CSW formalism.
Of course, many things have to be checked far more explicitly than done here. However, as we saw in the previous subsection, at least in principle we can follow the same steps needed to make the formalism run. One large caveat in all of the above is that we have not regulated the action very carefully since the light-cone formalism operates strictly in $4$ dimensions and it would be useful to do this properly. However, these problems are again just space-time ones. One obvious way around them is to write a $4-2\epsilon$ dimensional Yang-Mills action and apply lifting only to the four dimensional degrees of freedom in the spirit of a dimensional reduction.
Discussion
==========
In this exploratory article we have shown that there is a class of gauges in which the twistor action formulation of Yang-Mills theory makes sense as a quantum theory in the usual perturbative approach. A partial gauge fixing shows that the quantum effective action of twistor Yang-Mills in this gauge is equivalent to the twistor lift of the quantum effective action of the Chalmers and Siegel action calculated in the background field approach. In particular the divergence structure is the same and in this class of gauges twistor Yang-Mills is as renormalizable as ordinary Yang-Mills in the Chalmers and Siegel formulation. Although it is fully expected that that formulation is equivalent to ordinary Yang-Mills at the quantum level, this is not completely obvious. In particular it would be nice to have a renormalizability proof for the Chalmers and Siegel action, which should be a straightforward extension of results in the literature. In addition a full proof of unitarity would be nice, although again this is expected to hold by the close relationship between the perturbation series of twistor Yang-Mills and the space-time version exposed here. We also have formulated a conjecture on the non-divergence within perturbation theory of local terms in the twistor action based on the twistor structure and a counting argument. This certainly deserves some further study.Unfortunately, apart from the indirect argument that the self-dual Yang-Mills theory generates all all plus amplitudes, we were unable to connect the above vertex to the calculation of the amplitudes directly.
An obvious question remains as to what other gauge choices within the twistor framework are possible and/or interesting. In particular one would like to move away from the space-time oriented background gauge employed here and move toward more twistorial ones. The probably most well-behaved gauge of all for instance, the ‘generalised Lorenz’ gauge (${\bar\partial}^{\dagger} a=0={\bar\partial}^{\dagger} b $), is a natural possibility to consider. Besides choosing a metric on ${\mathbb{CP}}^3$, this requires however a better understanding of twistor propagators beyond the half-Fourier transform technique employed up to now and in particular their regularization at loop level. It would also still be interesting to find a way to make sense of CSW gauge directly, although there it remains a problem to see how to make systematic sense of the divergence structure. There are indications however that techniques currently being employed in the light-cone approach to MHV diagrams also should be applicable here.
The background field method as presented in this article can quite readily be employed in any theory for which a twistor action description is available. The general lifting procedure in the form described in this article is actually applicable to large classes of four dimensional gauge theories, amongst which $\mathcal{N}=4$ SYM [@us] and the full standard model [@usII]. Of course, in the latter case one would also like to have a better understanding of CSW gauge results. This is under study.
One research direction which might be interesting from this article is the question of twistor geometry within the context of renormalization: is there a natural geometric twistor interpretation of renormalization? In the twistor string context, the Yang-Mills coupling constant is related to the size of the ${\mathbb{CP}}^1$ instantons in the disconnected prescription. This suggests that the natural direction to look for ‘renormalization geometry’ is actually the non-projective twistor space $\mathbb{C}^4$. However, based on the results in this article it is not yet quite obvious how this might be achieved.
Another interesting avenue to pursue concerns questions of integrability: it is known that the self-dual Yang-Mills equations are in a real sense integrable, see e.g. [@lionelwoodhouse]. In fact, the transform to twistor space can in some sense be viewed as an explicit transformation to the free theory (the ‘action/angle’ variables) underlying the integrability. The twistor action approach to full Yang-Mills can then be understood as a perturbation around the self-dual, integrable sector. It is a very interesting question to what extend techniques employed in the study of classical integrable systems may be imported to the full theory.
However, the most important point to take from this article is that there is a clear indication that at least part of the structure which makes Yang-Mills perturbation theory at tree level so simple extends to loop level in a consistent way. The goal is that exploiting this observation at a much deeper level than here leads to a better understanding of Yang-Mills theory, both perturbatively and non-perturbatively. The study of perturbation theory in this paper is intended to be a stepping stone in that direction, although even in this form it does apparently furnish a completely regularized, well-behaved, off-shell quantum completion of the MHV formalism. This of course needs further work. One of the things to aim for are for instance (analogs of) Witten’s twistor space localization arguments as these make precise what kind of ‘hidden’ structure perturbation theory might have. It would be very interesting to see how these arise within the twistor action framework as this seems to be a natural starting point to try to derive them. This carries the great promise of being able to calculate complete generating functionals of loop amplitudes, similar to how MHV amplitudes are used at tree level. We hope to come back to this issue in future work.
*Note added in proof*
After this paper was submitted to the archive, two other preprints appeared which also deal with the problem of quantum completions of the CSW rules for non-supersymmetric Yang-Mills theory [@durhamsouthhampton; @queenmaryII]. Both of these propose a more direct solution to the problem and use Mansfield’s canonical transformation technique. By the results in this article, especially section \[sec:towardssmatrix\], this is *very* closely related to the twistor approach.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank Lionel Mason and David Skinner for comments, feedback and collaboration in an early stage of this work. In addition, many thanks to Ruth Britto, Freddy Cachazo, Paul Mansfield, Gabrielle Travaglini and Costas Zoubos for discussions. The work of RB is supported by the European Community through the FP6 Marie Curie RTN [*ENIGMA*]{} (contract number MRTN-CT-2004-5652).
An alternative CSW-like gauge {#app:difgaugcond}
=============================
In four dimensional quantum field theory one can use ’t Hooft’s trick to ‘square’ the gauge condition and arrive at a family of gauge fixings. The normal gauge fixing condition and ghost terms follow from the BRST variation of the gauge fixing fermion, $$S_{\textrm{gaugefix}} = Q_{\textrm{BRST}} \left(\bar{c} G \right)$$ where $G$ is the gauge condition to be imposed and $\bar{c}$ is the anti-ghost for which $\delta_{Q} \bar{c} = \lambda$ with $\lambda$ the Lagrange multiplier field. This is replaced by $$S_{\textrm{gaugefix}} = Q_{\textrm{BRST}} \left (\bar{c} G - \frac{\alpha}{2} \bar{c} \lambda \right).$$ Integrating out the Lagrange multiplier from the resulting action gives $$S_{\textrm{gaugefix}} = \frac{1}{2 \alpha} G^2 + \textrm{ghosts}.$$ In the context of the twistor action this is slightly difficult as the gauge conditions have weight: $\eta^{\alpha} a_\alpha$ for instance has weight $1$ if $\eta$ is weightless. Therefore, squaring this condition does not make sense as an integral on the projective space. However, one can easily normalise the gauge fixing vector $\eta$ to have weight $0$: $$\eta^{\alpha} \rightarrow \frac{\xi^{\dot\alpha} \hat{\pi}_{\dot \alpha}}{\pi \hat{\pi}} \eta^{\alpha}$$ Here $\xi^{\dot\alpha}$ and $\eta^{\alpha}$ are two constant spinors which taken together form a light-like vector on space-time. In the main text of this article $\xi$ arose as the basepoint of the holomorphic frame $H$. With this redefinition of $\eta$ one can now square the gauge fixing condition for $A$. In the limit $\alpha$ is taken to zero this gauge reduces to the CSW gauge employed in [@usII].
One can now calculate the propagators in this gauge in the standard way. It is expected when also $\eta B =0$ that in the limit $\alpha \rightarrow 0$ the CSW propagators are recovered. This turns out to be untrue, surprisingly. Instead, one needs to impose $$\hat{\pi}^{\alpha} \frac{\partial}{\partial x^{\alpha \dot\alpha}} b^{\alpha}=0$$ to obtain the same CSW propagators in the limit $\alpha \rightarrow 0$. In addition one obtains $$:\!B_{\alpha} A_{0}\!: = \frac{\hat{p}_{\alpha}}{(\pi \hat\pi) p^2}$$ At tree level the gauge constructed in this appendix is equivalent to the CSW gauge. At loop level however, there is now a ghost term and a host of non-zero diagrams connected to the vertex in the Chern-Simons part of the action. These seem unattractive however, since their dependence on $\pi$ indicates that propagators connecting *to* a loop can contribute loop momentum factors. In addition, it is hard to see how these contributions could lead to the all plus helicity amplitudes which are missing in the CSW rules.
Background field calculation for the Chalmers and Siegel action {#app:modelcalc}
===============================================================
In this appendix we briefly describe a background field calculation for Yang-Mills theory as formulated as the BF-like Chalmers and Siegel action. This appendix took some inspiration from a calculation in actual BF theory in 4 dimensions from [@martellini], although both our action and the technique applied are different.
Yang-Mills theory can be formulated as an action with an ‘auxiliary’ self-dual tensor field $cB_{\alpha \beta}$, $$\label{BFaction}
S = \frac{1}{2} {\, \mathrm{tr}\;}\int d^4\!x ({\mathbf{B}}^a_{{\dot{\alpha}}{\dot{\beta}}} F_a^{{\dot{\alpha}}{\dot{\beta}}} - \frac{1}{2} {\mathbf{B}}^a_{{\dot{\alpha}}{\dot{\beta}}} {\mathbf{B}}_a^{{\dot{\alpha}}{\dot{\beta}}} ),$$ where $F$ is the self-dual part of the Yang-Mills curvature: $$F_{{\dot{\alpha}}{\dot{\beta}}} = \frac{1}{2} \epsilon^{\beta \alpha} F_{\alpha {\dot{\alpha}}\beta {\dot{\beta}}}$$ In particular, $F_{{\dot{\alpha}}{\dot{\beta}}}$ is a symmetric tensor. The normalisation of this term is chosen to have $$F_{\alpha {\dot{\alpha}}\beta {\dot{\beta}}} = \epsilon_{\alpha \beta} F_{{\dot{\alpha}}{\dot{\beta}}} + \epsilon_{{\dot{\alpha}}{\dot{\beta}}} F_{\alpha \beta}$$ which is nothing but the usual observation that the curvature splits naturally in self and anti-self-dual parts in spinor coordinates. Note that, in the Abelian case, integrating out ${\mathbf{A}}$ yields the electromagnetic dual action. Integrating out ${\mathbf{B}}$ from the above action yields the usual Yang-Mills action up to the topological term. More specifically, if just ${\mathbf{A}}$ fields are inserted into the path integral, it is clear that the vev calculated in this way will be just the standard Yang-Mills answer (perturbatively). In particular, the $\beta$ function of this theory should be the same. Below we will show this explicitly through the background field method. Note that the path-integral contains an integration over a self-dual *auxiliary* field and a standard gauge field. Hence we do not expect anomalies to arise from the path-integral measure.
Setup
-----
Split the fields into a background and quantum part $$\begin{aligned}
\label{eq:splitfieldsBF}
\tilde{{\mathbf{A}}} = {\mathbf{A}}+ {{\mathbf a}}\nonumber \\
\tilde{{\mathbf{B}}} = {\mathbf{B}}+ {{\mathbf b}}\end{aligned}$$ indicated by capital letters and lower case letters respectively. We are going to calculate the quantum effective action by integrating out ${{\mathbf b}},{{\mathbf a}}$ in perturbation theory. As the action is invariant under $$\begin{aligned}
\tilde{{\mathbf{A}}} &\rightarrow& \tilde{{\mathbf{A}}} + d_{g \tilde{{\mathbf{A}}}} \chi\\
\tilde{{\mathbf{B}}} &\rightarrow& \tilde{{\mathbf{B}}} + g [\tilde{{\mathbf{B}}}, \chi]\end{aligned}$$ there are two obvious (disjoint) choices one can make for the symmetry transformations of quantum and background field: $$\begin{aligned}
{\mathbf{A}}&\rightarrow {\mathbf{A}}+ d_{g {\mathbf{A}}} \chi & {\mathbf{A}}&\rightarrow {\mathbf{A}}\nonumber \\
{\mathbf{B}}&\rightarrow {\mathbf{B}}+ g [{\mathbf{B}}, \chi] & {\mathbf{B}}&\rightarrow {\mathbf{B}}\nonumber \\
{{\mathbf a}}&\rightarrow {{\mathbf a}}+ g [{{\mathbf a}}, \chi] & {{\mathbf a}}&\rightarrow {{\mathbf a}}+ d_{g ({\mathbf{A}}+ {{\mathbf a}})} \nonumber \chi\\
{{\mathbf b}}&\rightarrow {{\mathbf b}}+ g [{{\mathbf b}}, \chi] & {{\mathbf b}}&\rightarrow {{\mathbf b}}+ g [{\mathbf{B}}+{{\mathbf b}}, \chi]\end{aligned}$$ The objective in a background field calculation is to fix symmetry number two and keep explicit symmetry number one. One might be tempted to impose Lorenz gauge. As the quantum field ${{\mathbf a}}$ transforms in the adjoint of the background field transformation, a pure Lorenz gauge would break the background symmetry. This is easily remedied by the background gauge condition: $$\label{eq:backgroundlorenzBF}
\partial_\mu {{\mathbf a}}^{\mu} + g [{\mathbf{A}}_{\mu}, {{\mathbf a}}^{\mu}] =0.$$ This gauge condition transforms in the adjoint of the background gauge transformation, so including this condition with a Lagrange multiplier which also transforms in the adjoint, a nice invariant term can be constructed. Following the usual steps we insert the split \[eq:splitfieldsBF\] and the appropriate gauge fixing and ghost terms into the action, discard the linear terms as only 1 PI diagrams contribute to the quantum effective action and obtain: $$\begin{aligned}
S[{\mathbf{A}}+{{\mathbf a}}, {\mathbf{B}}+{{\mathbf b}}] &=& S[{\mathbf{A}},{\mathbf{B}}] + S[{{\mathbf a}},{{\mathbf b}}] + \frac{g}{4} \int d^4\!x f_{abc} {\mathbf{B}}^a_{{\dot{\alpha}}{\dot{\beta}}} {{\mathbf a}}_{\alpha}^{{\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\beta}},c}
+ \frac{g}{4} \int d^4\!x f_{abc} {{\mathbf b}}^a_{{\dot{\alpha}}{\dot{\beta}}} {\mathbf{A}}_{\alpha}^{ \{{\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\beta}}\},c}\nonumber \\
& & - \frac{1}{2 \alpha} \left((\partial_\mu {{\mathbf a}}^{\mu,a})^2 + 2 g f_{abc} (\partial_\mu {{\mathbf a}}^{\mu,a}) {\mathbf{A}}_{\nu}^{b} {{\mathbf a}}^{\nu,c} + g^2 f_{abc} f^a_{ef} {\mathbf{A}}_{\mu}^{b} {{\mathbf a}}^{\mu,c} {\mathbf{A}}_{\nu}^{e} a^{\nu,f} \right) \nonumber \\
& & + \bar{c} \overleftarrow{D}_{\mu} (\overrightarrow{D}^{\mu}+{{\mathbf a}}^{\mu}) c\end{aligned}$$ Note that the ghosts inherit their symmetry properties from equation \[eq:backgroundlorenzBF\] by a simple BRST argument: they transform in the adjoint of both symmetry transformations.
At this point one could either work with the above action directly or one could integrate out the quantum field ${{\mathbf b}}_{{\dot{\alpha}}{\dot{\beta}}}$. The first option, which we also explored, leads generically to more diagrams to be calculated, although results do not change. Also, one would like to stay as close to the original Yang-Mills calculation as possible. Therefore the field ${{\mathbf b}}$ will be integrated out using its field equation, $${{\mathbf b}}_{{\dot{\alpha}}{\dot{\beta}}}^a = F_{{\dot{\alpha}}{\dot{\beta}}}^a + \frac{g}{2} f^{a}_{bc} {\mathbf{A}}_{\alpha \{{\dot{\alpha}}}^{b} {{\mathbf a}}_{{\dot{\beta}}\}}^{\alpha,c}$$ Note that $F_{{\dot{\alpha}}{\dot{\beta}}}$ is a functional of the quantum field ${{\mathbf a}}$ only. This equation leads to $$\begin{aligned}
\label{eq:backgroundChalSieg}
S[{\mathbf{A}}+{{\mathbf a}}, & {\mathbf{B}}+{{\mathbf b}}] = S[{\mathbf{A}},{\mathbf{B}}]+ \int d^4\!x \frac{1}{4} F_{{\dot{\alpha}}{\dot{\beta}}} F^{{\dot{\alpha}}{\dot{\beta}}} - \frac{1}{2 \alpha} (\partial_\mu {{\mathbf a}}^\mu)^2 + \bar{c} \overleftarrow{D}_{\mu} \overrightarrow{D+{{\mathbf a}}}^{\mu} c \nonumber \\
& - \frac{g}{4 \alpha} ( f_{abc} \partial_{\beta {\dot{\beta}}} {{\mathbf a}}^{a \beta {\dot{\beta}}} {\mathbf{A}}_{\alpha {\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\alpha}},c}) + \frac{g}{8} \left(f_{abc} \left( \partial_{\beta \{{\dot{\alpha}}} {{\mathbf a}}^{\beta,a}_{{\dot{\beta}}\}} + g f^{a}_{de}{{\mathbf a}}^d_{\beta \{{\dot{\alpha}}} {{\mathbf a}}^{\beta,e}_{{\dot{\beta}}\} } \right) {\mathbf{A}}_{\alpha}^{\{{\dot{\alpha}},b} {{\mathbf a}}^{{\dot{\beta}}\} \alpha,c} \right) \nonumber \\
& - \frac{g^2}{8 \alpha} f_{abc} f^a_{ef} {\mathbf{A}}_{\alpha {\dot{\alpha}}}^{b} {{\mathbf a}}^{\alpha {\dot{\alpha}},c} {\mathbf{A}}_{\beta {\dot{\beta}}}^{e} {{\mathbf a}}^{\beta {\dot{\beta}},f} + \frac{g^2}{16} f_{abc} f^{a}_{ef} {\mathbf{A}}_{\alpha}^{\{{\dot{\alpha}},b} {{\mathbf a}}^{{\dot{\beta}}\} \alpha,c} {\mathbf{A}}_{\beta \{{\dot{\alpha}}}^{e} {{\mathbf a}}_{{\dot{\beta}}\}}^{\beta,f} \\
& + \frac{g}{4} \int d^4\!x f_{abc} {\mathbf{B}}^a_{{\dot{\alpha}}{\dot{\beta}}} {{\mathbf a}}_{\alpha}^{{\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\beta}},c}
\nonumber \end{aligned}$$ The above action admits an intriguing simplification beyond the simple form of the propagator if $\alpha=1$. In that case the terms quadratic in quantum fields in the second and third lines combine to form $$\frac{g}{2} f_{abc} \partial_{\beta {\dot{\alpha}}} {{\mathbf a}}^{\beta}_{{\dot{\beta}},a} {\mathbf{A}}_{\alpha}^{{\dot{\alpha}},b} {{\mathbf a}}^{{\dot{\beta}}\alpha,c} + \frac{g^2}{4} f_{abc} f^{a}_{ef} {\mathbf{A}}_{\alpha}^{{\dot{\alpha}},b} {{\mathbf a}}^{\alpha {\dot{\beta}},c} {\mathbf{A}}_{\beta {\dot{\alpha}}}^{e} {{\mathbf a}}_{{\dot{\beta}}}^{\beta,f}$$ which can be proven by decomposing the primed tensor structure in this term into symmetric and antisymmetric parts. Hence we will fix $\alpha=1$ in the following. Note that decomposing the above expression in symmetric and antisymmetric parts in the *unprimed* tensor structure leads to the type of terms which might be derived from the anti-Chalmers and Siegel action (the parity conjugate action), $$S_{\textrm{anti-Chalmers and Siegel}} = \frac{1}{2} {\, \mathrm{tr}\;}\int d^4\!x ({\mathbf{B}}^a_{\alpha \beta} F_a^{\alpha \beta} - \frac{1}{2} {\mathbf{B}}^a_{\alpha \beta} {\mathbf{B}}_a^{\alpha \beta} ),$$ treated in the background field method. It would be interesting to understand this connection further as parity invariance is obscured in the Chalmers and Siegel action. The kinetic term $F_{{\dot{\alpha}}{\dot{\beta}}} F^{{\dot{\alpha}}{\dot{\beta}}}$ term can be written as a sum of (minus) the usual Yang-Mills action and a topological term. It therefore follows that perturbatively the quantum field ${{\mathbf a}}$ can be treated as a standard $d$-dimensional vector field in dimensional regularization with the ’t Hooft-Veltman prescription which only continues fields inside the loop.
Self-energies
-------------
Using \[eq:backgroundChalSieg\] we can calculate the self-energies of the fields in the theory, $$<{\mathbf{A}}{\mathbf{A}}> \quad <{\mathbf{A}}{\mathbf{B}}> \quad <{\mathbf{B}}{\mathbf{B}}> \quad <\bar{c} c>$$ As this does not affect the $\beta$ function we will ignore the ghost self-energy here. There is a slight irritation with the normalisation $g_{\mu\nu} = \frac{1}{2} \epsilon_{\alpha \beta} \epsilon_{{\dot{\alpha}}{\dot{\beta}}}$ in the calculations below: when the quantum fields are rewritten as actual Lorentz-vector fields contracted into a space-time tensor, one picks up a factor of $2$. Note that this rewriting has to be performed in order to employ dimensional regularization.
### ${\mathbf{B}}{\mathbf{B}}$ {#mathbfbmathbfb .unnumbered}
By straightforward calculation $$<{\mathbf{B}}{\mathbf{B}}> = -\frac{1}{2}\frac{g^2 C_A}{(4 \pi)^{d/2}} \frac{\Gamma[1-\epsilon]^2}{\Gamma[2-2\epsilon]} \Gamma[\epsilon] \int \frac{d^4\!q}{(2 \pi)^{d}} {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}}^a(q) {\mathbf{B}}^{{\dot{\alpha}}{\dot{\beta}}a}(-q) \left(\frac{1}{q^2} \right)^{\epsilon}$$ is obtained. This can be expanded as $$<{\mathbf{B}}{\mathbf{B}}> = -\frac{1}{2} \frac{g^2 C_A}{(4 \pi)^{2}} (\frac{1}{\epsilon})\int \frac{d^4\!q}{(2 \pi)^{d}} {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}}^a(q) {\mathbf{B}}^{{\dot{\alpha}}{\dot{\beta}}a}(-q) + \mathcal{O}(\epsilon^0)$$
### ${\mathbf{B}}{\mathbf{A}}$ {#mathbfbmathbfa .unnumbered}
This self-energy vanishes. This is a consequence of the fact that ${\mathbf{B}}$ is a symmetric tensor, as the only contribution to this self-energy which is not a tadpole is $$\sim f_{abc} f_{def} B^a_{{\dot{\alpha}}{\dot{\beta}}} A_{\alpha}^{{\dot{\gamma}},e} \int d^4p < :a_{\alpha}^{{\dot{\alpha}},b} a^{\alpha {\dot{\beta}},c}: :\partial_{\beta {\dot{\gamma}}} a^{\beta}_{\dot{\delta},d} a^{\dot{\delta} \alpha,f}: > .$$ Note that one type of vertex has primed indices contracted, while the other has unprimed indices contracted. Working out the contractions with the usual $\alpha=1$ propagator, it quickly emerges that a consequence of this is that the (four dimensional) tensor structure of the primed indices is $\sim \epsilon^{{\dot{\alpha}}{\dot{\beta}}}$. Contracted into ${\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}}$ this gives zero. It is obvious a similar argument applies to many terms in other one-(and higher-)loop diagrams, and at one loop there seem to be no diagrams which contain one $B$ and the rest $A's$ on the external lines. However, there might be sub-leading contributions in $\epsilon$ as for this reasoning to work we must be able to treat the quantum field ${{\mathbf a}}$ as a pure $4$ dimensional object which clashes with dimensional regularization: one should do all index contractions on the quantum fields in $d$ dimensions and then couple to the four dimensional background fields.
The above observation does show that in some sense ${\mathbf{B}}$ and ${\mathbf{A}}$ seem to couple to disjoint gluons in the loop and it would be very interesting to make this sense precise. An argument in favour of a ‘decoupling’ scenario is the fact that by writing the kinetic term as $$F^2 = - \frac{1}{2} \left(F_+^2 + F_{-}^2\right)$$ one can argue (at one loop) that the term with the background field $B$ might be part of a determinant which only features self-dual connection, while the terms with the gauge field $A$ might be part of a determinant with the anti-self-dual connection. Of course, this does not touch upon ghost terms and higher loop effects, but it does seem suggestive.
### ${\mathbf{A}}{\mathbf{A}}$ {#mathbfamathbfa .unnumbered}
There are two contributions, one with a gluon in the loop and one with a ghost loop. Note that in the usual Yang-Mills theory these are actually the only two diagram topologies contributing to the $\beta$ function calculation at one loop. The ghost loop is the same as in Yang-Mills and gives $$<{\mathbf{A}}{\mathbf{A}}>_{\textrm{ghost}} = -\frac{g^2 C_A}{(4 \pi)^{d/2}} \frac{1}{\epsilon-1}\Gamma[\epsilon] \frac{\Gamma[2-\epsilon]^2}{\Gamma[4-2 \epsilon]} \int \frac{d^4\!q}{(2\pi)^4} {\mathbf{A}}_{\mu}^a(q) (q^2 g^{\mu \nu} - q^{\mu} q^{\nu}) {\mathbf{A}}_{\nu}^a(-q) \left(\frac{1}{q^2} \right)^{\epsilon}$$ which yields $$<{\mathbf{A}}{\mathbf{A}}>_{\textrm{ghost}} = \frac{1}{6} \frac{g^2 C_A}{(4 \pi)^{2}} \frac{1}{\epsilon} \int \frac{d^4\!q}{(2\pi)^4} {\mathbf{A}}_{\mu}^a(q) (q^2 g^{\mu \nu} - q^{\mu} q^{\nu}) {\mathbf{A}}_{\nu}^a(-q) + \ldots (\epsilon^0)$$
The gluon loop yields $$<{\mathbf{A}}{\mathbf{A}}>_{\textrm{gluon}} = 4 \frac{g^2 C_A}{(4 \pi)^{d/2}} \frac{\Gamma[2-\epsilon]^2}{\Gamma[4-2\epsilon]} \Gamma[\epsilon] \int \frac{d^4\!q}{(2\pi)^4} {\mathbf{A}}_{\mu}^a(q) \left( q^2 g^{\mu \nu} - q^{\mu} q^{\nu}\right) {\mathbf{A}}_{\nu}^a(-q) \left(\frac{1}{Q^2} \right)^{\epsilon}$$ which can be expanded as $$<{\mathbf{A}}{\mathbf{A}}>_{\textrm{gluon}} = \frac{2}{3} \frac{g^2 C_A}{(4 \pi)^{2}} \int \frac{d^4\!q}{(2\pi)^4} {\mathbf{A}}_{\mu}^a(q) (q^2 g^{\mu \nu} - q^{\mu} q^{\nu}) {\mathbf{A}}_{\nu}^a(-q) + \ldots (\epsilon^0)$$
$\beta$ function
----------------
In order to renormalize the theory, renormalization $Z$ factors for the different background fields in the problem are introduced which preserve the Lorentz and gauge symmetries. The latter are preserved as it is known that the regularization procedure employed here does not break gauge invariance and we have set up our calculation explicitly to preserve it. In principle one could introduce renormalization factors for the quantum fields as well, but this never matters as those $Z$ factors cancel between propagators and vertices. As explained in [@abbott] the only exception to this is a possible renormalization of the gauge fixing parameter $\alpha$, but this will only contribute at higher loop orders. Since this appendix is only concerned with a one loop calculation, this will be ignored. We get $$\begin{aligned}
\label{eq:renormfacs}
{\mathbf{A}}^0 &= Z_{\mathbf{A}}{\mathbf{A}}^R \nonumber \\
{\mathbf{B}}^0 &= Z_{\mathbf{B}}{\mathbf{B}}^R + Z_{{\mathbf{B}}{\mathbf{A}}} F^R \nonumber \\
g^0 &= Z_g g^R\end{aligned}$$ The symmetries permit an extra field mixing renormalization term for ${\mathbf{B}}$ since $F$, like ${\mathbf{B}}$, transforms in the adjoint of the gauge group and is a (self-dual) 2-form. In the following the extra superscript $R$ will be suppressed in order to streamline the presentation. Plugging \[eq:renormfacs\] into the classical action we obtain $$S_{\textrm{ren}} = \frac{1}{2} \int d^4\!x (Z_{\mathbf{A}}Z_{\mathbf{B}}- Z_{\mathbf{B}}Z_{{\mathbf{B}}{\mathbf{A}}}) {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}} F^{{\dot{\alpha}}{\dot{\beta}}} + \frac{1}{2} (Z_{\mathbf{A}}Z_{{\mathbf{B}}{\mathbf{A}}} - \frac{1}{2} Z^2_{{\mathbf{B}}{\mathbf{A}}}) F_{{\dot{\alpha}}{\dot{\beta}}} F^{{\dot{\alpha}}{\dot{\beta}}} - \frac{Z_{\mathbf{B}}^2}{4} {\mathbf{B}}_{{\dot{\alpha}}{\dot{\beta}}} {\mathbf{B}}^{{\dot{\alpha}}{\dot{\beta}}}$$ Here the field $F$ is the self-dual part of the usual curvature tensor, which is renormalised to $$\sim d_{[\mu} {\mathbf{A}}_{\nu]} + Z_g Z_{\mathbf{A}}[{\mathbf{A}}_{\mu},{\mathbf{A}}_{\nu}]$$ In order for the renormalized action to be background gauge invariant, $$Z_g Z_{\mathbf{A}}=1$$ must hold. Calculating $Z_{\mathbf{A}}$ therefore determines $Z_g$, which can be used to determine the $\beta$ function through [@abbott]: $$\label{eq:calcbetafromZ}
\beta(g) = - g^2 \frac{\partial}{\partial g} Z^1_{A}$$ Here $Z^1_{{\mathbf{A}}}$ is the residue at the pole $\frac{1}{\epsilon}$ in the Laurent expansion of $Z_{\mathbf{A}}$. In the background field formalism one can therefore calculate the beta function from a self-energy calculation which is usually much simpler than the 3 point function one needs to calculate otherwise.
The $Z$-factors can be used to cancel the divergences calculated above. Note that from the divergent parts of the self-energy calculations we get equations which yield (temporarily restoring the loop counting parameter) $$\begin{aligned}
Z_{\mathbf{A}}= 1 + \frac{11}{6} \frac{g^2 C_A}{(4 \pi^2) \epsilon} (\hbar) + \textrm{h.o.}\\
Z_{{\mathbf{B}}{\mathbf{A}}} = \frac{5}{6} \frac{g^2 C_A}{(4 \pi^2) \epsilon}(\hbar) + \textrm{h.o.}\\
Z_{\mathbf{B}}= 1 - \frac{g^2 C_A}{(4 \pi^2) \epsilon} (\hbar) +\textrm{h.o.}\end{aligned}$$ which in turn yields the well-known one-loop Yang-Mills $\beta$ function through \[eq:calcbetafromZ\] $$\beta(g) = - \frac{11}{3} \frac{g^3 C_A}{(4 \pi^2)}$$
[^1]: We expect that the BCFW rules can be derived from an action written on ambi-twistor space [@lioneldavid] by direct derivation of the intriguing twistor diagram formalism of [@hodges] from the ambitwistor action. An alternative avenue of attack is the technique of [@leenair]. Both these approaches will involve a gauge choice similar to the CSW gauge.
[^2]: It has recently been pointed out to the author by K. Stelle that this can be made more precise. The reader averse to the words ‘twistor space’ is therefore encouraged to read ‘non-supersymmetric Lorentzian harmonic superspace’ [@sokatchev] instead at occurring places.
[^3]: There are of course non-trivial connections on the ${\mathbb{CP}}^1$ with vanishing curvature. As we will only be interested in perturbation theory in this paper these are avoided by the smallness assumption on $A$.
[^4]: In the recent paper [@BSTrecent] these diagrams were shown to give the correct amplitude in the four point case. Since these diagrams are actually equivalent to Yang-Mills in light-cone gauge, this is expected.
[^5]: but initially not to the author. A discussion can be found for instance in [@zinnjustin].
[^6]: If one wants to fix $\alpha=1$ in a Lorenz-like gauge fixing term $\sim \frac{1}{\alpha} (\partial A)^2$ one needs an extra renormalization constant for this extra coupling, $Z_\alpha$ [@abbott].
[^7]: This form of the ’all-plus vertex’ was first written down by Lionel Mason.
|
---
abstract: |
We consider the possibility that a mass of $\sim 10^{-5}-10^{-3}
M_\odot$ flows back from the dense shell of planetary nebulae and is accreted by the central star during the planetary nebula phase. This backflowing mass is expected to have a significant specific angular momentum even in (rare) spherical planetary nebulae, such that a transient accretion disk might be formed. This mass might influence the occurrence and properties of a very late thermal pulse (VLTP), and might even trigger it. For example, the rapidly rotating outer layer, and the disk if still exist, might lead to axisymmetrical mass ejection by the VLTP. Unstable burning of accreted hydrogen might result in a mild flash of the hydrogen shell, also accompanied by axisymmetrical ejection.
author:
- Adam Frankowski and Noam Soker
title: VERY LATE THERMAL PULSES INFLUENCED BY ACCRETION IN PLANETARY NEBULAE
---
INTRODUCTION {#sec:intro}
============
A planetary nebula (PN) is an expanding ionized circumstellar cloud that was ejected during the asymptotic giant branch (AGB) phase of the stellar progenitor. The nebula is ionized by the hot central star. After several thousands years the nuclear reactions in the central star cease, when more or less that star starts its cooling track toward the the WD track (e.g., Kovetz & Harpaz 1981; Schönberner 1981, 1983). The PN shell is very dense, while the region between the central star and the shell has a very low density. One of the questions at the early history of PN modelling was why part of the shell mass does not flow back to toward the central star? This was answered with the suggestion (Kwok et al. 1978) and discovery (Heap 1979; Cerruti-Sola & Perinotto 1985) of a fast and tenuous wind blown by the central stars of PNs. The fast wind exerts pressure on the PN shell, mainly by forming a hot low density bubble, accelerating the shell outward, and preventing mass from flowing back.
However, under some circumstances small amounts of mass might flow back toward the central star. Accretion of backflowing material during the post-AGB phase was mentioned before by Mathis & Lamers (1992), Bujarrabal et al. (1998), and Zijlstra et al. (2001), and studied by Soker (2001). Most of these studies were interested in substantial mass, $\sim 0.001-0.1 M_\odot$, backflowing from the nebular shell and accreted by the central star. A much smaller mass, $\sim 10^{-6} M_\odot$, is required in the scenario, involving segregation of gas and dust in the outflow, invoked by Mathis & Lamers (1992) to explain chemical abundance peculiarities observed in some post-AGB stars.
The backflowing mass might influence the evolution by forming accretion disk and jets, and by supplying fresh material to the core and by that extending the post-AGB (pre-PN) phase. Soker (2001) showed that such a process requires that some of the material ejected during the AGB phase has a very low outflow speed $\la 1 \km \s^{-1}$, such that it can slow down and fall within a few thousand years. According to Soker (2001), the back flowing material falls because of the gravitational attraction of the central star (or binary system), and falls from a distance of $\sim 400 \AU$. A binary interaction can lead to such an outflow.
In the present paper we consider a different kind of backflow, where gravity is not important. We consider a relatively small mass, $\sim 10^{-5}-10^{-3} M_\odot$, that falls back during the PN phase rather than the pre-PN phase, and is pushed back to the center by the thermal pressure of the ionized shell rather than by gravity. This is discussed in section \[sec:fallback\].
The small amount of mass that falls on the central star at the PN phase might influence a very late thermal pulse (VLTP), or even trigger it. Hajduk et al. (2007) considered the possibility that mass accretion can induce a VLTP. They discuss this for the ‘old nova’ CK Vul, where the mass is assumed to come from a companion. We instead, discuss the case where the accreted mass originates in the nebula (Sect. \[sec:fallback\]).
A VLTP is a helium shell flash occurring so late in the post-AGB evolution that the hydrogen burning shell is already extinct (e.g., Fujimoto 1977; Iben et al. 1983; Herwig et al. 1999). This allows the H-rich matter to be ingested directly into the convective flash zone and leads to a rapid, looped evolution on H-R diagram (Herwig et al. 1999). Other “late” thermal pulse scenarios, occurring earlier in the course of the post-AGB evolution, before complete exhaustion of the H-burning shell, are also considered in Sect. \[sec:pulse\], but it appears that VLTP pertains the most to our fall-back scenario. The possible influence of the accreted mass on the VLTP is discussed in section \[sec:pulse\]. A short summary follows in section \[sec:summary\].
BACKFLOW DURING THE PLANETARY NEBULA PHASE {#sec:fallback}
==========================================
Formation of backflowing blobs
------------------------------
We envision the following process that might lead to the formation of blobs that flow toward the central star.
The initial interaction of the fast wind blown by the post-AGB star with the dense shell blown during the end of the AGB phase is momentum-driven, since the shocked fast wind material cools very fast and the acceleration of the shell is determined by momentum conservation along each direction. When the fast wind speed is high enough the post shock radiative cooling time is longer than the flow time and a hot bubble is formed. Radiative cooling rate is small, and most of the energy is retained in the flow. The shell is in the energy-driven phase; see early works by Volk & Kwok (1985) and Kahn & Breitschwerdt (1990). The transition from momentum-driven to energy-driven shell occurs when the fast wind speed is $v_f \sim 160 \km \s^{-1}$ (Kahn & Breitschwerdt 1990). The interaction is prone to instabilities: the thin shell instability during the momentum-driven phase (Dwarkadas & Balick 1998), and the Rayleigh-Taylor instabilities during the energy-driven phase (Kahn & Breitschwerdt 1990; Dwarkadas & Balick 1998).
The energy-driven phase is likely to start before the ionization phase starts, or at about the same time. To show this we notice the following relation between the escape speed from the central star $v_{\rm esc}$, the central star effective temperature $T_{\ast}$, its mass $M_{\ast}$, and its luminosity $L_\ast$ $$v_{\rm esc} = 170
\left(\frac {T_\ast}{1.5 \times 10^4 \K} \right)
\left(\frac {M_\ast}{0.6 M_\odot} \right)
\left(\frac {L_\ast}{3000 L_\odot} \right)^{-1/4}
\km \s^{-1}.
\label{teff}$$ Assuming that the wind speed is $v_f \simeq v_{\rm esc}$, we find that the transition to the energy-driven shell occurring at $v_f \simeq 160-200 \km \s^{-1}$ (Kahn & Breitschwerdt 1990), happens at about the same time that ionization starts to be significant. This consideration is in accord with the central star model of Perinotto et al. (2004), who used a star with luminosity of $L_\ast= 6300 L_\odot$, and where the wind speed of $v_f \simeq 200 \km \s^{-1}$ is reached when the central star temperature is $T_{\ast} = 1.6 \times 10^4 \K$.
The order of events implies that the hot bubble develops before the dense shell of AGB wind is fully ionized, and with it the instabilities (Kahn & Breitschwerdt 1990; Dwarkadas & Balick 1998). Before ionization starts the pressure in the dense shell is of the order of the pressure in the hot bubble. When ionization becomes important it rapidly propagates through the shell, and heats it to a temperature of $\sim 10^4 \K$ (Perinotto et al. 2004). The shell pressure increases because both temperature and the number density of particles increase. The temperature increases from several$\times 100 \K$ to $\sim 10^4 \K$, so we can take a pressure jump of $ \sim 100$. The shell starts to expand into the hot bubble. Because the initial pressure of the ionized shell is much larger than that of the counter pressure of the hot bubble, the front of the shell expands at a velocity $v_{\rm exp}$ relative to the shell given by (Landau & Lifshits 1987, $\S 99$) $v_{\rm exp} \simeq {2 c_0}/({\gamma-1})$, where $\gamma$ is the adiabatic exponent and $c_0 \simeq 10 \km \s^{-1}$ is the sound speed in the ionized shell. For $\gamma= 5/3$ and $\gamma= 1.1$ one finds $v_{\rm exp}= 3 c_0$ and $v_{\rm exp}= 20 c_0$, respectively.
The ionizing radiation will keep the inflowing mass at a temperature of $\sim 10^4 \K$; a value of $\gamma$ close to unity is appropriate for the flow. The front of the flow does not contain much mass, and therefore the relevant flow speed is a lower velocity, say $(0.1-0.5)\times v_{\rm exp} = (0.2-1)\times c_0/({\gamma-1}) \simeq 20-100 \km \s^{-1}$, where we took $\gamma\simeq 1.1$. However, the acceleration will be less efficient than the one dimensional flow derived in Landau &Lifshits (1987, $\S 99$), because any blob formed by instability will expand to the sides (transversely) as well.
In any case, we conclude that dense blobs formed by instability will start to flow inward at a speed of $\sim 10-50 \km \s^{-1}$. This speed is larger than the expansion velocity of the shell $\sim 10-20 \km \s^{-1}$, and there will be a backflow at speed $v_{\rm back} \sim 3-30 \km \s^{-1}$ made up of blobs. In the 1D calculation of Perinotto et al. (2004) the initial expansion velocity of the shell is $10 \km \s^{-1}$. After ionization starts a zone to the main shell is formed with an expansion velocity of only $\sim 4 \km \s^{-1}$. Namely, the inner zone of the main shell was substantially slowed down. The pressure of the hot bubble prevents further acceleration inward, and later accelerates the entire shell outward. However, if a blob is disconnected from the main shell, it will have the hot bubble material all around it. The pressure of the hot bubble is more or less isotropic and it will not slow the blob down. This effect cannot be studied with a 1D code.
The end product of the initial evolution is a large number of dense (relative to the hot bubble) blobs at a temperature of $\sim 10^4 \K$ that are immersed in the hot low density gas of the hot bubble. Many of these will be flowing inward due to inward acceleration by the pressure of the ionized shell during the early ionization stage. If a blob does not slow down much, its backflow time from an initial radius $r_0$ is $$\tau_{\rm flow} \simeq \frac {r_0}{v_b} = 3000
\left( \frac {r_0}{10^{17} \cm} \right)
\left(\frac {v_b}{10 \km \s^{-1}} \right)^{-1} \yr.
\label{tauflow}$$ This is about the time required for the mass to be accreted during the PN phase, but before a VLTP occurs.
The properties of the blobs {#Sect:blob_properties}
---------------------------
For the blob not to slow much it should encounter a mass $M_e$ not much larger than its initial mass $M_b= 4 \pi a_b^3 \rho_b/3$, where $a_b$ is the radius of the blob and $\rho_b$ is its density. For simplicity we assume the blob is in the shape of a sphere. After flowing a distance $\sim r_0$ the blob encounters a mass $$M_e \simeq \pi a_b^2 r_0 \rho_h,
\label{me1}$$ and the condition reads $$1 \la \frac {M_b}{M_e} \simeq 400 \left(\frac {a_b}{r_0}\right)
\left(\frac {\rho_b}{300 \rho_h} \right).
\label{me2}$$ The density ratio of $300$ comes from pressure equilibrium between the hot bubble and the blob, and a temperature ratio of $\sim 3 \times 10^6 \K /10^4 \K
= 300$. The blobs’ radius is therefore $a_b \ga 0.0025 r_0$ The corresponding mass of the blob is $$M_b \sim M_{\rm neb} \left( \frac{a_b}{r_0} \right)^{3} = 10^{-7}
\left(\frac {a_b}{0.01r_0} \right)^3
\left(\frac {M_{\rm neb}}{0.1 M_\odot} \right) M_\odot
\label{mb1}$$ where $M_{\rm neb}$ is the mass in the nebula, and in the first equality we assume that the blob density is about equal to the density in the nebula.
To survive, the blob must be thermally isolated from the surrounding (otherwise it will be evaporated). We assume that tangled magnetic fields within the blob isolate the surviving blobs. We cannot estimate the number of surviving blobs, but the requirements of ($i$) an initial perturbation that will form a blob with negative radial velocity of $v_b \sim 10 \km \s^{-1}$, ($ii$) a large enough blob to maintain its speed, and ($iii$) suppression of heat conduction by magnetic fields, suggest that their number will be limited. We scale the number of blobs by $N_b = 1000$.
Constraints on the fast wind evolution {#Sect:fast_wind}
--------------------------------------
As long as the dense blob flows inward inside the hot bubble it does not feel the ram pressure of the fast wind. When it gets out of the hot bubble through the inner boundary (reverse shock), the fast wind hits the blob directly, exerting a force of $F_{\rm ram}= \rho_f v_f^2 \pi a_b^2$, where $\rho_f$ is the density of the fast wind. The deceleration is $a_{\rm ram}= F_{\rm ram}/M_b$. Using this value we find the distance over which the blob will stop, $r_{\rm stop} \simeq v_b^2/2 a_{\rm ram}$, to be $$r_{\rm stop} \simeq \frac{2}{3} \left( \frac{v_b}{v_f} \right)^2
\left(\frac {\rho_b}{\rho_f} \right) a_b .
\label{rstop1}$$ While inside the hot bubble the density in the blob is determined by pressure equilibrium with the hot bubble. Close to the inner boundary of the hot bubble, $r_{\rm bubble}$, the pressure is about equal to the ram pressure of the fast wind there, and so is the blob pressure. This gives $\rho_b \simeq \rho_f (v_f/c_0)^2$, where $c_0 \simeq 10 \km
\s^{-1}$ is the sound speed in the blob. In our scenario $c_0 \simeq {v_b}$, and we find from equation (\[rstop1\]) the condition for the blob to reach the center to be $a_b \ga r_{\rm bubble}$. Namely, the blob must be large when it gets out of the hot bubble; we’ll come back to this point later.
Equating the ram pressure to the pressure in the blob $\rho_f v_f^2
\simeq \rho_b c_0^2$, and substituting for the fast wind density $\rho_f = \dot M_f/(4 \pi r^2
v_f)$, and for the density in the blob $M_b/(4 \pi a_b^3/3)$, we can isolate $a_b$. Inserting this expression for the blob radius $a_b$ in the condition $a_b \ga r_{\rm bubble}$ we derive the condition $$r_{\rm bubble} \la 3 \frac {M_b c_0^2}{\dot M_f v_f} = 2.5 \times
10^{15} % 2.367 \times 10^{15}
\left( \frac {M_b}{10^{-7} M_\odot} \right)
\left( \frac {\dot M_f}{10^{-11} M_\odot \yr^{-1}} \right)^{-1}
\left( \frac {v_f}{4000 \km \s^{-1}} \right)^{-1} \cm,
\label{rstop2}$$ where in the second equality we have substituted $c_0= 10 \km \s^{-1}$.
When the fast wind properties are constant with time, the inner shock (the inner boundary of the hot bubble $r_{\rm bubble}$) moves outward as the nebula expands. We require, therefore, that within a short time the fast wind momentum discharge, defined as ${\dot M_f v_f}$, will decrease by a large factor. In that case $r_{\rm bubble} \propto ({\dot M_f v_f})^{1/2} $ . In the model of Perinotto et al. (2004), $\dot M_f$ drops from $\sim
10^{-8} M_\odot$ to $\sim 10^{-10} M_\odot \yr^{-1}$ within $\sim 1000 \yr$ at an age of $t\simeq 7000 \yr$. The fast wind speed then is $\sim 10^4 \km \s^{-1}$. Taking values for the inner hot bubble boundary from their model, as well as other parameters, we crudely scale the parameters as $$r_{\rm bubble} \sim 2 \times 10^{15}
\left( \frac {\dot M_f}{10^{-11} M_\odot \yr^{-1}} \right)^{1/2}
\left( \frac {v_f}{4000 \km \s^{-1}} \right)^{1/2} \cm.
\label{rstop3}$$ Substituting this crude relation in equation (\[rstop2\]) we find for our demand for mass accretion to occur $$\left( \frac {\dot M_f}{10^{-11} M_\odot \yr^{-1}} \right)
\left( \frac {v_f}{4000 \km \s^{-1}} \right) \la
\left( \frac {M_b}{10^{-7} M_\odot} \right)^{2/3}.
\label{rstop4}$$
The condition $a_b \ga r_{\rm bubble}$ for the mass to reach the center as derived above, implies that the backflowing gas covers a large solid angle. Let the solid angle covered by the backflowing gas be $4 \pi \beta$. The ratio of the force due to the ram pressure, $F_f= \beta \dot M_f
v_f$ of the fast wind to the gravitational force of the central star $F_G= G M_\ast M_b
/r^2$ is $$\frac{F_f}{F_g} =
\left( \frac {r}{5 \times 10^{14} \cm} \right)^2
\left( \frac {\dot M_f}{10^{-11} M_\odot \yr^{-1}} \right)
\left( \frac {v_f}{4000 \km \s^{-1}} \right)
\left( \frac {M_b}{10^{-7} M_\odot} \right)^{-1}
\left( \frac {M_\ast}{0.6 M_\odot} \right)^{-1}
\left( \frac {\beta}{0.25} \right).
\label{force1}$$
From the derivations above we learn the following. Equations (\[rstop4\]) shows that for the backflowing blob to reach the center the fast wind should be extremely weak, e.g., the mass loss rate is an order of magnitude below that in the model of Perinotto et al. (2004) at the age of 8000 years. Considering the many unknown in the evolution of the central stars of PNs, we consider this constraint on the fast wind not unreasonable. Namely, we conjecture that it applies in many cases at the age of several thousand years. From equation (\[force1\]) it turns out that gravity becomes important when the backflowing warm gas reaches a distance of $\sim 100 \AU$ under these conditions. The constraint on the blob mass is $M_b \ga 10^{-7}- 10^{-6} M_\odot$.
Angular momentum {#Sect:angular_momentum}
----------------
### Single central star
In section 2.1 we considered the radial acceleration by the nebular thermal pressure of small amount of mass toward the center. Let us assume that because of stochastic motion in the unstable layer the pressure gradient exerts a small azimuthal (tangential) acceleration in addition to the radial acceleration inward. Let the typical azimuthal velocity be $\delta v_b \sim 0.01
(\delta/0.001) \km \s^{-1}$, such that the typical angular momentum is $j_b= \delta v_b r_0$. The combined specific angular momentum of $N_b$ blobs with randomly oriented angular momentum is $j \simeq N_b^{-1/2} j_b$. This angular momentum corresponds to a Keplerian disk at a radius $r_d$ around the central star of mass $M_\ast$ given by $$r_d \simeq 10^{11}
\left(\frac {N_b}{1000} \right)^{-1}
\left(\frac {\delta}{0.001} \right)^2
\left(\frac {v_b}{10 \km \s^{-1}} \right)^2
\left(\frac {r_0}{10^{17} \cm} \right)^2
\left(\frac {M_\ast}{0.6 M_\odot} \right)^{-1} \cm.
\label{rd1}$$ This radius is larger than the radius of the central star during the PN phase, e.g., the VLTP occurs when the radius of the central star might be as small as $\sim 10^9 \cm$ (e.g., Iben et al. 1983). We conclude that an accretion disk might be formed around the central star. The accretion is likely to occur over hundreds to thousands of years, with an average mass accretion rate of $\dot M_{\rm acc} \sim 10^{-9}-10^{-6} M_\odot \yr^{-1}$. Jets might be launched. In any case, if the disk exists during the VLTP, the mass ejection, perhaps resembling a nova eruption, will be axisymmetrical. If the PN is not spherical, then the nebula itself might have some amount of initial angular momentum, making an accretion disk formation more likely.
### Interaction with a wide companion
An interaction with a wide companion is another process by which the backflowing material can acquire angular momentum. We will take the companion to be of low mass, e.g., a low mass main sequence star, a brown dwarf, or a massive planet. This process is more general, and can be relevant to inflow from smaller radii, and at earlier phases of evolution. For example, during the final AGB and early post-AGB phases if mass falls back from the ’effervescent zone’ (Soker 2008). The effervescent zone is an extended zone conjectured to exist above evolved AGB stars (those with high mass loss rates) extending to $\sim 100-300 \AU$. In addition to the escaping wind, in this zone there are parcels of gas that do not reach the escape velocity. These parcels of dense gas rise slowly and then fall back.
Let the companion orbit the central star at radius $a$. Gas falling to the center from radius $r_0>a$ will have an infall velocity of $$v_{\rm in} = \left[ \frac {2G M_\ast}{a} \left( 1-\frac {a}{r_0} \right)
\right]^{1/2}.
\label{vin1}$$ The orbital velocity of the companion is $v_2=(G M_\ast/{a})^{1/2}$. The relative velocity of the inflowing gas and the companion is given by $v_{\rm rel}^2=v_2^2+v_{\rm in}^2 \simeq {2G M_\ast}/{a}$, where we assume that $r_0 >1.5 a$. The accretion radius of the companion is the Bondi-Hoyle-Lyttleton one $$R_{a2} = \frac {2G M_2}{v_{\rm rel}^2} \simeq
\frac{M_2}{M_\ast}a,
\label{racc1}$$ where $M_2$ is the mass of the companion.
The mass with an impact parameter of $b \la R_{a2}$ is accreted by the companion. Mass at larger distances applies drag force on the secondary, with a magnitude of (Alexander et al. 1976) $$F=\pi R_{a2}^2 \rho v_{\rm rel}^2 \ln (R_{\rm max}/R_{a2}).
\label{f1}$$ The same force is applied by the star on the gas. The azimuthal component of this force is $F_{\theta} = Fv_2/v_{\rm rel}$. We also take the maximum radius of influence to be $R_{\rm max}\simeq a$, which by equation (\[racc1\]) gives $\ln (R_{\rm max}/R_{a2}) \simeq \ln(M_\ast/{M_2}) \simeq 1$, for $M_2 \simeq 0.1 M_\ast \simeq 0.1 M_\odot$.
The angular momentum imparted to the inflowing gas by this gravitational interaction is $$\frac{dJ}{dt} = F_\theta a \simeq \pi R_{a2}^2 \rho v_{\rm rel} v_2 a .
\label{jt1}$$ The total inflow rate through radius $a$ is $\dot M_{\rm in} = 4 \pi a^2 \rho v_{in}$. The specific angular momentum of the inflowing gas is $j=({dJ}/{dt})/\dot M_{\rm in}$. Using our approximations, the value of $j$ is $$j_{\rm in} \simeq \left( \frac{M_2}{M_\ast} \right)^2 v_2 a .
\label{j1}$$ To examine the possibility of disk formation around the central star, $j_{\rm in}$ should be compared with the specific angular momentum of a Keplerian circular orbit on the star equator $j_\ast=(G M_\ast R_\ast)^{1/2}$. We find, after using the expression for $v_2$, $$\frac{j_{\rm in}}{j_\ast} \simeq \left( \frac{M_2}{M_\ast} \right)^2
\left( \frac{a}{R_\ast} \right)^{1/2} \simeq
\left( \frac{M_2}{0.1 M_\ast} \right)^2
\left( \frac{a}{100 \AU} \right)^{1/2}
\left( \frac{R_\ast}{1 R_\odot} \right)^{-1/2}
\label{jj}$$ where in the second equality we have substituted values appropriate for a central star of a PN in our studied case of a wide low mass companion.
We see that a binary companion can spin-up the inflowing gas to a degree that will lead to the formation of a disk. If this occurs at early stages of the post-AGB phase, it can lead to the formation of accretion disk that launches two jets, and by that shapes the PNs. We note that the low mass companion, $M_2 \simeq 0.1 M_\ast$, will not accrete much mass by itself.
To summarize this entire section, we did not demonstrate that a backflow forms in the PN phase; this will require intensive 3D numerical simulations, because the dynamical range from perturbations to a backflow to the center is very large. We only showed here that it is quite plausible that dense blobs will flow from the nebula back to the central star, and will cause the star outer layers to rotate fast; it might even form an accretion disk. These effects will result in an axisymmetrical mass ejection if a VLTP take place.
THE INFLUENCE OF ACCRETION ON A VERY LATE THERMAL PULSE {#sec:pulse}
=======================================================
Various objects have their appearance (in particular, chemical abundances) explained as due to a final thermal pulse occuring during the transition from the AGB to the PN phase. To name a few examples (see e.g. recent reviews by Werner & Herwing 2006 and Kimeswenger et al. 2008): FG Sagittae (the prototypical born-again AGB object), Abell 30, Abell 78, CK Vul, PG 1159 (and the class of pulsating white dwarfs named after it), and WR-like central stars of PNe. If the above description of PN fall-back is correct, it is worth checking if a connection can be drawn between the fall-back and VLTP.
In general, three types of “late” thermal pulse scenarios can be distinguished (e.g., Herwig et al. 1999, Blöcker 2003). (1) AGB Final Thermal Pulse (AFTP) takes place at the very end of the AGB phase, right before the star leaves this evolutionary stage. (2) Late Thermal Pulse (LTP) occurs when a star evolves off the AGB at phase $\gtrsim 0.85$ of the thermal pulse cycle and it is caught by a TP on the horizontal part of its post-AGB evolution in the H-R diagram. (3) Very Late Thermal Pulse occurs still later, when the hydrogen burning shell is extinguished and a star is descending along the white dwarf cooling track towards lower luminosities. In the VLTP case, the H-rich envelope is penetrated by the pulse-driven convective zone which leads to an additional short-lived H-flash. The abundance patterns produced in these three cases are somewhat different, but they all produce H-deficient ejecta and H-deficient post-AGB objects.
Can the accretion of back-falling PN matter, as described in the previous section, affect when and how a “late” TP occurs? By definition, it cannot affect an AFTP (no PN yet at this stage), and interfering with an LTP seems precluded by the requirement that the fast wind should practically cease for fall-back to become possible (Sect. \[Sect:fast\_wind\]). That does not happen before moving below the high temperature knee on the HR diagram (e.g., Perinotto et al. 2004). Unlike for gravity driven backflows considered by Soker (2001, 2008), this late PN backflow cannot prolong the post-AGB lifetime at the horizontal part of the track and increase the chances for an LTP (as proposed in Zijlstra et al. 2001 for CK Vul). Note, however, that many main sequence late O dwarfs seem to have winds about two orders of magnitude weaker than expected from radiative winds theory (see Marcolino et al. 2009 for a recent update on the topic). If a similar phenomenon would exist among post-AGB stars, it would make fall back to the central star more likely, be it before or after the hight temperature knee in HR diagram. We will now focus on the VLTP.
According to sections \[Sect:blob\_properties\] and \[Sect:fast\_wind\], fall-back would add $10^{-5}-10^{-3}M_{\odot}$ of H-rich matter to the envelope of a PN central star before a VLTP occurs. Burning of this material can be either stable on unstable, depending on the exact conditions (accretion rate, metallicity, central star mass and intrinsic luminosity, see Shen & Bildsten 2007, esp. their Fig. 3). The relatively high intrinsic luminosity of the young pre-WD (a few hundred $L_\odot$) has a stabilising effect on H-shell burning and should be taken into account in the stability considerations, unlike in classical novae. In the $0.6 M_{\odot}$ core mass model used by Perinotto et al. (2004), at the time when fall-back is expected to occur, i.e, at the age of $8000-10\,000$yrs, the intrinsic luminosity is $\sim 10^{36} \erg \s^{-1}$. According to the results of Shen & Bildsten (2007), this leads to unstable H-shell burning only in a narrow range of accretion rates, $0.3-6 \times 10^{-8} M_{\odot} \yr^{-1}$.
In the case of a stable H-shell burning, the effect of the accreted matter would mainly be to prolong the residual H-shell burning and keep the star longer in the evolutionary stage not far below the high-temperature knee on the H-R diagram. Accreting $10^{-5}-10^{-3} M_{\odot}$ of H-rich matter on a $0.6 M_{\odot}$ pre-WD at a stable-burning rate of $\sim 10^{-9} M_{\odot}$ (Shen & Bildsten 2007) would take $10^{4}-10^{6}$ yrs. This is comparable to the duration of a TP cycle ($\sim 10^{5}$ yrs for a $0.6 M_{\odot}$ core mass) and processes a similar amount of matter (an AGB star processes $10^{-2}-10^{-3} M_{\odot}$ between two consecutive TPs, with the higher values characterizing lower mass cores, $\lesssim 0.6
M_{\odot}$). Effective fall-back would therefore increase the number of post-AGB objects experiencing a VLTP above the standard estimate of $\sim 10\%$ (Iben 1984). The increase would be more significant for the more massive pre-WD objects, as they require less mass processed and accumulated in the He-shell for a TP to occur. Notice also that after accretion the star will rotate due to the angular momentum transferred, which may affect the flash intrinsic geometry (in addition to constrainig the ejecta, as described in Sect. \[Sect:angular\_momentum\]) and possibly other characteristics. This is different from “normal” late TPs, in which the core is rather not expected to rotate rapidly.
On the other hand, if the fall-back accretion process is slow enough (but not as slow as to be stabilized by the residual core luminosity), H-shell burning becomes unstable. In this case, fall-back would also switch on a thermonuclear instability on the pre-WD, but in the form of H-shell flashes. They would be similar to weak nova eruptions, with the H-shell igniting rather than the He-shell (as is in a VLTP). However, from the external appearance – new ejecta within an old PN – they could easily be taken for a VLTP. In the past, Iben & MacDonald (1986) introduced a term [*self-induced nova*]{} for a final flash of H-burning they found to happen late in the evolution of a cooling WD with a very thin He layer buffering between the CO core and the H-rich envelope. In their case the ignition was due to diffusive mixing of C and H through this thin buffer and occured very late in the WD cooling stage ($2\times 10^{7}$yrs after the PN stage, when the luminosity of their model dropped to $0.3 L_\odot$). The fall-back induced flash described here would also be a [*self-induced nova*]{}, but of a distinct kind, less violent and much earlier in the evolution, when the pre-WD is still hot and luminous and its ejected PN is still around.
SUMMARY {#sec:summary}
=======
We have presented a mechanism for a backflow of matter to occur relatively late during PN evolution. The proposed fall-back consists of dense blobs that are formed from instabilities at the discontinuity between the slow and the fast winds of the central star. They detach from the dense shell of the slow AGB wind at the beginning of the PN ionization stage and fall back into the hot bubble with a velocity of several $\km \s^{-1}$. They can reach the PN central star only when the fast wind mass loss rate drops to $\la 10^{-11} M_\odot \yr^{-1}$.
The accreted mass is likely to posses enough angular momentum to spin-up the central star, and even to form an accretion disk. The estimated accretion rates of $\dot M_{\rm acc} \sim 10^{-9}-10^{-6} M_\odot \yr^{-1}$, lasting for hundreds to thousands of years, may lead to the formation of an accretion disk and to launching of jets, even in the case of single central stars of PNe. The accreted mass, $\sim 10^{-5}-10^{-3} M_\odot$, would be enough to increase the chance of a VLTP or even induce an H-shell flash, leading to a mild [*self-induced nova*]{} within a PN. The VLTP (or H-flash) ejecta would exhibit bipolar morphology, offering a binary-independent explanation of point-symmetric inner nebular structures prevalent among purported VLTP objects.
The estimates provided in this paper do not constitute a proof of the existence of such PN fall-back, but they show that it is a reasonable possibility. Numerical simulations of 3D radiative hydrodynamics, detailed enough to resolve the blobs, will be required for a definite statement. In any case, this scenario, together with other types of post-AGB backflow (as described by Soker 2001 and 2008), serves to extend the theoretical argument for the reality and importance of backflow in the PN phase.
This research was supported by the Asher Space Research Institute in the Technion and by the Israel Science foundation.
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6.2in -0.45in -0.9in
**CP VIOLATION AND $B$ PHYSICS [^1]**
Michael Gronau [^2]
*Stanford Linear Accelerator Center*
*Stanford University, Stanford, CA 94309*
**ABSTRACT**
> This is a quick review of CP non-conservation in $B$ physics. Several methods are described for testing the Kobayashi-Maskawa single phase origin of CP violation in $B$ decays, pointing out some limitations due to hadronic uncertainties. A few characteristic signatures of new physics in $B$ decay asymmetries are listed.
[*Invited talk given at the 1999 Chicago Conference on Kaon Physics\
Chicago, IL, June 21$-$26, 1999*]{}
The CKM Matrix
==============
In the standard model of electroweak interactions CP violation is due to a nonzero complex phase [@KM] in the Cabibbo-Kobayashi-Maskawa (CKM) matrix $V$, describing the weak couplings of the charged gauge boson to quarks. The unitary matrix $V$, given by three mixing angles $\theta_{ij} (i<j=1,2,3)$ and a phase $\gamma$, can be approximated by ($s_{ij}\equiv \sin\theta_{ij}$) [@PDG; @peccei] \[V\] V () . Within this approximation, the only complex elements are $V_{ub}$, with phase $-\gamma$ and $V_{td}$, the phase of which is denoted $-\beta$.
The measured values of the three mixing angles and phase are [@peccei] \[angles\] s\_[12]{} = 0.2200.002 , s\_[23]{} = 0.0400.003 , s\_[13]{} = 0.0030.001 , \[gamma\] 35\^0(V\^\*\_[ub]{})145\^0 . First evidence for a nonzero phase $\gamma$ came 35 years ago with the measurement of $\epsilon$, parameterizing CP violation in $K^0-\bar K^0$ mixing. The second evidence was obtained recently through the measurement of ${\rm Re}(\epsilon'/\epsilon)$ [@eps'; @eps'2] discussed extensively at this meeting.
Unitarity of $V$ implies a set of 6 triangle relations. The $db$ triangle, \[unit\] V\_[ud]{}V\^\*\_[ub]{} + V\_[cd]{}V\^\*\_[cb]{} + V\_[td]{}V\^\*\_[tb]{}=0 , is unique in having three comparable sides, which were measured in $b\to u\ell\nu,~b\to c\ell \nu$ and $\Delta M_{d,s}$, respectively. Whereas $V_{cb}$ was measured quite precisely, $V_{ub}$ and $V_{td}$ are rather poorly known at present. The three large angles of the triangle lie in the ranges $35^{\circ}\le \alpha\le 120^{\circ},~10^{\circ}\le\beta\le 35^{\circ}$ and Eq. (\[gamma\]). As we will show in the next sections, certain $B$ decay asymmetries can constrain these angles considerably beyond present limits.
For comparison with $K$ physics, note that due to the extremely small $t$-quark side of the $ds$ unitarity triangle V\_[ud]{}V\^\*\_[us]{}+V\_[cd]{}V\^\*\_[cs]{}+V\_[td]{}V\^\*\_[ts]{}=0 , this triangle has an angle of order $10^{-3}$, which accounts for the smallness of CP violation in $K$ decays. The area of this triangle, which is equal to the area of the $db$ triangle [@jarl], can be determined by fixing its tiny height through the rate of $K_L\to\pi^0\nu\bar\nu$. This demonstrates the complementarity of $K$ and $B$ physics in verifying or falsifying the assumption that CP violation originates solely in the single phase of the CKM matrix.
As we will show, the advantage of $B$ decays in testing the KM hypothesis is the large variety of decay modes. This permits a detailed study of the phase structure of the CKM matrix through various interference phenomena which can measure the two phases $\gamma$ and $\beta$. New physics can affect this interference in several ways to be discussed below.
CP violation in $B^0-\bar B^0$ mixing
=====================================
The wrong-sign lepton asymmetry A\_[sl]{} , measures CP violation in $B^0-\bar B^0$ mixing. Top-quark dominance of $B^0-\bar B^0$ mixing implies that this asymmetry is of order $10^{-3}$ or smaller [@bigi]. A\_[sl]{} = 4[Re]{}\_B = [Im]{}() = () () ( ) (10\^[-3]{}) . Present limits are at the level of 5$\%$ [@LEP].
Writing the neutral $B$ mass eigenstates as |B\_L> = p|B\^0> + q ||B\^0> , |B\_H> = p|B\^0> - q ||B\^0> , one has $2{\rm Re\epsilon_B}\approx 1 - |q/p| \leq {\cal O}(10^{-3})$. Thus, to a very high accuracy, the mixing amplitude is a pure phase = e\^[2i[Arg]{}(V\_[td]{})]{} = e\^[-2i]{} .
The asymmetry in $B^0(t)\to \psi K_S$
=====================================
When an initially produced $B^0$ state oscillates in time via the mixing amplitude which carries a phase $e^{-2i\beta}$, |B\^0(t)> = |B\^0>(mt/2) + ||B\^0> ie\^[-2i]{}( mt/2) , the $B^0$ and $\bar B^0$ components decay with equal amplitudes to $\psi K_S$. The interference creates a time-dependent CP asymmetry between this process and the corresponding process starting with a $\bar B^0$ [@sanda] A(t)= = -(2)(mt) . The simplicity of this result, relating a measured asymmetry to an angle of the unitarity triangle, follows from having a single weak phase in the decay amplitude which is dominated by $b\to c\bar c s$. This single phase approximation holds to better than 1$\%$ [@MG] and provides a clean measurement of $\beta$.
A recent measurement by the CDF collaboration at the Tevatron [@kroll], $\sin(2\beta) = 0.79\pm 0.39\pm 0.16$, has not yet produced a significant nonzero result. It is already encouraging however to note that this result prefers positive values, and is not in conflict with present limits, $0.4\leq\sin 2\beta\leq
0.8$.
Penguin pollution in $B^0\to\pi^+\pi^-$
=======================================
By applying the above argument to $B^0\to \pi^+\pi^-$, in which the decay amplitude has the phase $\gamma$, one would expect the asymmetry in this process to measure $\sin 2(\beta+\gamma)=-\sin(2\alpha)$. However, this process involves a second amplitude due to penguin operators which carry a different weak phase than the dominant current-current (tree) amplitude [@MG; @LP]. This leads to a more general form of the time-dependent asymmetry, which includes a new term due to direct CP violation in the decay [@MG] \[asym\] A(t) = a\_[dir]{}(mt) + 2(+ )(mt) . Both $a_{\rm dir}$ and $\theta$, the correction to $\alpha$ in the second term, are given roughly by the ratio of penguin to tree amplitudes, $a_{\rm dir}\sim 2({\rm Penguin}/{\rm Tree})\sin\delta,~
\theta\sim ({\rm Penguin}/{\rm Tree})\cos\delta$, where $\delta$ is an unknown strong phase. A crude estimate of the penguin-to-tree ratio, based on CKM and QCD factors, is 0.1. Recently, flavor SU(3) was applied [@DGR] to relate $B\to \pi\pi$ to $B\to K\pi$ data, finding this ratio to be in the range 0.3$ \pm $0.1. Precise knowledge of this ratio could provide very useful information about $\alpha$ [@MG; @P/T].
One way of eliminating the penguin effect is by measuring also the time-integrated rates of $B^0\to\pi^0\pi^0$, $B^+\to\pi^+\pi^0$ and their charge-conjugates [@GRLO]. The three $B\to\pi\pi$ amplitudes obey an isospin triangle relation, \[iso\] A(B\^0\^+\^-)/+ A(B\^0\^0\^0) = A(B\^+\^+\^0) . A similar relation holds for the charge-conjugate processes. One uses the different isospin properties of the penguin ($\Delta I=1/2$) and tree ($\Delta I=1/2, 3/2$) contributions and the well-defined weak phase ($\gamma$) of the tree amplitude. This enables one to determine the correction to $\sin2\alpha$ in the second term of Eq.(\[asym\]) by constructing the two isospin triangles.
Electroweak penguin contributions could spoil this method [@DH] since they involve $\Delta I=3/2$ components. This implies that the amplitudes of $B^+\to\pi^+\pi^0$ and its charge-conjugate differ in phase, which introduces a correction at the level of a few percent in the isospin analysis. It was shown recently [@GPY] that this small correction can be taken into account analytically in the isospin analysis, since the dominant electroweak contributions are related by isospin to the tree amplitude. Other very small corrections can come from isospin breaking in strong interactions [@gardner].
The major difficulty of measuring $\alpha$ without knowing the ratio Penguin/Tree is experimental rather than theoretical. The first signal for $B^0\to\pi^+\pi^-$ reported this summer [@poling; @pipi], ${\rm BR}(B^0\to\pi^+\pi^-)=
[0.47^{+0.18}_{-0.15}\pm 0.06)\times 10^{-5}$, is somewhat weaker than expected. Worse than that, the branching ratio into two neutral pions is expected to be at most an order of magnitude smaller. This estimate is based on color-suppression, a feature already observed in CKM-favored $B\to \bar D\pi$ decays. Here it was found that [@PDG], ${\rm BR}(B^0\to \bar D^0\pi^0)/{\rm BR}(B^0\to D^-\pi^+)
< 0.04$. If the same color-suppression holds in $B\to\pi\pi$, then ${\rm BR}(B^0\to \pi^0\pi^0)< 3\times 10^{-7}$, which would be too small to be measured with a useful precision. Constructive interference between a color-suppressed current-current amplitude and a penguin amplitude can increase the $\pi^0\pi^0$ rate somewhat. Limits on this rather rare mode can be used to bound the uncertainty in determining $\sin(2\alpha)$ from $B^0\to\pi^+\pi^-$ [@GQ] () Other ways of treating the penguin problem were discussed in [@other].
$B$ decays to three pions
=========================
The angle $\alpha$ can also be studied in the processes $B\to\pi \rho$ [@rhopi], which have already been seen with branching ratios larger than those of $B\to\pi\pi$ [@CLEOrhopi], ${\rm BR}(B^0\to \pi^{\pm}\rho^{\mp})=
(3.5^{+1.1}_{-1.0}\pm 0.5)\times 10^{-5}$, ${\rm BR}(B^\pm\to \pi^{\pm}\rho^0)=
(1.5\pm 0.5 \pm 0.4)\times 10^{-5}$. An effective study of $\alpha$, which can eliminate uncertainties due to penguin corrections, requires
- A separation between $B^0$ and $\bar B^0$ decays.
- Time-dependent rate asymmetry measurements in $B\to\pi^{\pm}\rho^{\mp}$.
- Measuring the rates of processes involving neutral pions, including the color-suppressed $B^0\to \pi^0\rho^0$.
This will not be an easy task.
$\gamma$ from $B \to K\pi$ and other processes
==============================================
While discussing $B^\pm$ decays to three charged pions, we note that these decays are of high interest for a different reason [@EGM]. When two of the pions form a mass around the charmonium $\chi_{c0}(3415)$ state, a very large CP asymmetry is expected between $B^+$ and $B^-$ decays. In this case the direct decay amplitude into three pions ($b\to u u\bar d$) interferes with a comparable amplitude into $\chi_{c0}\pi^\pm$ ($b\to c\bar c d$) followed by $\chi_{c0}\to \pi^+\pi^-$. The large asymmetry (proportional to $\sin\gamma$), of order several tens of percent, follows from the $90^{\circ}$ strong phase obtained when the two pion invariant mass approaches the charmonium mass.
A method for determining the angle $\gamma$ through $B^{\pm}\to D K^{\pm}$ decays [@GW], which in principle is completely free of hadronic uncertainties, faces severe experimental difficulties. It requires measuring separately decays to states involving $D^0$ and $\bar D^0$. Tagging the flavor of a neutral $D$ by the charge of the decay lepton suffers from a very large background from $B$ decay leptons, while tagging by hadronic modes involves interference with doubly Cabibbo-suppressed $D$ decays. A few variants of this method were suggested [@ADS], however, due to low statistics, it seems unlikely that these variants can be performed effectively in near future facilities.
Much attention was drawn recently to studies of $\gamma$ in $B\to K\pi$, motivated by measurements of charge-averaged $B\to K\pi$ decay branching ratios [@poling; @pipi] (B\^K\^) & = & (1.82\^[+0.46]{}\_[-0.40]{}0.16) 10\^[-5]{} ,\
(B\^K\^\^0) & = & (1.21\^[+0.30+0.21]{}\_[-0.28-0.14]{}) 10\^[-5]{} ,\
(B\^0K\^\^) & = & (1.88\^[+0.28]{}\_[-0.26]{}0.13) 10\^[-5]{} ,\
(B\^0K\^0\^0) & = & (1.48\^[+0.59+0.24]{}\_[-0.51-0.33]{}) 10\^[-5]{} . The first suggestion to constrain $\gamma$ from $B\to K\pi$ was made in [@GRL], where electroweak penguin contributions were neglected. The importance of electroweak penguin terms was noted in [@DHF], which was followed by several ideas about controlling these effects [@EWP]. In the present discussion we will focus briefly on very recent work along these lines [@GPY; @NR; @GP; @BF], simplifying the discussion as much as possible.
Decomposing the $B^+\to K\pi$ amplitudes into contributions from penguin ($P$), color-favored tree ($T$) and color-suppressed tree ($C$) terms [@GHLR], \[PTC\] A(B\^+K\^0\^+)=P , A(B\^+K\^+\^0)=-(P+T+C)/ , $P$ has a weak phase $\pi$, while $T$ and $C$ each carry the phase $\gamma$. Some information about the relative magnitudes of these terms can be gained by using SU(3) and comparing these amplitudes to those of $B\to \pi\pi$ [@DGR]. This implies r = 0.240.06 . Defining the ratio of charge-averaged rates [@NR] R\^[-1]{}\_\*= , one has \[noEWP\] R\^[-1]{}\_\* = 1 - 2r+ r\^2 , where $\delta$ is the penguin-tree strong phase-difference. Any deviation of this ratio from one would be a clear signal of interference between $T+C$ and $P$ in $B^+\to K^+\pi^0$ and could be used to constrain $\gamma$.
So far, electroweak penguin contributions have been neglected. These terms can be included in the above ratio by relating them through flavor SU(3) to the corresponding tree amplitudes. This is possible since the two types of operators have the same (V-A)(V-A) structure and differ only by SU(3). Hence, in the SU(3) limit, the dominant electroweak penguin term and the tree amplitude have the same strong phase, and the ratio of their magnitudes is given simply by a ratio of the corresponding Wilson coefficients multiplied by CKM factors [@GPY; @NR] \[del\] \_[EW]{} & &\
& = & - = 0.6 0.2 , where the error comes from $|V_{ub}|$. Consequently, one finds instead of (\[noEWP\]) R\^[-1]{}\_\* = 1 - 2r (- \_[EW]{}) + [O]{}(r\^2) , implying |- \_[EW]{}| $R^{-1}_*\ne 1$, this constraint can be used to exclude a region around $\gamma=50^{\circ}$. The present value of $R^{-1}_*$ is consistent with one. Experimental errors must be substantially reduced before drawing any conclusions.
The above constraint is based only on charge-averaged rates. Further information on $\gamma$ can be obtained by measuring separately $B^+$ and $B^-$ decay rates. The $B^+ \to K\pi$ rates obey a triangle relation with $B^+\to\pi^+\pi^0$ [@GPY; @GRL; @NR] 2 A(B\^+K\^+\^0) + A(B\^+K\^0\^+) = r\_u A(B\^+\^+\^0) (1 - \_[EW]{} e\^[-i]{}) , where $\tilde r_u = (f_K/f_{\pi})\tan\theta_c\simeq 0.28$ contains explicit SU(3) breaking. This relation and its charge-conjugate permit a determination of $\gamma$ which does not rely on $R^{-1}_*\ne 1$.
This analysis involves uncertainties due to errors in $r$ and $\delta_{EW}$, which are expected to be reduced to the level of 10$\%$. Additional uncertainties follow from SU(3) breaking in (\[del\]) and from rescattering effects in $B^+\to K^0\pi^+$ which introduce a term with phase $\gamma$ in this process. The latter effects can be bounded by the U-spin related rate of $B^+\to K^+ \bar K^0$ [@rescat]. Present limits on rescattering corrections are at a level of 20$\%$ and can be reduced to 10$\%$ in future high statistics experiments. Such rescattering corrections introduce an error of about $10^{\circ}$ in determining $\gamma$ [@GP]. Summing up all the theoretical uncertainties, and neglecting experimental errors, it is unlikely that this method will determine $\gamma$ to better than $\pm 20^{\circ}$. Nevertheless, this would be a substantial improvement relative to the present bounds (\[gamma\]).
We conclude this section with a simple observation [@MJR], which enables an early detection of a CP asymmetry in $B\to K\pi$. Using $A(B^0\to K^+\pi^-)=-P-T$, the hierarchy among amplitudes [@GHLR], $|P|\gg |T| \gg |C|$, implies ${\rm Asym}(B^{\pm}\to K^{\pm}\pi^0) \approx
{\rm Asym}(B\to K^{\pm}\pi^{\mp})$. This may be used to gain statistics by measuring the combined asymmetry in these two modes. The magnitude of the asymmetry depends on an unknown final state strong phase. Very recently a 90$\%$ confidence level upper limit was reported ${\rm Asym}(B\to K^{\pm}\pi^{\mp}) < 0.35$ [@poling; @asym].
Signals of new physics
======================
The purpose of future $B$ physics is to over-constrain the unitarity triangle. $|V_{ub}|$ can at best be determined to 10$\%$ [@liget] and $|V_{td}|$ relies on future measurements of the higher order $B^0_s-\bar B^0_s$ mixing [@kroll] and $K^+\to \pi^+\nu\bar\nu$ [@red]. Constraining the angles $\alpha,~\beta$ and $\gamma$ by CP asymmetries is complementary to these CP conserving measurements. The asymmetry measurements involve discrete ambiguities in the angles, which ought to be resolved [@wolf].
Hopefully, these studies will not only sharpen our knowledge of the CKM parameters but will eventually show some inconsistencies. In this case, the first purpose of $B$ physics will be to identify the source of the inconsistencies in a model-independent way. Let us discuss this scenario briefly by considering a few general possibilities.
Physics beyond the standard model can modify CKM phenomenology and predictions for CP asymmetries by introducing additional contributions in three types of amplitudes:
- $B^0-\bar B^0$ and $B^0_s-\bar B^0_s$ mixing amplitudes.
- Penguin decay amplitudes.
- Tree decay amplitudes.
The first case is the most likely possibility, demonstrated by a large variety of models [@nir]. New mixing terms, which can be large and which often also affect the rates of electroweak penguin decays, modify in a universal way the interpretation of asymmetries in terms of phases of $B^0-\bar B^0$ and $B^0_s-\bar B^0_s$ mixing amplitudes. These contributions can be identified either by measuring asymmetries which lie outside the allowed range, or by comparison with mixing-unrelated constraints. On the other hand, new contributions in decay amplitudes [@gw] are usually small, may vary from one process to another, and can be detected be comparing asymmetries in different processes. Processes in which the KM hypothesis implies extremely small asymmetries are particularly sensitive to new amplitudes.
To conclude this brief discussion, let us list a few examples of signals for new physics.
- $A_{sl} \geq {\cal O}(10^{-2})$.
- Sizable asymmetries in $b \to s \gamma$ or $B_s\to \psi\phi$.
- “Forbidden” values of angles, $|\sin 2\beta - 0.6| > 0.2,~~
\sin\gamma < 0.6$.
- Different asymmetries in $B^0(t)\to \psi K_S,~\phi K_S,~\eta' K_S$.
- Contradictory constraints on $\gamma$ from $B\to K\pi,~
B\to D K,~B_s\to D_sK$.
- Rate enhancement beyond standard model predictions for electroweak penguin decays, $B\to X_{d,s}\ell^+\ell^-,~B^0/B_s\to\ell^+\ell^-$.
Conclusion
==========
The CP asymmetry in $B\to \psi K_S$ is related cleanly to the weak phase $\beta$ and can be used experimentally to measure $\sin 2\beta$. In other cases, such as in $B^0\to\pi^+\pi^-$ which measures $\sin 2\alpha$ and $B \to DK$ which determines $\sin\gamma$, the relations between the asymmetries, supplemented by certain rates, and the corresponding weak phases are free of significant theoretical uncertainties. However, the application of these methods are expected to suffer from experimental difficulties due to the small rates of color-suppressed processes.
While one expects qualitatively that color-supression is affected by final-state interactions, these long distance phenomena are not understood quantitatively. The case of $B\to K\pi$ demonstrates the need for a better undersanding of these features, and the need for a reliable treatment of SU(3) breaking. That is, whereas the short distance effects of QCD in weak hadronic $B$ decays are well-understood [@BBL], we are in great need of a theoretical framework for studying long distance effects. An interesting suggestion in this direction was made very recently in [@beneke].
We discussed mainly the very immediate $B$ decay modes, for which CP asymmetries can provide new information on CKM parameters. Asymmetries should be searched in [*all $B$ decay processes*]{}, including those which are plagued by theoretical uncertainties due to unknown final state interactions, and those where the KM framework predicts negligibly small asymmetries. Afterall, our understanding of the origin of CP violation is rather limited and surprises may be right around the corner.
: I thank the SLAC Theory Group for its very kind hospitality. I am grateful to Gad Eilam, David London, Dan Pirjol, Jon Rosner and Daniel Wyler for collaborations on topic discussed here. This work was supported in part by the United States $-$ Israel Binational Science Foundation under research grant agreement 94-00253/3, and by the Department of Energy under contract number DE-AC03-76SF00515.
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[99]{} M. Kobayashi and T. Maskawa, . C. Caso , . R. Peccei, these proceedings, also discusses the Wolfenstein parameterization in terms of $\lambda,~A,~\rho$ and $\eta$. Y. B. Hsiung, these proceedings. M. Sozzi, these proceedings. C. Jarlskog, . I. I. Bigi , in [*CP Violation*]{}, ed. C. Jarlskog (World Scientific, Singapore, 1992). OPAL Collaboration, K. Ackerstaff , . A. B. Carter and A. I. Sanda, ; ; I. I. Bigi and A. I. Sanda, . M. Gronau, . J. Kroll, these proceedings. D. London and R. D. Peccei, ; B. Grinstein, . A. Dighe, M. Gronau and J. L. Rosner, . F. DeJongh and P. Sphicas, ; P. S. Marrocchesi and N. Paver, . M. Gronau and D. London, . N. G. Deshpande and X. G. He, [*Phys. Rev. Lett.*]{} [**74**]{}, 26, 4099(E) (1995). M. Gronau D. Pirjol and T. M. Yan, . S. Gardner, . R. Poling, Rapporteur talk at the 19th International Lepton-Photon Symposium, Stanford, CA, August 9$-$14, 1999. CLEO Collaboration, Y. Kwon , hep-ex/9908029. Y. Grossman and H. R. Quinn, . J. Charles, ; D. Pirjol, ; R. Fleischer, hep-ph/9903456. H. J. Lipkin, Y. Nir, H. R. Quinn and A. Snyder, ; M. Gronau, ; H. R. Quinn and A. Snyder, . CLEO Collaboration, M. Bishai , hep-ex/9908018. G. Eilam, M. Gronau and R. R. Mendel, ; N. G. Deshpande , ; I. Bediaga, R. E. Blanco, C. Gobel and R. Mendez-Galain, ; B. Bajc , . M. Gronau and D. Wyler, . D. Atwood, I. Dunietz and A. Soni, ; M. Gronau, ; M. Gronau and J. L. Rosner, . M. Gronau, J. L. Rosner and D. London, . N. G. Deshpande and X. G. He, . Electroweak penguin effects in other $B$ decays were studied earlier by R. Fleischer, . R. Fleischer, ; A. J. Buras and R. Fleischer, ; M. Gronau and J. L. Rosner, ; A. S. Dighe, M. Gronau and J. L. Rosner, ; R. Fleischer and T. Mannel, . M. Neubert and J. L. Rosner, ; ; M. Neubert, JHEP [**9902**]{}, 014 (1999). M. Gronau and D. Pirjol, hep-ph/9902482. A. J. Buras and R. Fleischer, hep-ph/9810260. M. Gronau, O. Hernández, D. London and J. L. Rosner, ; . A. Falk, A. L. Kagan, Y. Nir and A. A. Petrov, ; M. Gronau and J. L. Rosner, ; [**58**]{}, 113005 (1998); R. Fleischer, ; . M. Gronau and J. L. Rosner, . CLEO Collaboration, T. E. Coan , hep-ex/9908029. Z. Ligeti, these proceedings. G. Redlinger, these proceedings. Y. Grossman and H. R. Quinn, ; L. Wolfenstein, . C. O. Dib, D. London and Y. Nir, ; M. Gronau and D. London, . Y. Grossman and M. P. Worah, ; D. London and A. Soni, . G. Buchalla, A. J. Buras and M. E. Lautenbacher, . M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, hep-ph/9905312.
[^1]: Supported in part by the Department of Energy under contract number DE-AC03-76SF00515.
[^2]: Permanent Address: Physics Dept., Technion – Israel Institute of Technology, 32000 Haifa, Israel.
|
---
abstract: |
We study the effect of the charm quark mass in the CTEQ global analysis of parton distribution functions (PDFs) of the proton. Constraints on the $\overline{\rm MS}$ mass of the charm quark are examined at the next-to-next-to-leading order (NNLO) accuracy in the S-ACOT-$\chi$ heavy-quark factorization scheme. The value of the charm quark mass from the hadronic scattering data in the CT10 NNLO fit, including semiinclusive charm production in DIS at HERA collider, is found to agree with the world average value. Various approaches for constraining $m_{c}$ in the global analysis and impact on LHC cross sections are reviewed.\
**Keywords**: Global Analysis, Charm Quark, Parton Distributions
**PACS**: 14.65.Dw, 12.38.-t
author:
- Jun Gao
- Marco Guzzi
- Pavel Nadolsky
title: Charm quark mass dependence in a global QCD analysis
---
Preprint SMU-HEP-13-09
Introduction
============
Quark masses are free parameters of the QCD Lagrangian that parametrize explicit breaking of the chiral symmetry. For quarks heavier than 1 GeV, quark masses arise as independent hard scales $m_{Q}$ in perturbative QCD calculations for particle cross sections. Quarks are not observed freely because of color confinement, hence their properties and masses are established indirectly by comparing theory calculations against experimental data on hadronic reactions. In a global analysis of parton distribution functions (PDFs) of the proton, the method by which the heavy-quark masses are included in experiments at energies comparable to $m_{Q}$ has non-negligible impact on the extracted PDFs [@Tung:2006tb]. Variations in the PDFs associated with the treatment of heavy quarks have phenomenological consequences for electroweak precision measurements at the Large Hadron Collider [@Nadolsky:2008zw].
In this paper we explore constraints on the charm quark mass from the global hadronic data in the CTEQ NNLO PDF analysis. The study is motivated by publications of combined cross sections on inclusive deep-inclusive scattering (DIS) and semi-inclusive DIS charm production at the $ep$ collider HERA [@Aaron:2009aa; @Abramowicz:1900rp]. Among all experimental data sets included in the global fit, the DIS experiments have the best potential to constrain the charm mass. On the theory side, PQCD calculations for neutral-current deep inelastic scattering have been extended to the 2-loop level in the QCD coupling strength $\alpha_{s}$ both for massless [@SanchezGuillen:1990iq; @vanNeerven:1991nn; @Zijlstra:1991qc] and massive [@Laenen:1992zk; @Riemersma:1994hv; @Harris:1995tu] quarks, while massless [@Moch:2004xu; @Vermaseren:2005qc] and some massive [@Blumlein:2006mh; @Bierenbaum:2009mv; @Ablinger:2010ty; @Blumlein:2012vq; @Ablinger:2011pb; @Ablinger:2012qj; @Ablinger:2012qm; @Ablinger:2012sm] coefficient functions were also obtained at the 3-loop level. With such accuracy, it becomes possible to determine the charm quark mass and its uncertainty from the DIS data.
The current world average of the charm mass in the $\overline{\textrm{MS}}$ renormalization scheme is $m_{c}(m_{c})=1.275\pm 0.025$ GeV [@Beringer:1900zz]. This value is derived primarily from measurements in timelike scattering processes and lattice simulations, using analyses that include up to four orders in perturbative QCD. The precise DIS data from HERA in principle allows us to extract $m_{c}(m_{c})$ from a spacelike scattering process and compare to other determinations.
In many previous PDF analyses, the heavy-quark masses have been treated as effective parameters rather than fundamental constants. They were anticipated to deviate from the $\overline{\rm MS}$ masses or even be fully independent. Besides entering the exact matrix elements, the heavy-quark masses control various approximations in DIS calculations, as will be reviewed below. The approximations affect extraction of the mass values from the hadronic data, but their effect is of a higher order in the QCD coupling strength according to the QCD factorization theorem. Thus, although the bias due to the approximations can be important at low orders, it is expected to subside as more loops are included in PQCD calculations. At high enough order of $\alpha_s$, such as the NNLO, a comparison to the world-average $\overline{\rm MS}$ quark mass becomes feasible.
There exist several theoretical approaches, or heavy-quark schemes, for computations involving massive quarks [@Aivazis:1993pi; @Buza:1996wv; @Chuvakin:1999nx; @Thorne:1997uu; @Thorne:1997ga; @Thorne:2006qt; @Forte:2010ta]. The extracted $m_{c}$ value depends on the heavy-quark scheme as well as the order of the PQCD calculation. In this work, we adopt a two-loop implementation [@Guzzi:2011ew] of the general-mass scheme S-ACOT-$\chi$ [@Aivazis:1993pi; @Kramer:2000hn; @Tung:2001mv] employed in the CT10 NNLO analysis [@Gao:2013xoa]. We discuss physics assumptions affecting the extracted value of $m_c(m_c)$ and comparisons to other recent extractions of $m_{c}$ [@Martin:2010db; @Abramowicz:1900rp; @Alekhin:2012vu]. Within this scheme, we find the range of input $m_{c}(m_{c})$ values providing the best description of the CT10 fitted data and compare it to the world-average value. Finally, we analyze the impact of the uncertainty in the charm mass on benchmark LHC predictions.
Charm mass definitions and NNLO predictions for DIS\[sec:Charm-mass-definitions\]
=================================================================================
Overview\[sec:TheoryOverview\]
------------------------------
Quantum field theory operates with two common definitions of the quark mass, the pole mass and the $\overline{\textrm{MS}}$ mass. The pole mass is defined as the position of the pole in the renormalized quark propagator. The pole mass is infrared-safe, gauge-invariant, and is derived in the on-shell renormalization scheme. It is often close to the experimental mass definition [@Gray:1990yh; @Chetyrkin:1999qi; @Melnikov:2000qh; @Marquard:2007uj], but, as the pole charm mass value of 1.3-1.8 GeV borders the nonperturbative region, accuracy of its determination is limited by significant radiative contributions associated with renormalons [@Bigi:1994em; @Beneke:1994sw; @Beneke:1998ui]. Because of large perturbative coefficients arising even at three or four loops in the QCD coupling $\alpha_{s}$, the pole $m_{c}$ value cannot be determined to better than a few hundred MeV.
The $\overline{\textrm{MS}}$ mass $m_{c}(\mu)$, on the other hand, is the renormalized quark mass in the modified-minimal-subtraction scheme, defined as a short-distance mass that is not affected by nonperturbative ambiguities. It is evaluated at a momentum scale $\mu$ typical for the hard process, frequently taken to be the mass $m_{c}$ itself. Precise determinations of $m_{c}(m_{c})$ achieve a smaller uncertainty of order 30 MeV or less. The $\overline{\textrm{MS}}$ mass starts to differ from the pole mass beginning at order ${\cal O}(\alpha_{s})$. The conversion between the $\overline{\textrm{MS}}$ mass to the pole mass is required in the PDF analysis, as the massive 2-loop Wilson coefficients and operator matrix elements in DIS are available in terms of the pole mass. The conversion procedure will be reviewed in the next section.
In parton-level diagrams for deep-inelastic scattering, external massive quarks may arise both as the final and initial states. For the quarks that are heavier than the proton, some factorization schemes introduce an effective PDF to describe their quasi-collinear production at high energy. The heavy-quark PDF can contribute to the hadronic cross section through a convolution with a hard-scattering matrix element with the heavy quark(s) in the initial state, also called a “flavor-excitation” matrix element. In contrast, the “flavor-creation” matrix elements include only light quarks and gluons in the initial state, while the heavy quarks are only in the final state. The “flavor-excitation” contributions commonly arise in the variable flavor number (VFN) schemes, such as the general-mass VFN (GM VFN) scheme. The alternative fixed-flavor number (FFN) scheme does not introduce a heavy-quark PDF operates with “flavor-creation” terms only.
For this study we employ the S-ACOT-$\chi$ general-mass scheme [@Aivazis:1993pi; @Kramer:2000hn; @Nadolsky:2009ge] implemented to the 2-loop (NNLO) accuracy [@Guzzi:2011ew]. This scheme includes exact massive flavor-creation contributions that dominate at low boson virtualities $Q$, as well as the approximate flavor-excitation terms that are important at high $Q$. Thus, the S-ACOT-$\chi$ scheme reduces to the FFN scheme at $Q^{2}\approx m_{c}^{2}$ and to the zero-mass VFN scheme at $Q^{2}\gg m_{c}^{2}$.
In a comprehensive factorization scheme such as GM-VFN, the charm mass plays several roles. First, the *exact* charm mass enters Feynman diagrams for charm particle creation in the final state, such as $\gamma^{*}g\rightarrow c\bar{c}$ in NC DIS. Second, auxiliary scales are introduced that are of order of the fundamental charm mass (either the pole mass or $\overline{\rm MS}$ mass), but need not to coincide with it.
One such scale sets the energy for switching from the 3-flavor to 4-flavor evolution in the running $\alpha_{s}(\mu)$, which is utilized by both FFN and VFN computations. A similar switching scale from 3-flavor evolution to 4-flavor evolution arises in the PDFs $f_{a/p}(x,\mu)$. The charm mass also defines characteristic energy scales in the flavor-excitation contributions, cf. Sec. \[sec:Details-of-implementation\]. Finally, there can be auxiliary scales associated with the final-state quark fragmentation into hadrons, present both in the FFN and GM-VFN schemes. The dependence on these auxiliary scales is reduced with each successive order of perturbation theory.
When the input mass is varied in the global fit, the response of the DIS cross sections reflects coordinated variations of all such scales. An important question arises when interpreting the outcome of the fit: which mass parameter controls the agreement with the data, the exact charm mass or the approximate mass in the auxiliary scales?
We have found that the global fit is sensitive to the exact charm mass, despite the introduced approximations. In one exercise, we have varied the input charm mass in the exact DIS coefficient functions for flavor-creation processes, while keeping it fixed in the above auxiliary mass scales. In a complementary exercise, we varied the input mass in all auxiliary scales, while keeping it fixed in the exact flavor-creation coefficient functions. In both cases, we examined the agreement with the data as a function of the varied mass parameter. We followed the fitting procedure outlined in the next section and assumed fixed PDF parametrizations for the best-fit $m_c(m_c)$ found in the main analysis.
The dependence of the figure-of-merit function $\chi^{2}$ on the varied $m_{c}(m_{c})$ in these exercises is shown for all fitted experiments, the combined HERA inclusive DIS data [@Aaron:2009aa], and the combined HERA semi-inclusive charm production data [@Abramowicz:1900rp] in the upper left, upper right, and lower panels of Fig. \[fig:chi2separate\]. To better visualize the comparison, $\chi^2$ is divided by the number $N_{pt}$ of data points for each data set. The solid blue line and dashed magenta line are for $\chi^2/N_{pt}$ for the varied mass parameter in the exact DIS coefficient functions and in the auxiliary mass scales, respectively.
With all PDF parameters fixed, variations in $\chi^{2}$ are more pronounced in the mass scan of the first type, when the charm mass is varied only in the exact DIS coefficient functions. In this case, $\chi^2$ in both inclusive and semiinclusive DIS shows a pronounced minimum as a function of $m_c(m_c)$.
In the second case, when $m_{c}(m_{c})$ is varied only in the auxiliary scales, the $\chi^{2}$ dependence is flatter and has a shallow minimum at most. This exercise indicates that both inclusive and semiinclusive DIS cross sections are more sensitive to the exact $m_{c}$ mass in the flavor-creation coefficient functions than to the auxiliary scales. The detailed constraints on $m_{c}$ are determined by the interplay of these two trends as well as by variations in the PDFs and other inputs.
![\[fig:chi2separate\] Dependence of $\chi^{2}/N_{pt}$ as a function of $m_{c}(m_{c})$ in the exact flavor-creation coefficient functions (solid blue lines) and auxiliary energy scales listed in the text (dashed magenta lines). ](chi2_mcscan_allexpts "fig:"){width="48.00000%"} ![\[fig:chi2separate\] Dependence of $\chi^{2}/N_{pt}$ as a function of $m_{c}(m_{c})$ in the exact flavor-creation coefficient functions (solid blue lines) and auxiliary energy scales listed in the text (dashed magenta lines). ](chi2_mcscan_exp159 "fig:"){width="48.00000%"}\
![\[fig:chi2separate\] Dependence of $\chi^{2}/N_{pt}$ as a function of $m_{c}(m_{c})$ in the exact flavor-creation coefficient functions (solid blue lines) and auxiliary energy scales listed in the text (dashed magenta lines). ](chi2_mcscan_exp140147 "fig:"){width="48.00000%"}
Details of implementation \[sec:Details-of-implementation\]
-----------------------------------------------------------
### Conversion to the pole mass
Our calculation proceeds by taking the $\overline{\rm MS}$ quark masses as the input for the whole calculation. The transition from the 3-flavor to 4-flavor evolution in $\alpha_{s}$ and PDFs is taken to occur at the scale equal to this input mass.[^1]
The massive 2-loop contributions to neutral-current DIS include matrix elements with explicit creation of $c\bar{c}$ pairs [@Riemersma:1994hv] and operator matrix elements $A_{ab}^{(k)}$ [@Buza:1996wv] for collinear production of massive quarks. Their available expressions take the pole mass as the input. For these parts, the $\overline{\rm MS}$ mass is converted to the pole mass according to the perturbative relation in Eq. (17) of [@Chetyrkin:2000yt], $$\begin{aligned}
m^{pole}_Q&=&m_Q(m_Q)\ \Biggl\{ 1 +
\frac{\alpha_s(m_Q(m_Q),N_f)}{\pi}\frac{4}{3} \nonumber\\
&+&
\frac{\alpha_s^2 (m_Q(m_Q),N_f)}{\pi^2}\Bigl[13.4434-1.04137 N_f+
\frac{4}{3}\sum_{i=1}^{N_f}\Delta(m_i(m_i)/m_Q(m_Q))\Bigr] \Biggr\},
\label{msbar2pole}\end{aligned}$$ where $\Delta (x)= 1.2337\ x - 0.597\ x^2 +0.23 \ x^3$, and the coefficients are shown up to order $\alpha_s^2$.
For the charm quark, the $\alpha_s$ and $\alpha_s^2$ contributions in the conversion formula have comparable magnitudes,[^2] the procedure for $m_c^{pole}$ truncation is numerically important. We implement two conversion methods. In one method, the $\overline{\rm
MS}$ mass is always converted to the pole mass via the full 2-loop relation. Alternatively, the 2-loop (1-loop) conversion is performed within the ${\cal O}(\alpha_s)$ and ${\cal O}(\alpha_s^2)$ terms of the Wilson coefficient functions and OME’s, respectively. This is argued to be equivalent to calculating DIS structure functions directly in terms of the $\overline{{\rm MS}}$ mass and improve perturbative convergence of the $m_{c}(m_c)$ fit [@Alekhin:2010sv]. Numerically, it is not obvious yet that the second (truncated) method improves perturbative convergence at the implemented orders: the effect of including an approximate 3-loop correction in the second method [@Alekhin:2012vu] (producing the change $\delta m_c(m_c)\approx 0.09 $ GeV compared to the 2-loop result) exceeds the difference between two conversion methods at two loops ($\delta m_c(m_c)\approx 0.07 $ GeV).
In charged-current DIS, for which only 1-loop expressions are included, we can use either the $\overline{\rm MS}$ mass or pole mass and find almost no sensitivity to the choice.
### Rescaling variable
The general-mass schemes [@Buza:1996wv; @Chuvakin:1999nx; @Thorne:1997uu; @Thorne:1997ga; @Thorne:2006qt; @Forte:2010ta] differ primarily in the form of approximation for flavor-excitation coefficient functions (with incoming heavy quarks) at $Q$ comparable to $m_{c}$. When $m_{c}$ is negligible, the coefficient functions in all GM-VFN schemes reduce to unique zero-mass expressions, but, near the threshold, they may differ by powerlike contributions $(m_{c}^{2}/Q^{2})^{p}$ with $p>0$ in the approximate flavor-excitation terms. The powerlike corrections are suppressed in the full result by an additional power of $\alpha_{s}$, *i.e.*, they are of order $\alpha_{s}^{3}$ in an NNLO ($\alpha_{s}^{2}$) calculation. In the S-ACOT-$\chi$ scheme, the form of the powerlike contributions is selected based on the general consideration of energy-momentum conservation [@Tung:2001mv; @Tung:2006tb; @Nadolsky:2009ge]. As a result, the flavor-excitation contributions are suppressed at energies close to the massive quark production threshold, producing better description of the DIS data. The flavor-excitation coefficient functions are constructed from the respective zero-mass coefficient functions $c_{ZM}(\chi/\xi,Q/\mu)$, where the parton’s momentum fraction $\chi$ is rescaled with respect to Bjorken $x$ by a factor dependent on the total mass $M_f$ of heavy quarks in the final state. The general behavior of $\chi$ is determined from the condition of the threshold suppression, while its detailed form can be varied to estimate the associated higher-order uncertainty in the extracted $m_{c}$. This approach is readily demonstrated [@Guzzi:2011ew] to be compliant with the QCD factorization theorem for DIS cross sections to all orders [@Collins:1998rz], which is one of the advantages of using the S-ACOT-$\chi$ scheme.
In the default implementation, the momentum fraction in NC DIS charm production is given by $\chi=x(1+4m_{c}^{2}/Q^{2})$, corresponding to $M_{f}=2m_{c}$ for the lightest final state ($c\bar{c}$). [^3] In charged-current DIS, we set $\chi=x\left(1+m_{c}^{2}/Q^{2}\right)$. The rescaling ratio $\chi/x$ is thus independent of $x$. On general grounds, it may be expected that the threshold suppression is less pronounced at $W^{2}=Q^{2}(1/x-1)\rightarrow\infty$ for fixed $Q$, corresponding to $x\rightarrow0$. In this limit, it may be desirable to reduce or even eliminate the rescaling altogether, as quasi-collinear production of heavy quarks becomes more feasible.
To allow for this possibility, a generalized rescaling variable $\zeta$ can be implicitly defined by [@Nadolsky:2009ge] $$x=\zeta\, \left(1+\zeta^{\lambda}M_{f}^{2}/Q^{2}\right)^{-1},$$ where $\lambda$ is a positive parameter, typically $0\leq
\lambda \lesssim 1$. The S-ACOT-$\chi$ scheme is reproduced with $\lambda=0$, and the rescaling is fully turned off for $\lambda\gg 1$. For $\lambda\neq0$, the mass-threshold constraints are enforced at small $W$ (large $x$), but the standard $x$ variable is recovered at large $W$ (small $x$) in a controllable way.
The sensitivity of the CTEQ global fit to $\lambda$ is of the same order as the sensitivity to the $m_c^{pole}$ truncation method. The changes in the preferred value of $m_{c}$ that we observe provide an estimate of the uncertainty due to the powerlike corrections.
Charm mass scans \[sec:Results\]
================================
Setup
-----
Using the theoretical setup reviewed in the previous section and the general procedure of the CT10 NNLO PDF analysis [@Gao:2013xoa], a scan of the log-likelihood function $\chi^{2}$ over the input $\overline{\rm MS}$ charm mass was carried out in the range $1\leq m_{c}(m_{c})\leq1.4$ GeV. Besides the combined HERA data on inclusive DIS and semiinclusive charm production, we include experimental data from DIS measurements by BCDMS [@Benvenuti:1989rh; @Benvenuti:1989fm], NMC [@Arneodo:1996qe], CDHSW [@Berge:1989hr], and CCFR [@Yang:2000ju; @Seligman:1997mc]; NuTeV and CCFR dimuon production [@Goncharov:2001qe; @Mason:2006qa]; $F_{2c}$ measurements at HERA [@Adloff:2001zj] that are not included in the combined set; fixed-target Drell-Yan process [@Moreno:1990sf; @Towell:2001nh; @Webb:2003ps]; vector boson and inclusive jet production at the Tevatron [@Abe:1996us; @Acosta:2005ud; @Abazov:2008qv; @Abazov:2007pm; @Abazov:2006gs; @Aaltonen:2010zza; @Aaltonen:2008eq; @Abazov:2008ae]. We also include inclusive jet production at the LHC [@Aad:2011dm; @Aad:2011fc], which slightly reduces the uncertainty in the gluon PDF.
Depending on the candidate fit, the QCD coupling strength was taken to be either the world average, $\alpha_{s}(M_{Z})=0.118$ [@Beringer:1900zz], or a lower $\alpha_{s}(M_{Z})=0.115$, which is preferred by the CT10 NNLO analysis when $\alpha_s(M_Z)$ is allowed to vary. The factorization/renormalization scale $\mu$ in DIS was set equal to the momentum transfer $Q$. To test the sensitivity to the PDF parametrization form, the initial scale $Q_{0}$ at which the input PDFs are provided was either set to be $Q_{0}=1$ GeV independently of $m_{c}$ or varied in the scan together with $m_c$ as $Q_{0}=m_{c}-0.005$ GeV. Several forms of the gluon PDF parametrization were considered, since DIS charm production is sensitive to the gluon PDF $g(x,Q)$. At the initial scale $Q_0$, we either constrained $g(x,Q_0)$ to be positive at all $x$ or allowed it to be negative at small $x$, provided the negative gluon did not lead to unphysical predictions. In the latter case of the negative gluon, the fit included the H1 data on the longitudinal structure function $F_{L}(x,Q)$ [@Collaboration:2010ry] and an additional theoretical constraint to ensure positivity of $F_{L}(x,Q)$ at $x>10^{-5}$.
Sensitivity of individual experiments
-------------------------------------
![\[fig:ind\] Individual $\chi^{2}$ contributions of the combined HERA-1 inclusive DIS data, NMC $F_{2}^{p}$ data, and combined HERA charm quark production data as a function of $m_{c}(m_c)$.](indchi1 "fig:"){width="30.00000%"} ![\[fig:ind\] Individual $\chi^{2}$ contributions of the combined HERA-1 inclusive DIS data, NMC $F_{2}^{p}$ data, and combined HERA charm quark production data as a function of $m_{c}(m_c)$.](indchi2 "fig:"){width="30.00000%"} ![\[fig:ind\] Individual $\chi^{2}$ contributions of the combined HERA-1 inclusive DIS data, NMC $F_{2}^{p}$ data, and combined HERA charm quark production data as a function of $m_{c}(m_c)$.](indchi3 "fig:"){width="30.00000%"}
Among all experiments included in the mass scan, the tightest constraints on $m_{c}$ are imposed by the inclusive DIS HERA data [@Aaron:2009aa] and HERA charm production [@Abramowicz:1900rp]. Weaker constraints also arise from the other DIS experiments, notably the NMC measurement of $F_{2}^{p}(x,Q)$.
Fig. \[fig:ind\] shows a sample of behavior of $\chi^{2}$ in each of three experiments as a function of $m_{c}$. The inclusive DIS data from HERA (upper left inset) and NMC (upper right inset) broadly prefer an $m_{c}$ in the range 1.1-1.3 GeV, while the charm HERA data (upper right inset) prefers lower $m_{c}$ values of order 1 GeV. The NMC shown in the lower inset prefers a higher $m_{c}$ of order 1.2 GeV in accord with the inclusive HERA data, but with flatter $\chi^2$ dependence. While in the shown mass scan the inclusive DIS data prefer a higher $m_{c}$ value than in DIS charm production, the $\chi^{2}$ minimum in inclusive DIS may shift to lower values of about 1.05-1.1 GeV depending on the parametrization forms of the PDFs and other fit assumptions.
We note that the semiinclusive reduced cross sections on HERA charm production were derived from charm differential distributions by applying a significant acceptance correction computed with the program <span style="font-variant:small-caps;">HVQDIS</span> [@Harris:1997zq] in the FFN scheme. Due to the mismatch in the schemes, the $m_{c}(m_{c})$ preferred by the HERA charm data may be biased when determined in the S-ACOT-$\chi$ general-mass scheme. Nevertheless, we see from Fig. \[fig:ind\] that the $m_{c}(m_{c})$ values that are separately preferred by the inclusive DIS and charm DIS data sets are about the same. In another cross check, we used separate, rather than combined, data sets for HERA charm production [@Chekanov:2008yd; @Chekanov:2009kj; @Aaron:2009af; @Aaron:2009jy; @Aktas:2006py; @Breitweg:1999ad; @Chekanov:2003rb; @Aaron:2011gp]. Such fit did not differ much from the fit based on the combined HERA charm data set.
The global $\chi^2$ and PDF uncertainty \[tol\]
-----------------------------------------------
![\[gchi2\] Global $\chi^{2}$ of the S-ACOT-$\chi$ NNLO fit as a function of the $\overline{{\rm MS}}$ charm mass. Lines with left/right arrows indicate 90% C.L. intervals obtained with different tolerance criteria.](chi2_massconv1 "fig:"){width="45.00000%"} ![\[gchi2\] Global $\chi^{2}$ of the S-ACOT-$\chi$ NNLO fit as a function of the $\overline{{\rm MS}}$ charm mass. Lines with left/right arrows indicate 90% C.L. intervals obtained with different tolerance criteria.](chi2_massconv2 "fig:"){width="45.00000%"}
The constraints from various experiments are generally compatible and produce a well-defined minimum in the global $\chi^2$. The plots of $\chi^{2}$ for all experiments in the above $m_{c}(m_{c})$ scans is shown in Fig. \[gchi2\] for the full $\overline{\rm MS}\rightarrow\mbox{pole}$ mass conversion (left inset) and truncated conversion (right inset). The PDF parameters were refitted for every $m_{c}(m_c)$. The functional form for $\chi^2$ can be found in Ref. [@Gao:2013xoa]. The default rescaling parameter $\lambda=0$ was used. The scattered points are for individual fits at discrete $m_c(m_c)$ values, while the continuous line indicates smooth interpolation across the individual fits.
In the figure, the preferred value of the $\overline{{\rm MS}}$ charm mass, corresponding to the minimum of $\chi^{2}$, is $m_{c}(m_{c})=1.12$ GeV for the full conversion and 1.18 GeV for the truncated conversion. The optimal charm mass is below the world-average $m_{c}(m_{c})=1.275\pm0.025$ GeV, and this general trend is observed in all fits.
From the mass dependence of $\chi^{2}$ in the figure, we can compute the uncertainty on $m_c(m_c)$ due to the PDF parameters. Since the CT analysis traditionally operates with the 90% confidence level (90% C.L.) uncertainty, we compared 3 different criteria for defining it for $m_{c}(m_{c})$: 1) the uniform $\chi^{2}$ tolerance, in which one assigns a 90% C.L. to a $\Delta\chi^{2}\leq100$ variation as in CTEQ6 [@Pumplin:2002vw]; 2) the CT10-like criterion, which supplements the uniform $\chi^{2}$ tolerance condition by additional $\chi^{2}$ penalties to prevent strong disagreements with individual experiments *on average* [@Lai:2010vv]; 3) and the MSTW-like criterion, which does not introduce the uniform tolerance, but requires the $\chi^{2}$ value for *every* individual experiment to lie within the specified confidence interval. In the latter two methods, deviations of the PDF parameters are additionally constrained so as not to trigger a strong disagreement with one of the fitted experiments. This condition effectively reduces the PDF uncertainty in methods 2 and 3 compared to the uniform tolerance criterion (method 1), see further details in [@Lai:2010vv].
The PDF uncertainties on $m_{c}(m_{c})$ obtained with the three definitions are reported in Table \[TAB1\]. The uncertainty according to the uniform tolerance definition is larger than with the other two definitions, as anticipated.
[|c|c|]{} [\
]{} $\Delta\chi^{2}\leq100$ & $\delta m_{c}=_{-0.22}^{+0.30}$ [\
]{} CT10-like & $\delta m_{c}=_{-0.17}^{+0.11}$[\
]{} MSTW-like & $\delta m_{c}=_{-0.18}^{+0.12}$[\
]{}
![\[fig:dchi\]Total $\chi^2$ and its parabolic fit as functions the charm mass $m_c(m_c)$.](abm1_AllExpts_chi2mc){width="50.00000%"}
The rest of the paper interchangeably operates with the 68% and 90% C.L. intervals, with the former taken to be 1.65 times smaller than the latter. It is insightful to compare the above tolerance criteria to the procedure in the FFN analyses, [@Alekhin:2012vu; @Abramowicz:1900rp] that assigns the 68% C.L. PDF uncertainty to the increase in the global $\chi^2$ by one unit. This definition of the PDF error [@Collins:2001es; @BevingtonRobinson] is applicable under ideal conditions and leads to a smaller PDF error compared to the CT10 criterion. We have examined the $\Delta\chi^{2}=1$ criterion as an alternative to the CT10 criterion and found that it does not realistically describe the observed probability distribution. The reason is that the $\Delta\chi^2 =1$ criterion is strictly valid when the experimental errors are Gaussian, which implies quadratic dependence of $\chi^2$ on $m_c(m_c)$. The actual $\chi^2$ distribution is not perfectly Gaussian and exhibits asymmetry as well as some random fluctuations.
In Fig. \[fig:dchi\] we compare the observed distribution of $\chi^2(m_{c,i})$ (scattered circles) with a fit by a second-degree polynomial $\chi^2_{parabola}(m_c)=A m_c^2+B m_c+ C$ shown by the solid line. Deviations from the perfect quadratic behavior can be characterized by $$X^2\equiv\sum_{i=1}^{N_{m_c}}(\chi^2_{parabola}(m_{c,i})-\chi^2(m_{c,i}))^2,
\label{X2}$$ where $N_{m_c}$ is the number of discrete $m_c(m_c)$ values in the scan. In the ideal Gaussian case, when the individual $\chi^2(m_{c,i})$ follow neatly the parabola in the scanned $m_c(m_c)$ region, $X^2$ is much less than $N_{m_c}-1$, and $\delta m_c=\sqrt{1/A}$ (about 0.025 GeV in our fits) provides a fair estimate of the $1\sigma$ error. But in the actual fits of the kind indicated by the circles in Fig. \[fig:dchi\], $X^2/(N_{m_c}-1)$ is of order 2.5, hence $\sqrt{1/A}$ underestimates the $m_c$ error by a factor of about 2.5. This reflects the probability distribution that is broader than the ideal parabola and hence contains less than 68% of the net probability in the $m_c$ region where the parabolic growth results in $\Delta\chi^2=1$.
A partial remedy is procured by symmetric rescaling, if one defines $\widetilde X^{2}\equiv X^2/C$ with a constant $C\approx 2.5$ so that $\widetilde X^{2}\approx 1$ for the observed distribution of $\chi^2(m_{c,i})$. When the PDF uncertainty is derived from the rescaled $\widetilde X^{2}$ statistic, it is larger by a factor $C$ compared to the idealized assumption. Symmetric rescaling increases the 68% C.L. error but does not fix the shape of $\chi^2(m_c)$. After the rescaling, the PDF error of 0.06 GeV gets closer to the one obtained by the CT10-like criterion, which is of order 0.09 GeV at 68% C.L. if the asymmetric errors are averaged over.[^4]
Our main conclusion is that, in the contest of our NNLO extraction of $m_c(m_c)$, the quadratic assumption fails to describe the actual $\chi^2(m_c(m_c)$ distribution to the extent needed to justify the $\Delta\chi^2=1$ prescription. The actual distribution is flatter near the minimum, asymmetric, and has occasional fluctuations. A number of reasons can explain this behavior. It has been observed in the earlier work [@Lai:2010vv; @Pumplin:2009nm; @Pumplin:2009sc] that moderate disagreements between the fitted experiments broaden the probability distribution around the global minimum of $\chi^2$ compared to the ideal Gaussian case. The net effect of these disagreements may be approximated, to the first order, by increasing the $\Delta \chi^2=1$ PDF error by a factor of two or three, [*i.e.*]{}, about the same factor as in the $m_c(m_c)$ fit.
Systematic uncertainties
------------------------
Theoretical systematic uncertainty $m_c^{pole}$ conversion DIS scale $\alpha_{s}(M_{Z})$ $\lambda$ $\chi^{2}$ definition
------------------------------------ ------------------------- ---------------------- -------------------------- -------------------- -----------------------
Parameter range – $[Q/2,\ 2Q]$ [\[]{}0.116, 0.120[\]]{} [\[]{}0, 0.2[\]]{} –
$\delta m_c(m_c)$ (GeV) 0.07 ${}^{+0.02}_{-0.02}$ ${}^{+0.01}_{-0.01}$ ${}^{+0.14}_{-0}$ 0.06
: \[unc\] Shifts of the optimal value of the charm mass $m_{c}(m_{c})$ obtained by varying theoretical inputs.
In addition to the PDF uncertainty associated with experimental errors, Table \[unc\] summarizes systematic uncertainties on $m_{c}(m_c)$ associated with theoretical inputs, including the $\overline{\rm MS}\rightarrow\mbox{pole}$ conversion procedure, the factorization/renormalization scale, $\alpha_{s}(M_{Z})$, the $\lambda$ parameter in the rescaling variable, and implementation of experimental correlated systematic errors. The last source of uncertainty arises from the existence of several prescriptions for including correlated systematic errors from the fitted experiments into the figure-of-merit function $\chi^{2}$ [@Ball:2012wy; @Gao:2013xoa]. Depending on the prescription, the relative correlated errors published by the experiments can be interpreted as fractions of either central data values or theoretical values. These methods are designated as “extended T” and “D” methods in Ref. [@Gao:2013xoa]. This leads to numerical differences in absolute correlated errors, which may affect the outcomes of the fit, as discussed in the above references. We estimate the associated uncertainty by alternating between the two normalization methods for the correlation matrices of the DIS processes.
![\[fig:bench\] Comparison of the $\overline{\textrm{MS}}$ charm mass and its uncertainty extracted with various methods. Illustrated are also $m_c(m_c)$ values obtained in an FFN analysis by Alekhin et al. [@Alekhin:2010sv].](bench){width="50.00000%"}
The values of $m_c(m_c)$ obtained under various assumptions are illustrated by Fig. \[fig:bench\]. At order $\alpha_s^2$, the highest fully implemented order in our calculation, we show $m_c(m_c)$ found with four methods. Methods 1 and 2 correspond to the “extended $T$” and “experimental” $\chi^2$ definitions respectively [@Gao:2013xoa], both using the full $\overline{{\rm MS}}\rightarrow\mbox{pole}$ mass conversion formula, and $\lambda=0$. Methods 3 and 4 are the same as 1 and 2, but with truncated mass conversion equivalent to computing the coefficient functions in the $\overline{\rm MS}$ scheme. The resulting $m_c(m_c)$ values in the four methods are $1.12^{+0.05}_{-0.11}$, $1.18^{+0.05}_{-0.11}$, $1.19^{+0.06}_{-0.15}$ and $1.24^{+0.06}_{-0.15}$ GeV, respectively. Here and in the figure we quote the 68% C.L. PDF uncertainties defined as in the CT10 analysis, cf. Sec. \[tol\].
The dependence on the $\lambda$ parameter is illustrated by Fig. \[fig:con\], showing boundaries of the 68% and 90% C.L. regions when $\lambda$ takes values on the horizontal axis, for the “extended $T$” (solid lines) and “experimental” (dashed lines) definitions of $\chi^2$, and using the full $\overline{\rm MS}\rightarrow\mbox{ pole}$ conversion.[^5] The red empty triangle and black diamond symbols are the best-fit values of $m_{c}(m_{c})$ obtained with the two $\chi^2$ prescriptions, equal to 1.12 and 1.18 GeV, and reached when $\lambda\approx 0$ in both cases. Values of $\lambda$ above 0.14 and 0.20 are disfavored at 68% (90%) C.L.[^6] Finally, the horizontal blue band indicates the world-average interval $m_{c}(m_{c})=1.275\pm 0.025$ GeV.
![\[fig:con\] Preferred regions for $m_{c}(m_{c})$ vs. the rescaling parameter $\lambda$. The best-fit values and confidence intervals are shown for two alternative methods for implementation of correlated systematic errors.](contour){width="40.00000%"}
As we see, there is some spread in the $m_c$ values depending on the adopted $\overline{\rm MS}\rightarrow\mbox{pole}$ conversion and $\chi^2$ definition. In addition, moderate dependence exists on the rescaling parameter $\lambda$, associated with missing higher-order corrections. We can estimate the projected range for the ${\cal O}(\alpha_s^3)$ value of $m_c(m_c)$ in method 3 (or any other method) according to the CT10-like criterion from the $\chi^2$ dependence for a range of $\lambda$ values. This produces $1.19^{+0.08}_{-0.15}$ GeV for the estimated ${\cal O}(\alpha_s^3)$ value in method 3, as shown in line 6 in Fig. \[fig:bench\]. Here the errors are estimated from the 68% C.L. contour for $\chi^2$ vs. $\lambda$, and the scale and $\alpha_s$ errors are added in quadrature.
The central $m_c(m_c)$ is consistent with the PDG value of $1.275\pm 0.025$ GeV [@Beringer:1900zz] within the errors. A tendency of the fits to undershoot the PDG value may be attributable to the missing ${\cal O}(\alpha_s^3)$ contribution [@Alekhin:2012vu]. The results of our fit are compatible with $m_c(m_c)$ determined in the (FFN) scheme [@Alekhin:2012vu] both at the exact ${\cal O}(\alpha_s^2)$ and approximate ${\cal O}(\alpha_s^3)$, cf. lines 5 and 7 in Fig. \[fig:bench\]. However, our PDF error of about 0.08 GeV is 2.7 times larger than the one quoted in the FFN study. The main reason is that the 68% C.L. PDF uncertainty of the FFN analysis corresponds to $\Delta\chi^{2}=1$ and hence is smaller than the CT10-like uncertainty. As discussed above, our $\chi^2$ dependence on $m_c(m_c)$ is not compatible with the ideal Gaussian behavior and could be accommodated by increasing the PDF error by a factor 2-3 compared to the $\Delta\chi^{2}=1$ definition. Besides this difference in the PDF uncertainty, the results for $m_c(m_c)$ from the S-ACOT-$\chi$ and FFN fits are in general agreement.
![\[fig:pdf\] Relative changes in select PDFs $f_{a/p}(x,Q)$ at $Q=2$ GeV obtained in a series of PDF fits with $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV, plotted as ratios to the respective PDFs for $m_{c}(m_{c})=1$ GeV. Default rescaling ($\lambda=0$) in DIS coefficient functions is assumed. Darker colors correspond to larger $m_{c}(m_{c})$ values.](mcpdf8 "fig:"){width="48.00000%"}![\[fig:pdf\] Relative changes in select PDFs $f_{a/p}(x,Q)$ at $Q=2$ GeV obtained in a series of PDF fits with $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV, plotted as ratios to the respective PDFs for $m_{c}(m_{c})=1$ GeV. Default rescaling ($\lambda=0$) in DIS coefficient functions is assumed. Darker colors correspond to larger $m_{c}(m_{c})$ values.](mcpdf1 "fig:"){width="48.00000%"}\
![\[fig:pdf\] Relative changes in select PDFs $f_{a/p}(x,Q)$ at $Q=2$ GeV obtained in a series of PDF fits with $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV, plotted as ratios to the respective PDFs for $m_{c}(m_{c})=1$ GeV. Default rescaling ($\lambda=0$) in DIS coefficient functions is assumed. Darker colors correspond to larger $m_{c}(m_{c})$ values.](mcpdf2 "fig:"){width="48.00000%"}![\[fig:pdf\] Relative changes in select PDFs $f_{a/p}(x,Q)$ at $Q=2$ GeV obtained in a series of PDF fits with $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV, plotted as ratios to the respective PDFs for $m_{c}(m_{c})=1$ GeV. Default rescaling ($\lambda=0$) in DIS coefficient functions is assumed. Darker colors correspond to larger $m_{c}(m_{c})$ values.](mcpdf4 "fig:"){width="48.00000%"}
Implications for PDFs and collider observables\[sec:Implications\]
------------------------------------------------------------------
As $m_{c}(m_{c})$ changes in the mass scan, parametrizations of the PDFs are adjusted so as to maximize agreement with the data. Representative best-fit PDFs from a charm mass scan are plotted in Fig. \[fig:pdf\] at $Q=2$ GeV. In this example, we vary the charm mass in the interval $1.0\leq m_{c}(m_{c})\leq1.36$ GeV that is about the same as the 90% C.L. CT10-like PDF uncertainty determined in the previous section. For each $m_{c}(m_{c})$ from this range, we refit the PDFs, while keeping the other theoretical inputs at their default values, in particular assuming $\alpha_{s}(M_{Z})=0.118$ and $\lambda=0$ in the rescaling variable. Darker color points in Fig. \[fig:pdf\] correspond to larger mass values.
In general, as the charm mass is increased, both the charm PDF (upper left subfigure) and charm contributions to DIS cross sections are suppressed. Consequently, the gluon PDF (upper right subfigure) is enhanced in the intermediate $x$ region, $10^{-3}\sim10^{-2}$ so as to partly compensate for this reduction. This is accompanied by moderate enhancements in the up and down quark PDFs (lower subfigures) at $x\approx10^{-2}-0.5$ and slight suppression of the same PDFs at $x\approx10^{-3}$.
![\[fig:wz1\] Plot of NNLO cross sections for $W^{\pm}$, $Z^{0}$, Higgs boson production through gluon fusion, and top quark pair production at the LHC (8 TeV) for charm quark mass $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV and $\lambda=\{0,\ 0.02,\ 0.05,\ 0.1,\ 0.15,\ 0.2\}$. Darker color corresponds to larger mass values. The black boxes represent the cross sections evaluated by using $m_{c}(m_{c})=1.28$ (close to world average) GeV and the explored $\lambda$ values. The empty triangle and ellipse indicate the central predictions and its 90% C.L. interval based on the CT10NNLO fit.](wpwm8 "fig:"){width="40.00000%"} ![\[fig:wz1\] Plot of NNLO cross sections for $W^{\pm}$, $Z^{0}$, Higgs boson production through gluon fusion, and top quark pair production at the LHC (8 TeV) for charm quark mass $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV and $\lambda=\{0,\ 0.02,\ 0.05,\ 0.1,\ 0.15,\ 0.2\}$. Darker color corresponds to larger mass values. The black boxes represent the cross sections evaluated by using $m_{c}(m_{c})=1.28$ (close to world average) GeV and the explored $\lambda$ values. The empty triangle and ellipse indicate the central predictions and its 90% C.L. interval based on the CT10NNLO fit.](zgws8 "fig:"){width="41.00000%"}\
![\[fig:wz1\] Plot of NNLO cross sections for $W^{\pm}$, $Z^{0}$, Higgs boson production through gluon fusion, and top quark pair production at the LHC (8 TeV) for charm quark mass $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV and $\lambda=\{0,\ 0.02,\ 0.05,\ 0.1,\ 0.15,\ 0.2\}$. Darker color corresponds to larger mass values. The black boxes represent the cross sections evaluated by using $m_{c}(m_{c})=1.28$ (close to world average) GeV and the explored $\lambda$ values. The empty triangle and ellipse indicate the central predictions and its 90% C.L. interval based on the CT10NNLO fit.](zgh8 "fig:"){width="40.00000%"} ![\[fig:wz1\] Plot of NNLO cross sections for $W^{\pm}$, $Z^{0}$, Higgs boson production through gluon fusion, and top quark pair production at the LHC (8 TeV) for charm quark mass $m_{c}(m_{c})$ ranging from 1 to 1.36 GeV and $\lambda=\{0,\ 0.02,\ 0.05,\ 0.1,\ 0.15,\ 0.2\}$. Darker color corresponds to larger mass values. The black boxes represent the cross sections evaluated by using $m_{c}(m_{c})=1.28$ (close to world average) GeV and the explored $\lambda$ values. The empty triangle and ellipse indicate the central predictions and its 90% C.L. interval based on the CT10NNLO fit.](zgtt8 "fig:"){width="40.00000%"}
![\[fig:wz2\] Same as in Fig. \[fig:wz1\], but with $\sqrt{S}=14$ TeV.](wpwm14 "fig:"){width="40.00000%"} ![\[fig:wz2\] Same as in Fig. \[fig:wz1\], but with $\sqrt{S}=14$ TeV.](zgws14 "fig:"){width="40.00000%"}\
![\[fig:wz2\] Same as in Fig. \[fig:wz1\], but with $\sqrt{S}=14$ TeV.](zgh14 "fig:"){width="39.50000%"} ![\[fig:wz2\] Same as in Fig. \[fig:wz1\], but with $\sqrt{S}=14$ TeV.](zgtt14 "fig:"){width="41.00000%"}
PDF variations of such magnitude may have impact on collider observables [@Nadolsky:2008zw]. For example, in Fig. \[fig:wz1\] and \[fig:wz2\] we show the dependence of NNLO total cross sections for $W,$ $Z,$ Higgs boson production through gluon fusion, and top quark pair production at the LHC at $\sqrt{S}=8$ and $14$ TeV. The NNLO cross sections for $W$ and $Z$ production are computed using FEWZ2.1 [@Gavin:2010az; @Gavin:2012sy]. The NNLO cross sections for Higgs and top quark pair production are obtained from iHixs1.3 [@Anastasiou:2011pi] and Top++2.0 [@Baernreuther:2012ws; @Czakon:2013goa] with $m_h=125\ {\rm GeV}$, $m_t=173.3\ {\rm GeV}$, and the QCD scales set to the corresponding mass values. For each pair of total cross sections, we show the central CT10NNLO prediction and an ellipse corresponding to the 90% C.L. PDF uncertainty interval of the CT10NNLO set [@Gao:2013xoa]. In the same figure, scattered points indicate the cross sections obtained with various combinations of $m_{c}(m_{c})$ and $\lambda$ in the intervals $1\leq m_{c}(m_{c})\leq1.36$ GeV and $0\leq\lambda\leq0.2$. Here the darker colors represent larger values of $m_{c}$, as in Fig. \[fig:pdf\]. The $W$, $Z$, and Higgs production cross sections increase with the charm mass by a few percent in the mass range considered. The trend is opposite for $t\bar t$ production. The changes are contained within the CT10NNLO PDF uncertainty ellipse, however, and do not modify the PDF-induced (anti-)correlations observed between the shown total cross sections.
The shown uncertainty in the LHC cross sections due to $m_{c}$ is comparable to the experimental PDF uncertainty and in principle should be included independently from the latter. Let us outline one possibility for reducing the $m_c$ uncertainty. Instead of allowing $m_{c}(m_c)$ to vary in the whole interval $1 - 1.36$ GeV of its PDF uncertainty, we could set it to be at the world-average $m_{c}(m_{c})$ value of 1.275 GeV or in the $1\sigma$ interval $\pm0.025$ GeV around it. This would reduce the associated uncertainty in the PDFs and QCD observables. The corresponding predictions for the LHC $W,$ $Z,$ Higgs, and top quark pair production cross sections, obtained for $m_{c}(m_{c})=1.28$ GeV and five explored $\lambda$ values, are shown in Figs. \[fig:wz1\] and \[fig:wz2\] by black boxes, with the size of the boxes increasing with the value of $\lambda$. The spread in these predictions constitutes only a part of the full span covered by the scattered points for the interval $1\leq m_{c}\leq1.36$ GeV. It reflects only the uncertainty due to the form of the rescaling variable, controlled by the $\lambda$ parameter. Theoretical predictions are better clustered in this case.
Conclusions
===========
We explored the charm quark mass dependence in the CTEQ NNLO global PDF analysis that includes the recently published combined data on charm quark production at the $ep$ collider HERA. This analysis, carried out in the S-ACOT-$\chi$ heavy-quark scheme at order $\alpha_s^2$, renders an optimal $\overline{\rm MS}$ charm mass that is compatible with the world-average value $m_{c}(m_{c})=1.275\pm 0.025$ GeV. For example, with the $\overline{\rm MS}$ Wilson coefficient functions for NC DIS, we obtain $m_c(m_c)=1.19^{+0.08}_{-0.15}
\mbox{ GeV}$, where the errors indicate the 68% C.L. uncertainty due to the PDFs and variation of the rescaling variable, as well as the scale and $\alpha_s$ uncertainties added in quadrature.
In QCD predictions for massive-quark DIS, one draws a distinction between the $\overline{\rm MS}$ charm mass, the fundamental parameter of the QCD Lagrangian, and auxiliary energy scales of order of the physical charm mass. The auxiliary scales that can contribute are specified by the factorization scheme. They can be associated with the evolution of the QCD coupling strength, evolution of PDFs, heavy-quark fragmentation, and powerlike contributions $(m_{c}/Q)^{p}$ in DIS coefficient functions with initial-state charm quarks.
We argue that the sensitivity to the auxiliary scales is reduced as the order of the PQCD calculation increases. In support of this argument, Fig. \[fig:chi2separate\] demonstrates that the DIS data are sensitive mostly to the physical mass parameter in the exact DIS coefficient functions and less to the auxiliary mass scales in the other parts of the calculation. Thus, the hadronic cross sections in DIS and other processes at NNLO and beyond become increasingly suitable for the determination of the fundamental charm mass.
The main findings of our fit are summarized in Fig. \[fig:bench\] showing the 68% C.L. intervals for the $\overline{\rm MS}$ charm mass obtained under various assumptions explained in the text. The central value of $m_{c}(m_{c})$ in the S-ACOT-$\chi$ scheme at two loops tends to undershoot the world-average value according to the figure, but is compatible with the latter within the uncertainty.
The uncertainties in the $m_{c}(m_{c})$ determination of both experimental and theoretical origin were explored in Sec. \[sec:Results\]. If $m_{c}(m_{c})$ is varied as an independent parameter in the full range of order 1-1.4 GeV allowed by the NNLO PDF fit, it increases the uncertainty on the resulting PDFs, compared to a fixed $m_c$.
For comparison, the accuracy of the world-average $m_{c}(m_{c})$, at about $0.025$ GeV, is smaller than the NNLO fit uncertainty. By using a constant value of the $m_{c}(m_{c})$ parameter, for example by setting it equal to its world-average value, one can suppress the corresponding uncertainty in the PDFs. This strategy is similar to implementing the QCD coupling dependence in the global analysis [@Lai:2010nw], when using the world-average value of $\alpha_{s}(M_{Z})=0.118$ results in tighter constraints on the PDFs than in a fit with a free $\alpha_{s}(M_{Z})$. When the input $m_{c}(m_{c})$ value is held constant instead of being fitted, one suppresses the associated uncertainty in the PDFs and LHC cross sections, see sample calculations in Figs. \[fig:wz1\] and \[fig:wz2\]. The residual theoretical uncertainty for a fixed $m_{c}(m_{c})$ then arises only from variations in the auxiliary scales and powerlike contributions and can be suppressed by including higher orders in $\alpha_{s}$.
This work was supported by the U.S. DOE Early Career Research Award DE-SC0003870 by Lightner-Sams Foundation. We thank Achim Geiser for the critical reading of the manuscript and appreciate detailed discussions with Karin Daum, Joey Huston, Hung-Liang Lai, Katerina Lipka, Fred Olness, Jon Pumplin, Carl Schmidt, Dan Stump, and C.-P. Yuan. P. N. thanks DESY (Hamburg) for hospitality and financial support of his visit during the work on this project.
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[^1]: The evolution of $\alpha_{s}$ and PDFs is carried out using the HOPPET computer code [@Salam:2008qg], configured so that transitions from $N_{f}$ to $N_{f+1}$ flavors occur at the $\overline{\rm MS}$ masses.
[^2]: For example, $m_c(m_c)=1.15$ GeV translates into $m_c^{pole}=1.31, 1.54, 1.86$ GeV using one, two, three loops in the conversion formula with $\alpha_s(M_Z)=0.118$.
[^3]: Starting from $O$($\alpha_{s}^{2}$), contributions with up to four massive quarks in the final state can appear. In such terms, we still use $\chi=x\left(1+4m_{c}^{2}/Q^{2}\right)$, given their smallness in the total result [@Guzzi:2011ew].
[^4]: A broadscale argument is also available that the probability distribution ${\cal P}(m_c)\propto \exp\left(-\chi^2(m_c)/2\right)$ on which the $\Delta \chi^2=1$ criterion is based underestimates the confidence levels in PDF fits [@Ball:2011gg].
[^5]: Similar $\lambda$ dependence is observed for the truncated conversion.
[^6]: The CT10 or MSTW-like tolerance criteria lead to about the same boundaries.
|
---
abstract: 'Let $X$ be a smooth projective surface with Picard number 1. Let $L$ be the ample generator of the Néron-Severi group of $X$. Given an integer $r\ge 2$, we prove lower bounds for the Seshadri constant of $L$ at $r$ very general points in $X$.'
address: 'Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103, India'
author:
- Krishna Hanumanthu
date: 'October 17, 2016'
title: Seshadri constants on surfaces with Picard number 1
---
[^1]
Introduction
============
Seshadri constants are a local measure of positivity of a line bundle on a projective variety. They arose out of an ampleness criterion of Seshadri, [@Har70 Theorem 7.1] and were defined by Demailly, [@Dem]. They have become an important area of research with interesting connections to other areas of mathematics. For a comprehensive account of Seshadri constants, see [@primer].
Let $X$ be a smooth projective variety and let $L$ be a nef line bundle on $X$. Let $r \ge 1$ be an integer. For $x_1,\ldots,x_r \in
X$, the [*Seshadri constant*]{} of $L$ at $x_1,\ldots,x_r$ is defined as follows. $$\varepsilon(X,L,x_1,\ldots,x_r):= \inf\limits_{C \cap \{x_1,\ldots,x_r\}
\ne \emptyset} \frac{L\cdot C}{\sum\limits_{i=1}^r {\rm
mult}_{x_i}C}.$$
We remark that the infimum above is the same as the infimum taken over irreducible, reduced curves $C$ such that $C \cap \{x_1,\ldots,x_r\}
\ne \emptyset$. Indeed, we have the inequality $$\frac{L\cdot
(C+D)}{\sum_{i=1}^r{\rm mult}_{x_i}(C)+\sum_{i=1}^r {\rm mult}_{x_i}(D)} \ge {\rm min} \left(\frac{L\cdot
C}{\sum_{i=1}^r{\rm mult}_{x_i}(C)}, \frac{L\cdot
D}{\sum_{i=1}^r{\rm mult}_{x_i}(D)}\right).$$
The Seshadri criterion for ampleness says that a line bundle $L$ on a projective variety is ample if and only if $\varepsilon(X,L,x)>0$ for every $x \in X$.
The following is a well-known upper bound for Seshadri constants. Let $n$ be the dimension of $X$. Then for any $x_1,\ldots,x_r \in X$, $$\begin{aligned}
\label{wellknown}
\varepsilon(X,L,x_1,\ldots,x_r) \le
\sqrt[n]{\frac{L^n}{r}}.\end{aligned}$$
Next one defines $$\varepsilon(X,L,r) : = \max\limits_{x_1,\ldots,x_r \in X}
\varepsilon(X,L,x_1,\ldots,x_r).$$
It is known that $\varepsilon(X,L,r)$ is attained at a very general set of points $x_1,\ldots,x_r \in X$; see [@Ogu]. Here [*very general*]{} means that $(x_1,\ldots,x_r)$ is outside a countable union of proper Zariski closed sets in $X^r = X
\times X \times \ldots \times X$.
In this paper we will be concerned with lower bounds for $\varepsilon(X,L,r)$ when $X$ is a smooth projective surface with Picard number 1, $L$ is the ample generator of the Néron-Severi group of $X$ (see definitions below) and $r\ge 2$ is an integer.
There has been extensive work on lower bounds for $\varepsilon(X,L,1)$.
Let $X$ be an arbitrary smooth complex projective surface and let $L$ be an ample line bundle on $X$. It is easy to see that if $L$ is very ample then $\varepsilon(L,x) \ge 1$ for any point $x \in X$. On the other hand, if $L$ is ample, but not very ample, it is possible that $\varepsilon(L,x) < 1$. In fact, Seshadri constants can be arbitrarily small. Miranda has shown that given any [*rational*]{} number $\varepsilon > 0$, there exist a surface $X$, an ample line bundle $L$ and a point $x \in X$ such that $\varepsilon(X,L,x) = \varepsilon$; see [@L Example 5.2.1]. In Miranda’s construction, the surface $X$ typically has a large Picard number. It is possible that $\varepsilon(X,L,x) <
1$ even on a surface with Picard number 1; see [@BS Example 1.2].
However, in an important paper [@EL], Ein and Lazarfeld proved that $\varepsilon(X,L,1) \ge 1$, if $L$ is ample. In fact, they prove the following theorem.
Let $X$ be a smooth projective surface and let $L$ be an ample line bundle on $X$. Then $\varepsilon(X,L,x) \ge 1$ for all except possibly countably many points $x \in X$. Further, if $L^2 > 1$, the set of exceptional points is actually finite.
There are many other results calculating $\varepsilon(X,L,1)$ or giving bounds for it. We will just mention a few that are of relevance to us here. Bauer [@Bau] considered several cases of surfaces including abelian surfaces of Picard number 1. Szemberg, in [@Sze] and [@Sze08], dealt with surfaces of Picard number 1. Szemberg has a conjecture for a lower bound for $\varepsilon(X,L,1)$ for surfaces with Picard number 1; see [@Sze Conjecture]. [@FSST] gives new lower bounds for $\varepsilon(X,L,1)$ for an arbitrary surface and an ample line bundle $L$. In the process, it verifies the conjecture in [@Sze] in many cases.
Contrary to the case of $\varepsilon(X,L,1)$, there are not too many lower bounds for $\varepsilon(X,L,r)$ in the literature when $r \ge 2$. We mention some results in this case.
Küchle [@K] considers multi-point Seshadri constants on an arbitrary projective variety. Syzdek and Szemberg [@SS] prove a lower bound for multi-point Seshadri constants on any surface.
For an arbitrary surface $X$ and a nef and big line bundle $L$ on $X$, [@HR08 Theorem 1.2.1] gives good lower bounds for $\varepsilon(X,L,r)$ in terms of the degree of the least degree curve passing through $r$ general points with a given multiplicity $m$. Using this theorem, [@HR08 Corollary 1.2.2] has a bound for $\varepsilon(X,L,r)$ when $L$ is the ample generator the Néron-Severi group of $X$.
For an arbitrary surface $X$, when $L$ is very ample, [@Har Theorem I.1] gives lower bounds for $\varepsilon(X,L,r)$.
When $X$ has Picard number 1 and $L$ is the ample generator of the Néron-Severi group, [@Sze Theorem 3.2] also gives lower bounds for $\varepsilon(X,L,r)$. [@Bau Section 8] considers multi-point Seshadri constants for abelian surfaces.
Another method of obtaining lower bounds for $\varepsilon(X,L,r)$ comes from the following bound (see [@Bir; @Roe; @RR]): $$\begin{aligned}
\label{biran}
\varepsilon(X,L,r) \ge
\varepsilon(X,L,1)\varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1),r).\end{aligned}$$
The well-known Nagata Conjecture is a statement about $\varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1),r)$. It says that, for $r \ge 10$, $\varepsilon({\mathbb{P}}^2, {\mathcal{O}}_{{\mathbb{P}}^2}(1),r) = \frac{1}{\sqrt{r}}.$ This was proved by Nagata when $r$ is a square and is open in all other cases. But there are several results which give good bounds close to the bound expected by the Nagata Conjecture (see [@Har; @HR]). Using known bounds for $\varepsilon(X,L,1)$, some of which were mentioned above, and bounds on $\varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1),r)$, one can get bounds on $\varepsilon(X,L,r)$ by .
Let $X$ be a smooth projective surface over ${\mathbb C}$. The [*Picard group*]{} of $X$, denoted $\rm{Pic}(X)$, is the group of isomorphism classes of line bundles on $X$. The [*divisor class group*]{} of $X$, denoted $\rm{Div}(X)$, is the group of divisors on $X$ modulo linear equivalence. Then $\rm{Pic}(X)$ and $\rm{Div}(X)$ are isomorphic as abelian groups. The [*Néron-Severi*]{} group of $X$ is defined as $$\rm{NS}(X) :=
\rm{Pic}(X)/\rm{Pic}^0(X),$$ where $\rm{Pic}^0(X)$ is the subgroup of $\rm{Pic}(X)$ consisting of line bundles which are algebraically equivalent to the trivial line bundle. The Néron-Severi group $\rm{NS}(X)$ is a finitely generated abelian group and the [*Picard number*]{} of $X$ is defined to be $$\rho(X) := \rm{rank ~
NS}(X).$$
Our main result Theorem \[main\] gives lower bounds for $\varepsilon(X,L,r)$ when $X$ has Picard number 1 and $L$ is the ample generator of $\rm{NS}(X)$. Some examples of surfaces with Picard number 1 include ${\mathbb{P}}^2$, general K3 surfaces, and general hypersurfaces in ${\mathbb{P}}^3$ of degree at least 4.
Our bounds are not always easily comparable to bounds given previously in the literature, but in many cases, we get better bounds. We give some examples comparing our result with known bounds.
We work throughout over the complex number field ${\mathbb C}$. By a [ *surface*]{}, we mean a nonsingular projective variety of dimension 2. We denote the Seshadri constant $\varepsilon(X,L,r)$ simply by $\varepsilon(L,r)$ when the surface $X$ is clear from context.
[**Acknowledgements:**]{} We sincerely thank Brian Harbourne and Tomasz Szemberg for carefully reading this paper and making many useful suggestions. We also thank the referee for making numerous corrections and suggestions that improved the paper.
Main theorem
============
The following is the main theorem of this paper.
\[main\] Let $X$ be a smooth projective surface with Picard number 1. Let $L$ be the ample generator of $\rm{NS}(X)$. Let $r \ge 2$ be an integer. Then we have the following.
1. If $(r,L^2) = (2,6)$ then $\varepsilon(L,r) \ge
\frac{3}{2}$. If the equality holds, $\varepsilon(L,2)$ is achieved by a curve $C
\in |L|$ which passes through two very general points with multiplicity two each. \[3\]
2. If $(r,L^2) \ne (2,6)$ then one of the following holds:
1. $\varepsilon(L,r) \ge
\sqrt{\frac{r+2}{r+3}}\sqrt{\frac{L^2}{r}}$, or \[1\]
2. if $\varepsilon(L,r) <
\sqrt{\frac{r+2}{r+3}}\sqrt{\frac{L^2}{r}}$ then, for some $d \ge 1$, there exists an irreducible, reduced curve $C\in |dL|$ which passes through $s\le r$ very general points $x_1,\ldots,x_s$ with multiplicity one each, $s-1 \le C^2$, and $\varepsilon(L,r) = \frac{C\cdot L}{s} = \frac{dL^2}{s}$. \[2\]
If $\varepsilon(L,r)$ is optimal $\Big{(}$namely, equal to $\sqrt{\frac{L^2}{r}}\Big{)}$, there is nothing to prove. So we assume that $\varepsilon(L,r) < \sqrt{\frac{L^2}{r}}$. In this case, there is in fact an irreducible and reducible curve $C$ which computes $\varepsilon(L,r)$. See [@Sze01 Proposition 4.5], [@HR08 Lemma 2.1.2], or [@BS Proposition 1.1].
Since the Picard number of $X$ is 1, $C$ is algebraically equivalent, and hence numerically equivalent, to $dL$ for some $d \ge 1$. Let $k=L^2$. Then $L\cdot C = kd$ and $C^2
= kd^2$.
Let $x_1,\ldots,x_r$ be very general points of $X$ so that $\varepsilon(L,r) = \varepsilon(L,x_1,\ldots,x_r)$. Denote the multiplicity of $C$ at $x_i$ to be $m_i$. We rearrange the points so that $m_1
\ge m_2 \ge \ldots \ge m_r$. Let $s \in \{1,\ldots,r\}$ be such that $m_s > 0$ and $m_{s+1} = \ldots = m_r = 0$.
Since the points $x_1,\ldots,x_r$ are very general, there is a non-trivial one-parameter family of irreducible and reduced curves $\{C_t\}_{t\in T}$ parametrized by some smooth curve $T$ and containing points $x_{1,t},\ldots,x_{r,t} \in C_t$ with mult$_{x_{i,t}}(C_t) \ge
m_i$ for all $1\le i \le r$ and $t \in T$.
By a result of Ein-Lazarsfeld [@EL] and Xu [@X1 Lemma 1], we have $$\begin{aligned}
\label{el}
d^2k = C^2 \ge
m_1^2+\ldots+m_s^2-m_s.\end{aligned}$$
First assume that $m_1 = 1$. Then $m_i = 1$ for all $i=1,\ldots,s$ and $m_{s+1}=\ldots=m_r=0$. By , we have $s-1 \le C^2$. Since $C$ computes the Seshadri constant, $\varepsilon(L,r) = \frac{C
\cdot L}{s} = \frac{dk}{s}$. So we are in case .
We assume now that $m_1 \ge 2$.
The desired inequality in case is $\frac{L\cdot C}{\sum_{i=1}^r m_i} = \frac{dk}{\sum_{i=1}^rm_i} \ge \sqrt{\frac{r+2}{r(r+3)}}\sqrt{k}$. It is equivalent to $$\begin{aligned}
\label{desired1}
d^2k \ge \frac{r+2}{r(r+3)} \left(\sum_{i=1}^r
m_i\right)^2.\end{aligned}$$
Note that it suffices to prove $$\begin{aligned}
\label{desired}
d^2k \ge \frac{s+2}{s(s+3)} \left(\sum_{i=1}^r
m_i\right)^2.\end{aligned}$$ Indeed, it is clear that $\frac{s+2}{s(s+3)} \ge \frac{r+2}{r(r+3)}$ for $s \le r$.
We use the following inequality which is a special case of [@Han Lemma 2.3]. Suppose $m_1 \ge 2$, $s \ge 3$ or if $s=2$ then $(m_1,m_2) \ne (2,2)$. $$\begin{aligned}
\label{ineq}
\frac{(s+3)s}{s+2}\left(\sum_{i=1}^s m_i^2 -m_s\right) \ge
\left(\sum_{i=1}^s m_i\right)^2.\end{aligned}$$
If $s \ge 3$ then is applicable and together with , it easily gives . Suppose next that $s=2$. If $(m_1,m_2) \ne (2,2)$ we again use to obtain .
If $(m_1,m_2) = (2,2)$, the right hand side of is $\frac{32}{5}$. By , $d^2k \ge 6$. So fails only if $d=1$ and $k=6$. So we have case of the theorem.
Finally, if $s=1$ then we have $d^2k \ge m_1^2-m_1$ by . Since $r \ge 2$, we have $m_1^2-m_1 \ge \frac{r+2}{r(r+3)}m_1^2$ if $m_1\ge 2$. So holds. If $m_1=1$ then we are in case .
This completes the proof.
\[nagata-biran-szemberg\] Let $X$ be any surface and $L$ an ample line bundle on $X$. The Nagata-Biran-Szemberg Conjecture predicts that $\varepsilon(X,L,r) = \sqrt{\frac{L^2}{r}}$ for $r\ge r_0$, for some $r_0$ depending on $X$ and $L$ (see [@Sze01]). Our bound in Theorem \[main\] shows that on a surface $X$ with Picard number 1, $\varepsilon(X,L,r)$ is arbitrarily close to the optimal value of $\sqrt{\frac{L^2}{r}}$ as $r$ tends to $\infty$, as we explain below.
If Case of Theorem \[main\] holds then $\varepsilon(X,L,r)
= \frac{dk}{s}$ where $k=L^2$ and $C$ is a curve numerically equivalent to $dL$ which passes through $s \le r$ general points. Since we always have $\varepsilon(X,L,r) \le \sqrt{\frac{k}{r}}$, we get $d^2 \le
\frac{s^2}{rk}$. On the other hand, for such a curve $C$ to exist, we must have $s < h^0(X,dL)$. By the Riemann-Roch theorem applied asymptotically, the dimension of the space of global sections of $dL$ is bounded by $\frac{d^2L^2}{2} =
\frac{d^2k}{2}$. Hence $s \le \frac{d^2k}{2}$. Putting the above two inequalities together, we get $d^2 \le s \frac{s}{rk} \le
\frac{d^2k}{2}\frac{s}{rk}$. This in turn gives $2r \le s$, which is a contradiction.
Hence Case of Theorem \[main\] does [ *not*]{} hold for large $r$. This means that Case does. So we conclude that $\varepsilon(X,L,r)$ is arbitrarily close to the optimal value of $\sqrt{\frac{L^2}{r}}$ as $r$ tends to infinity.
\[HR08\] Let $X$ be any surface and $L$ a nef and big line bundle. In [@HR08 Theorem 1.2.1] Harbourne and Roé give lower bounds for $\varepsilon(X,L,r)$. When $X$ has Picard number 1 and $L$ is an ample line bundle on $X$, [@HR08 Corollary 1.2.2] gives similar bounds. The bounds stated in [@HR08] are implicit and therefore difficult to compare with our bounds. In all situations when the statements in [@HR08] can be made effective, the bounds in [@HR08] turn out to be better than our bounds in Theorem \[main\].
As an application, Harbourne and Roé give good lower bounds when $X = {\mathbb{P}}^2$; see [@HR08 Corollary 1.2.3]. In order to use these bounds, one needs information about the smallest degree curves passing through $r$ general points with a given multiplicity $m$. For specific classes of surfaces, it may be possible to obtain this information and in turn get good lower bounds for $\varepsilon(X,L,r)$.
Let $X$ be a surface with Picard number 1 and let $L$ be the ample generator of $\rm{NS}(X)$. Let $r \ge 1$. Szemberg [@Sze Theorem 3.2] obtains the following bound: $$\begin{aligned}
\label{szemberg}
\varepsilon(L,r) \ge \left\lfloor \sqrt{\frac{L^2}{r}}\right\rfloor.\end{aligned}$$
We compare our result Theorem \[main\] with this bound.
If $\frac{L^2}{r}$ is the square of an integer, shows that the Seshadri constant is optimal, namely equal to $\sqrt{\frac{L^2}{r}}$, while our bound is sub-optimal. So the bound in is always better in this situation.
On the other hand, our bound is always better if $L^2 < r$, because the bound in is 0 in this case.
If $L^2$ is large compared to $r$, the bound in is better. More precisely, for a fixed $r$, there is some $N_r$ such that if $L^2 \ge N_r$, uniformly gives a better bound. However, if $L^2 < N_r$, our bound is better in some cases and is better in some cases.
We give an example to illustrate this. Before considering the example, we note that given any integer $k\ge 4$, a general hypersurface $X$ of degree $k$ in ${\mathbb{P}}^3$ has Picard number 1. In fact, if $L = {\mathcal{O}}_X(1)$, the Picard group of $X$ is isomorphic to ${\mathbb Z}\cdot L$, by Noether-Lefschetz. Further, $L^2 = k$.
Let $r=10$. For $L^2 = 150$, the the bound in [@Sze] is 3, while our bound is 3.72. For $L^2 = 1050$, the bound in [@Sze] is 10, while ours is 9.84. If $L^2 = 2500$, then the bound in [@Sze] is 15, and our bound is 15.19. Starting at about $L^2=5000$, the bound in [@Sze] is uniformly better than ours.
\[harbourne\] Let $X$ be a surface and let $L$ be a very ample line bundle. Let $k =
L^2$. Let $\varepsilon_{r,k}$ be the maximum element in the following set: $$\Bigg\{\frac{\lfloor d\sqrt{rk}\rfloor}{dr}\Bigg|1 \le d \le
\sqrt{\frac{r}{k}}\Bigg\} \cup \Bigg\{ \frac{1}{\lceil
\sqrt{\frac{r}{k}} \rceil} \Bigg\} \cup \Bigg\{ \frac{dk}{\lceil
d\sqrt{rk}\rceil} \Bigg| 1 \le d \le \sqrt{\frac{r}{k}}\Bigg\}.$$
Harbourne [@Har Theorem I.1] proves that $\varepsilon(L,r) \ge
\varepsilon_{r,k}$, unless $k\le r$ and $rk$ is a square, in which case $\sqrt{\frac{k}{r}} = \varepsilon_{r,k}$ and $\varepsilon(L,r) \ge
\sqrt{\frac{k}{r}}-\delta$, for every positive rational number $\delta$.
Suppose now that $X$ has Picard number 1 and let $L$ be a very ample line bundle on $X$.
When $r < L^2$ our bound is better than [@Har Theorem I.1]. If $r \ge L^2$, our result is not always uniformly comparable with this result. The two bounds are in general very close to each other. We give an example to illustrate this. Let $r=10$. Then for $L^2 =6$ our bound is 0.744 and the bound in [@Har] is 0.75. On the other hand, for $L^2=7$, our bound is 0.803 while the bound in [@Har] is 0.8.
It should be noted however that [@Har Theorem I.1] is a general result proved for a very ample line bundle on *any* surface.
\[from-biran\] We compare our bounds with the bounds that can be derived from .
Let $X$ be a surface with Picard number 1 and let $L$ be the ample generator of $\rm{NS}(X)$. [@Sze Conjecture] says that $\varepsilon(X,L,1) \ge \frac{p_o}{q_o}k$ where $k = L^2$ and $(p_o,q_0)$ is a primitive solution to Pell’s equation: $q^2-kp^2=1$. [@FSST] verifies this conjecture whenever $k$ is of the form $n^2-1$ or $n^2+1$ for a positive integer $n$. In all cases (not just if $X$ has Picard number one) it is proved that $\varepsilon(X,L,1) \ge \frac{p_o}{q_o}k$ or $\varepsilon(X,L,1)$ belongs to a finite set of exceptional values.
Let $r=101$, $L^2=35$. Then our bound is 0.5858. Assuming $\varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1),r) = \sqrt{\frac{1}{r}}$ (which is the value predicted by the Nagata Conjecture) and using the bound $\varepsilon(X,L,1) \ge \frac{35}{6}$ in [@FSST Theorem 4.1], gives us $\varepsilon(X,L,r) \ge 0.5804$. If $L$ is very ample, [@Har Theorem I.1] gives the bound $0.5833$. Note that the optimal value of $\varepsilon(X,L,r)$ is $\sqrt{\frac{35}{101}} = 0.5886$ in this case.
Bauer [@Bau Theorem 6.1] explicitly calculates the Seshadri constants $\varepsilon(X,L,1)$, where $X$ is an abelian surface with Picard number 1, in terms of the primitive solution to Pell’s equation. These values, together with bounds on $\varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1),r)$ can be used (via ) to obtain bounds for $\varepsilon(X,L,r)$. These bounds are sometimes better than our bound, but not always.
Examples
--------
\[p2\]
Let $(X,L) = ({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1))$.
The values of $\varepsilon = \varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1,r))$ for $2 \le r \le
9$ are given below:
: Here $\varepsilon = 1/2$. In all three cases the Seshadri constant is achieved by a line through two points. Note that the bound expected in Case in Theorem \[main\] is 0.63, 0.52 and 0.4 when $r=2,3,4$ respectively. So when $r=2$ or $r=3$, Case in Theorem \[main\] is realized.
: In this case $\varepsilon = 2/5$, achieved by a conic through five general points. The bound predicted by Case in Theorem \[main\] is 0.41 which is more than 2/5. So this is also an instance where Case in Theorem \[main\] is achieved.
: In these cases $\varepsilon = 2/5,3/8,6/17,3$ respectively. In each case, the bound of Case of Theorem \[main\] holds.
For $r\ge 10$, the Nagata Conjecture says that $\varepsilon({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1,r)) = \frac{1}{\sqrt{r}}$.
\[k3\] Let $X$ be a general K3 surface. It is well-known that Picard number is 1 for such surfaces. Let $L$ be the ample generator of $\rm{NS}(X)$ with $k = L^2$. Let $r \ge 3$. We will show in this case that $\varepsilon(X,L,r) \ge \sqrt{\frac{r+2}{r+3}}\sqrt{\frac{k}{r}}$.
If not, by Theorem \[main\], for some $d \ge 1$, there exists a curve $C\in |dL|$ passing through $s\le r$ very general points with multiplicity one at each point and such that $\varepsilon(X,L,r) = \frac{L \cdot C}{s}$.
We first suppose that $C^2 = d^2k \ge s$. Then $\varepsilon(X,L,r) = \frac{dk}{s}\ge \sqrt{\frac{k}{r}}$. But this violates the assumption that $\varepsilon(X,L,r) < \sqrt{\frac{r+2}{r+3}}\sqrt{\frac{k}{r}}$
Now we compute $h^0(dL)$ and show that $C^2 \ge s$ always holds. By the Kodaira vanishing, $h^1(dL) = h^2(dL) = 0$. It follows from the Riemann-Roch theorem that $h^0(dL) = \frac{d^2k}{2}+2$ for $d \ge 1$. Since the linear system $|dL|$ contains a curve through $s$ general points, its dimension is at least $s$. On the other hand, if its dimension is more than $s$, then $|dL|$ contains a curve $D$ through at least $s+1$ points. Then by inequality , $d^2k = C^2 = D^2 \ge s$. In this case, we are done by the argument in the previous paragraph. So suppose that $h^0(dL) = \frac{d^2k}{2}+2 = s+1$. Hence $C^2 = d^2k = 2s-2$. If $s\ge 2$, then $C^2 \ge s$. If $s=1$, then anyway $C^2 = d^2k \ge 1$. So in both cases, we are done by the argument in the previous paragraph.
If $\frac{d^2k}{2}+2 \ge 1+\sum_{i=1}^r \frac{m_i^2+m_i}{2}$, then $|dL|$ contains a curve $C$ which passes through $r$ general points with multiplicity at least $m_i$ for $i = 1,\ldots,r$. Note that the converse is true when $m_1=m_2=\ldots=m_r=1$. This is because general points impose independent conditions and the linear system $|dL|$ contains a curve $C$ passing through $r$ general points if and only if the linear system $|dL|$ has dimension at least $r$.
Suppose that the converse in fact holds in general. In other words, the existence of a curve $C$ passing through $r$ general points with multiplicity at least $m_i$ for $i = 1,\ldots,r$, implies that $C$ moves in a linear system of dimension at least 1+$\sum_{i=1}^r
\frac{m_i^2+m_i}{2}$. Then it is easy to check that the Seshadri constant $\varepsilon(X,L,r)$ attains the maximal value $\sqrt{\frac{L^2}{r}}$. Indeed, our assumption implies that $d^2k \ge \sum_{i=1}^r (m_i^2+m_i) -2 =
(\sum_{i=1}^r m_i^2) + (\sum_{i=1}^r m_i)-2$. We can assume that $m_i
> 0$ for at least two different $i$, since we are only interested in $\varepsilon(L,r)$ for $r \ge 2$. So $d^2k \ge \sum_{i=1}^r m_i^2 \ge
\frac{(\sum_{i=1}^r m_i)^2}{r}$. This inequality is equivalent to $\frac{dk}{\sum_{i=1}^r m_i} \ge \sqrt{\frac{k}{r}}$.
Another class of surfaces which can have Picard number 1 are abelian surfaces. Bauer [@Bau] studies both single and multi-point Seshadri constants on such surfaces generally (not just when the Picard number is 1). In [@Bau Section 8], lower bounds for multi-point Seshadri constants are obtained for [*arbitrary*]{} points $x_1,\ldots,x_r$. These bounds are naturally smaller than our bounds for $\varepsilon(L,r)$, which is the Seshadri constant at $r$ [*general*]{} points.
Let $X$ be a general surface of degree $d \ge 5$ in ${\mathbb{P}}^3$. Then $X$ is a surface of general type and has Picard number 1. As in the case of Example \[k3\], one can check that if there are no curves that are not predicted by the Riemann-Roch theorem, then the Seshadri constant $\varepsilon(L,r)$ is optimal.
We do not know of any example where Case of Theorem \[main\] is realized.
Though the Nagata-Biran-Szemberg Conjecture (see Remark \[nagata-biran-szemberg\]) predicts that for a fixed polarized surface $(X,L)$, the Seshadri constants $\varepsilon(X,L,r)$ are optimal for large enough $r$, we can ask the following question.
\[question\] Given an integer $r \ge 1$, is there a pair $(X,L)$ where $X$ is a surface with Picard number 1 and $L$ is the ample generator of $\rm{NS}(X)$ such that $\varepsilon(X,L,r)$ is sub-optimal, that is, $\varepsilon(X,L,r) < \sqrt{\frac{L^2}{r}}$?
For $r=1$, an affirmative answer is given by Bauer [@Bau Theorem 6.1]. This result shows that for a polarized abelian surface $(X,L)$ of type $(1,d)$, the Seshadri constant $\varepsilon(X,L,1)$ is sub-optimal if $L^2 = 2d$ is not a square. For $r=2,3,5,6,7$ and $8$, $(X,L)=({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1))$ (see Example \[p2\]) gives an affirmative answer. For other values of $r$, we do not know an answer to Question \[question\].
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[^1]: Author was partially supported by a grant from Infosys Foundation
|
---
abstract: 'We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane $\mbox{Re}(s)>\frac12$ with $\overline{f(\overline{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann’s classical explicit formula, these representations use integrals along the critical line $\mbox{Re}(s)=\frac12$ and Blaschke zeta zeroes.'
author:
- |
[Matthias Kunik [^1]]{}\
Institute for Analysis and Numerics, Otto-von-Guericke-Universität\
Postfach 4120 . D-39106 Magdeburg, Germany
title: |
Logarithmic Fourier integrals for the\
Riemann Zeta Function
---
[**Key words:**]{}
Zeta function, explicit formulas, Fourier analysis, symmetric Poisson-Schwarz formulas.\
[**Mathematics Subject Classification (2000):**]{}
11M06, 11N05, 42A38, 30D10, 30D50.\
Preprint No. 01/2008,\
Otto-von-Guericke University of Magdeburg,\
Faculty of Mathematics.\
Introduction
============
Due to a theorem of Balazard-Saias-Yor [@BSY], the Riemann-hypothesis is true if and only if the integral $$\begin{aligned}
\label{bsyintegral}
\Omega_{\zeta}:=\frac{1}{2\pi}\int \limits_{\mbox{Re}(w)=1/2}
\frac{\log |\zeta(w)|}{|w|^2}\,|dw|\end{aligned}$$ vanishes. These studies have their origin in Beurlings work [@Beu], see also [@BS]. Using that $(s-1)\zeta(s)/s^2$ belongs to the Hardy space $H_2(\mbox{Re}(s)>1/2)$ one can also show that the logarithmic integral is absolutely convergent, see Burnol [@Br1], [@Br2]. The general result about absolute convergence of the logarithmic integral for Hardy spaces $H_p(\mbox{Im}(s)>0)$ is given for example in the textbooks of Koosis [@Koo1] and Garnett [@Gar].
The following factorisation formula presented in [@Br1] is valid for $\mbox{Re}(s)>1/2$ $$\begin{aligned}
\label{burnfact0}
\frac{s-1}{s} \zeta(s)=
\zeta_B(s) \cdot B(s)\,,\end{aligned}$$ with the function $\zeta_B$ in the right half plane $\mbox{Re}(s)>1/2$ given by $$\begin{aligned}
\label{burnfact1}
\zeta_B(s):=
\exp \left[
\frac{1}{2 \pi} \int \limits_{\mbox{Re}(w)=1/2}
\log \left| \zeta(w) \right|
\frac{s+w-2sw}{s-w} \, \frac{|dw|}{|w|^2}
\right] \,,\end{aligned}$$ and the Blaschke product $$\begin{aligned}
\label{burnfact2}
B(s):=
\prod \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ Re(\rho) > 1/2}}
\left\{
\frac{1-
\begin{displaystyle}\frac{s}{\rho}
\end{displaystyle}}{
\begin{displaystyle}
1-
\begin{displaystyle}
\frac{s}{1-\overline{\rho}}
\end{displaystyle}
\end{displaystyle}}^{~}
\cdot
\left|\frac{\rho}{1-\rho} \right|
\right\}
\,.\end{aligned}$$ It follows than by putting $s=1$ into , , that $$\begin{aligned}
\label{burnfact3}
B(1)=
\prod \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ Re(\rho) > 1/2}}
\left| \frac{1-\rho}{\rho} \right|\,, \quad \Omega_{\zeta}=-\log B(1) \geq 0\,,\end{aligned}$$ which gives the Balazard-Saias-Yor Theorem.
By using the symmetry of $\log |\zeta(1/2+iu)|$ we can also rewrite $\zeta_B$ in the form $$\begin{aligned}
\label{zetabpoisson}
\zeta_B(s) =
\exp \left[
\frac{2}{\pi} \left(s-\frac12\right)
\int \limits_{0}^{\infty}
\frac{\log \left| \zeta(\frac12+iu) \right|}
{u^2+(s-\frac12)^2} \, du\,
\right] \,, \quad \mbox{Re}(s)>1/2\,.\end{aligned}$$ Thus the function $\zeta_B$ results from $\zeta$ by multiplication with $\frac{s-1}{s}$ and replacing the Blaschke zeros $\rho$ of the zeta function with $\mbox{Re}(\rho)>\frac12$ by $1-\overline{\rho}$, i.e. by reflecting these zeros on the critical line $\mbox{Re}(s)=1/2$ into the left half plane $\mbox{Re}(s)<1/2$.
Blaschke products and factorisation formulas play an important role in the theory of Hardy spaces, see for example the textbooks of Koosis [@Koo1], Hoffmann [@Hoff] and Garnett [@Gar]. In the monographs [@Koo2; @Koo3] of Koosis the meaning of logarithmic integrals like is highlighted by an important theoretical background including results of Beurling, Malliavin and others, but the Riemann zeta function is not considered there.
In Section 2 we derive another symmetric Poisson-Schwarz formula by using the Hadamard product decomposition of $(s-1) \zeta(s)$ and the Riemann-von-Mangoldt function $N(t)$ counting the nontrivial zeta zeroes. The resulting formula is a counterpart of . Then we prove a variant of the Balazard-Saias-Yor Theorem.
In Section 3 we apply the Fourier transform or Mellin’s inversion formula on $$\label{logzetaa}
-\frac{\log \left((s-1)/s \, \zeta(s) \right)}{s(s-1)} \,,
\quad \mbox{Re}(s)>1\,,$$ and relate it to the Fourier transforms of $$\label{logzetab}
\frac{\log |\zeta(\frac12+it)|}{t^2+\frac14}\,, \quad
\frac{\pi N(t) - \vartheta(t)-2 \arctan(2t)}{t^2+\frac14}\,, \quad t \in {{\mathbb R}}\,,$$ by using the Blaschke zeta zeroes and a variant of Riemann’s explicite prime counting formula. If the Riemann-hypothesis is true, then can be interpreted as the trace of the real- and imaginary part of the analytical continuation of on the critical line, respectively.
Conversely, some important number theoretic functions are obtained in terms of integrals along the critical line by using the two expressions in .
A further factorisation of the Riemann zeta function with a symmetric Poisson-Schwarz integral
==============================================================================================
In this section we first prove a Poisson-Schwarz representation which turns out to be a dual counterpart of equations and . In the next section we will combine these results with Fourier- or Mellin transform techniques in order to obtain integrals along the critical line for the representation of interesting explicite prime number formulas.
The desired Poisson-Schwarz formula results from the Hadamard factorisation of $\zeta(s)$. For this purpose we need the following theorem, which provides some information about the vertical distribution of the zeros of the zeta function in the critical strip. The following result was used by Riemann in [@Rm], it was first shown by von Mangoldt [@Mn2] and then simplified by Backlund [@Bck1], see the Edwards textbook [@Ed].
[**Theorem (2.1)**]{}
*For $t \geq 0$ let $N(t)$ be the number of zeros $\rho = \sigma +i\tau$ of the $\zeta$-function in the critical strip $0 < \sigma <1$ with $0 \leq \tau \leq t$, regarding the multiplicity of the roots. For $t < 0$ we put $N(t):=-N(-t)$, and define the function $\vartheta: {{\mathbb R}}\to {{\mathbb R}}$ by $$\begin{aligned}
\vartheta(t) := -\arctan(2t)-\frac{t}{2}(\gamma+\log \pi) +
\sum \limits_{k=1}^{\infty}
\left\{
\frac{t}{2k}- \arctan\frac{t}{2k+\frac12}
\right\}\,.\label{theta}\end{aligned}$$*
Then we have for $t \to \infty$ the asymptotic relations $$\label{nt1}
\begin{split}
N(t) = \frac{1}{\pi} \vartheta(t) + O(\log t) \,,\\
\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ 0 < Re \, \rho < 1\\
|Im \, \rho| \leq t}}\,\frac{1}{|\rho|} = \frac{1}{2 \pi}\,(\log t)^2
+ O(\log t)\,,\\
\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ 0 < Re \, \rho < 1\\
|Im \, \rho| > t}}\,\frac{1}{|Im \, \rho|^2}
= \frac{1}{\pi}\,\frac{\log t}{t} + O(1/t)\,.
\end{split}$$
[**Remark:**]{} The exact form of $\vartheta(t)$ is needed for later purposes beside its asymptotic approximation $$\begin{aligned}
\label{asymp}
\theta(t) \sim \frac{t}{2} \log \frac{t}{2 \pi}-
\frac{t}{2} -\frac{\pi}{8} + o(1)\,, \quad t \to \infty\,.\end{aligned}$$
[**Theorem (2.2)**]{} [*We define for $\mbox{Re}(s)>1/2$ the analytic function $\zeta_C$ by $$\begin{aligned}
\label{zetacpoisson}
\zeta_C(s) :=
\exp \left[
s(s-1)\frac{2}{\pi}
\int \limits_{0}^{\infty}
\frac{u \left( \pi N(u)-\vartheta(u)-2 \arctan(2u) \right) }
{(u^2+(s-\frac12)^2) \, (u^2+\frac14)}\, du\,
\right] \,,\end{aligned}$$ as well as the symmetric counterpart $C(s)$ of the Blaschke product, $$\begin{aligned}
\label{cproduct}
C(s):=
\prod \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ Re(\rho) > 1/2}}
\frac{
\left(1-
\begin{displaystyle}
\frac{s}{\rho}
\end{displaystyle}
\right) \left(1-
\begin{displaystyle}
\frac{s}{1-\overline{\rho}}
\end{displaystyle}
\right)
}
{\begin{displaystyle}
\left(1 -
\frac{2s}{1-{\overline{\rho}}+\rho}
\right)^{\large 2}
\end{displaystyle}
}
\,.\end{aligned}$$ Then the following representation is valid for $\mbox{Re}(s)>1/2$, $$\begin{aligned}
\label{logzetaim}
\frac{s-1}{s}\zeta(s)=\zeta_C(s) \cdot C(s)\,.\end{aligned}$$* ]{}
[**Remarks:**]{} Note that the integral in is well defined by the Riemann-von Mangoldt Theorem (2.1). The function $\zeta_C$ results from $\zeta$ by multiplication with $\frac{s-1}{s}$ and projecting the Blaschke zeros $\rho$ of the zeta function with $\mbox{Re}(\rho)>1/2$ on the critical line $\mbox{Re}(s)=1/2$. The Riemann-Hypothesis is equivalent to $B(s)=C(s)\equiv 1$.
[**Proof:**]{} We define the integration kernel $$\begin{aligned}
\label{KCkern}
K_C(s,u):=\frac{2}{\pi}\frac{s(s-1)\cdot u}
{(u^2+(s-\frac12)^2) \, (u^2+\frac14)}\,,\end{aligned}$$ and for all $\alpha \in {{\mathbb R}}$ the characteristic function $\chi_{\alpha}:{{\mathbb R}}\to {{\mathbb R}}$ by
$$\label{randpuls}
\chi_{\alpha}(u) := \left\{
\begin{array}{ccl}
1 & , & u \geq \alpha \\
0 & , & u < \alpha \,. \\
\end{array} \right.$$
The following integrals can be obtained for $\mbox{Re}(s)>\frac12$ from elementary theory of the Poisson-Schwarz integral representations,
$$\begin{aligned}
\label{quadfactor}
\exp \left[\pi
\int \limits_{0}^{\infty}
\chi_{\alpha}(u) K_C(s,u)\, du\,
\right]
=
\left(
1-\frac{s}{\frac12+i\alpha}
\right)
\left(
1-\frac{s}{\frac12-i\alpha}
\right)\,, \quad \alpha \geq 0\,,\end{aligned}$$
$$\begin{aligned}
\label{gammafactor}
\exp \left[
\int \limits_{0}^{\infty}
\arctan\left( \frac{u}{a}\right) K_C(s,u)\, du\,
\right]
=1+\frac{s-1}{a+\frac12}\,, \quad \mbox{Re}(a) > 0\,,\end{aligned}$$
$$\begin{aligned}
\label{sminus1}
\int \limits_{0}^{\infty}
u K_C(s,u)\, du = s-1\,.\end{aligned}$$
Using and , we obtain
$$\label{thetaKCur}
\begin{split}
\exp \left[
\int \limits_{0}^{\infty}
\vartheta(u) K_C(s,u)\, du\,
\right] =
\frac{\pi^{-\frac{s-1}{2}}}{s}
\begin{displaystyle}
e^{-\gamma \frac{s-1}{2}}
\end{displaystyle}
\prod \limits_{k=1}^{\infty}
\frac{
\begin{displaystyle}
e^{\frac{s-1}{2k}}
\end{displaystyle}
}{1+\frac{s-1}{2k+1}}\,.
\end{split}$$
Note that the integral on the left hand side in is absolutely convergent by the asymptotic relation .
By using the general product representation $$\label{gammaprod}
\prod \limits_{k=1}^{\infty}
\left\{
\left( 1 + \frac{z}{k} \right) e^{-\frac{z}{k}}
\right\}=
\frac{e^{-\gamma z}}{\Gamma(z+1)}\,, \quad z \in {{\mathbb C}}\,,$$ we can rewrite the right hand side of as $$\label{thet2}
\begin{split}
& \frac{
\pi^{-\frac{s-1}{2}}
\begin{displaystyle}
e^{-\gamma \frac{s-1}{2}}
\end{displaystyle}}{2^{s-1}}
\prod \limits_{k=0}^{\infty}
\frac{
\begin{displaystyle}
e^{\frac{s-1}{2k+1}}
\end{displaystyle}
}{1+\frac{s-1}{2k+1}} \nonumber\\
= &
\frac{
\pi^{-\frac{s-1}{2}}
\begin{displaystyle}
e^{-\gamma \frac{s-1}{2}}
\end{displaystyle}}{2^{s-1}}
\frac{e^{- \gamma \frac{s-1}{2}}}{\Gamma\left( \frac{s-1}{2} + 1 \right)}
\prod \limits_{k=1}^{\infty}
\frac{
\begin{displaystyle}
e^{\frac{s-1}{k}}
\end{displaystyle}
}{1+\frac{s-1}{k}} \nonumber \\
= & \frac{\pi^{-\frac{s-1}{2}} e^{-\gamma (s-1)}}{2^{s-1}
\Gamma \left( \frac{s+1}{2}\right)}
\frac{\Gamma(s)}{e^{-\gamma (s-1)}}
\nonumber \\
= & \Gamma\left(\frac{s}{2} \right)\, \pi^{-\frac{s}{2}}\,,
\end{split}$$ where the last step follows from the duplication formula for the $\Gamma$ function, see for example the textbook of Andrews et al. [@AAR].
We obtain from for $\mbox{Re}(s)> \frac12$ that $$\label{thetaKC}
\exp \left[
\int \limits_{0}^{\infty}
\vartheta(u) K_C(s,u)\, du\,
\right]
= \Gamma\left(\frac{s}{2} \right)\, \pi^{-\frac{s}{2}}\,.$$ From it also follows with $a=\frac12$ that $$\label{atanKC}
\exp \left[
\int \limits_{0}^{\infty}
2 \arctan(2u) K_C(s,u)\, du\,
\right]
= s^2\,, \quad \mbox{Re}(s) > \frac12\,.$$ If we denote the positive imaginary parts of the nontrivial zeta zeros by $$\label{roots1}
0 < t_1 \leq t_2 \leq t_3 \leq \cdots\,,$$ where the imaginary parts are listed according to the multiplicity of each root, then we can rewrite $N(u)$ for $u>0$ as $$\label{roots2}
N(u) = \sum \limits_{n=1}^{\infty} \chi_{t_n}(u)\,,$$ and obtain from that $$\label{NKC}
\exp \left[
\int \limits_{0}^{\infty}
\pi N(u) K_C(s,u)\, du\,
\right]
= \prod \limits_{n=1}^{\infty}
\left\{
\left(
1-\frac{s}{\frac12+it_n}
\right)
\left(
1-\frac{s}{\frac12-it_n}
\right)
\right\}\,.$$ Now we use the definition of the analytic functions $\zeta_C$ in and $C$ in and calculate from , and for $\mbox{Re}(s)>\frac12$ the product $$\label{zetaccprod}
\begin{split}
\zeta_C(s)\,C(s) & =
\frac{
\prod \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\
Re(\rho), Im(\rho) > 0}}
\left\{
\left(
1-\frac{s}{\rho}
\right)
\left(
1-\frac{s}{1-\rho}
\right)
\right\}}
{\Gamma\left(\frac{s}{2} \right)\, \pi^{-\frac{s}{2}}\, s^2}\,.
\end{split}$$ The Theorem results from with Hadamard’s product decomposition of Riemann’s function $\xi : {{\mathbb C}}\to {{\mathbb C}}$ with $$\label{xihadamard}
\xi(s) := \frac{s(s-1)}{2}\,\pi^{-\frac{s}{2}}\,\Gamma(\frac{s}{2})\,\zeta(s)
= \frac12 \, \lim \limits_{T \to \infty}
\prod \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ 0 < Re \, \rho < 1\\
|Im \, \rho| \leq T}}\,\left( 1-\frac{s}{\rho} \right)\,.$$
[**Corollary (2.3)**]{} [*With the integration kernel we have $$\begin{aligned}
\label{xipoisson}
\xi(s) = \frac{C(s)}{2} \cdot \exp \left[
\int \limits_{0}^{\infty}
\pi N(u) K_C(s,u)\, du\,
\right] \,, \quad \mbox{Re}(s)>\frac12\,.\end{aligned}$$*]{}
[**Remark:**]{} For the proof of Theorem (2.2) we have directly used the Riemann von Mangoldt Theorem (2.1) and the Hadamard products for $\zeta$ and $\xi$. In contrast, the integral is divergent if we replace there $\zeta(w)$ by $\Gamma(w/2)$ or by Riemann’s function $\xi(w)$. In this case even the Cauchy limit $$\lim \limits_{T \to \infty} \int \limits_{-T}^{T}
\frac{\log |\xi(1/2+it)|}{1/4+t^2}\,dt$$ is divergent due to the asymptotic behaviour $$\lim \limits_{|t| \to \infty}
\frac{\log |\Gamma(1/4+\frac{it}{2})|}{\frac{\pi}{4}|t|}=-1$$ of the Gamma function. Thus it is not possible to evaluate or by applying Hadamard’s product representation for $\zeta(w)$ directly on $\log |\zeta(w)|$.
We have obtained Theorem (2.2) completely independent from the theory of Hardy spaces and the results of Balazard-Saias-Yor. Note that for all $t \in {{\mathbb R}}$ $$\begin{aligned}
\label{spur1}
\zeta(\frac12 +it)=\exp \left[ \log |\zeta(\frac12 +it)|+
i \left(\pi N(t) - \vartheta(t)- \pi \, \mbox{sign}(t)\right) \right]\,.\end{aligned}$$ Moreover, if the Riemann-hypothesis is true, then an analytical logarithm of $\frac{s-1}{s} \zeta(s)$ is defined for $\mbox{Re}(s)>\frac12$ with real values for $s>1/2$ and trace $$\begin{aligned}
\label{spur2}
\log |\zeta(\frac12 +it)|+
i \left(\pi N(t) - \vartheta(t)-2 \arctan(2t)\right)\end{aligned}$$ on $s=\frac12+it$. Now we obtain from Theorem (2.2) the following counterpart of the Balazard-Saias-Yor theorem, namely
[**Theorem (2.4)**]{} [*We have the two inequalities $$\begin{aligned}
\label{BSY1counter}
-\frac{1}{\pi}
\int \limits_{0}^{\infty}
\frac{\log \left| \zeta(\frac12 + iu) \right|}
{(u^2+\frac14)^2}\, du \leq \gamma-1\,,\end{aligned}$$ $$\begin{aligned}
\label{BSY2counter}
\frac{2}{\pi}
\int \limits_{0}^{\infty}
\frac{u \left( \pi N(u)-\vartheta(u)-2 \arctan(2u) \right) }
{(u^2+\frac14)^2}\, du \geq \gamma -1\,.\end{aligned}$$ In both cases equality holds if and only if the Riemann hypothesis is valid.*]{}
[**Proof:**]{} If we denote the left hand side of the inequalities , by $J_1$ and $J_2$, respectively, and put $s=1$ in the logarithmic derivatives of and , then we obtain $$\begin{aligned}
\gamma-1=
\lim \limits_{s \to 1}
\left[
\frac{1}{s-1}-\frac{1}{s}+\frac{\zeta'(s)}{\zeta(s)}
\right] = J_1+2\Omega_{\zeta} +\frac{B'(1)}{B(1)}
= J_2+\frac{C'(1)}{C(1)}\,.\end{aligned}$$ We will first prove that $2\Omega_{\zeta}+\frac{B'(1)}{B(1)} \geq 0$, and that $2\Omega_{\zeta}+\frac{B'(1)}{B(1)}=0$ is equivalent to the Riemann hypothesis. Put $$\begin{aligned}
\label{crho}
f_{\rho}:=
2 \log \left| \frac{\rho}{1-\rho} \right| + \frac{1}{1-\rho}-
\frac{1}{\overline{\rho}}\end{aligned}$$ for any $\rho = \sigma + i \tau$ with $\frac12 < \sigma \leq 1$ and $\tau \in {{\mathbb R}}$ with $|\tau| >1$. Then we obtain $$\begin{aligned}
\label{frhoeval}
f_{\rho}+f_{\overline{\rho}}= 2\int \limits_{\frac12}^{\sigma}
\frac{x^2(1-x)^2+\tau^2(\tau^2-6x^2+6x-1)}
{(x^2+\tau^2)^2 \left((1-x)^2+\tau^2\right)^2}
\,dx > 0\,,\end{aligned}$$ such that the first part of the Theorem results from $$\begin{aligned}
\label{frhoeval2}
\frac{B'(1)}{B(1)}+2\Omega_{\zeta}=
\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ Re(\rho) > 1/2}} f_{\rho}\,.\end{aligned}$$ It remains to show that $C'(1)/C(1)\leq 0$ and that the equality $C'(1)/C(1)= 0$ holds if and only if the Riemann hypothesis is valid. Put $$\begin{aligned}
\label{crho}
C_{\rho}(s):=
\frac{
\left(1-
\begin{displaystyle}
\frac{s}{\rho}
\end{displaystyle}
\right) \left(1-
\begin{displaystyle}
\frac{s}{1-\overline{\rho}}
\end{displaystyle}
\right)
}
{\begin{displaystyle}
\left(1 -
\frac{2s}{1-{\overline{\rho}}+\rho}
\right)^{\large 2}
\end{displaystyle}
}\end{aligned}$$ for any $\rho = \sigma + i \tau$ with $\frac12 < \sigma \leq 1$ and $\tau \in {{\mathbb R}}$. Then we obtain $$\label{logcrho}
\frac{C'_{\rho}(1)}{C_{\rho}(1)}+
\frac{C'_{\overline{\rho}}(1)}{C_{\overline{\rho}}(1)}=
-\frac{2\left(3\tau^2-\sigma(1-\sigma)\right) \left(\sigma-\frac12\right)^2}
{((1-\sigma)^2+\tau^2)(\sigma^2+\tau^2)(\frac14+\tau^2)}\,.$$ For $\begin{displaystyle}|\tau|>\frac{1}{\sqrt{12}}\end{displaystyle}$ and $\frac12<\sigma \leq 1$ this expression is always negative. Thus we have shown the Theorem.\
[**Theorem (2.5)**]{} [*For $t \geq 0$ except on a discrete singular set we define the number $N_B(t)$ of Blaschke zeta zeros $\rho$ with $\mbox{Re}(\rho)>\frac12$ and $|\mbox{Im}(\rho)| \leq t$ as well as the quantities $$\begin{aligned}
\label{N1}
N_1(t):=\frac{1}{\pi}
\left[
\vartheta(t)+2 \arctan(2t)
\right]\,,\end{aligned}$$ $$\begin{aligned}
\label{N2}
N_2(t):=-\frac{1}{2\pi^2}
\frac{d}{dt}\,\int \limits_{-\infty}^{\infty}
\log \left| 1- \frac{t^2}{u^2} \right| \,
\log \left| \zeta(\frac12+iu) \right|\, du\,,\end{aligned}$$ $$\begin{aligned}
\label{N3}
N_3(t):=\frac{1}{2\pi}
\int \limits_{-t}^{t}
\frac{B'(\frac12+iu)}{B(\frac12+iu)}\, du\,.\end{aligned}$$ If we extend $N_B$ and $N_1, N_2, N_3$ as odd functions to the whole real axis, then we obtain for almost all $t \in {{\mathbb R}}$ that $$\begin{aligned}
\label{Nsumme}
N(t) - N_B(t) = N_1(t) + N_2(t) + N_3(t)\,. \end{aligned}$$*]{} [**Remarks:**]{}
\(a) Without using any information about the horizontal distribution of the zeta zeroes in the critical strip $0 < \mbox{Re}(s)<1$, one can employ Theorem (2.1) and estimations for the logarithmic derivative of the Blaschke product on the critical line to prove the following asymptotic relations for $t \to \infty$, $$\begin{aligned}
\label{NBN3}
N_2(t) = O(\log^2 t)\,, \quad
N_3(t) + N_B(t) = O(\log^2 t)\,. \end{aligned}$$ (b) If the Riemann hypothesis is valid, then we obtain from Theorem (2.1) with $N_3=N_B \equiv 0$ the better result $N_2(t) = O(\log t)$ for $t \to \infty$.
[**Proof:**]{} For $s \in {{\mathbb C}}\setminus \{1\}$ with $\zeta(s) \neq 0$ we obtain from $$\begin{aligned}
\label{N49a}
\frac{\xi'(s)}{\xi(s)}=\frac{1}{s-1}-\frac12(\gamma+\log \pi)
+\sum \limits_{n=1}^{\infty} \left(\frac{1}{2n}-\frac{1}{s+2n}\right)
+ \frac{\zeta'(s)}{\zeta(s)}\,, \end{aligned}$$ and note the functional equation $$\begin{aligned}
\label{N49d}
\frac{\xi'(1-s)}{\xi(1-s)}= -\frac{\xi'(s)}{\xi(s)}\,. \end{aligned}$$ The logarithmic derivative of , is given for $\mbox{Re}(s)>\frac12$ with $s \neq 1$ and $\zeta(s) \neq 0$ by $$\begin{aligned}
\label{N49b}
\frac{\zeta'(s)}{\zeta(s)}=\frac{1}{s}-\frac{1}{s-1}
-\frac{1}{\pi}
\int\limits_{-\infty}^{\infty}
\frac{\log \left|\zeta(\frac12\pm iu)\right|}{(s-\frac12-iu)^2}\,du
+\frac{B'(s)}{B(s)}\,. \end{aligned}$$ Thus we obtain for $\mbox{Re}(s)>\frac12$ from that $$\begin{aligned}
\label{N49c}
\frac{\xi'(s)}{\xi(s)}&=\frac{1}{s}-\frac12(\gamma+\log \pi)
+\sum \limits_{n=1}^{\infty} \left(\frac{1}{2n}-\frac{1}{s+2n}\right)
\nonumber\\
&-\frac{1}{\pi}
\int\limits_{-\infty}^{\infty}
\frac{\log \left|\zeta(\frac12\pm iu)\right|}{(s-\frac12-iu)^2}\,du
+ \frac{B'(s)}{B(s)}\,. \end{aligned}$$ For $\eta,t >0$ we define the rectangular positive oriented integration path $\it{P}_{\eta,t}$ consisting on the straight line segments $[\frac12+\eta-it,\frac12+\eta+it]$, $[\frac12+\eta+it,\frac12-\eta+it]$, $[\frac12-\eta+it,\frac12-\eta-it]$, $[\frac12-\eta-it,\frac12+\eta-it]$. It is centred around $s=\frac12$, is the boundary of the rectangle $|\mbox{Re}(s)-\frac12| \leq \eta$, $|\mbox{Im}(s)| \leq t$ and contains the critical strip for any $\eta > \frac12$. We decompose $\it{P}_{\eta,t}$ into two parts $\it{P}_{\eta,t}^{\pm}$ for the complex numbers $s$ with $\mbox{Re}(s)\geq\frac12$ and $\mbox{Re}(s)\leq\frac12$ respectively, $$\begin{aligned}
\label{pathpm}
{\it P}_{\eta,t}^{+}:=[\frac12-it,\frac12+\eta-it] \oplus
[\frac12+\eta-it,\frac12+\eta+it]\oplus
[\frac12+\eta+it,\frac12+it]\,,\nonumber\\
{\it P}_{\eta,t}^{-}:=[\frac12+it,\frac12-\eta+it] \oplus
[\frac12-\eta+it,\frac12-\eta-it] \oplus
[\frac12-\eta-it,\frac12-it]\,.\end{aligned}$$ Let be $t>0$ given with $\zeta(\frac12+it)\neq0$. Since $\zeta$ and $\xi$ have only isolated zeros, we can choose $0 < \eta < \frac12$ small enough such that the compact rectangle $|\mbox{Re}(s)-\frac12| \leq \eta$, $|\mbox{Im}(s)| \leq t$ contains only zeros of the $\zeta$-function on the critical line. Then we obtain due to and that $$\begin{aligned}
\label{N49g}
\frac{1}{2\pi i} \int \limits_{\it{P}_{\eta,t}^{\pm}}
\frac{\xi'(s)}{\xi(s)}\,ds
=N(t)-N_B(t)\,.\end{aligned}$$ The direct integration of is problematic due to the presence of the pole singularities from the $\zeta$-zeros on the critical line. Instead of this we evaluate the primitive $\int \limits_0^t ( N(\tau)-N_B(\tau) )\,d\tau$. We have assumed $\zeta(\frac12 \pm it) \neq 0$, and the integral on the left hand side in does not depend on $\eta$ for sufficiently small $\eta >0$, and therefore $$\begin{aligned}
\label{N49l}
\int \limits_0^t ( N(\tau)-N_B(\tau) )\,d\tau
=\frac{1}{2 \pi}\,\lim \limits_{\eta \to 0^+}
\int \limits_{0}^{t} \int \limits_{-\tau}^{\tau}
\frac{\xi'(\frac12+\eta+i\vartheta)}{\xi(\frac12+\eta+i\vartheta)}
\,d \vartheta \,d\tau\end{aligned}$$ by using for the short integration paths $[\frac12-it,\frac12+\eta-it]$ and $[\frac12+\eta+it,\frac12+it]$ of ${\it P}_{\eta,t}^{+}$ the well known standard estimates.
First we regard for each $u \in {{\mathbb R}}$ the double integral $$\label{doppelint}
\begin{split}
&\frac{1}{2 \pi} \int \limits_{0}^{t}
\int \limits_{-\tau}^{\tau}
\frac{d \vartheta}{(\eta+i(\vartheta-u))^2} d \tau \\
&=\frac{1}{2 \pi}
\left[\,
\log(\eta+i(t-u))+\log(\eta-i(t+u))-2\log(\eta-iu)\,
\right]\,.
\end{split}$$ Its real part is an even function on $u \in {{\mathbb R}}$, and its imaginary part an odd function on $u$. In the limit $\eta \to 0^+$ we obtain for $u \neq 0$ and $|u| \neq t$ that $$\label{doppelintlim}
\begin{split}
&\lim \limits_{\eta \to 0^+} \frac{1}{2 \pi}
\left[\,
\log(\eta+i(t-u))+\log(\eta-i(t+u))-2\log(\eta-iu)\,
\right]\\
&=\frac{1}{2 \pi} \log \left|1-\frac{t^2}{u^2} \right| + \frac{i}{4}
\left[
\mbox{sign}(t-u)-\mbox{sign}(t+u)+ 2 \, \mbox{sign}(u)
\right]
\,.
\end{split}$$ Moreover, we have $$\begin{aligned}
\label{N49k}
\frac{1}{2\pi i} \int \limits_{0}^{t}
\int \limits_{\frac12+\eta-i\tau}^{\frac12+\eta+i\tau}
\,ds\,d\tau
=\frac {t^2}{2 \pi}\,,\end{aligned}$$ and for all $\alpha \geq 0$ $$\begin{aligned}
\label{N49j}
\frac{1}{2\pi i} \int \limits_{0}^{t}
\int \limits_{\frac12+\eta-i\tau}^{\frac12+\eta+i\tau}
\frac{ds}{s+\alpha}\,d\tau
=\frac {1}{\pi}\,
\left[
t \arctan\frac{t}{\frac12+\alpha+\eta}\right. \nonumber \\
\left. -\frac{\frac12+\alpha+\eta}{2}
\log \left(1+\frac{t^2}{(\frac12+\alpha+\eta)^2} \right)
\right]\,.\end{aligned}$$ Thus we obtain by direct calculation from the last two equations and $$\label{N1form}
\begin{split}
&\lim \limits_{\eta \to 0^+}\frac{1}{2\pi i} \int \limits_{0}^{t}
\int \limits_{\frac12+\eta-i\tau}^{\frac12+\eta+i\tau}
\left[
\frac{1}{s}-\frac12(\gamma+\log \pi)
+\sum \limits_{n=1}^{\infty} \left(\frac{1}{2n}-\frac{1}{s+2n}\right)
\right]
\,ds\,d\tau\\
&= \int \limits_0^t N_1(\tau)\,d \tau\,.
\end{split}$$ Next we use , and obtain from Fubini’s theorem and the Lebesgue dominated convergence theorem that $$\label{N2form}
\begin{split}
\lim \limits_{\eta \to 0^+}\frac{1}{2\pi i} \int \limits_{0}^{t}
\int \limits_{\frac12+\eta-i\tau}^{\frac12+\eta+i\tau}
\left[-\frac{1}{\pi}
\int\limits_{-\infty}^{\infty}
\frac{\log \left|\zeta(\frac12\pm iu)\right|}{(s-\frac12-iu)^2}\,du
\right]
\,ds\,d\tau
= \int \limits_0^t N_2(\tau)\,d \tau\,.
\end{split}$$ Finally we recall that for $\eta > 0$ sufficiently small the rectangle bounded by $\it{P}_{\eta,t}^{\pm}$ does not contain Blaschke $\zeta$-zeros, such that $$\label{N3form}
\begin{split}
\lim \limits_{\eta \to 0^+}\frac{1}{2\pi i} \int \limits_{0}^{t}
\int \limits_{\frac12+\eta-i\tau}^{\frac12+\eta+i\tau}
\frac{B'(s)}{B(s)}
\,ds\,d\tau
= \int \limits_0^t N_3(\tau)\,d \tau\,.
\end{split}$$ From , and , , we conclude .
Logarithmic Fourier integrals and their relation to the distribution of prime numbers
=====================================================================================
In this section we derive Fourier-Mellin transforms from the boundary integral formulas obtained in Section 2 which give interesting relations to the distribution of prime numbers.
As a first basic building block we need a representation theorem for the Mellin transforms of certain functions involving the exponential integral, which is generally useful for the study of Hadamard’s product decomposition of entire functions with appropriate growth conditions.
One of the various representations of the exponential integral is $$\begin{aligned}
\label{intro1}
{\rm Ei}(z) := \gamma + \log z + {\rm Ei}_0(z)\end{aligned}$$ with Euler’s constant $\begin{displaystyle}
\gamma = \lim \limits_{n \to \infty}
\left(
\sum \limits_{k=1}^n \frac{1}{k} - \log n
\right)
\end{displaystyle}$ and the entire function $$\begin{aligned}
\label{intro2}
{\rm Ei}_0(z) := \sum \limits_{k=1}^{\infty}\frac{z^k}{k \cdot k!} =
\int \limits_{0}^{z} \frac{e^t - 1}{t}\,dt =
\int \limits_{0}^{1} \frac{e^{u z} - 1}{u}\,du\,.\end{aligned}$$ In contrast to the entire function ${\rm Ei}_0$, the functions ${\rm Ei}$ and $\log$ are only defined on the cut plane $\begin{displaystyle}
{{\mathbb C}}_{-} := {{\mathbb C}}\setminus (-\infty,0]\,.
\end{displaystyle}$
The representation theorem formulated below was proved in [@Ku1], which also contains its application to Riemann’s explicit formula for the number of primes less than a given limit $x > 1$ and related formulas.\
[**Theorem (3.1)**]{}
*Let $s$ be any complex number with ${\rm Re}(s)>0$.*
- The expression $
(\gamma + \log(\log x))/x^{s+1}
$ is Lebesgue integrable on the interval $(1,\infty)$, and there holds the relation $$\frac{1}{s} = \exp \left( s\,
\int \limits_1^{\infty} \frac{\gamma + \log(\log x)}{x^{s+1}}\,dx \right)\,.$$
- The expression $
{\rm Ei}(\log x)/x^{s+1}
$ is Lebesgue integrable on the interval $(1,\infty)$, and with ${\rm Li}(x):={\rm Ei}(\log x)$ there holds the relation $$\begin{aligned}
\frac{1}{s-1} = \exp \left( s\,
\int \limits_1^{\infty} \frac{{\rm Li}(x)}{x^{s+1}}\,dx \right)\,.\end{aligned}$$
- Assume that $\rho \in {{\mathbb C}}\setminus [0,\infty)$ and ${\rm Re}(\rho) < {\rm Re}(s)$. Then $$\begin{aligned}
1-\frac{s}{\rho} = \exp \left( - s\,
\int \limits_1^{\infty}
\frac{\gamma + \log(\log x) + \log(-\rho) + {\rm Ei}_0({\rho} \log x)}
{x^{s+1}}\,dx \right)\,.\end{aligned}$$
If we note that for $u>0$ and $\mbox{Re}(s)>\frac12$ $$\label{sinlogint}
\begin{split}
\int \limits_{1}^{\infty} \frac{\sqrt{x} \, \sin(u \log x)}{x^{s+1}}\,dx
=\frac{u}{u^2+(s-\frac12)^2}
\end{split}$$ and define $f_C : (1,\infty) \to {{\mathbb R}}$ by $$\label{fC}
\begin{split}
f_C(x):=
\frac{2 \sqrt{x}}{\pi}\int \limits_{0}^{\infty}
\frac{ \pi N(u) - \vartheta(u)-2 \arctan(2u)
}{u^2+\frac14} \cdot \sin(u \log x) \, du \,,
\end{split}$$ then we obtain from and with Fubini’s theorem $$\label{sincosint}
\begin{split}
\zeta_C(s)=
\exp\left[ s(s-1)\int \limits_{1}^{\infty} \frac{f_C(x)}{x^{s+1}}\,dx \right]\,.
\end{split}$$ Next we define for $-\alpha \notin {{\mathbb C}}_{-}$ the functions $\varphi_{\alpha}\,, \Phi_{\alpha} : (1, \infty) \to {{\mathbb C}}$ by $$\label{phialpha}
\begin{split}
\varphi_{\alpha}(x) &:= \gamma + \log(\log x) + \log(-\alpha)
+\mbox{Ei}_0(\alpha \log x)\,,\\
\Phi_{\alpha}(x) &:= x \int \limits_{1}^{x} \frac{\varphi_{\alpha}(y)}{y^2}dy\,.
\end{split}$$ From Theorem (3.1) above we obtain by partial integration that for $\mbox{Re}(s)>1$ and $\mbox{Re}(s) > \mbox{Re}(\alpha) > 0$ $$\label{Phimellin}
1-\frac{s}{\alpha}= \exp \left[
-s(s-1) \int \limits_{1}^{\infty} \frac{\Phi_{\alpha}(x)}{x^{s+1}} \, dx\right]\,.$$ The integral expression for $\Phi_{\alpha}$ in can be solved explicitely. For any $\alpha \in {{\mathbb C}}$ with $\mbox{Im}(\alpha) \neq 0$ and for all $x>1$ we obtain that $$\label{phialphamod}
\begin{split}
\Phi_{\alpha}(x) &= x \cdot \varphi_{\alpha -1}(x)-\varphi_{\alpha}(x) +
x \cdot \left[\,\log\left(-\alpha\right)-\log\left(-(\alpha-1)\right)\,\right] \,.
\end{split}$$ Equation can be checked easily by forming the derivative of $\Phi_{\alpha}(x)/x$ with respect to $x$ and by regarding that ${{\displaystyle}}\lim \limits_{x \to 1}\Phi_{\alpha}(x)=0$. We have not expressed ${\Phi}_{\alpha}(x)$ in terms of the exponential integral, because $\varphi_{\alpha}(x)$ in has a better asymptotic behaviour than $\mbox{Ei}(\alpha \log x)$ for $\mbox{Im}(\alpha) \neq 0$ and $x>1$, $$\label{phiasym1}
\begin{split}
\varphi_{\alpha}(x) &= \mbox{Ei}(\alpha \log x) -
i \, \pi \, \mbox{sign}(\mbox{Im}(\alpha))\\
&= \frac{x^{\alpha}}{\alpha \log x}+
x^{\alpha} \int \limits_0^{\infty}
\frac{e^{- y}}{(\alpha \log x - y)^2}\,dy\,,\\
& \left|\, x^{\alpha} \int \limits_0^{\infty}
\frac{e^{- y}}{(\alpha \log x - y)^2}\,dy \,\right|
\leq \frac{x^{\mbox{Re}(\alpha)}}{|\mbox{Im}(\alpha)|^2 \, {\log}^2 x}\,.
\end{split}$$ A proof of the following theorem can be found in the textbook [@Ed] of Edwards.\
[**Theorem (3.2)**]{} [*For any integer number $n \geq 1$ we define the von Mangoldt function $$\Lambda(n):=\left\{
\begin{array}{c@{\quad,\quad}l}
\ln p & \mbox{for}~ n=p^m, ~m \geq 1, ~ p ~ \mbox{prime}\\
0 & \mbox{otherwise}\,,
\end{array}
\right.$$ and for $x \geq 1$ the functions $$\psi(x) := \sum \limits_{n \leq x} \Lambda(n)\,, \quad
\pi_*(x) :=
\sum \limits _{1 < n \leq x}\frac{\Lambda(n)}{\ln n} =
\sum \limits_{n=1}^{\infty}\frac{\pi(\sqrt[n]{x})}{n} \,,$$ where $\pi(x)$ is the number of primes $\leq x$. Then we obtain for ${\rm Re}(s)>1$ $$\begin{aligned}
\zeta(s) &= \exp(\,\sum \limits _{n=2}^{\infty}\frac{\Lambda(n)}
{\ln n}n^{-s}\,)
= \exp(\, s\int \limits_{1}^{\infty}\frac{\pi_*(x)}{x^{s+1}}\,dx \,)
\label{zetapi}\\
-\frac{\zeta'(s)}{\zeta(s)} &=
\sum \limits _{n=2}^{\infty}\Lambda(n) \, n^{-s} =
s\,\int \limits_{1}^{\infty}\frac{\psi(x)}{x^{s+1}}\,dx\,. \label{zetapsi}\end{aligned}$$*]{}
From Theorem (3.1)(a,b) and we obtain with the entire function $\mbox{Ei}_0$ defined in for $\mbox{Re}(s)>1$ that $$\label{zetap1}
\frac{s-1}{s}\,\zeta(s)=
\exp\left[\, s\int \limits_{1}^{\infty}
\frac{\pi_*(x)-\mbox{Ei}_0(\log x)}{x^{s+1}}\,dx \,
\right]\,.$$ If we apply partial integration on the last integral, we can also rewrite in the form $$\label{zetap2}
\frac{s-1}{s}\,\zeta(s)=
\exp\left[\, s(s-1) \int \limits_{1}^{\infty}\frac{f_{*}(x)}{x^{s+1}}\,dx \,
\right]\,,$$ where the function $f_{*}: (1, \infty) \to {{\mathbb R}}$ is given by $$\label{zetap3}
\begin{split}
f_{*}(x)&=x\,\int \limits_{1}^{x} \frac{\pi_*(y)-\mbox{Ei}_0(\log y)}{y^2}\,dy\\
&=x\,\left(
\sum \limits_{n \leq x}
\frac{\Lambda(n)}{n \log n}+\mbox{Ei}_0(-\log x)\right)-
\left( \pi_*(x)-\mbox{Ei}_0(\log x) \right)
\,.
\end{split}$$
Now we are able to prove the following result,
[**Theorem (3.3)**]{}
*For $\alpha \in {{\mathbb C}}$ with $\mbox{Im}(\alpha) \neq 0$ we define $\tilde{\Phi}_{\alpha} : (1, \infty) \to {{\mathbb C}}$ by $$\label{tildephialpha}
\tilde{\Phi}_{\alpha}(x) := x \, \varphi_{\alpha-1}(x)-\varphi_{\alpha}(x)\,.$$ Then we obtain for all $x>1$ for the function $f_*$ in*
- $$\label{fsternsin}
\begin{split}
f_*(x) & =\frac{2 \sqrt{x}}{\pi}\int \limits_{0}^{\infty}
\frac{ \pi N(u) - \vartheta(u)-2 \arctan(2u)}{u^2+\frac14}
\cdot \sin(u \log x) \, du\\
&-\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ \mbox{Re}(\rho) > \frac12}}
\left[
\tilde{\Phi}_{\rho}(x)+\tilde{\Phi}_{1-\overline{\rho}}(x)
-2\tilde{\Phi}_{\frac12(1+\rho-\overline{\rho}\,)}(x)
\right]\,,
\end{split}$$
- $$\label{fsterncos}
\begin{split}
f_*(x) &=-\frac{2 \sqrt{x}}{\pi}\int \limits_{0}^{\infty}
\frac{ \log |\zeta(\frac12+iu)|}{u^2+\frac14}
\cdot \cos(u \log x) \, du\\
&-\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ \mbox{Re}(\rho) > \frac12}}
\left[
\tilde{\Phi}_{\rho}(x)-\tilde{\Phi}_{1-\overline{\rho}}(x)
\right]\,.
\end{split}$$
In equations (a) and (b), the integrals as well as the sums (or series) with respect to the Blaschke zeroes all converge in the absolute sense.
[**Proof:**]{} We define the functions $f_{11}, f_{12}, f_{21}, f_{22} : (1,\infty) \to {{\mathbb R}}$ by $$\label{fmn}
\begin{split}
f_{11}(x) & := \frac{2 \sqrt{x}}{\pi}\int \limits_{0}^{\infty}
\frac{ \pi N(u) - \vartheta(u)-2 \arctan(2u)}{u^2+\frac14}
\cdot \sin(u \log x) \, du\,, \\
f_{12}(x) & := -\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ \mbox{Re}(\rho) > \frac12}}
\left[
\tilde{\Phi}_{\rho}(x)+\tilde{\Phi}_{1-\overline{\rho}}(x)
-2\tilde{\Phi}_{\frac12(1+\rho-\overline{\rho}\,)}(x)
\right]\,, \\
f_{21}(x) & := -\frac{2 \sqrt{x}}{\pi}\int \limits_{0}^{\infty}
\frac{ \log |\zeta(\frac12+iu)|}{u^2+\frac14}
\cdot \cos(u \log x) \, du\,, \\
f_{22}(x) & := -\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ \mbox{Re}(\rho) > \frac12}}
\left[
\tilde{\Phi}_{\rho}(x)-\tilde{\Phi}_{1-\overline{\rho}}(x)
\right]\,. \\
\end{split}$$ The absolute convergence of the $f_{21}$-integral was already mentioned for the formulation of the Balazard-Saias-Yor Theorem, and the absolute convergence of the $f_{11}$-integral results from the first equation in . For the absolute convergence of the sums $f_{12}, f_{22}$ we notice the third equation in and for $x>1$, $\mbox{Im}(\alpha) \neq 0$ the asymptotic law , which implies that $$\label{phiasym2}
\begin{split}
\tilde{\Phi}_{\alpha}(x) =
\frac{1}{\alpha (\alpha -1)}\,\frac{x^{\alpha}}{\log x}
+ R(\alpha,x)\,,\quad
|R(\alpha,x)| \leq \,
\frac{2\,x^{\mbox{Re}(\alpha)}}{|\mbox{Im}(\alpha)|^2 \, {\log}^2 x}\,.
\end{split}$$ We also obtain that the convergence of the expressions in is uniform on each compact interval $0 < x_0 \leq x \leq x_1$, such that the functions $f_{11}, f_{12}, f_{21}, f_{22}$ are continuous.
Using the following asymptotic behaviour for $x \to \infty$, $$\label{pnt}
\pi_*(x)-\mbox{Li}(x) = O(x\,e^{-c \sqrt{\log x}})\,, \quad
c>0 \mbox{~constant}\,,$$ we conclude from in the limit $s \to 1$ for real $s>1$ that $$\label{fsternasym}
\lim \limits_{x \to \infty} \frac{f_{*}(x)}{x}=
\int \limits_1^{\infty} \frac{\pi_*(t)-\mbox{Ei}_0(\log t)}{t^2}\,dt = 0\,.$$ In accordance with this we also obtain from that $$\label{fmnasymptotik}
\lim \limits_{x \to \infty} \frac{f_{mn}(x)}{x}=0\,, \quad m, n \in \{1,2\}\,.$$ On the other hand we have from the definition of $f_*$ $$\label{fsternlim1}
\lim \limits_{x \to 1} f_{*}(x)=0\,,$$ and the symmetry of the zeta zeroes with respect to complex conjugation and , , imply that $$\label{fmnlim1}
\begin{split}
\lim \limits_{x \to 1} f_{11}(x)=0\,, \quad
\lim \limits_{x \to 1} f_{12}(x)= 0\,,\\
\lim \limits_{x \to 1} f_{21}(x)= -2 \Omega_{\zeta}\,, \quad
\lim \limits_{x \to 1} f_{22}(x)= 2 \Omega_{\zeta}\,.\\
\end{split}$$ From Theorem (2.2), and , we obtain for $\mbox{Re}(s)>1$ with a constant $K \in {{\mathbb C}}$ that $$\label{hilf1}
1=\exp \left[\, s(s-1)\int \limits_{1}^{\infty}
\frac{f_*(x)-\left(f_{11}(x)+f_{12}(x) + K \cdot x \right)}{x^{s+1}}\,dx \,
\right]\,.$$ We conclude for $x>1$ from to with Mellin’s inversion formula $$\label{hilf2}
f_*(x)=f_{11}(x)+f_{12}(x)\,.$$ Thus we have shown the first part of Theorem (3.3).\
In order to prove the second part, we first recall that the function $\zeta_{B}$ in and is analytic for $\mbox{Re}(s)>\frac{1}{2}$ with $$\label{expzb1}
\begin{split}
B(1) \zeta_{B}(s) & = \frac{s-1}{s}\,\zeta(s) \, \frac{B(1)}{B(s)}\\
& = \exp \left[ -\Omega_{\zeta} +
\frac{2}{\pi} \left(s-\frac12\right)
\int \limits_{0}^{\infty}
\frac{\log \left| \zeta(\frac12+iu) \right|}
{u^2+(s-\frac12)^2} \, du\,
\right] \,.\\
\end{split}$$ Since we have the limit $$\label{expzb1}
B(1) \lim \limits_{s \to 1}
\zeta_{B}(s) = \frac{B(1)}{B(1)}=1\,,$$ the real valued function $Q : (1/2, \infty) \to {{\mathbb R}}$ with $$\label{expzb2}
Q(s) := \frac{\log \left( B(1)\zeta_{B}(s) \right)}{s(s-1)}\\$$ has an analytical continuation $Q_*$ in $\mbox{Re}(s)> \frac12$ with $$\label{expzb3}
\begin{split}
Q_*(s) & = \frac{2\Omega_{\zeta}}{s} +
\frac{2}{\pi}\frac{s-\frac12}{s(s-1)}
\int \limits_0 ^{\infty} \left[
\frac{\log \left| \zeta(\frac12+iu) \right|}{(u^2+(s-\frac12)^2)} -
\frac{\log \left| \zeta(\frac12+iu) \right|}{(u^2+\frac14)}
\right] \, du\\
& = \frac{2\Omega_{\zeta}}{s}-\frac{2}{\pi}(s-\frac12)
\int \limits_0 ^{\infty} \frac{\log \left| \zeta(\frac12+iu) \right|}
{(u^2+\frac14) \, (u^2+(s-\frac12)^2)} \, du\\
& = \int \limits_1^{\infty}
\frac{2 \Omega_{\zeta} + f_{21}(x)}{x^{s+1}}\,dx\,.\\
\end{split}$$ For the last equation in we have used Fubini’s theorem with $$\label{expzb4}
\int \limits_1^{\infty}
\frac{\sqrt{x} \cos(u \log x)}{x^{s+1}}\,dx =
\frac{s-\frac12}{u^2+(s-\frac12)^2}\,, \quad \mbox{Re}(s)>\frac12\,,$$ because all integrals converge in the absolute sense.
Next we obtain from and for $\mbox{Re}(s)>1$ that $$\label{Bsdurchb1}
\frac{B(s)}{B(1)}=\exp
\left[
s(s-1) \int \limits_1 ^{\infty}
\{
2 \Omega_{\zeta}\cdot (x-1) -
\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ \mbox{Re}(\rho) > \frac12}}
\left(
\Phi_{\rho}(x)-\Phi_{1-\overline{\rho}}(x)
\right)\,\}
\frac{dx}{x^{s+1}}
\right]\,,$$ and with , , , for $\mbox{Re}(s)>1$ $$\label{finale1}
\begin{split}
\frac{s-1}{s}\,\zeta(s) &=
\exp
\left[
s(s-1) \int \limits_1 ^{\infty}
\frac{f_*(x)}{x^{s+1}}\,dx
\right]\\
&= \left( \zeta_B(s) B(1) \right) \cdot \frac{B(s)}{B(1)}\\
&=\exp
\left[
s(s-1) \int \limits_1 ^{\infty}
\frac{2 \Omega_{\zeta}+f_{21}(x)}{x^{s+1}}\,dx
\right] \cdot \frac{B(s)}{B(1)}\,.
\end{split}$$ Note the relation between $\Phi_{\alpha}(x)$ and $\tilde{\Phi}_{\alpha}=x \cdot \varphi_{\alpha -1}(x)-\varphi_{\alpha}(x)$. Then we combine and and use - to conclude the second part of the theorem from Mellin’s inversion formula.\
Finally we mention without going into details that the mathematical technique developed here can be used as well in order to derive an integral form for other explicite formulas related to the distribution of prime numbers, for example for the following result\
[**Theorem (3.4)**]{} [*For $x>1$, $\alpha,r \in {{\mathbb C}}$ and $u \in {{\mathbb R}}$ we define*]{}
$$\label{pir}
\begin{split}
{\Theta}(x,{\alpha}) &:=
\left\{\begin{displaystyle}
\begin{array}{cl}
- {{\displaystyle}}\frac{x^{\alpha}-1}{\alpha}+\mbox{Ei\,}_0(\alpha \log x) \log x
\,, & \alpha \neq 0\,, \\
-\log x\,, & \alpha = 0\,,\\
\end{array}\end{displaystyle}
\right.\\
K(x,r,u) &:= \left \{
\begin{array}{cl}
\frac{1}{ \pi}\,\left[
\frac{x^{1/2+iu-r}-1}{(1/2+iu-r)^2}-\frac{\log x}{1/2+iu-r}
\right]\,, & r \neq 1/2+iu\,,\\
\frac{\log ^2 x}{2 \pi}\,, & r = 1/2+iu\,,
\end{array}\right.\\
\pi_{*,r}(x) &:= \sum \limits_{n \leq x} \frac{\Lambda(n)}{n^r \log n}\,,
\quad
\psi_{r}(x) := \sum \limits_{n \leq x} \frac{\Lambda(n)}{n^r}\,.\\
\end{split}$$
[*Then we have*]{} $$\label{pir}
\begin{split}
\int \limits_{1}^{x} \frac{\pi_{*,r}(y)}{y}\,dy
&=\pi_{*,r}(x) \log x - \psi_r(x)\\
&=\Theta(x,1-r)-\Theta(x,-r)\\
&+ \int \limits_{0}^{\infty}
\left( K(x,r,u) + K(x,r,-u) \right)
\log |\zeta(\frac12+iu)|\,du\\
&-\sum \limits_{\substack{\rho \,:\, \zeta(\rho)=0\\ \mbox{Re}(\rho) > \frac12}}
\left\{
\Theta(x,\rho-r)-\Theta(x,1-\overline{\rho}-r)
\right\}\,.
\end{split}$$ [*On the right hand side in the integral as well as the sum (or series) with respect to the Blaschke zeroes converge in the absolute sense.*]{}
[999]{} , [*Special Functions* ]{}, Cambridge University Press (2000). , [*Sur les zéros de la fonction $\zeta(s)$ de Riemann*]{}, C.R. Acad. Sci. Paris [**158**]{}, 1979-1982 (1914). , [*The Nyman-Beurling equivalent form for the Riemann hypothesis*]{}, Expositiones Mathematicae [**18**]{}, 131-138 (2000). , [*Notes sur la fonction $\zeta$ de Riemann, [**2**]{}*]{}, Advances in Math. [**143**]{}, 284-287 (1999). , [*A closure problem related to the Riemann zeta-function*]{}, Proc. Nat. Acad. Sci. [**41**]{}, 312-314 (1955). , [*A note on Nyman’s equivalent formulation of the Riemann hypothesis*]{}, Algebraic methods in statistics and probability (Notre Dame, IN, 2000), Contemp. Math., 287, Amer. Math. Soc., Providence, RI, 23–26 (2001). , [*An adelic causality problem related to abelian $L$-functions*]{}, J. Number Theory 87 [**2**]{}, 253-269 (2001). , [*Riemann’s Zeta Function* ]{}, Dover Publications, New York (2001). , [*Bounded analytic functions.*]{} Revised first edition. Graduate Texts in Mathematics, 236. Springer, New York (2007). , [*Banach spaces of analytic functions*]{}, Dover Publication, (1988). , [*Introduction to $H_p$-spaces*]{}, second edition, Cambridge University Press (1998). , [*The logarithmic integral. I.*]{} Corrected reprint of the 1988 original. Cambridge Studies in Advanced Mathematics, 12. Cambridge University Press (1998). , [*The logarithmic integral. II.*]{} Cambridge Studies in Advanced Mathematics, 21. Cambridge University Press (1992). , [*On the formulas of $\pi(x)$ and $\psi(x)$ of Riemann and von-Mangoldt*]{}, Preprint Nr. 09/2005, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (2005). , [*Zur Verteilung der Nullstellen der Riemannschen Funktion $\xi(t)$.*]{} Math. Ann [**60**]{}, 1-19 (1905). , [*Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse*]{} (1859), in “Gesammelte Werke”, Teubner, Leipzig (1892), wieder aufgelegt in Dover Books, New York (1953).
[^1]: [email protected]
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abstract: 'Using sheaves of ${{\mathbb A}}^1$-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be ${{\mathbb A}}^1$-local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of ${{\mathbb A}}^1$-invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize ${{\mathbb A}}^1$-connected reductive algebraic groups over a field of characteristic $0$.'
address:
- 'Department of Mathematics, Indian Institute of Science Education and Research (IISER), Knowledge City, Sector-81, Mohali 140306, India.'
- 'Mathematisches Institut, Ludwig-Maximilians Universität, Theresienstr. 39, D-80333 München, Germany.'
author:
- Chetan Balwe
- Anand Sawant
title: '${{\mathbb A}}^1$-connectedness in reductive algebraic groups'
---
Introduction {#section Introduction}
============
Let us fix a base field $k$ and let ${{\mathcal H}}(k)$ denote the ${{\mathbb A}}^1$-homotopy category of schemes over $k$ developed by Morel and Voevodsky [@Morel-Voevodsky]. This category is constructed by first enlarging the category of smooth (finite type, separated) schemes over $k$ to the category of simplicial sheaves on the big Nisnevich site of smooth schemes over $k$ and then taking a suitable localization. Given a smooth scheme $X$ over $k$, one may ask if the set of morphisms ${{\rm Hom}}_{{{\mathcal H}}(k)}(U, X)$ has a geometric description, at least when $U$ is a smooth henselian local scheme. In particular, one may ask if the set ${{\rm Hom}}_{{{\mathcal H}}(k)}(U, X)$ is in bijection with the equivalence classes of morphisms of schemes $U \to X$ by *naive* ${{\mathbb A}}^1$-homotopies. This question is closely related to the behaviour of the Morel-Voevodsky singular construction ${{\rm Sing_*^{{{\mathbb A}}^1}}}X$ (for precise definitions, see Section \[section preliminaries A1-connectedness\]). More precisely, the above question has an affirmative answer if ${{\rm Sing_*^{{{\mathbb A}}^1}}}X$ is *${{\mathbb A}}^1$-local*, that is, discrete as an object of ${{\mathcal H}}(k)$ (see Definition \[definition A1-local\]). However, there exist (even smooth, projective) varieties $X$ for which ${{\rm Sing_*^{{{\mathbb A}}^1}}}X$ is not ${{\mathbb A}}^1$-local [@Balwe-Hogadi-Sawant 4.1].
In this paper, we will study this question for a reductive algebraic group $G$ over an infinite perfect field $k$. Under a suitable isotropy hypothesis on $G$, it was shown by Asok, Hoyois and Wendt [@Asok-Hoyois-Wendt-2] that ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is ${{\mathbb A}}^1$-local. This isotropy hypothesis on reductive algebraic $k$-groups was introduced in [@Raghunathan-1989]: $$\begin{split}
(\ast)~ & \text{Every almost $k$-simple component of the derived group $G_{\rm der}$ of $G$ contains}\\
& \text{a $k$-subgroup scheme isomorphic to ${{\mathbb G}}_m$.}
\end{split}$$ Asok, Hoyois and Wendt obtain the ${{\mathbb A}}^1$-locality of ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ for a reductive algebraic group satisfying the hypothesis $(\ast)$ by showing affine homotopy invariance of Nisnevich locally trivial torsors under such groups. More precisely, they show that for every smooth affine scheme $U$ over $k$ and a group $G$ satisfying $(\ast)$, the natural map $$H^1_{\rm Nis}(U, G) \to H^1_{\rm Nis}(U \times {{\mathbb A}}^n, G)$$ is a bijection, for every $n \geq 0$. Special cases of this result (such as the case $G = GL_n$ [@Lindel] and the case where $G$ satisfies $(\ast)$ and $U = {{\rm Spec \,}}k$ [@Raghunathan-1989]) were known much earlier.
Examples of anisotropic groups $G$ for which ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ fails to be ${{\mathbb A}}^1$-local were obtained in [@Balwe-Sawant]. For these examples, the failure of affine homotopy invariance of $G$-torsors was noted in [@Asok-Hoyois-Wendt-2]. Explicit examples of failure of affine homotopy invariance of the presheaf $H^1_{\rm Nis}(-, G)$ were already known in many cases, see [@Ojanguren-Sridharan], [@Parimala] for the first examples; and [@Raghunathan-1989 Theorem B] for a general statement excluding certain groups that are not of classical type. See Section \[subsection failure A1-invariance\] for more details. In this paper, we generalize these results as follows:
\[intro theorem 1\] Let $G$ be a reductive algebraic group over an infinite perfect field $k$. Then the following conditions are equivalent:
- ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is ${{\mathbb A}}^1$-local;
- $G$ satisfies the isotropy hypothesis $(\ast)$ (Hypothesis \[hypothesis isotropy\]);
- The presheaf $H^1_{\rm Nis}(-, G)$ is ${{\mathbb A}}^1$-invariant on smooth affine schemes over $k$.
Thus, Theorem \[intro theorem 1\] compares a motivic statement, a group-theoretic statement and a cohomological statement. In view of [@Asok-Hoyois-Wendt-2], the main results of this paper are Theorem \[theorem failure of A1-locality\] and Proposition \[proposition failure of A1-invariance\]. Although Theorem \[intro theorem 1\] addresses all reductive groups that fail to satisfy the isotropy hypothesis $(\ast)$ uniformly, our proof of failure of affine homotopy invariance for groups that do not satisfy hypothesis $(\ast)$ is existential. The main ingredient in the proof of failure of ${{\mathbb A}}^1$-locality of the singular construction for such groups is the behaviour of ${{\mathbb A}}^1$-locality of the singular construction under central isogenies, which is described in Section \[subsection failure A1-locality\].
In [@Balwe-Sawant], it was shown that sections over fields of the sheaf of ${{\mathbb A}}^1$-connected components of a semisimple simply connected group agree with the group of its $R$-equivalence classes. However, this is not the case if we drop the hypothesis of simple connectedness. This can be seen via our following characterization of ${{\mathbb A}}^1$-connected reductive algebraic groups.
\[intro theorem 2\] Let $G$ be a reductive algebraic group over a field $k$ of characteristic $0$. Then $G$ is ${{\mathbb A}}^1$-connected if and only if $G$ is semisimple, simply connected and every almost $k$-simple factor of $G$ is $R$-trivial.
In particular, this shows that one cannot have ${{\mathbb A}}^1$-connected components of a non-simply connected group agree with its $R$-equivalence classes in general. Indeed, any split semisimple group is a rational variety [@Borel V.15.8] and hence $R$-trivial; however, its ${{\mathbb A}}^1$-connected components cannot be trivial unless the group is simply connected. As a consequence of Theorem \[intro theorem 2\], we observe that ${{\mathbb A}}^1$-connected components of a semisimple group with an ${{\mathbb A}}^1$-connected, simply connected central cover form a sheaf of abelian groups (see Proposition \[proposition pi0 abelian\]).
We now outline the contents of this paper. In Section \[section preliminaries A1-connectedness\], we recollect preliminaries about ${{\mathbb A}}^1$-connectedness. In Section \[section preliminaries algebraic groups\], we recall basic notions about reductive algebraic groups and describe ${{\mathbb A}}^1$-connected components of semisimple, simply connected algebraic groups as an immediate consequence of results of [@Balwe-Sawant]. Section \[section failure\] is devoted to the proof of Theorem \[intro theorem 1\]. In Section \[section characterization of A1-connectedness\], we characterize ${{\mathbb A}}^1$-connected reductive algebraic groups over a field of characteristic $0$. We then obtain the abelian-ness of the sheaf of ${{\mathbb A}}^1$-connected components of certain algebraic groups as an application.
Preliminaries on A1-connected components of schemes {#section preliminaries A1-connectedness}
===================================================
In this section, we briefly recall some definitions from the ${{\mathbb A}}^1$-homotopy theory and some basic properties, particularly regarding ${{\mathbb A}}^1$-connectedness. We will begin by briefly reviewing the construction of the *${{\mathbb A}}^1$-homotopy category* from [@Morel-Voevodsky].
Let $k$ be a field. Let $Sm/k$ denote the big Nisnevich site of smooth, separated, finite-type schemes over $k$. We begin with the category of simplicial sheaves over $Sm/k$. A morphism ${{\mathcal X}}\to {{\mathcal Y}}$ of simplicial sheaves is a *local weak equivalence* if it induces a weak equivalence of stalks ${{\mathcal X}}_x \to {{\mathcal Y}}_x$ at every point $x$ of the site. The *local injective model structure* on this category is the one in which the morphism of simplicial sheaves is a cofibration (resp. a weak equivalence) if and only if it is a monomorphism (resp. a local weak equivalence). The corresponding homotopy category is called the *simplicial homotopy category* and is denoted by ${{\mathcal H}}_s(k)$. The left Bousfield localization of the local injective model structure with respect to the collection of all projection morphisms ${{\mathcal X}}\times {{\mathbb A}}^1 \to {{\mathcal X}}$, as ${{\mathcal X}}$ runs over all simplicial sheaves, is called the *${{\mathbb A}}^1$-model structure*. The corresponding homotopy category is called the *${{\mathbb A}}^1$-homotopy category* and is denoted by ${{\mathcal H}}(k)$.
\[definition A1-local\] A simplicial sheaf ${{\mathcal X}}$ on $Sm/k$ is said to be *${{\mathbb A}}^1$-local* if for any $U \in Sm/k$, the projection map $U \times {{\mathbb A}}^1 \to U$ induces a bijection $${{\rm Hom}}_{{{\mathcal H}}_s(k)}(U, {{\mathcal X}}) \to {{\rm Hom}}_{{{\mathcal H}}_s(k)}(U \times {{\mathbb A}}^1, {{\mathcal X}}).$$ Following standard conventions, an ${{\mathbb A}}^1$-local scheme will be called *${{\mathbb A}}^1$-rigid*. A scheme $X \in Sm/k$ is ${{\mathbb A}}^1$-rigid if for every $U \in Sm/k$, any morphism $h: U \times {{\mathbb A}}^1 \to X$ factors through the projection map $U \times {{\mathbb A}}^1 \to U$.
Curves of genus $\geq 1$, abelian varieties and algebraic tori are some examples of ${{\mathbb A}}^1$-rigid schemes.
We now recall the singular construction ${{\rm Sing_*^{{{\mathbb A}}^1}}}$ in ${{\mathbb A}}^1$-homotopy theory defined by Morel-Voevodsky (see [@Morel-Voevodsky p.87]). For a simplicial sheaf ${{\mathcal X}}$ on $Sm/k$, define ${{\rm Sing_*^{{{\mathbb A}}^1}}}{{\mathcal X}}$ to be the simplicial sheaf given by $$({{\rm Sing_*^{{{\mathbb A}}^1}}}{{\mathcal X}})_n = \underline{{{\rm Hom}}}(\Delta_n,{{\mathcal X}}_n),$$ where $\Delta_{\bullet}$ denotes the simplicial sheaf $$\Delta_n = {{\rm Spec \,}}\left(\frac{k[x_0,...,x_n]}{(\sum_ix_i=1)}\right)$$ with natural face and degeneracy maps analogous to the ones on topological simplices. The functor ${{\rm Sing_*^{{{\mathbb A}}^1}}}$ commutes with limits; in particular, with products. Also, there exists a natural transformation $Id \to {{\rm Sing_*^{{{\mathbb A}}^1}}}$ such that for any simplicial sheaf ${{\mathcal X}}$, the morphism ${{\mathcal X}}\to {{\rm Sing_*^{{{\mathbb A}}^1}}}({{\mathcal X}})$ is an ${{\mathbb A}}^1$-weak equivalence.
There exists an *${{\mathbb A}}^1$-localization* endofunctor ([@Morel-Voevodsky 2, Theorem 1.66 and p.107]) on the simplicial homotopy category ${{\mathcal H}}_s(k)$, denoted by $L_{{{\mathbb A}}^1}$, such that for every simplicial sheaf ${{\mathcal X}}$, the simplicial sheaf $L_{{{\mathbb A}}^1}({{\mathcal X}})$ is ${{\mathbb A}}^1$-local. In [@Morel-Voevodsky 2, Theorem 1.66 and p. 107], an explicit description of $L_{{{\mathbb A}}^1}$ is given as follows: $$L_{{{\mathbb A}}^1} = Ex \circ (Ex \circ {{\rm Sing_*^{{{\mathbb A}}^1}}})^{{{\mathbb N}}} \circ Ex,$$ where $Ex$ denotes a simplicial fibrant replacement functor on ${{\mathcal H}}_s(k)$. There exists a natural transformation $Id \to L_{{{\mathbb A}}^1}$ which factors through the natural transformation $Id \to {{\rm Sing_*^{{{\mathbb A}}^1}}}$ mentioned above. For any object ${{\mathcal X}}$, the morphism ${{\mathcal X}}\to L_{{{\mathbb A}}^1}({{\mathcal X}})$ is an ${{\mathbb A}}^1$-weak equivalence.
Given a simplicial sheaf of sets ${{\mathcal X}}$ on $Sm/k$, we will denote by $\pi_0({{\mathcal X}})$ the presheaf on $Sm/k$ that associates with $U \in Sm/k$ the coequalizer of the diagram ${{\mathcal X}}_1(U) \rightrightarrows {{\mathcal X}}_0(U)$, where the maps are the face maps coming from the simplicial data of ${{\mathcal X}}$. We will denote by $\pi_0^s({{\mathcal X}})$ the Nisnevich sheafification of the presheaf $\pi_0({{\mathcal X}})$.
Now, let $n \geq 1$ be an integer and let $({{\mathcal X}}, x)$ be a pointed simplicial sheaf of sets on $Sm/k$. For $U \in Sm/k$, we will denote by $U_+$ the scheme $U \coprod {{\rm Spec \,}}k$, pointed at the added basepoint ${{\rm Spec \,}}k$. We will denote by $\pi_n^s({{\mathcal X}},x)$ the Nisnevich sheafification of the presheaf (of groups) on $Sm/k$ that associates with $U \in Sm/k$ the group ${{\rm Hom}}_{{{\mathcal H}}_s(k)}(\Sigma_n^s U_{+}, ({{\mathcal X}},x))$ of simplicial homotopy classes of pointed maps from the simplicial $n$-fold suspension of the pointed scheme $U_{+}$ into $({{\mathcal X}}, x)$.
We caution the reader that this notation is not to be confused with the similar notation used for the sheaves of *stable homotopy groups*. Although this choice of notation is unfortunate, we use it here nevertheless in order to be consistent with the notation in [@Balwe-Hogadi-Sawant] and [@Balwe-Sawant].
\[definition-S\] Let ${{\mathcal X}}$ be a simplicial sheaf on $Sm/k$. The sheaf of *${{\mathbb A}}^1$-chain connected components* of ${{\mathcal X}}$ is defined by $${{\mathcal S}}({{\mathcal X}}) := \pi_0^s({{\rm Sing_*^{{{\mathbb A}}^1}}}{{\mathcal X}}).$$
Let $X$ be a scheme over $k$. For any smooth scheme $U$ over $k$, we say that two morphisms $f,g: U \to X$ are *${{\mathbb A}}^1$-homotopic* if there exists a morphism $h: U \times {{\mathbb A}}^1 \to X$ such that $h|_{U \times \{0\}} = f$ and $h|_{U \times \{1\}} = g$. We say that $f,g: U \to X$ are *${{\mathbb A}}^1$-chain homotopic* if there exists a finite sequence $f_0=f, \ldots, f_n=g$ such that $f_i$ is ${{\mathbb A}}^1$-homotopic to $f_{i+1}$, for all $i$. Clearly, ${{\mathbb A}}^1$-chain homotopy is an equivalence relation. It is easy to see that ${{\mathcal S}}(X)$ is the sheafification in the Nisnevich topology of the presheaf on $Sm/k$ that associates with every smooth scheme $U$ over $k$ the set of equivalence classes in $X(U)$ under the relation of ${{\mathbb A}}^1$-chain homotopy.
\[definition pi0A1\] Let ${{\mathcal X}}$ be a simplicial sheaf on $Sm/k$. The sheaf of *${{\mathbb A}}^1$-connected components* of ${{\mathcal X}}$ is defined by $$\pi_0^{{{\mathbb A}}^1}({{\mathcal X}}) := \pi_0^s(L_{{{\mathbb A}}^1}({{\mathcal X}})).$$ For any smooth scheme $U$ over $k$, we will say that $f,g \in {{\mathcal X}}(U)$ are *${{\mathbb A}}^1$-equivalent* if they map to the same element of $\pi_0^{{{\mathbb A}}^1}({{\mathcal X}})(U)$. We say that ${{\mathcal X}}$ is *${{\mathbb A}}^1$-connected* if $\pi_0^{{{\mathbb A}}^1}({{\mathcal X}}) \simeq \ast$, the trivial point sheaf.
There is a canonical epimorphism ${{\mathcal S}}({{\mathcal X}}) \to \pi_0^{{{\mathbb A}}^1}({{\mathcal X}})$ [@Morel-Voevodsky 2, Corollary 3.22, p. 94]. This epimorphism is an isomorphism if ${{\rm Sing_*^{{{\mathbb A}}^1}}}{{\mathcal X}}$ is ${{\mathbb A}}^1$-local.
\[definition A1-homotopy sheaves\] Let $({{\mathcal X}}, x)$ be a pointed simplicial sheaf on $Sm/k$ (that is, $x$ is a morphism ${{\rm Spec \,}}k \to {{\mathcal X}}$). For every integer $n \geq 1$, the $n$th *${{\mathbb A}}^1$-homotopy sheaf* of ${{\mathcal X}}$ with basepoint $x$ is defined by $$\pi_n^{{{\mathbb A}}^1}({{\mathcal X}}, x) := \pi_n^s(L_{{{\mathbb A}}^1}{{\mathcal X}}, x),$$ where $L_{{{\mathbb A}}^1}({{\mathcal X}})$ is pointed by ${{\rm Spec \,}}k \xrightarrow{x} {{\mathcal X}}\to L_{{{\mathbb A}}^1}({{\mathcal X}})$, which we continue to denote by $x$. We will always suppress base-points for the sake of brevity, when the base-point is understood from notation.
The main difficulty in the study of $\pi_0^{{{\mathbb A}}^1}$ of schemes is that the explicit description of the ${{\mathbb A}}^1$-localization functor is cumbersome to handle. However, in the cases when $\pi_0^{{{\mathbb A}}^1}$ of a scheme is *${{\mathbb A}}^1$-invariant*, it can be studied with geometric methods using results of [@Balwe-Hogadi-Sawant].
The notion of *Weil restriction* of a simplicial sheaf will be very useful in what follows. We briefly recall it here. Let $F/k$ be a finite field extension and let $f: {{\rm Spec \,}}F \to {{\rm Spec \,}}k$ denote the morphism corresponding to the inclusion $k \hookrightarrow F$. The pushforward functor $f_*$ from the category of simplicial sheaves on $Sm/F$ into the category of simplicial sheaves on $Sm/k$ is defined by $$f_*({{\mathcal X}})(U) = {{\mathcal X}}(U \times_{{{\rm Spec \,}}k} {{\rm Spec \,}}F).$$ If $Ex$ denotes a simplicial fibrant replacement functor on the category of simplicial sheaves over $Sm/k$, one can show that the functor $f_* \circ Ex$ preserves simplicial weak equivalences. Thus, it induces a functor $\mathbf{R}f_*: {{\mathcal H}}_s(F) \to {{\mathcal H}}_s(k)$, which is the *right derived functor* of $f_*$. We recall that the functor $\mathbf{R}f_*$ preserves ${{\mathbb A}}^1$-local objects and thus induces the composition $\mathbf{R}f_* \circ L_{{{\mathbb A}}^1}$ induces a functor $\mathbf{R}^{{{\mathbb A}}^1}f_*: {{\mathcal H}}(F) \to {{\mathcal H}}(k)$ (see [@Morel-Voevodsky pages 92 and 108]). It follows from [@Morel-Voevodsky page 109, Proposition 2.12] that for any simplicial sheaf ${{\mathcal X}}$ on $Sm/F$, the canonical morphism $$\mathbf{R}f_*({{\mathcal X}}) \to \mathbf{R}^{{{\mathbb A}}^1}f_*({{\mathcal X}}) = \mathbf{R}f_* \circ L_{{{\mathbb A}}^1}({{\mathcal X}})$$ is an ${{\mathbb A}}^1$-weak equivalence. This induces an isomorphism $$L_{{{\mathbb A}}^1} \circ \mathbf{R}f_* ({{\mathcal X}}) \to \mathbf{R}f_* \circ L_{{{\mathbb A}}^1}({{\mathcal X}}).$$
Let $F$ be a finite field extension of $k$. For the finite map $f: {{\rm Spec \,}}F \to {{\rm Spec \,}}k$, we will denote $\mathbf{R}f_*$ by $R_{F/k}$.
The following is a straightforward consequence of the above discussion.
\[lemma weil restriction\] Let $F/k$ be a finite extension of fields. For every simplicial sheaf ${{\mathcal X}}$ over $Sm/F$, we have $R_{F/k} \pi_0^{{{\mathbb A}}^1}({{\mathcal X}}) = \pi_0^{{{\mathbb A}}^1}(R_{F/k}{{\mathcal X}})$.
Algebraic groups and their A1-connected components {#section preliminaries algebraic groups}
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In this section, we briefly recall the basic definitions and properties from algebraic group theory; for details, refer to [@SGA3.2], [@SGA3.3] and [@Conrad-Gabber-Prasad Appendix A].
We will always work over a field $k$. We will write $GL_n$ for the general linear group scheme and write ${{\mathbb G}}_m$ for $GL_1$. We recall the definitions of reductive and semisimple group schemes from [@SGA3.3 Exposé XIX, 1.6, 2.7]. A *reductive* algebraic group over $k$ is a smooth, affine $k$-group scheme with trivial unipotent radical. A *semisimple* algebraic group over $k$ is a smooth, affine $k$-group scheme with trivial radical. Over a field, reductive algebraic groups are *linear*, that is, they admit a finitely presented, closed immersion into $GL_n$ for some $n$, which is a group homomorphism. All the reductive algebraic groups considered in what follows will be assumed to be connected.
The *derived group* of $G$ [@SGA3.3 Exposé XXII, Theorem 6.2.1(iv)], will be denoted by $G_{\rm der}$. It is a normal, semisimple subgroup scheme of $G$ and the quotient $${\rm corad}(G) := G/G_{\rm der}$$ is a $k$-torus [@SGA3.3 Exposé XXII, 6.2] called the *coradical* of $G$.
The center of a reductive group is of multiplicative type [@SGA3.2 Exposé XII, Proposition 4.11]. There exists a central isogeny [@SGA3.3 Exposé XXII, Proposition 6.2.4] $$G_{\rm der} \times T \to G,$$ where $T$ is a torus, the radical of $G$. This is a faithfully flat, finitely presented morphism, whose kernel is a finite group of multiplicative type contained in the center of $G_{\rm der} \times T$.
An algebraic group is said to be *almost $k$-simple* if it is smooth, connected over $k$ and admits no infinite normal $k$-subgroup [@Tits-classification p.41]. An algebraic group $G$ over $k$ is said to be *absolutely almost simple* if $G_{\-k}$ is almost $\-k$-simple. An algebraic group $G$ is said to be the *almost direct product* of its algebraic subgroups $G_1, \ldots, G_n$ if the product map $$G_1 \times \cdots \times G_n \to G$$ is an isogeny. Semisimple algebraic $k$-groups are exactly those that occur as the almost direct product of their almost $k$-simple algebraic subgroups, called the *almost $k$-simple factors*.
A connected semisimple algebraic group $G$ over $k$ is said to be *simply connected* if every central isogeny $G' \to G$ is an isomorphism. Given a connected semisimple algebraic group $G$, there exists a simply connected group $G_{\rm sc}$ and a central isogeny $\pi: G_{\rm sc} \to G$. The pair $(G_{\rm sc}, \pi)$ is unique up to unique isomorphism and its formation respects base change by field extensions. $G_{\rm sc}$ is called the *simply connected central cover* of $G$. Every semisimple simply connected $k$-group is uniquely given by a direct product of almost $k$-simple simply connected groups. If $G$ is almost $k$-simple and simply connected, there exists a finite field extension $F/k$ and an absolutely almost simple, simply connected $F$-group $H$ such that $G = R_{F/k}(H)$ [@Tits-classification p. 41].
\[definition isotropic anisotropic\] A reductive algebraic group $G$ over a field $k$ is called *isotropic* if $G$ contains a non-central $k$-subgroup scheme isomorphic to ${{\mathbb G}}_m$. A reductive algebraic group $G$ over a field $k$ is called *anisotropic* if it contains no subgroup isomorphic to ${{\mathbb G}}_m$.
We now describe ${{\mathbb A}}^1$-connected components of algebraic groups over a field $k$. By [@Choudhury Theorem 4.18], for any algebraic group $G$, the sheaf $\pi_0^{{{\mathbb A}}^1}(G)$ is ${{\mathbb A}}^1$-invariant. Putting this together with [@Balwe-Hogadi-Sawant Theorem 1], we obtain the following description.
For an algebraic group $G$ over a field $k$, we have $$\pi_0^{{{\mathbb A}}^1}(G) \simeq \underset{n}{\varinjlim}~{{\mathcal S}}^n(G).$$
We end this section with an explicit description of ${{\mathbb A}}^1$-connected components of semisimple, simply connected groups in terms of other classical invariants of algebraic groups. This description is a straightforward consequence of the results of [@Balwe-Sawant]. We first recall the definitions of *$R$-equivalence* and *Whitehead groups*.
\[definition R-equivalence\] Let $G$ be an algebraic group over a field $k$. Two $k$-rational points $x, y$ of $G$ are said to be *$R$-equivalent* if there is a rational map $f: {{\mathbb P}}^1_k \dashrightarrow G$ defined at $0$ and $1$ such that $f(0)=x$ and $f(1)=y$.
The notion of $R$-equivalence was first studied in the context of algebraic groups in [@Colliot-Thelene-Sansuc-1979]. For the basic properties regarding $R$-equivalence, also see [@Gille-IHES Section II], [@Gille] and [@Voskresenskii Chapter 6]. It can be shown that $R$-equivalence gives an equivalence relation on $G(k)$. It is easy to see that elements of $G(k)$ that are $R$-equivalent to the identity form a normal subgroup of $G(k)$. The quotient of $G(k)$ by this normal subgroup is denoted by $G(k)/R$ and called the *group of $R$-equivalence classes* of $G$ over $k$.
\[definition R-trivial\] We say that an algebraic group $G$ over a field $k$ is *$R$-trivial* if the group $G(F)/R := G_F(F)/R$ is trivial, for every field extension $F/k$.
\[definition Whitehead group\] For an algebraic group $G$ over a field $k$ and a field extension $F$ of $k$, let $G(F)^+$ be the normal subgroup of $G(F)$ generated by the subsets $U(F)$ where $U$ varies over all $F$-subgroups of $G$ which are isomorphic to the additive group ${{\mathbb G}}_a$. The group $$W(F,G):= G(F)/G(F)^+$$ is called the *Whitehead group* of $G$ over $F$.
Evidently, there is a canonical surjection $$\label{equation surjections1}
W(k,G) \to G(k)/R,$$ for any algebraic group $G$ over a field $k$. This surjection is an isomorphism if $G$ is semisimple, simply connected, absolutely almost simple and isotropic (see [@Gille Théorème 7.2], for example). The above surjection is not an isomorphism in general for non-simply connected groups.
\[theorem IMRN\] Let $G$ be a semisimple, simply connected group over an infinite perfect field $k$. Let $F$ be a perfect field extension of $k$. Then there is a canonical isomorphism $$\pi_0^{{{\mathbb A}}^1}(G)(F) \to G(F)/R.$$
First assume that $G$ is a semisimple, simply connected and absolutely almost simple group over $k$. If $G$ is isotropic, then by [@Asok-Hoyois-Wendt-2 Theorem 4.3.1], ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is ${{\mathbb A}}^1$-local and it follows that $\pi_0^{{{\mathbb A}}^1}(G)(F) \simeq G(F)/R$, for every field extension $F/k$ (see [@Balwe-Sawant Theorem 3.4]). If $G$ is anisotropic, this is [@Balwe-Sawant Theorem 4.2] (although it is stated there with the assumption that the base field is of characteristic $0$, it is easy to see that the proof works over any infinite perfect field).
Now, let $G$ be an arbitrary semisimple, simply connected group. There exist almost $k$-simple algebraic groups $H_1, \ldots, H_r$ such that $G \simeq H_1 \times \cdots \times H_r$. For each $i$, there exists a finite field extension $k_i/k$ and an absolutely almost simple group $G_i$ such that $R_{k_i/k}(G_i) \simeq H_i$. Note that for any finitely generated field extension $F/k$, we have $$\pi_0^{{{\mathbb A}}^1}(H_i)(F) \simeq \pi_0^{{{\mathbb A}}^1}(R_{k_i/k}(G_i))(F) \simeq R_{k_i/k}(\pi_0^{{{\mathbb A}}^1}(G_i))(F) \simeq \pi_0^{{{\mathbb A}}^1}(G_i)(F \otimes_k k_i)$$ However, since $F \otimes_k k_i$ is a product of fields, by the special case of absolutely almost simple groups explained above, we have $$\pi_0^{{{\mathbb A}}^1}(G_i)(F \otimes_k k_i) \simeq G_i(F \otimes_k k_i)/R$$ Since $$G_i(F \otimes_k k_i)/R \simeq R_{k_i/k}(G_i)(F)/R \simeq H_i(F)/R,$$ for every $i$, we conclude that $$\pi_0^{{{\mathbb A}}^1}(G)(F) \simeq \prod_{i=1}^{r} ~ \pi_0^{{{\mathbb A}}^1}(H_i)(F) {\simeq} \prod_{i=1}^{r} ~ H_i(F)/R \simeq G(F)/R.$$
This immediately implies the failure of ${{\mathbb A}}^1$-locality of the singular construction on $G$ satisfying the hypotheses of Theorem \[theorem IMRN\] and having at least one anisotropic factor.
\[corollary IMRN\] Let $G$ be a semisimple, simply connected group over an infinite perfect field $k$. If $G$ has an anisotropic almost $k$-simple factor, then ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is not ${{\mathbb A}}^1$-local.
Since $G$ is a direct product of its almost $k$-simple factors, we are reduced to the case of an anisotropic, semisimple, almost $k$-simple group. Since $G$ is reductive over a perfect field $k$, it is unirational over $k$ (see [@Borel Theorem 18.2]). Therefore, there exists a pair of distinct $R$-equivalent elements in $G(k)$. Since $G$ is anisotropic, we have $G(k) \simeq {{\mathcal S}}(G)(k)$, by [@Balwe-Sawant Lemma 3.7]. Thus, the map ${{\mathcal S}}(G)(k) \to \pi_0^{{{\mathbb A}}^1}(G)(k)$ is not a bijection. This shows that ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ cannot be ${{\mathbb A}}^1$-local.
Failure of A1-locality of the singular construction and consequences {#section failure}
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A1-locality of Sing\* {#subsection failure A1-locality}
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The following isotropy hypothesis on reductive algebraic $k$-groups was introduced in [@Raghunathan-1989]. This is the isotropy hypothesis $(\ast)$ from the introduction.
\[hypothesis isotropy\] Each of the almost $k$-simple components of $G_{\rm der}$ contains a $k$-subgroup scheme isomorphic to ${{\mathbb G}}_m$.
We caution the reader that reductive groups satisfying Hypothesis \[hypothesis isotropy\] are called *isotropic reductive groups* in [@Asok-Hoyois-Wendt-2]. However, in this paper, we stick to the classical definitions and terminology [@Borel V.20.1].
In this section, we show that if a reductive algebraic group $G$ over an infinite perfect field does not satisfy Hypothesis \[hypothesis isotropy\], then the Morel-Voevodsky singular construction ${{\rm Sing_*^{{{\mathbb A}}^1}}}(G)$ is not ${{\mathbb A}}^1$-local. A key role in the proof will be played by the *fppf* classifying space $B_{\rm fppf} G$ of a reductive group $G$. We begin by briefly introducing this object.
\[definition t-local replacement\] Let $t$ be a Grothendieck topology on a small category $\mathbf{C}$. We say that a simplicial sheaf ${{\mathcal F}}$ on $\mathbf{C}$ is *$t$-local* if, for every $X \in \mathbf{C}$ and every $t$-covering sieve ${{\mathcal U}}$ of $X$, the restriction map $${{\mathcal F}}(X) \to \underset{(Y\to X) \in {{\mathcal U}}}{\rm holim}~ {{\mathcal F}}(Y)$$ is a weak equivalence.
The *Čech $t$-local injective model structure* on this category is the left Bousfield localization of the injective model structure with respect to the set of maps $\{{{\mathcal U}}\to X\}$, where $X$ runs over all objects of $\mathbf{C}$ and ${{\mathcal U}}$ runs over all covering sieves of $X$. In this model structure, an object ${{\mathcal F}}$ is fibrant if and only if it is $t$-local and also fibrant with respect to the injective model structure.
It can be proved (see the argument in [@Dugger-Hollander-Isaksen Example A.10]) that in the case of the category $Sm/k$, the Čech Nisnevich-local injective model structure is the same as local injective model structure described in Section \[section preliminaries A1-connectedness\].
We now apply this notion to the category $Sch/k$ of schemes of finite type over $k$ with the *fppf* topology. Thus, we have the model category of simplicial sheaves on $Sch/k$ with the Čech *fppf*-local injective model structure. Let $\mathbf{R}_{\rm fppf}$ denote the fibrant replacement functor for this model structure. The inclusion functor $i: Sm/k \to Sch/k$ induces a restriction functor $i^*$ from the category of simplicial fppf-sheaves on $Sch/k$ to the category of simplicial Nisnevich sheaves on $Sm/k$.
For a group sheaf $G$, we will denote by $BG$ the pointed simplicial sheaf whose $n$-simplices are $G^n$ with usual face and degeneracy maps.
Let $G$ be an $fppf$-sheaf of groups on $Sch/k$. Then we define $B_{\rm fppf}G$ to be the simplicial Nisnevich sheaf defined by $$B_{\rm fppf}G := i^* \circ \mathbf{R}_{\rm fppf}(BG).$$
We denote by $(Sm/k)_{\rm fppf}$ the site of faithfully flat, finitely presented smooth schemes over $k$ which are separated and of finite type. We will use simplicial and ${{\mathbb A}}^1$-fiber sequences of simplicial *fppf*-sheaves of sets. Following [@Asok-Hoyois-Wendt-2 2], by a *simplicial fiber sequence* of pointed simplicial presheaves, we mean a homotopy Cartesian square in which either the top-right or bottom-left corner is a point. One defines an *${{\mathbb A}}^1$-fiber sequence* similarly with appropriate modifications, see [@Asok 2.3, Definition 2.9]. As in topology, ${{\mathbb A}}^1$-fiber sequences of pointed simplicial sheaves induce long exact sequences of ${{\mathbb A}}^1$-homotopy sheaves. In what follows, we will suppress the basepoints for the sake of brevity.
The following lemma will be very useful in the proof of our main theorem.
\[lemma classifying spaces\] Let $G$ be an algebraic group of multiplicative type over a field $k$. Then the classifying space $B_{\rm fppf} G$ is ${{\mathbb A}}^1$-local.
We imitate the proof of [@Morel-Voevodsky 4.3, Proposition 3.1]. We abuse the notation and continue to denote by $B_{\rm fppf} G$ the restriction to $(Sm/k)_{\rm Nis}$ of the *fppf*-local replacement of the simplicial presheaf $BG$.
Since $B_{\rm fppf} G$ is etale-local in the sense of Definition \[definition t-local replacement\] (that is, $B_{\rm fppf} G$ satisfies étale descent), it suffices to show that the map $$\label{equation lemma multiplicative type}
(B_{\rm fppf} G)(S) \to (B_{\rm fppf} G)({{\mathbb A}}^1_S)$$ induced by the projection ${{\mathbb A}}^1_S \to S$ is a weak equivalence for every $S$ which is the strict henselization of a local ring of a smooth scheme over $k$. In order to show this, it suffices to show that the map induced by the map on every $\pi_i$ is a bijection. Since $\pi_i(B_{\rm fppf}G)$ is trivial for $i>1$, it suffices to examine the map on $\pi_i$ for $i = 0$ and $1$. Since $G$ is an algebraic group of multiplicative type, it follows that $G$ is diagonalizable over $S$ [@Conrad-SGA Proposition B.3.4]. Hence, $G_S$ is a product of group schemes of the form ${{\mathbb G}}_m$ or $\mu_n$ over $S$, for a natural number $n$. So without loss of generality, we may assume that $G={{\mathbb G}}_m$ or $G=\mu_n$.
By [@Asok-Hoyois-Wendt-2 Lemma 2.2.2], $\pi_0(B_{\rm fppf} G)(-) \simeq H^1_{\rm fppf}(-, G)$. Hence the map induced by on $\pi_0$ is the map $$H^1_{\rm fppf}(-, G) \to H^1_{\rm fppf}(-, G)$$ induced by the projection ${{\mathbb A}}^1_S \to S$. This map is a bijection by the ${{\mathbb A}}^1$-invariance of Picard group of schemes (over any normal base-scheme).
It remains to verify that the map is an isomorphism on $\pi_1$’s at the base point. However, this is just the map $G(S) \to G({{\mathbb A}}^1_S)$ (induced by the projection ${{\mathbb A}}^1_S \to S$), which is clearly an isomorphism, since $S$ is reduced and since $G$ is either ${{\mathbb G}}_m$ or a finite group.
We are now set to prove the main reduction step in the proof of our main theorem.
\[proposition central isogeny\] Let $G' \to G$ be a central isogeny of reductive algebraic groups. Suppose that ${{\rm Sing_*^{{{\mathbb A}}^1}}}G'$ is not ${{\mathbb A}}^1$-local. Then ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ cannot be ${{\mathbb A}}^1$-local.
Suppose, if possible, that ${{\rm Sing_*^{{{\mathbb A}}^1}}}(G)$ is ${{\mathbb A}}^1$-local. Let $\mu$ denote the kernel of the given central isogeny $G' \to G$. Then we have a sequence $$G' \to G \to \mathbf{R}_{\rm fppf}(B\mu)$$ which is a fiber sequence in the model category of simplicial *fppf*-sheaves on $Sch/k$ (with the Čech $fppf$-local injective model structure). The restriction functor $i^*$, from the category of simplicial sheaves on $Sch/k$ to the category of simplicial sheaves on $Sm/k$, preserves objectwise fiber sequences. An objectwise fiber sequence is a fiber sequence in ${{\mathcal H}}_s(k)$. Thus, we have a simplicial fiber sequence $$G' \to G \to B_{\rm fppf}\mu.$$ Note that $\mu$ is a group of multiplicative type, being contained in the center of the reductive group $G'$. Since $\mu$ is ${{\mathbb A}}^1$-rigid and since $B_{\rm fppf}\mu$ is ${{\mathbb A}}^1$-local by Lemma \[lemma classifying spaces\], it follows that $\pi_0^{{{\mathbb A}}^1}(\mu) \simeq \mu$ is a strongly ${{\mathbb A}}^1$-invariant sheaf, in the sense of [@Morel Definition 1.7]. Therefore, by [@Morel Theorem 6.50], the simplicial fiber sequence $$G' \to G \to B_{\rm fppf}\mu$$ is also an ${{\mathbb A}}^1$-fiber sequence. The associated long exact sequence of homotopy groups gives us the following commutative diagram with exact rows, for every $i\geq 0$:
$$\label{equation commutative diagram}
\minCDarrowwidth 10pt
\begin{CD}
\pi_{i+1}^s({{\rm Sing_*^{{{\mathbb A}}^1}}}G) @>>> \pi_{i+1}^s({{\rm Sing_*^{{{\mathbb A}}^1}}}B_{\rm fppf}\mu) @>>> \pi_{i}^s({{\rm Sing_*^{{{\mathbb A}}^1}}}G') @>>> \pi_{i}^s({{\rm Sing_*^{{{\mathbb A}}^1}}}G) @>>> \pi_{i}^s({{\rm Sing_*^{{{\mathbb A}}^1}}}B_{\text{fppf}}\mu) \\
@VV{\simeq}V @VV{\simeq}V @VVV @VV{\simeq}V @VV{\simeq}V \\
\pi_{i+1}^{{{\mathbb A}}^1}(G) @>>> \pi_{i+1}^{{{\mathbb A}}^1}(B_{\rm fppf}\mu) @>>> \pi_i^{{{\mathbb A}}^1}(G') @>>> \pi_i^{{{\mathbb A}}^1}(G) @>>> \pi_i^{{{\mathbb A}}^1}(B_{\text{fppf}}\mu) .
\end{CD}$$
Here the first row is obtained as follows: since $\pi_0(B_{\rm fppf} \mu)(-) \simeq H^1_{\rm fppf}(-, \mu)$ is an ${{\mathbb A}}^1$-invariant presheaf, the functor ${{\rm Sing_*^{{{\mathbb A}}^1}}}$ preserves simplicial fiber sequences [@Asok-Hoyois-Wendt-2 Proposition 2.1.1]; we then take the associated long exact sequence of simplicial homotopy groups. All the vertical maps are induced by the natural transformation of functors ${{\rm Sing_*^{{{\mathbb A}}^1}}}\to L_{{{\mathbb A}}^1}$. In the diagram , the second and the last vertical arrows are isomorphisms since $B_{\rm fppf}\mu$ is ${{\mathbb A}}^1$-local (Lemma \[lemma classifying spaces\]); and the first and fourth vertical arrows are isomorphisms since ${{\rm Sing_*^{{{\mathbb A}}^1}}}(G)$ is ${{\mathbb A}}^1$-local. It follows from five lemma that the map $\pi_{i}^s({{\rm Sing_*^{{{\mathbb A}}^1}}}G') \to \pi_{i}^{{{\mathbb A}}^1}(G')$ is an isomorphism for all $i\geq 0$. This shows that the natural map ${{\rm Sing_*^{{{\mathbb A}}^1}}}G' \to L_{{{\mathbb A}}^1}(G')$ is a weak equivalence, by the ${{\mathbb A}}^1$-Whitehead theorem [@Morel-Voevodsky 3, Proposition 2.14, p. 110]. Consequently, ${{\rm Sing_*^{{{\mathbb A}}^1}}}G'$ is ${{\mathbb A}}^1$-local, contradicting the hypothesis.
We now prove the main theorem of this section.
\[theorem failure of A1-locality\] Let $G$ be a reductive algebraic group over an infinite perfect field $k$ that does not satisfy Hypothesis \[hypothesis isotropy\]. Then ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is not ${{\mathbb A}}^1$-local.
Since $G$ is reductive, there exists a central isogeny $$G_{\rm der} \times T \to G,$$ where $G_{\rm der}$ is a semisimple group (the derived group of $G$) and $T$ is a torus (the radical of $G$). By Proposition \[proposition central isogeny\], it suffices to show that ${{\rm Sing_*^{{{\mathbb A}}^1}}}(G_{\rm der} \times T)$ is not ${{\mathbb A}}^1$-local. Note that ${{\rm Sing_*^{{{\mathbb A}}^1}}}$ commutes with products and that ${{\rm Sing_*^{{{\mathbb A}}^1}}}T$ is ${{\mathbb A}}^1$-local. Therefore, we are reduced to showing that ${{\rm Sing_*^{{{\mathbb A}}^1}}}G_{\rm der}$ is not ${{\mathbb A}}^1$-local.
Let $G_{\rm sc}$ denote the simply connected cover of $G_{\rm der}$. There exists a central isogeny $G_{\rm sc} \to G_{\rm der}$. Again by Proposition \[proposition central isogeny\], it suffices to prove that ${{\rm Sing_*^{{{\mathbb A}}^1}}}(G_{\rm sc})$ is not ${{\mathbb A}}^1$-local. Let $G_1, \ldots, G_n$ be the almost $k$-simple factors of $G_{\rm sc}$; we have an isomorphism $G_1 \times \cdots \times G_n \stackrel{\sim}{\to} G_{\rm sc}$. If $G$ does not satisfy Hypothesis \[hypothesis isotropy\], then $G_{\rm der}$ has at least one anisotropic almost $k$-simple factor. Therefore, there exists $i \in \{1, \ldots, n\}$ such that $G_i$ is anisotropic. By Corollary \[corollary IMRN\], we conclude that ${{\rm Sing_*^{{{\mathbb A}}^1}}}G_i$ is not ${{\mathbb A}}^1$-local. Hence, ${{\rm Sing_*^{{{\mathbb A}}^1}}}(G_1 \times \cdots \times G_n) \simeq {{\rm Sing_*^{{{\mathbb A}}^1}}}G_{\rm sc}$ cannot be ${{\mathbb A}}^1$-local. Thus, if $G$ does not satisfy Hypothesis \[hypothesis isotropy\], then ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ cannot be ${{\mathbb A}}^1$-local.
We end the section by recording a proof of Theorem \[intro theorem 1\], stated in the intoduction, by putting together Theorem \[theorem failure of A1-locality\] and relevant results from [@Asok-Hoyois-Wendt-2].
\[theorem equivalence\] Let $G$ be a reductive algebraic group over an infinite perfect field $k$. Then the following conditions are equivalent:
- ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is ${{\mathbb A}}^1$-local;
- $G$ satisfies the isotropy hypothesis $(\ast)$ (Hypothesis \[hypothesis isotropy\]);
- The presheaf $H^1_{\rm Nis}(-, G)$ is ${{\mathbb A}}^1$-invariant on smooth affine schemes over $k$.
The implication $(1) \Rightarrow (2)$ follows from Theorem \[theorem failure of A1-locality\]. The implication $(2) \Rightarrow (3)$ is proved in [@Asok-Hoyois-Wendt-2 Theorem 3.3.3], whereas the implication $(3) \Rightarrow (1)$ is proved in [@Asok-Hoyois-Wendt-2 Theorem 2.3.2].
Failure of affine homotopy invariance for G-torsors {#subsection failure A1-invariance}
---------------------------------------------------
In [@Asok-Hoyois-Wendt-2 Theorem 3.3.6], it is shown that if $G$ is a reductive algebraic group over an infinite field $k$ satisfying Hypothesis \[hypothesis isotropy\] and $A$ is a smooth affine $k$-algebra, then the map $$H^1_{\rm Nis}({{\rm Spec \,}}A, G) \to H^1_{\rm Nis}({{\rm Spec \,}}A[t_1, \ldots, t_n], G)$$ induced by the projection ${{\rm Spec \,}}A[t_1, \ldots, t_n] \to {{\rm Spec \,}}A$ is a bijection for all $n \geq 0$. In view of the Grothendieck-Serre conjecture (see [@Colliot-Thelene-Ojanguren-1995], [@Raghunathan-1994], [@Fedorov-Panin]), Nisnevich locally trivial $G$-torsors are Zariski locally trivial, where $G$ is a connected reductive group over an infinite perfect field $k$. Therefore, [@Asok-Hoyois-Wendt-2 Theorem 3.3.6] can be seen as a generalization of the results of Lindel [@Lindel] (the case $G = GL_n$) and Raghunathan [@Raghunathan-1989] (the case where $G$ satisfies Hypothesis \[hypothesis isotropy\] and $A=k$). Counterexamples to affine homotopy invariance of $G$-torsors were found in case the group $G$ does not satisfy Hypothesis \[hypothesis isotropy\] by Ojanguren-Sridharan [@Ojanguren-Sridharan] and Parimala [@Parimala]. A general result about failure of affine homotopy invariance is due to Raghunathan [@Raghunathan-1989 Theorem B], where it is shown that if $G$ is an anisotropic, absolutely almost simple group not of type $F_4$ or $G_2$ and satisfying a technical condition (there exists a group $G'$ in the central isogeny class of $G$ and an embedding of $G'$ in a connected reductive group $H$ as a closed normal subgroup such that $H$ is a $k$-rational variety and such that $H/G'$ is a torus) which holds for groups of classical type, then there are infinitely many mutually non-isomorphic $G$-bundles on ${{\mathbb A}}^2_k$ that are not extended from ${{\rm Spec \,}}k$.
A straightforward application of Theorem \[theorem equivalence\] shows that torsors for reductive groups not satisfying Hypothesis \[hypothesis isotropy\] fail to be ${{\mathbb A}}^1$-invariant on smooth affine schemes over an infinite perfect field. In the case of semisimple, simply connected, absolutely almost simple anisotropic groups, this was shown by Asok, Hoyois and Wendt in [@Asok-Hoyois-Wendt-2 Proposition 3.3.7] using [@Balwe-Sawant Corollary 3]. Theorem \[theorem failure of A1-locality\] generalizes [@Balwe-Sawant Corollary 3] to all reductive algebraic groups not satisfying Hypothesis \[hypothesis isotropy\] and hence generalizes [@Asok-Hoyois-Wendt-2 Proposition 3.3.7] to all such groups using the same method. We end this section by formally stating the result for the sake of completeness.
\[proposition failure of A1-invariance\] Let $G$ be a reductive algebraic group over an infinite perfect field $k$, which does not satisfy Hypothesis \[hypothesis isotropy\]. Then the presheaf $H^1_{\rm Nis}(-, G)$ cannot be ${{\mathbb A}}^1$-invariant on smooth affine schemes over $k$.
Characterization of A1-connectedness in reductive groups and applications {#section characterization of A1-connectedness}
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A1-connected reductive algebraic groups
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In this section, we characterize ${{\mathbb A}}^1$-connected reductive algebraic groups over a field of characteristic $0$. We first treat the case where the base field is algebraically closed.
\[theorem algebraically closed\] Let $G$ be a reductive algebraic group over an algebraically closed field $k$. Then $G$ is ${{\mathbb A}}^1$ -connected if and only if it is semisimple and simply connected.
First, we assume that $G$ is semisimple and simply connected. Since $G$ is semisimple and $k$ is algebraically closed, $G$ is an almost direct product of absolutely almost simple groups $H_1, \ldots, H_r$ over $k$. Since $G$ is simply connected, it has no nontrivial isogenies, thus giving an isomorphism $\prod_{i=1}^{r} ~ H_i \xrightarrow{\sim}G$.
Since $k$ is algebraically closed, each of the $H_i$’s is a rational variety [@Borel 14.14, Remark] and hence, is $R$-trivial. By Theorem \[theorem IMRN\], we have $$\pi_0^{{{\mathbb A}}^1}(H_i)(F) \simeq H_i(F)/R \simeq \ast,$$ for every field extension $F$ of $k$. By [@Morel-connectivity Lemma 6.1.3], it follows that $G$ is ${{\mathbb A}}^1$-connected.
We now prove the converse. Let $G$ be an ${{\mathbb A}}^1$-connected reductive algebraic group. We have the exact sequence $$1 \to G_{\rm der} \to G \to \rm corad (G) \to 1,$$ where $G_{\rm der}$ is a semisimple group (the derived group of $G$) and ${\rm corad}(G)$ is a torus (the coradical of $G$). Since $k$ is algebraically closed, $G$ clearly satisfies Hypothesis \[hypothesis isotropy\]. Therefore, by [@Asok-Hoyois-Wendt-2 Theorem 4.3.1], ${{\rm Sing_*^{{{\mathbb A}}^1}}}G$ is ${{\mathbb A}}^1$-local and we have ${{\mathcal S}}(G) \simeq \pi_0^{{{\mathbb A}}^1}(G)$. Since tori are ${{\mathbb A}}^1$-rigid, the surjective map $G(k) \to {\rm corad}(G)(k)$ induces a surjective map ${{\mathcal S}}(G)(k) \to {\rm corad}(G)(k)$. Since ${{\mathcal S}}(G)= \ast$ by hypothesis, it follows that ${\rm corad}(G)$ is trivial. Thus, $G_{\rm der} \simeq G$, that is, $G$ is semisimple.
Let $G_{\rm sc}$ denote the simply connected central cover of $G$ so that we have a central isogeny $G_{\rm sc} \to G$, whose kernel will be denoted by $\mu$. As we noted before in the proof of Proposition \[proposition central isogeny\], the simplicial fiber sequence $$G_{\rm sc} \to G \to B_{\rm fppf} \mu$$ is also an ${{\mathbb A}}^1$-fiber sequence. Thus, the long exact sequence of ${{\mathbb A}}^1$-homotopy gives us the following exact sequence of pointed sheaves: $$\cdots \to \pi_1^{{{\mathbb A}}^1}(B_{\rm fppf} \mu) \to \pi_0^{{{\mathbb A}}^1}(G_{\rm sc}) \to \pi_0^{{{\mathbb A}}^1}(G) \to \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu).$$ Since $G_{\rm sc}$ is a split semisimple simply connected group, we have $\pi_0^{{{\mathbb A}}^1}(G_{sc})(F) \simeq {{\mathcal S}}(G_{\rm sc})(F) \simeq W(F, G_{\rm sc}) = \ast$, for every finitely generated field extension $F$ of $k$, by [@Tits-Bourbaki 1.1.2].
Let $k(G)$ and $k(G_{\rm sc})$ denote the function fields of $G$ and $G_{\rm sc}$ respectively. Let $\eta: {{\rm Spec \,}}k(G) \to G$ denote the generic point of $G$. By Lemma \[lemma classifying spaces\], $B_{\rm fppf} \mu$ is ${{\mathbb A}}^1$-local and so $\pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)$ is the sheafification of the pointed presheaf $H^1_{\rm fppf}(-, \mu)$ (where the base-point corresponds to the trivial torsor). Thus, $\pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)(k(G)) = H^1_{\rm fppf}(k(G), \mu)$ and the image of $\eta$ in $H^1_{\rm fppf}(k(G), \mu)$ under the composition $$G(k(G)) \to \pi_0^{{{\mathbb A}}^1}(G)(k(G)) \to \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)(k(G))$$ corresponds to the class of the $\mu$-torsor $G_{\rm sc} \times_{G, \eta} {{\rm Spec \,}}k(G) \to {{\rm Spec \,}}k(G)$. Since $\pi_0^{{{\mathbb A}}^1}(G) = \ast$, the morphism of pointed sheaves $\pi_0^{{{\mathbb A}}^1}(G) \to \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)$ in the above ${{\mathbb A}}^1$-fiber sequence is trivial. Hence, the image of $\eta$ under the composition $$G(k(G)) \to \pi_0^{{{\mathbb A}}^1}(G)(k(G)) \to \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)(k(G))$$ is equal to the base-point of $\pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)(k(G))$, that is, the class of the trivial torsor. Therefore, the $\mu$-torsor $G_{\rm sc} \times_{G, \eta} {{\rm Spec \,}}k(G) \to {{\rm Spec \,}}k(G)$ is trivial and thus admits a section. Hence, the morphism $\eta: {{\rm Spec \,}}k(G) \to G$ can be lifted to a morphism $\eta': {{\rm Spec \,}}k(G_{\rm sc}) \to G_{\rm sc}$. The image of this morphism must be the generic point of $G_{\rm sc}$. As a result, $\eta'$ induces a morphism ${{\rm Spec \,}}k(G) \to {{\rm Spec \,}}k(G_{\rm sc})$ which is a section of the morphism ${{\rm Spec \,}}k(G_{\rm sc}) \to {{\rm Spec \,}}k(G)$ induced by the isogeny $G_{\rm sc} \to G$. This gives us a sequence $k(G) \to k(G_{\rm sc}) \to k(G)$ of homomorphisms of fields such that the composition is the identity homomorphism on $k(G)$. Thus, we see that the homomorphism $k(G) \to k(G_{\rm sc})$ induced by the isogeny $G_{\rm sc} \to G$ is an isomorphism. We conclude that the isogeny $G_{\rm sc} \to G$ is a finite morphism of degree $1$ and hence it is an isomorphism. Thus, we see that $G$ is simply connected.
Using Theorem \[theorem algebraically closed\] and Lemma \[lemma weil restriction\], we now treat the general case.
\[theorem reductive connectedness\] Let $G$ be a reductive algebraic group over a field $k$ of characteristic $0$. Then the following are equivalent:
- $G$ is ${{\mathbb A}}^1$-connected;
- $G$ is semisimple, simply connected and the almost $k$-simple factors of $G$ are $R$-trivial.
\(1) $\Rightarrow$ (2): Let $\-k$ denote an algebraic closure of $k$. By Theorem \[theorem algebraically closed\], $G_{\-k}$ is semisimple and simply connected. Therefore, $G$ is semisimple and simply connected. Now, there exist almost $k$-simple algebraic groups $H_1, \ldots, H_r$ such that $G \simeq H_1 \times \cdots \times H_r$. Since $G$ is ${{\mathbb A}}^1$-connected, each of the $H_i$’s is ${{\mathbb A}}^1$-connected. For each $i$, there exists a finite field extension $k_i/k$ and an absolutely almost simple group $G_i$ such that $R_{k_i/k}(G_i) \simeq H_i$. Note that $$H_i(F)/R = R_{k_i/k}(G_i)(F)/R \simeq G_i(F {\otimes}_k k_i)/R$$ By Theorem \[theorem IMRN\], we have $G_i(F {\otimes}_k k_i)/R \simeq \pi_0^{{{\mathbb A}}^1}(G_i)(F {\otimes}_k k_i)$ for each $i$, since each $G_i$ is semisimple, simply connected, absolutely almost simple. Hence, for every $i$, we have $$H_i(F)/R \simeq \pi_0^{{{\mathbb A}}^1}(G_i)(F {\otimes}_k k_i) \simeq \pi_0^{{{\mathbb A}}^1}(R_{k_i/k}(G_i))(F) \simeq \pi_0^{{{\mathbb A}}^1}(H_i)(F) \simeq \ast.$$
\(2) $\Rightarrow$ (1): Since $G$ is semisimple, it is an almost direct product of almost $k$-simple groups $H_1, \ldots, H_r$. For each $i$, there exists a finite field extension $k_i/k$ and an absolutely almost simple group $G_i$ such that the Weil restriction $R_{k_i/k}(G_i)$ is isomorphic to $H_i$. Since $G$ is simply connected, it has no nontrivial isogenies, thus giving isomorphisms $$\prod_{i=1}^{r} ~ R_{k_i/k}(G_i) \xrightarrow{\simeq} \prod_{i=1}^{r} ~ H_i \xrightarrow{\simeq}G.$$ By Theorem \[theorem IMRN\], $\pi_0^{{{\mathbb A}}^1}(G_i)(F) = G_i(F)/R$. Since for every $i$, the group $H_i=R_{k_i/k}(G_i)$ is $R$-trivial by hypothesis, we have $$\pi_0^{{{\mathbb A}}^1}(R_{k_i/k}(G_i))(F) \simeq \pi_0^{{{\mathbb A}}^1}(G_i)(F {\otimes}_k k_i) \simeq G_i(F {\otimes}_k k_i)/R \simeq R_{k_i/k}(G_i)(F)/R \simeq \ast.$$ By [@Morel-connectivity Lemma 6.1.3], it follows that every $R_{k_i/k}(G_i)$ is ${{\mathbb A}}^1$-connected. Consequently, $G$ is ${{\mathbb A}}^1$-connected. This completes the proof of the theorem.
Abelian-ness of A1-connected components of certain reductive algebraic groups
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Recall from Section \[section preliminaries algebraic groups\] that ${{\mathbb A}}^1$-connected components of semisimple, simply connected groups agree with their $R$-equivalence classes (Corollary \[corollary IMRN\]). In this subsection, as an application of Theorem \[theorem reductive connectedness\], we show that the sheaf of ${{\mathbb A}}^1$-connected components of certain reductive groups is a sheaf of abelian groups.
Let $G$ be a semisimple algebraic group over a field of characteristic $0$. The simply connected cover $G_{\rm sc}$ of $G$ gives rise to a central isogeny $$G_{\rm sc} \to G,$$ whose kernel $\mu$ is a finite abelian group. By Lemma \[lemma classifying spaces\], we get an ${{\mathbb A}}^1$-fiber sequence $$G_{\rm sc} \to G \to B_{\text{fppf}}\mu$$ The associated long exact sequence of ${{\mathbb A}}^1$-homotopy groups yield the following exact sequence: $$\cdots \to \pi_0^{{{\mathbb A}}^1}(G_{\rm sc}) \to \pi_0^{{{\mathbb A}}^1}(G) \to \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu).$$ We first show that the morphism $\pi_0^{{{\mathbb A}}^1}(G) \to \pi_0^{{{\mathbb A}}^1}(B_{\text{fppf}} \mu)$ is a homomorphism of group sheaves. Given a smooth scheme $U$ over $k$, and an element $s \in G(U)$, we obtain a $\mu$-torsor $G'_s:= G'\times_{G,s} U \to U$, which defines an element of $H^1_{\text{fppf}}(U,\mu) = \pi_0(B_{\text{fppf}}\mu)(U)$. Thus, we obtain a morphism of sheaves $G \to \pi_0(B_{\text{fppf}}\mu)$. This morphism factors through the quotient morphism $G \to \pi_0^{{{\mathbb A}}^1}(G)$, inducing the morphism $\pi_0^{{{\mathbb A}}^1}(G) \to \pi_0(B_{\text{fppf}}\mu)$ mentioned above. Thus, it suffices to show that the map $G(U) \to H^1_{\text{fppf}}(U,\mu)$ is a homomorphism.
For an element $s \in G(U)$, suppose $\{U_i \to U\}_{i\in I}$ is an fppf cover which trivializes the torsor $G'_s$. Thus, for every $i$, there exists an element $s'_i \in G'(U_i)$ such that the map $G'(U) \to G(U)$ maps $s'_i$ to $s|_{U_i}$. For any two indices $i,j \in I$, we write $U_{ij}:= U_i \times_U U_j$ and define $s_{ij} = (s'_i|_{U_{ij}})(s'_j|_{U_{ij}})^{-1}$. Then, the collection $(s_{ij})_{i,j}$ is a $1$-cocycle which represents the isomorphism class $[G_s]$ of $G_s$ in $H^1_{\text{fppf}}(U,\mu)$.
Given elements $s,t \in G(U)$, we may choose an fppf cover $\{U_i \to U\}_{i \in I}$ that trivializes both $G'_s$ and $G'_t$. It is easy to see (using the fact that $\mu$ is abelian) that the collection $(s_{ij}t_{ij})_{i,j}$ is a $1$-cocycle which defines a $\mu$-torsor over $U$, which we denote by $G'_{s \ast t}$. The binary operation $([G'_s],[G'_t]) \mapsto [G'_{s \ast t}]$ is precisely the one that defines the group structure on $H^1_{\text{fppf}}(U,\mu)$. If $st$ denotes the product of $s$ and $t$ in $G(U)$, we wish to prove that $[G'_{st}] = [G'_{s \ast t}]$. In other words, we wish to prove that the torsor $G_{st}$ can be represented by the $1$-cocycle $(s_{ij}t_{ij})_{i,j}$.
For every $i \in I$, the element $s'_i t'_i \in G'(U_i)$ maps to $s|_{U_i} t|_{U_i} \in G(U_i)$. Thus, the cover $\{U_i \to U\}_{i \in I}$ trivializes the torsor $G'_{st}$. So, the class $[G'_{st}]$ is represented by the $1$-cocycle $\{u_{ij}\}_{i,j}$ where $$\begin{aligned}
u_{ij} & := (s'_i|_{U_{ij}})(t'_i|_{U_{ij}})(t'_j|_{U_{ij}})^{-1}(s'_j|_{U_{ij}})^{-1} \\
& = (s'_i|_{U_{ij}})(s'_j|_{U_{ij}})^{-1}(t'_i|_{U_{ij}})(t'_j|_{U_{ij}})^{-1} \\
& = s_{ij}t_{ij},\end{aligned}$$ where the second equality follows from the fact that $(t'_i|_{U_{ij}})^{-1}(t'_j|_{U_{ij}})^{-1}$ lies in $\mu(U_{ij})$ and hence commutes with $(s'_j|_{U_{ij}})^{-1}$.
We are now ready to show the main result of this subsection.
\[proposition pi0 abelian\] Let $G$ be a semisimple algebraic group over an infinite perfect field such that its simply connected cover $G_{\rm sc}$ is $R$-trivial. Then $\pi_0^{{{\mathbb A}}^1}(G)$ is a sheaf of abelian groups.
The simply connected cover $G_{\rm sc}$ of $G$ gives rise to an ${{\mathbb A}}^1$-fiber sequence $$G_{\rm sc} \to G \to B_{\text{fppf}}\mu$$ as described in the discussion above. The associated long exact sequence of ${{\mathbb A}}^1$-homotopy groups yield the following exact sequence: $$\cdots \to \pi_0^{{{\mathbb A}}^1}(G_{\rm sc}) \to \pi_0^{{{\mathbb A}}^1}(G) \to \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu).$$ Since $G_{\rm sc}$ is $R$-trivial, so are its almost $k$-simple components. By Theorem \[theorem reductive connectedness\], we then have $\pi_0^{{{\mathbb A}}^1}(G_{\rm sc}) = \ast$. Hence we have an injection of sheaves $\pi_0^{{{\mathbb A}}^1}(G) \hookrightarrow \pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)$. Since $\pi_0(B_{\rm fppf} \mu)(-) = H^1_{\rm fppf}(-, G)$ by [@Asok-Hoyois-Wendt-2 Lemma 2.2.2] and since $B_{\text{fppf}}\mu$ is ${{\mathbb A}}^1$-local, we conclude that $\pi_0^{{{\mathbb A}}^1}(B_{\rm fppf} \mu)$ is a sheaf of abelian groups. Since $\pi_0^{{{\mathbb A}}^1}(G) \to \pi_0^{{{\mathbb A}}^1}(B_{\text{fppf}} \mu)$ is a homomorphism of group sheaves, we conclude that $\pi_0^{{{\mathbb A}}^1}(G)$ is a sheaf of abelian groups.
\[remark pi0(G) abelian\] It is an open question whether $\pi_0^{{{\mathbb A}}^1}(G)$ is always a sheaf of abelian groups, for a reductive group $G$ over a field.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful to Marc Hoyois for a very helpful correspondence; especially, for pointing out the need to use the *fppf* topology and for suggesting that Lemma \[lemma classifying spaces\] can be proven by mimicking the proof of [@Morel-Voevodsky 4.3, Proposition 3.1]. We also thank Aravind Asok for comments and discussions and the referee for a careful reading of the paper as well as for a number of suggestions that improved the presentation. Finally, we warmly thank Fabien Morel for stimulating discussions, suggestions and encouragement. Part of this work was done when the second-named author was visiting Tata Institute of Fundamental Research, Mumbai, India; he thanks the institute for hospitality.
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abstract: 'We consider the orbital magnetic properties of non-interacting charge carriers in graphene-based nanostructures in the low-energy regime. The magnetic response of such systems results both, from bulk contributions and from confinement effects that can be particularly strong in ballistic quantum dots. First we provide a comprehensive study of the magnetic susceptibility $\chi$ of bulk graphene in a magnetic field for the different regimes arising from the relative magnitudes of the energy scales involved, [*i.e.*]{} temperature, Landau level spacing and chemical potential. We show that for finite temperature or chemical potential, $\chi$ is not divergent although the diamagnetic contribution $\chi_{0}$ from the filled valance band exhibits the well-known $-B^{-1/2}$ dependence. We further derive oscillatory modulations of $\chi$, corresponding to de Haas-van Alphen oscillations of conventional two-dimensional electron gases. These oscillations can be large in graphene, thereby compensating the diamagnetic contribution $\chi_{0}$ and yielding a net paramagnetic susceptibility for certain energy and magnetic field regimes. Second, we predict and analyze corresponding strong, confinement-induced susceptibility oscillations in graphene-based quantum dots with amplitudes distinctly exceeding the corresponding bulk susceptibility. Within a semiclassical approach we derive generic expressions for orbital magnetism of graphene quantum dots with regular classical dynamics. Graphene-specific features can be traced back to pseudospin interference along the underlying periodic orbits. We demonstrate the quality of the semiclassical approximation by comparison with quantum mechanical results for two exemplary mesoscopic systems, a graphene disk with infinite mass-type edges and a rectangular graphene structure with armchair and zigzag edges, using numerical tight-binding calculations in the latter case.'
author:
- Lisa Heße
- Klaus Richter
bibliography:
- 'mybib.bib'
nocite: '[@*]'
title: |
Orbital Magnetism of Graphene Nanostructures:\
Bulk and Confinement Effects
---
\[sec:Introduction\]Introduction
================================
Since the seminal work of Landau [@springerlink:10.1007/BF01397213] it is known that a conventional free electron gas exihbits a weak diamagnetic orbital magnetic response. In two dimensions (2d) and at low magnetic field, its magnetic susceptibility $\chi$ is just a constant, i.e. independent of Fermi energy and $B$-field. For Dirac fermions in 2d, e.g. charge cariers in graphene close to the charge neutrality point, the situation is different: As McClure showed nearly 50 years ago [@PhysRev.104.666], a non-interacting 2d system of massless Dirac fermions features a Curie-type $1/k_{\rm B}T$ behavior[@PhysRevB.75.115123] at finite temperature $T$ that merges, for vanishing temperature, into a peculiar dependence on the chemical potential[@PhysRev.104.666; @PhysRevB.20.4889; @PhysRevB.75.235333; @PhysRevB.76.113301; @PhysRevLett.102.177203; @JPSJ.80.114705; @1751-8121-44-27-275001; @PhysRevB.83.235409; @PhysRevB.80.075418] : $\chi \sim \delta(\mu)$, i.e. a magnetic response that is divergent in the undoped limit and otherwise zero.
In this work we pose the question how orbital magnetism in graphene-based nano- and mesoscale systems is altered through the presence of the confinement. Similar questions had been intensively discussed in the early nintees for small disordered metallic rings [@Webb], quasi ballistic micron-sized rings [@Mailly] and square cavities [@Levy] based on conventional 2d electron systems. The magnetic response of (ensembles of) these mesoscopic systems, namely the observed persistent current in the rings and the susceptibility of the cavities, turned out to exceed the bulk Landau diamagnetism by one to two orders of magnitude. These original experimental findings triggered broad theoretical activities (for reviews see [@Schwab; @Richter; @Ullmo3]) investigating in particular also the role of non-interacting versus interacting contributions to the orbital magnetism. While twenty years ago further progress in the field had been hindered by experimental limitations, recent new high-precision cantilever magnetization (persistent current) measurements of ensembles of rings proved [@Harris] the feasibility to reliably measure orbital magnetism of nanoscale objects. The results of these recent experiments are essentially in line with earlier theory based on non-interacting systems [@Oppen]. Given the peculiar orbital magnetic behavior of bulk graphene, and in view of the above mentioned possibility to observe confinement-enhanced magnetism in nanostructures [@Harris], it hence is of interest to explore also orbital magnetism in graphene nanostructures, a topic that has been barely addressed in the literature.
Here, we employ a trajectory-based semiclassical path integral formalism to compute the orbital magnetic susceptibility. As recently shown, such an approach is suitable for the quantitative description and interpretation of the density of states [@PhysRevB.84.075468] and conductance [@PhysRevB.84.205421] of graphene-based cavities. This approach allows for the incorporation of graphene-specific boundary effects (zigzag, armchair and infinite mass). The confinement geometry and the type of edge is then encoded in the amplitudes and phases of paths (hitting the boundaries) that enter into the respective semiclassical trace formulae. We combine this approach with an earlier semiclassical treatment of orbital magnetism in conventional ballistic electron cavities [@PhysRevLett.74.383; @Richter]. We show that the susceptibility of graphene cavities of linear system size $R$ exhibits confinement-induced oscillations in $k_{\rm F} R$ where $k_{\rm F}$ is the Fermi momentum. For integrable geometries and at low temperatures their amplitude is parametrically larger by a factor of $\sqrt{k_{\rm F}R}$ than the corresponding bulk susceptibility. However, graphene cavities additionally carry features of bulk graphene. Hence, in the first part of the paper we include a comprehensive discussion of graphene bulk orbital magnetism. While a number of previous works addressed various parameter regimes separately we aim at a systematic presentation of the various bulk regimes.
This is simplistically sketched in Fig. \[fig:overview\]. It shows an overall diamagnetic behavior up to the energy region governed by de Haas-van Alphen oscillations[@onsager; @Peierls; @PhysRevB.60.R11277; @PhysRevB.69.075104; @0953-8984-22-11-115302] for $k_{\rm B}T < \Delta_{\rm LL} <
\mu$, with $\Delta_{\rm LL}$ proportional to the Landau level spacing. However, the diamagnetic regions exhibit interesting parametrical dependences that we will derive and review. For instance, the afore mentioned divergent behavior of $\chi$ at $T\!=\!0$ is smoothed out if $k_{\rm B}T$ is bigger than the mean level spacing.
The paper is organized as follows: After summarizing the necessary thermodynamic formalism in Sec. \[sec:Basic\], we first give a comprehensive account of bulk magentism in graphene in Sec. \[sec:Main\_1\], addressing the various parameter regimes mentioned above. This also involves introducing our numerical approach and our scheme to extract bulk results from the numerics performed for finite systems. In the other main Sec. \[sec:Main\_2\] we consider in detail finite-size effects in the orbital magnetic response of nanostructued graphene. There we generalize the existing semiclassical approaches to quantitatively describe and interprete oscillatory effects in the susceptibility. These semiclassical predictions are compared to corresponding quantum calculations for disc-like and rectangluar geometries. We focus on integrable structures since chaotic or diffusive geometries are expected to exhibit a parametically weaker magnetic response.
\[sec:Basic\]Basic thermodynamic quantities
===========================================
In order to investigate the orbital magnetic properties of a quasi-two dimensional solid in general, it is convenient to start from the total grand potential in the presence of a perpendicular magnetic field of strength $B$, $$\begin{aligned}
\label{eq:i1}
\hspace{-0.3cm}\Omega(\mu, B) = -\frac{1}{\beta} \! \int\limits_{-\infty}^{\infty} \! \mathrm{d}E \,
\rho(E, B) \ln \left[ 1 + {\mathrm{e}}^{-\beta \left(E-\mu\right) } \right], \end{aligned}$$ where denotes the thermal energy. The chemical potential $\mu$ is assumed to be $B$-independent. The total density of states [@enderlein1997fundamentals] (DOS), $\rho(E, B)=\rho_{v}(E, B) + \rho_{c}(E, B)$, comprises conduction and valence band states simultaneously as well as the field-dependence of the energy spectrum of the solid. Defining $E_{v / c}$ as the energy of the band edge of the valence/conduction band, the corresponding densities of states fullfil , even for a vanishing energy gap as in the case of graphene. Without loss of generality, $\mu$ is chosen to be larger than $E_v$. Due to the properties of the total DOS, the grand potential can be decomposed as $\Omega = \Omega_{v} + \Omega_{c}$, where $$\begin{aligned}
\label{eq:i2}
\Omega_{v}(\mu, B) = - \frac{1}{\beta} \int\limits_{-\infty}^{E_{v}} \! \mathrm{d}E\,
\rho_{v}(E, B) \ln \left[ 1 + {\mathrm{e}}^{-\beta \left(E-\mu\right) } \right],\end{aligned}$$ $$\begin{aligned}
\label{eq:i3}
\Omega_{c}(\mu, B) = - \frac{1}{\beta}\int\limits_{E_{c}}^{\infty} \! \mathrm{d}E \,
\rho_{c}(E, B) \ln \left[1 + {\mathrm{e}}^{-\beta \left(E-\mu\right) } \right].\end{aligned}$$ Equation (\[eq:i3\]) contains the contribution to $\Omega$ from electrons in the conduction band for Fermi energies $\mu > E_{v}$ or thermal excitation. In the limit $T\rightarrow 0$ only states with energy $E_{c} \leq E \leq \mu$ are occupied. In view of $$\begin{aligned}
\label{eq:i4}
-\lim\limits_{\beta \rightarrow \infty}\frac{1}{\beta} \ln\left(1 + {\mathrm{e}}^{-\beta x}\right) = x \, \theta\left(-x\right),\end{aligned}$$ and taking the limit $T \! \rightarrow \! 0$ in , yields the contribution to $\Omega$ from the completely filled valence band: $$\begin{aligned}
\label{eq:i5}
\Omega_{0}(\mu, B) = \int\limits_{-\infty}^ {E_{v}} \! \mathrm{d}E \, \rho_{v}(E, B) (E - \mu).\end{aligned}$$ In general, the integral (\[eq:i5\]) can diverge, if the particular model assumes a valence band without lower boundary. As we will discuss in for bulk graphene in the low energy approximation, $\Omega_{0}$ can be decomposed into a $B$-field-dependent and a divergent part, which does not include any field-dependence and therefore has no effect on the magnetic properties. By pulling a factor $\exp{\left[-\beta(E-\mu)\right]}$ out of the logarithm in $\Omega_{v}$ can be represented as $$\begin{aligned}
\hspace{-0.5cm}\label{eq:i6}
\begin{split}
\Omega_{v}(\mu, B) =\ & \Omega_{0}(\mu, B)\\
&- \frac{1}{\beta} \! \int\limits_{-\infty}^{E_{v}} \! \mathrm{d}E \,
\rho_{v}(E, B) \ln \left[ 1 + {\mathrm{e}}^{\beta \left(E-\mu\right) } \right]\!.
\end{split}\hspace{-0.5cm}\end{aligned}$$ The second term in contains a similar contribution to $\Omega$ as $\Omega_{c}$ corresponding to electron vacancies at finite temperature. As a first conclusion, $\Omega$ can be decomposed into the $T$-independent part $\Omega_{0}$, coming from the filled part of the valence band, and a contribution $$\begin{aligned}
\hspace{-0.5cm}\label{eq:i7}
\Omega_{T}(\mu, B) =&\ \Omega(\mu, B) - \Omega_{0}(\mu, B)\\
\label{eq:i8}
\begin{split}
=& - \frac{1}{\beta} \! \int\limits_{-\infty}^{\infty} \! \! \mathrm{d}E \,
\left\{
\rho_{v}(E, B) \ln \left[ 1 + {\mathrm{e}}^{\beta \left(E-\mu\right) } \right]
\right.\\
&\left. \qquad \qquad +
\rho_{c}(E, B) \ln \left[ 1 + {\mathrm{e}}^{-\beta \left(E-\mu\right) } \right]
\right\}
\end{split}\hspace{-0.5cm}\end{aligned}$$ due to excited electrons in the conduction band and holes in the valence band. Within the relevant temperature range the integral (\[eq:i8\]) converges fast due to the exponential decay of the integrand at both integration limits.
The total magnetic susceptibility is defined as $$\begin{aligned}
\label{eq:i9}
\chi(\mu, B) =- \frac{\mu_{0}}{\mathcal{A}} \left(\frac{\partial^{2}\Omega(\mu, B)}{\partial B^2} \right)_{T,
\mu} \, .\end{aligned}$$ In view of , it can be decomposed into $$\begin{aligned}
\label{eq:i10}
\chi(\mu, B) =\ \chi_{0}(\mu, B) + \chi_{T}(\mu, B),\end{aligned}$$ with $$\begin{aligned}
\label{eq:i11}
\hspace{-0.5cm}\chi_{\mathrm{x}}(\mu, B) = - \frac{\mu_{0}}{\mathcal{A}} \left(\frac{\partial^{2}
\Omega_{\mathrm{x}}(\mu, B)}{\partial B^2} \right)_{T, \mu}\!\!\!\!,\ \
\mathrm{x} = 0,\, T.\end{aligned}$$ Here, $\mathcal{A}$ denotes the area of the system and $\mu_0$ is the vacuum permeability. As will be shown in for bulk graphene, $\chi_{0}$, which is of similar origin as the Landau susceptibility [@springerlink:10.1007/BF01397213; @Richter] of non-relativistic electron gases, represents a smooth, diamagnetic contribution $\propto 1/\sqrt{B}\ $ [@PhysRevB.69.075104; @PhysRevB.75.115123; @PhysRevB.80.075418; @PhysRevB.75.235333] to the total susceptibility. Contrarily, $\chi_{T}$ in Eq. (\[eq:i10\]) can yield an oscillatory contribution to $\chi$ for certain energy regimes. In bulk systems this oscillatory behavior refers to the de Haas-van Alphen effect [@springerlink:10.1007/BF01338364; @onsager], whereas in finite systems additional modulations in $\chi$ occur as signatures of the confinement, see .
\[sec:Main\_1\]Bulk orbital susceptibility
==========================================
\[sec:Main\_1\_dens\]Spectral properties of Landau quantized charge carriers with linear dispersion
---------------------------------------------------------------------------------------------------
In this section the orbital magnetic properties of bulk graphene in the energy range of linear dispersion are discussed. The graphene sheet is assumed to lie in the $x$-$y$-plane perpendicular to an external, homogeneous $B$-field. Then the energies of the charge carriers are Landau quantized [@springerlink:10.1007/BF01397213]. The Landau levels of massless Dirac-Weyl particles in 2D describing bulk graphene read [@JPSJ.74.777; @PhysRevB.76.081406; @10.1038/nphys653] $$\begin{aligned}
\label{eq:landau}
E_{n} = \mathrm{sgn}(n)\frac{ \sqrt{2}\hbar v_{F}}{l_{B}}\sqrt{\left|n\right|} \, ,\end{aligned}$$ with . Here, denotes the magnetic length with the magnetic flux quantum . Every Landau level $E_{n}$ has a twofold spin degeneracy $g_{s}$ and valley degeneracy $g_{v}$ as well as a -fold degeneracy which can be, e.g., deduced from phase space arguments [@onsager; @kittel] and Bohr-Sommerfeld quantization [@messiah1991quantenmechanik] of the corresponding cyclotron orbits. Thus the orbital degeneracy in graphene is identical to that of Landau levels of ordinary 2D electron gases[@springerlink:10.1007/BF01397213], , with effective mass $m^{*}$ and . In this case the lowest Landau level has the finite value while for graphene attains zero and lies precisely at the touching point of conduction and valence band. In the presence of a magnetic field conduction and valence band states occupy the zeroth Landau level equally leading to an increase of the total energy of the filled valence band. Thus the contribution $\chi_0$ from the filled valence band is expected to be diamagnetic as discussed in detail in . Whether the total susceptibility $\chi$, , is para- or diamagnetic depends on the contribution $\chi_{T}$ of excited electrons and holes in the particular energy regime.
The single-particle DOS of bulk graphene, $$\begin{aligned}
\label{eq:m1_12}
\rho(E, B) = g\,\varphi \sum\limits_{n=-\infty}^{\infty} \delta\left(E - E_{n}(B)\right),\end{aligned}$$ can be decomposed into a smooth and an oscillatory part with respect to $E$ and $B$. By means of Poisson summation[@brack_sc] of the Landau index $n$ one obtains $$\begin{aligned}
\label{eq:m1_13}
\hspace{-0.5cm}\rho(E, B) &= C \left|E\right| \left[
1 + 2 \sum\limits_{m=1}^{\infty}\cos\left(\pi m \left(\frac{E\, l_{B}}{\hbar v_{F}}\right)^{2}\right)\right]\\
\label{eq:m1_14}
&= \bar{\rho}(E) + \rho^{\mathrm{osc}}(E, B)\end{aligned}$$ with and . Note that each term in and thereby the total DOS reflects particle-hole symmetry, i.e. , due to the nearest neighbor hopping approximation underlying the effective Dirac hamiltonian. The smooth part is $B$-independent and identical to the bulk DOS of the field free system [@RevModPhys.81.109]. Hence the entire contribution to $\chi$ arises from the oscillatory part $\rho^{\mathrm{osc}}(E, B)$ that can be rewritten as $$\begin{aligned}
\label{eq:m1_15}
\begin{split}
\hspace{-0.5cm} \rho^{\mathrm{osc}}(E, B) = g\,\varphi \sum_{n = 1}^{\infty}\left[\delta\left(E - E_{n}(B)
\right) + \delta\left(E + E_{n}(B)\right)\right]\\
\hspace{-1cm} + g\,\varphi\,\delta\left(E\right) - C \left|E\right|.
\end{split}\end{aligned}$$ This represention clearly indicates that the orbital magnetism arises only from Landau levels with $n \neq 0$. The zeroth Landau level leads to a $\varphi$-linear contribution to $\Omega$ and thus does not contribute to $\chi$. As for the DOS the related thermodynamic potentials can be decomposed into $$\begin{aligned}
\label{eq:m1_16}
\Omega(\mu, B)
= \bar{\Omega}(\mu) + \tilde{\Omega}(\mu, B).
$$ Each term in can be further split as shown in , i.e. , where . Note that $\bar{\Omega}$ arises directly from the field independent bulk DOS $\bar{\rho}(E)$, and hence . We will show below that though $\tilde{\Omega}$ arises from the oscillatory part of the DOS, it yields not only an oscillatory but also a smooth contribution to the susceptibility.
\[sec:Main\_1\_comp\]Comparability of numerical results with analytical bulk DOS calculations
---------------------------------------------------------------------------------------------
The comparison between the analytical results for bulk graphene, to be discussed in and \[sec:Main\_1\_therm\], with the numerical tight-binding data of confined graphene quantum dots will demonstrate the importance of bulk effects in finite structures. Moreover, vice versa, we will employ the numerical calculations, restricted to finite gemetries, to confirm the results from the effective bulk theory based on the Dirac equation. For such a comparison we need to extract the bulk contribution from the numerical results in an appropriate way as discussed below.
The finite systems considered have an equilateral triangular geometry with either pure armchair or zigzag boundaries. This particular choice of geometry enables also a distinct analysis of edge effects due to zigzag boundaries. Each system has mesoscopic dimensions, i.e. the triangle side lengths are , where $a$ is the graphene lattice constant, such that the region of linear dispersion contains enough energy levels to require good comparability with the theory. The eigenenergies of the triangles are calculated within tight-binding approximation[@epub12142; @Wimmer] including only nearest neighbor hopping $t$ and using the Lanczos algorithm [@arpack1; @ARPACK]. Figure \[fig:spectrum\_qd\] shows the resulting energy spectrum for conduction and valence band energies as a function of the normalized magnetic flux . One can clearly see the condensation of the eigenenergies into Landau levels[@PhysRevB.81.245411; @PhysRevB.84.245403]for fluxes $\phi > 5\,\phi_{0}$. This is the regime where bulk effects should be distinctly observable in the finite systems.
In we calculate the contribution $\chi_{0}$ from the valence band in Dirac approximation, which corresponds to the Landau susceptibility\cite{} of electron gases. Therefore we assume an unbounded valence band with linear dispersion which does not reflect the real band structure of graphene further away from the Dirac point. For this reason, $\chi_{0}$ is not accessible within tight-binding approximation even if one would go to very large system sizes. Thus the comparison of the analytic theory with numerical data for the quantum dots is restricted to the temperature dependent part of $\tilde{\Omega}$ and $\chi$, respectively, where only the energy levels close to the Fermi level contribute.
As one can deduce from the mean level spacings of the finite systems differ from the mean Landau level spacing of the bulk system. In Fig. \[fig:mean\_level\_spacing\]a) we compare explicitly (), calculated from the first $600$ electronic states for each case, as a function of the normalized magnetic flux. When dealing with thermodynamic potentials and related observables, finite temperature $T$, encoded in the Fermi-Dirac statistics, implies an effective broadening $1/\beta$ of each energy level as can be seen from Eq. (\[eq:i8\]). For an appropriate comparison of the Dirac-type bulk theory with the tight-binding results for the finite-size structures the thermal energies chosen should obey $$\begin{aligned}
\label{eq:m1_17}
\hspace{-0.1cm}\beta^{(\mathrm{bulk})}\Delta\bar{E}^{(\mathrm{bulk})}(\phi) \approx \beta^{(\mathrm{x})}\Delta\bar{E}^{(\mathrm{x})}(\phi), \ \ \mathrm{x} = \mathrm{ac}, \mathrm{zz}.\end{aligned}$$ To get reliable values from this expression the edge states are not considered in the case of the zigzag system since they lead to underestimating the mean level spacing.
To compare the properties of the bulk system with those of the quantum dots for finite magnetic flux it is necessary to average Eq. (\[eq:m1\_17\]) over the flux interval considered. The resulting level spacings averaged over $\phi \in \left[0, 30\,\phi_{0}\right]$ can be read off from . The above procedure, providing an adequate comparison between all three systems, refers to the entire thermodynamic potentials and related properties. Independently, the individual energy levels for each of the considered systems can be written as $E^{(\mathrm{x})}_{i} = \bar{E}^{(\mathrm{x})} + \delta E^{(\mathrm{x})}_{i}$, where denotes the mean energy of the $N$ valence band states considered. shows the flux dependence of $\bar{E}^{(\mathrm{x})}$ for each system averaged over the lowest $N=600$ electron states. In all three cases the mean energies are of the same order of magnitude. Due to the contribution of edge states, .
The grand potential for each system reads $$\begin{aligned}
\label{eq:m1_18}
\hspace{-0.2cm}\Omega^{(\mathrm{x})}(\mu, B) = - \frac{1}{\beta} \sum\limits_{i} \ln\!\left[1 + {\mathrm{e}}^{-\beta\left(\bar{E}^{(\mathrm{x})} + \delta E^{(\mathrm{x})}_{i} - \mu\right)}\right]\!\!.\end{aligned}$$ The properties of the exponential function and the logarithm yield a rough scaling behavior of for each system reflecting in first approximation $$\begin{aligned}
\label{eq:m1_19}
\beta^{\mathrm{ac, zz}}\bar{E}^{(\mathrm{ac, zz})} -\beta^{\mathrm{bulk}}\bar{E}^{(\mathrm{bulk})} \gtrless 0.\end{aligned}$$ Resulting differences in the absolute value of $\Omega$ and $\chi$, respectively, for fixed $\mu$ and $\varphi$ can be approximately compensated by rescaling the bulk value with the factor $\gamma^{(\mathrm{ac}, \mathrm{zz})}$. The factors were obtained by fitting using the Levenberg-Marquardt [@springerlink:10.1007/BFb0067700] algorithm.
As a consequence of , , and we expect the susceptibility contribution $\chi_{T}$ for a zigzag triangular quantum dot to be smaller than for the corresponding armchair system at same temperature corresponding to . This behavior is also confirmed in , where the orbital magnetic properties of hexagonal and triangular graphene nanostructures are numerically studied within tight-binding approximation.
------------------------------------------------------------------------------- ------- ------- ---------------
$\langle\Delta \bar{E}^{(\mathrm{x})}\rangle_{\phi}\ \left[10^{-3}\,t\right]$ 3.514 1.828 2.124 (1.780)
0.349 0.368 0.306
------------------------------------------------------------------------------- ------- ------- ---------------
: \[tab:table1\] Flux average of the mean energy and mean level spacing for the first $600$ electron states of bulk graphene, an armchair and a zigzag triangular quantum dot, same as Fig. \[fig:spectrum\_qd\]. The considered flux interval amounts to . The number in parenthesis comprises the edge states.
\[sec:Main\_1\_cond\]Susceptibility contribution from filled valence band
-------------------------------------------------------------------------
As discussed in , the susceptibility contribution $\chi_{T}$ from the filled valence band, , can be evaluated from by using only the field-dependent part of the DOS, $\rho^{\mathrm{osc}}(E, B)$: $$\begin{aligned}
\hspace{-0.5cm}\label{eq:m1_20}
\tilde{\Omega}_{0}(\mu, B) =& \int\limits_{-\infty}^{0}\mathrm{d}E \, \rho^{\mathrm{osc}}(E, B) (E-\mu)\\
\label{eq:m1_21}
\begin{split}
=& -2C\sum\limits_{m=1}^{\infty}\mathrm{Re}\Bigg[
\lim\limits_{\eta \rightarrow 0}
\int\limits_{0}^{\infty}\mathrm{d}E \left(E^{2} + \mu E\right)\\
&\times{\mathrm{e}}^{-\left[\eta - {\mathrm{i}}\pi m \left(\frac{l_{B}}{\hbar v_{F}}\right)^{2}\right]\,E^{2}}\Bigg].
\end{split}\hspace{-0.5cm}\end{aligned}$$ Solving this integral and taking the limit yields $$\begin{aligned}
\label{eq:m1_22}
\tilde{\Omega}_{0}(B) = \frac{K}{2}\varphi^{3/2} \sum\limits_{m=1}^{\infty}\frac{1}{m^{3/2}}
= \frac{K}{2}\varphi^{3/2} \zeta\left(\frac{3}{2}\right)\!,\end{aligned}$$ where all prefactors are absorbed in the constant $$\begin{aligned}
\label{eq:m1_23}
K = 4 \sqrt{\pi}\, C \left(\frac{\hbar v_{F}}{\sqrt{\mathcal{A}}}\right)^{3}
= 2 g\frac{\hbar v_{F}}{\sqrt{\mathcal{A} \pi}}.\end{aligned}$$ Indeed, $\Omega_{0}$ and thereby the corresponding susceptibility $$\begin{aligned}
\label{eq:m1_24}
\hspace{-0.5cm}
\chi_{0}(B) = - \frac{\mu_{0}g}{\phi_{0}^{2}}\hbar v_{F} \sqrt{\frac{\mathcal{A}}{\pi}}\frac{3\zeta\left(\frac{3}{2}\right)}{4} \frac{1}{\sqrt{\varphi}}
\propto -\frac{1}{\sqrt{B}}\end{aligned}$$ are independent of the chemical potential. $\chi_0(B)$ is diamagnetic because the grand potential of the valence band, , increases in the presence of a perpendicular magnetic field, i.e. . The susceptibility $\chi_{0}$ diverges as $1/\sqrt{B}$ implying that small variations of the flux cause huge changes in the magnetization of bulk graphene in the low-field regime. The scaling behavior (\[eq:m1\_24\]) of $\chi_{0}$ was first discovered by McClure in 1956 within his studies of the diamagnetic properties of graphite[@PhysRev.104.666] and confirmed by various research groups[@PhysRevB.69.075104; @PhysRevB.75.115123; @PhysRevB.80.075418; @PhysRevB.75.235333; @PhysRevB.86.125440] for monolayer graphene. In we show that this singularity of $\chi_{0}$, however, need not lead to a divergence of the total susceptibility .\
In the case of a bulk 2DEG the quantity corresponding to $\chi_{0}$ is the Landau susceptibility [@Richter] $$\begin{aligned}
\label{eq:m1_25}
\chi_{L} = - \mu_{0} g_{s} \frac{\pi}{6}\frac{\hbar^{2}}{\phi_{0}^{2} m^{*}} \, .\end{aligned}$$ It is also independent of $\mu$ but moreover does not depend on $B$. To estimate the relative strength of graphene diamagnetism we consider the ratio $\chi_{0}/\chi_{L}$ which reads $$\begin{aligned}
\label{eq:m1_26}
\frac{\chi_{0}(B)}{\chi_{L}} \approx 0.2\; \frac{m^{*}}{m_{\mathrm{e}^{-}}} \sqrt{ \frac{\mathcal{A}}{\varphi}}\;\mathrm{nm}^{-1}.\end{aligned}$$ To give an explicit example, consider GaAs and a graphene flake with a typical length , such that bulk effects dominate over finite-size signatures in $\chi$. Choosing typical values and , Eq. (\[eq:m1\_26\]) yields , i.e. the diamagnetic contribution from the valence band in graphene is comparable to the Landau susceptibility of a 2DEG for a magnetic field of $B \approx 0.5\,\mathrm{T}$.
\[sec:Main\_1\_therm\]Susceptibility contribution from thermally excited charge carriers
----------------------------------------------------------------------------------------
To investigate the contribution to $\chi$ from excited electrons and holes we start from considering only the field-dependent part $\rho^{\mathrm{osc}}(E, B)$ of the DOS in : $$\begin{aligned}
\label{eq:m1_27}
&&\tilde{\Omega}_{T}(\mu, B) = -\frac{1}{\beta}\int\limits_{-\infty}^{\infty}\mathrm{d}E \, \rho^{\mathrm{osc}}(E, B)\\
&&\times \left\{\theta(-E)\ln\left[1+{\mathrm{e}}^{\beta\left(E-\mu\right)}\right]
+ \theta(E)\ln\left[1+{\mathrm{e}}^{-\beta\left(E-\mu\right)}\right]\right\}.\nonumber\end{aligned}$$ Due to the integration over energy and the temperature dependence of $\tilde{\Omega}_{T}$ the corresponding susceptibility $\chi_{T}$ can contain a smooth as well as an oscillatory part, which is directly accessible within the semiclassical description of finite-size contributions to $\chi$ as shown in Sec. \[sec:Main\_2\].
For the following considerations it is useful to integrate Eq. (\[eq:m1\_27\]) twice by parts yielding $$\begin{aligned}
\label{eq:m1_28}
\tilde{\Omega}_{T}(\mu, B) = \int\limits_{-\infty}^{\infty}\mathrm{d}E \, \mathcal{N}(E, B) f'\left(E-\mu\right),\end{aligned}$$ with the integral over particle number fluctuations, $$\begin{aligned}
\label{eq:m1_29}
\mathcal{N}(E, B) =& \int\limits_{0}^{E}\mathrm{d}E'\int\limits_{0}^{E'}\mathrm{d}E'' \rho^{\mathrm{osc}}(E, B)\\
\label{eq:m1_30}
=& K\varphi^{3/2}\sum\limits_{m=1}^{\infty}
\frac{\mathrm{S}\left(\sqrt{\pi m} \frac{\left|E\right|\,l_{B}}{\hbar
v_{F}}\right)}{m^{3/2}} \, .\end{aligned}$$ Here, is the Fresnel integral[@gradshtein] and $K$ is defined by . In $$\begin{aligned}
\label{eq:m1_31}
f'\left(x\right) =
-\frac{\beta}{4} \mathrm{sech}^{2}\left(\frac{\beta}{2}x\right) \xrightarrow{\beta \rightarrow \infty} -\delta(x)\end{aligned}$$ denotes the derivative of the Fermi distribution function . We rewrite as $$\begin{aligned}
\label{eq:m1_32}
\tilde{\Omega}_{T}(\mu, B) = K\varphi^{3/2}\sum\limits_{m = 1}^{\infty} \frac{\omega_{m, T}(\mu, B)}{m^{3/2}},\end{aligned}$$ where $\omega_{m, T}$ is defined as the energy integral $$\begin{aligned}
\label{eq:m1_33}
\hspace{-0.5cm}\omega_{m, T}(\mu, B) = \!\!\!\int\limits_{-\infty}^{\infty}\!\!\!\mathrm{d}E\,\mathrm{S}\!\left(\!\sqrt{\pi m}\frac{\left|E\right|\,l_{B}}{\hbar v_{F}}\!\right)f'\left(E - \mu\right)\!.\end{aligned}$$ Since the integral (\[eq:m1\_33\]) cannot generally be solved analytically, it is convenient to discuss separately different regimes defined through the ratios between the relevant length scales entering the problem, namely the magnetic length $l_{B}$, the Fermi wavelength , and the thermal wavelength $\lambda_{T} = \hbar v_{F}\,\beta$. For the sake of simplicity we define the two dimensionless parameters $$\begin{aligned}
\label{eq:m1_34}
\alpha = \frac{\lambda_{F}}{l_{B}} \propto \frac{\Delta_{LL}}{\mu},\quad \gamma= \frac{\lambda_{T}}{l_{B}} \propto \frac{\Delta_{LL}}{k_{B}T},\end{aligned}$$ where denotes the energy spacing between adjacent Landau levels.
### \[sec:Main\_1\_therm\_1\]Regime: $\gamma > 1 > \alpha$
In this parameter range the Landau level spacing is larger than or comparable to the thermal energy, but smaller than the chemical potential. The resulting temperature dependent contribution to $\Omega$ is therefore expected to show an oscillatory modulation as a function of $\mu$ or $\varphi$ known as de Haas-van Alphen effect in electron gases [@onsager; @Peierls; @PhysRevB.60.R11277]. Moreover we will show that the $1/\sqrt{B}$ singularity in Eq. (\[eq:m1\_24\]) is cancelled. Hence it is useful to decompose the Fresnel integral in Eq. (\[eq:m1\_33\]) into its smooth and oscillatory part, i.e. . The function $\tilde{\mathrm{S}}(x)$ oscillates around zero and can be written in terms of the hypergeometric function or its integral representation as shown in , $$\begin{aligned}
\label{eq:m1_36}
\tilde{\mathrm{S}}(x)=& -\frac{1}{\sqrt{2\pi} }\mathrm{Im} \left[\int\limits_{0}^{\infty}\!\mathrm{d}u \frac{{\mathrm{e}}^{-\left(u - {\mathrm{i}}\right)x^{2}}}{\sqrt{\pi\,u}\left(u - {\mathrm{i}}\right)}\right].\end{aligned}$$ Then the energy integral (\[eq:m1\_33\]) reads $$\begin{aligned}
\label{eq:m1_37}
\omega_{m, T} = -\frac{1}{2} + \!\!\int\limits_{-\infty}^{\infty}\!\!\mathrm{d}E\,\tilde{\mathrm{S}}\!\left(\!\sqrt{\pi m}\frac{\left|E\right|\,l_{B}}{\hbar v_{F}}\!\right)f'\left(E - \mu\right).\end{aligned}$$ Note that the remaining integral directly leads to the $B$ field-dependent part of the total grand potential, , since the first term in exactly cancels with $\tilde{\Omega}_{0}$, Eq. (\[eq:m1\_27\]), after inserting it into . Then $\tilde{\Omega}$ can be cast into the form $$\begin{aligned}
\label{eq:m1_38}
\hspace{-0.125cm}\tilde{\Omega}(\mu, B) =\sum\limits_{m=1}^{\infty}\!\frac{K\varphi^{3/2}}{\sqrt{2\pi\,m^{3}}}
\mathrm{Im}\!\left[\int\limits_{0}^{\infty}\mathrm{d}u\frac{\mathrm{Y}_{T}(\mu, B, u)}{\sqrt{\pi\,u}(u - {\mathrm{i}})}\right]\!\!,\end{aligned}$$ with $$\begin{aligned}
\label{eq:m1_39}
\mathrm{Y}_{T}(\mu, B, u) =\!\!\!\int\limits_{-\infty}^{\infty}\!\!\mathrm{d}E\,f'(E - \mu) \,
{\mathrm{e}}^{-\left(u - {\mathrm{i}}\right)\pi m\left(\frac{E\,l_{B}}{\hbar v_{F}}\right)^{2}} .\end{aligned}$$ As shown in App. A of for a similar situation, $$\begin{aligned}
\label{eq:m1_41}
\mathrm{Y}_{T}(\mu, B, u) \approx \mathrm{Y}_{0}(\mu, B, u)\,\mathrm{R}_{T}\left[\mathrm{\phi}'(\mu, B, u)\right],\end{aligned}$$ where the temperature damping factor $\mathrm{R}_{T}$ is defined as $$\begin{aligned}
\label{eq:m1_42}
\mathrm{R}_{T}\left[\mathrm{\phi}'(\mu, B, u)\right] = \frac{\frac{\pi}{\beta}\mathrm{\phi}'(\mu, B, u)}{
\sinh\left[\frac{\pi}{\beta}\mathrm{\phi}'(\mu, B, u)\right]} \xrightarrow{\beta \rightarrow \infty} 1\end{aligned}$$ and results from the derivative of the Fermi-Dirac distribution and . From Eqs. (\[eq:m1\_41\], \[eq:m1\_42\]) follows $$\begin{aligned}
\label{eq:m1_43}
\mathrm{Y}_{T}(\mu, B, u) \approx {\mathrm{e}}^{-\left(u - {\mathrm{i}}\right)\pi m/\alpha^{2}} \mathrm{R}_{T}\left(\beta\frac{2\pi m}{\alpha \gamma}\right),\end{aligned}$$ so the field-dependent part of $\Omega$ finally reads $$\begin{aligned}
\label{eq:m1_44}
\tilde{\Omega}(\mu, B)
&\approx& \!\!\!\!K\sum\limits_{m=1}^{\infty}\frac{\varphi^{3/2}}{m^{3/2}} \tilde{\mathrm{S}}\left(\frac{\sqrt{\pi\,m}}{\alpha}\right) \mathrm{R}_{T}\left(\beta\frac{2\pi m}{\alpha \gamma}\right)\!.\end{aligned}$$ Compared to the rapid magneto oscillations of $\tilde{\mathrm{S}}$, the factor $\mathrm{R}_{T}$ only slowly varies on the relevant scales so that its magnetic field derivatives can be neglected in the calculation of the total magnetic susceptibility: $$\begin{aligned}
\label{eq:m1_45}
\chi(\mu, B) =& -\frac{\mu_{0} g}{\phi_{0}^{2}} \hbar v_{F}\frac{3\sqrt{\mathcal{A}}}{2\pi} \sum\limits_{m = 1}^{\infty}
\frac{\mathrm{R}_{T}\left(\beta\frac{2\pi m}{\alpha\gamma}\right)}{m^{3/2}}\frac{\mathrm{J}\left(\frac{\sqrt{\pi m}}{\alpha}\right)}{\sqrt{\varphi}}\\
\label{eq:m1_46}
=&\chi_{0}(B)\times\frac{2}{\sqrt{\pi}\zeta\left(\frac{3}{2}\right)}\sum\limits_{m = 1}^{\infty}
\frac{\mathrm{R}_{T}\left(\beta\frac{2\pi m}{\alpha\gamma}\right)}{m^{3/2}}\mathrm{J}\left(\frac{\sqrt{\pi m}}{\alpha}\right),\end{aligned}$$ with $\chi_{0}(B)$ defined in . At finite temperatures the sum in is exponentially damped due to $\mathrm{R}_{T}$, ensuring convergence of the corresponding expression. The function $\mathrm{J}(x)$ is defined as $$\begin{aligned}
\label{eq:m1_47}
\mathrm{J}(x) = \tilde{\mathrm{S}}\left(x\right) + \sqrt{\frac{2}{\pi}} x \left[\sin\left(x^2\right)- \frac{2 x^2}{3}\cos\left(x^2\right)\right],\end{aligned}$$ yielding $\mu^{2}$- as well as $1/\phi$-periodic oscillations of $\chi$, respectively , which can be extracted from and . This becomes more obvious by transforming the expression (\[eq:m1\_47\]) for $\mathrm{J}(x)$ into $$\begin{aligned}
\label{eq:m1_48}
\mathrm{J}(x) \!=\! -\!\frac{\cos\!\left(x^{2}\right)}{\sqrt{2\pi}}\left[\mathrm{\Sigma}_{1}\!\left(x^{2}\right)
\! +\! \frac{4}{3} x^{3}\right]
\!-\!\frac{\sin\!\left(x^{2}\right)}{\sqrt{2\pi}}\!\left[\mathrm{\Sigma}_{2}\left(x^{2}\right) \!-\! 2 x\right].
$$ by defining and rewriting $\tilde{\mathrm{S}}(x)$, . For the magnetization of bulk graphene an expression similar to is derived in considering additionally a band gap and impurity scattering, whereas in the effect of an additional in-plane electric field is studied.
In the oscillatory behavior of $\chi_{T}$ is demonstrated. In panel a) $\chi_{T}$ exhibits equidistant extrema when plottet as a function of $\mu^{2}$ at . Panel b) shows the $1/\phi$-periodicity of $\chi_{T}$ at . In both cases the thermal energy is chosen such that . The amplitude of the $\chi_{T}$ oscillations is about one order of magnitude larger than $|\chi_{0}|$, implying that the full orbital susceptibility $\chi$ of graphene oscillates between strong diamagnetic but also paramagnetic behavior as a function of $\mu$ and $B$, respectively.
In we show the numerically calculated susceptibility contribution $\chi_{T}$ for a triangular armchair and zigzag quantum dot for the same value of the magnetic flux as in , i.e. $\phi = 15\,\phi_{0}$. The thermal energies are chosen as to satisfy . The levels of the finite systems are then well resolved leading to extra peaks with smaller amplitude inbetween those caused by level clustering in the vicinity of Landau levels (see ). The latter are indicated by red arrows in and coincide with the maxima in $\chi_{T}$ of the bulk system. These extra peaks are signatures of the confinement of the system and not captured within the bulk theory. Similar signatures are numerically observed in for triangular but also hexagonal graphene quantum dots. In we will show how one can interprete these finite-size signatures within a semiclassical approach using periodic orbit theory. The amplitudes of the susceptibility oscillations of the quantum dots exceed the contribution $\chi_{0}$ from the filled valence band as well, implying that for certain ranges of $\phi$ and $\mu$ the total orbital magnetic susceptibility can become paramagnetic. By raising the thermal energy to the finite-size features are smeared out and only extrema at the positions of the Landau levels survive as demonstrates.
compares the susceptibility contribution $\chi_{T}$ of the triangular quantum dots with the bulk system as a function of $\phi$ at and , respectively , such that finize-size effects are smeared out. For flux values the peak positions coincide very well. This corresponds to the spectral regime of the finite systems (see ) where the levels cluster in the vicinity of Landau levels and the influence of the boundaries becomes negligible.
### \[sec:Main\_1\_therm\_2\]Regime: $\alpha, \gamma > 1$
When the thermal energy and chemical potential are comparable to or smaller than the Landau level spacing the temperature dependent part of the susceptibility is expected to vanish[@PhysRevB.75.115123]. If the magnetic field is tuned to very high values such that the degeneracy of each Landau level rises accordingly, and all occupied states condense into the first or even to the zeroth level $E_{0}$. This yields a contribution to the temperature dependent part of the total grand potential $\Omega_{T}$ linear in $B$ as mentioned in . In order to calculate $\chi_{T}$ from it is useful to apply the representation (\[eq:m1\_15\]) for $\rho^{\mathrm{osc}}$. Then it is sufficient to consider only the sum over the Landau indices since the other terms do not contribute to $\chi$. To this end we write $$\begin{aligned}
\label{eq:m1_49}
\tilde{\Omega}_{T}(\mu, B) =\hat{\Omega}_{T}(\mu, B)- g\frac{\varphi}{\beta}\sum\limits_{s = \pm 1}\sum\limits_{n = 1}^{\infty}
\ln\left(1 + {\mathrm{e}}^{-\sqrt{2 n}\gamma + s\frac{\gamma}{\alpha}}\right)\end{aligned}$$ where the $B$-linear term $$\begin{aligned}
\label{eq:m1_50}
\hat{\Omega}_{T}(\mu, B) =
- \frac{g}{2}\frac{\varphi}{\beta}\sum\limits_{s = \pm 1}
\ln\left(1 + {\mathrm{e}}^{s\frac{\gamma}{\alpha}}\right) - \bar{\Omega}_{T}(\mu)\end{aligned}$$ does not contribute to $\chi_{T}$. $\bar{\Omega}_{T}$, defined through is only based on the average DOS . In order to get an appropriate expression for $\tilde{\Omega}_{T}$ in this parameter range we Taylor expand the logarithm and the exponential function in using the condition $\gamma \! > \! 1$. Resumming the resulting triple infinite sums yields $$\begin{aligned}
\label{eq:m1_51}
\begin{split}
\tilde{\Omega}_{T}(\mu, B) \approx &\, \hat{\Omega}_{T}(\mu, B) - g \frac{\varphi}{\beta}
\sum\limits_{s \pm 1}\mathrm{ln}\left(1+{\mathrm{e}}^{-\sqrt{2}\gamma + s\frac{\gamma}{\alpha}}\right),
\end{split}\end{aligned}$$ as shown in . Only the second term contributes to $\chi$. It is identical to the contribution from the first electron- and hole-like Landau level to $\tilde{\Omega}_{T}$ as a comparison with shows. The susceptibility contribution from then yields $$\begin{aligned}
\label{eq:m1_52}
\chi_{T}(\mu, B) = & - \frac{1}{8} \sqrt{\frac{\pi}{2}} \frac{\mu_{0} g}{\phi_{0}^{2}}\hbar v_{F}
\sqrt{\frac{\mathcal{A}}{\varphi}}\times \mathrm{F}\left(\alpha, \gamma\right)\\
\label{eq:m1_53}
= &\, \chi_{0}(B)\times\frac{\pi}{6\sqrt{2}\zeta\left(\frac{3}{2}\right)}\times\mathrm{F}\left(\alpha, \gamma\right).\end{aligned}$$ Here $$\begin{aligned}
\label{eq:m1_54}
\begin{split}
\mathrm{F}\left(\alpha, \gamma\right) = \sum\limits_{s = \pm 1}
\left[3\left(1+{\mathrm{e}}^{-\sqrt{2}\gamma + s\frac{\gamma}{\alpha}}\right) - \sqrt{2}\gamma\right]\\
\times\mathrm{sech}^{2}\left[\frac{1}{2}\left(\sqrt{2}\gamma - s\frac{\gamma}{\alpha}\right)\right]
\end{split}
$$ can assume positive or negative values hence yielding a dia- or paramagnetic susceptibility contribution. For , i.e. the level spacing is comparable to the thermal energy, $\mathrm{F}\left(\alpha, \gamma\right)$ takes positive values and hence $\chi_{T}$ is diamagnetic. In the same parameter regime is discussed for the special case $\mu = 0$ but treated in a slightly different way obtaining a diamagnetic result for $\chi_{T}$ which decays as a function of $\gamma$. In the range of validity of $\left|\chi_{T}\right|$ is at most half as large as $|\chi_{0}|$ as the following considerations show: In its validity range, $\mathrm{F}$ approaches a supremum . Together with the additional prefactors in this yields .
In the flux dependence of the bulk result, , is compared with the numerically calculated contribution from the conduction and valence band to $\chi$ of an (a) armchair and (b) zigzag triangular quantum dot at . The thermal energies of the bulk systems are chosen such that . By choosing lower thermal energies finite size effects gain importance and deviations from the bulk theory emerge as can be seen from : The susceptibilities $\chi_{T}$ of the quantum dots exhibit oscillatory behavior which becomes all the more pronounced, as the thermal energies tend to lower values. In this case all parameters are chosen as in but the thermal energy of the quantum dots is one order of magnitude smaller, i.e. . For these parameters, the function $\mathrm{F}(\alpha, \gamma)$, , reaches positive values only in the considered flux range. Therefore $\chi_{T}$, , is diamagnetic. This holds also true for the numerically calculated contribution $\chi_{T}$ of the triangular quantum dots. From the definition (\[eq:m1\_34\]) of $\alpha \propto \Delta_{LL}/\mu$ one expects the bulk effects to dominate over finite-size signatures and therefore good agreement of the numerical data with the bulk calculations for $\phi \gtrsim 15\ \phi_{0}$. This is confirmed by and \[fig:norm\_regime\_b\_a\_g\_1\_II\].
For lower values of $\phi$ is no longer valid yielding deviations from the tight-binding calculations as the oscillatory modulations of $\chi_{T}$ demonstrate in . These oscillations are smeared out due to the larger thermal energies chosen in . In both figures $\chi_{T}$ of the quantum dots reaches zero for $\varphi \approx 0$ and $\chi_{T}$ is morevoer suppressed on a finite flux interval, $\phi \lesssim 3\,\phi_{0}$ in and $\phi \lesssim 7\,\phi_{0}$ in b), respectively. This behavior can be understood in view of the energy spectra of the quantum dots, . In each case there is a small gap between and the first non-zero energy level as a signature of confinement. For thermal energies smaller than this gap and there are no occupied states above the Dirac point besides the edge states of the zigzag quantum dot contributing $\varphi$-linear to $\Omega_{T}$ and yielding $\chi_{T} = 0$. In the case of the armchair quantum dot $\Omega_{T}$ and therefore $\chi_{T}$ vanish completely in this specific parameter range.
### \[sec:Main\_1\_therm\_3\]Regime: $\gamma < 1$ and arbitrary $\alpha$
If the thermal energy of the system is larger than the level spacing, also states above the Fermi level are occupied implying that tuning the chemical potential or the magnetic field does not lead to a discontinuity of the corresponding contribution to the grand potential. As a consequence the susceptibility is expected to be a smooth function of these parameters. In this parameter range the magnetic flux and the thermal energy can be chosen in such a way that the Landau level clustering in the quantum dot spectra is pronounced enough to make the bulk theory valid, on the one hand, and effectively wash out the finite size signatures on the other hand. Hence one can expect good agreement of the bulk theory with the susceptibility of the quantum dots.
Using again the decomposition of the Fresnel integral into smooth and oscillatory part, one can start from representation of the field-dependent part of $\Omega$. Substituting gives $$\begin{aligned}
\hspace{-0.5cm}\label{eq:m1_55}
\begin{split}
\tilde{\Omega} =& \frac{K}{2 \sqrt{2}\pi}\varphi^{3/2}\sum\limits_{m=1}^{\infty} \frac{1}{\sqrt{m}^{3}}
\mathrm{Im}\!\left[\int\limits_{0}^{\infty}\mathrm{d}u \frac{1}{\sqrt{u}(u - {\mathrm{i}})}\right.\\
&\times\left.\int\limits_{-\infty}^{\infty}\mathrm{d}x\, \mathrm{sech}^{2}\left(x\right)
{\mathrm{e}}^{-u \pi m \left(2 \frac{x}{\gamma} + \frac{1}{\alpha}\right)^{2}}{\mathrm{e}}^{ {\mathrm{i}}\pi m \left(2 \frac{x}{\gamma} + \frac{1}{\alpha}\right)^{2}}\right].\hspace{-0.5cm}
\end{split}\end{aligned}$$ For $\gamma < 1$ the complex phase rapidly oscillates as a function of $x$ for all values of $\alpha$. Therefore the second integral can be solved within stationary phase approximation: $$\label{eq:m1_56}
\int\limits_{-\infty}^{\infty}\!\!\mathrm{d}x\, \mathrm{sech}^{2}\left(x\right)
{\mathrm{e}}^{{\mathrm{i}}\pi m \left(2 \frac{x}{\gamma} + \frac{1}{\alpha}\right)^{2}} \!
\approx \frac{\left|\gamma\right|}{2\sqrt{m}}\,\mathrm{sech}^{2}\left(\frac{\gamma}{2 \alpha}\right) {\mathrm{e}}^{{\mathrm{i}}\frac{\pi}{4}}.$$ Using in yields $$\begin{aligned}
\label{eq:m1_57}
\tilde{\Omega}\approx & \frac{K}{4 \sqrt{2}}{\sqrt{\varphi}^{3}}\left|\gamma\right|\,\mathrm{sech}^{2}\left(\frac{\gamma}{2 \alpha}\right)
\sum\limits_{m=1}^{\infty} \frac{1}{m^{2}} \\
\label{eq:m1_58}
=& \frac{ g}{\mathcal{A}} (\hbar v_{F})^{2}\frac{\pi^{2}}{12}\,\beta \,\mathrm{sech}^{2}\left(\frac{\gamma}{2 \alpha}\right)\,\varphi^{2}, \end{aligned}$$ where is used. The corresponding expression for the total orbital susceptibility reads $$\begin{aligned}
\label{eq:m1_59}
\chi(\mu) =&-\frac{\mu_{0} g}{\phi_{0}^{2}} (\hbar v_{F})^{2} \frac{\pi^{2}}{6} \,\beta \,\mathrm{sech}^{2}\left(\frac{\mu\,\beta}{2}\right)\\
\label{eq:m1_60}
=& \chi_{0}(B)\times \frac{\sqrt{2}\pi^{2}}{9\zeta\left(\frac{3}{2}\right)}\,\gamma\,\mathrm{sech}^{2}\left(\frac{\gamma}{2\alpha}\right).\end{aligned}$$
In this regime the divergent contribution $\chi_{0}$ of the filled valence band is compensated by the contribution $\chi_{T}$ of the thermally excited charge carriers leading to a distinctly diamagnetic and moreover flux independent magnetic response. This result can also be found in the literature[@PhysRev.104.666; @PhysRevB.20.4889; @PhysRevB.75.235333; @PhysRevB.75.115123]. The contribution can be extracted from Eq. (\[eq:m1\_60\]) reading $$\begin{aligned}
\label{eq:m1_61}
\hspace{-0.5cm}\chi_{T}(\mu)= -\chi_{0}(B)\, \left[1 -
\frac{\sqrt{2}\pi^{2}}{9\zeta\left(\frac{3}{2}\right)}\,\gamma\,\mathrm{sech}^{2}\left(\frac{\gamma}{2\alpha}\right)\right]
\, .\end{aligned}$$ Since and , the contribution $\chi_{T}$ exhibits paramagnetic behavior in this parameter range. The comparison of this bulk contribution with numerical data for the triangular armchair and zigzag quantum dot in and b) shows perfect agreement as expected at larger fluxes.
To fulifil $\gamma < 1$, i.e. $\sqrt{\mathcal{A}/(2 \pi)\,\varphi} < k_{B} T/(\hbar v_{F})$, in the limit of very low temperatures requires that , , tend to zero even for large Landau indices $n$. Hence a change in the magnetization of bulk graphene due to weakly thermally excited charge carriers can only occur for Fermi energies close to the Dirac point. For this leads to a sharply peaked susceptibility at $\mu = 0$. In view of , this can be deduced from yielding the well known expression [@PhysRev.104.666; @PhysRevB.20.4889; @PhysRevB.75.235333; @PhysRevB.76.113301; @PhysRevLett.102.177203; @JPSJ.80.114705; @1751-8121-44-27-275001; @PhysRevB.83.235409; @PhysRevB.80.075418] $$\begin{aligned}
\label{eq:m1_62}
\chi(\mu) \xrightarrow{\beta \rightarrow \infty}-\frac{\mu_{0} g}{\phi_{0}^{2}} (\hbar v_{F})^{2} \frac{2 \pi^{2}}{3} \,\delta\left(\mu\right).\end{aligned}$$ This limit is not truly reachable numerically for the finite systems considered since the Landau level structure is not pronounced enough as it can be seen from .
Another limit of physical relevance concerns $\mu\rightarrow 0$ or $\alpha \rightarrow \infty$. In this limit the total orbital susceptibility reads $$\label{eq:m1_63}
\chi(\mu) \xrightarrow{\mu \rightarrow 0}-\frac{\mu_{0} g}{\phi_{0}^{2}} (\hbar v_{F})^{2}
\frac{\pi^{2}}{6} \,\beta \propto - \frac{1}{k_{B} T} .$$
This typical temperature dependence, already known in the literature[@PhysRevB.75.115123], is also affirmed by the numerical data, see . The double logarithmic graphs in the insets show clearly the $1/k_B T$ dependence in the limit $\mu\rightarrow 0$ (for $\phi = 5\,\phi_{0}$). The difference between the bulk theory and the numerical data for small thermal energies reflects, on the one hand, the limit of validity of the analytical approximation for $\gamma < 1$; on the other hand, it is a signature of finite-size effects which gain importance in the low temperature limit.
\[sec:Main\_2\]Oscillatory finite-size effects for graphene nanostructures
==========================================================================
General semiclassical framework
-------------------------------
Semiclassical periodic orbit theories offer a distinguished way to analytically describe finite-size effects encoded in the energy spectra of spatially confined systems of arbitrary shape. Boundary effects are incorporated in the semiclassical approximation of the oscillatory part of the DOS, $\rho^{\mathrm{osc}}_{\mathrm{sc}}(E)$. One important criteria for applying such semiclassical approximations, *the Gutzwiller trace formula*[@gutzwiller1990chaos] for chaotic classical dynamics or the *Berry-Tabor trace formula*[@0305-4470-10-3-009] for regular classical dynamics, requires that the linear system size lies in a mesoscopic regime, , where $k = E/(\hbar v_{F})$ is the Fermi wave number. In general, $d^{\mathrm{osc}}_{\mathrm{sc}}(E)$ is of the form $$\begin{aligned}
\label{eq:sc_64}
d^{\mathrm{osc}}_{\mathrm{sc}}(E) &= \sum\limits_{\gamma}d^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E),\\
\label{eq:sc_64a}
d^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E)&\propto \mathrm{Re} D_{\gamma} {\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S_{\gamma}},\end{aligned}$$ where the sum runs over infinitely many classical periodic orbits $\gamma$ with classical action and length $\mathcal{L}_{\gamma}$. The exact form of the classical amplitude $D_{\gamma}$ sensitively depends on the specific geometry of the system and can be calculated either within the recipe given by Gutzwiller[@gutzwiller1990chaos] in the case of non-integrable classical dynamics or within the recipe of Berry and Tabor[@0305-4470-10-3-009] when the classical dynamics is integrable. In the latter case, relevant in the following, the summation over $\gamma$ in runs over families of degenerate orbits, as depicted in for a disk geometry. This degeneracy of orbits in a regular billiard can be described in terms of continuous symmetry groups $G$ such that the members of a specific orbit family are related to each other through the action of a group element $g$ of $\mathbb{G}$. This is already included in the Berry-Tabor trace formula[@0305-4470-10-3-009] for field-free regular systems. In the case of small symmetry breaking, as it is caused by an weak external magnetic field, one has to take these degeneracies separately into account as discussed in . Therefore, we will associate an orbit family $\gamma$ with the corresponding element $g$ of the underlying symmetry group $\mathbb{G}$ if necessary, i.e. .
In the authors show in a general way, how the trace formulas for "Schrödinger billiards" with classically regular or chaotic dynamics can be extended to an arbitrary shaped, field-free graphene flake including the most common types of boundaries, i.e. zigzag, armchair and infinite-mass-type edges. Resembling the semiclassical trance fromulas for graphene read $$\begin{aligned}
\label{eq:sc_65}
\rho^{\mathrm{osc}}_{\mathrm{sc}}(E) = \sum\limits_{\gamma}\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E),\quad \rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E)\propto d^{\mathrm{osc}}_{\mathrm{sc, \gamma}}(E)\mathrm{Tr}K_{\gamma},
$$ where $d^{\mathrm{osc}}_{\mathrm{sc, \gamma}}$ is given by , of the corresponding Schrödinger system. Hence, the $d^{\mathrm{osc}}_{\mathrm{sc}, \gamma}$ contain all information about the orbital dynamics in the graphene system. The additional factor $\mathrm{Tr}K_{\gamma}$ denotes a trace over the pseudospin propagator $K_{\gamma}$ of the orbit $\gamma$ and contains only graphene specific information about the boundary. In a general expression for $\mathrm{Tr}K_{\gamma}$ of an orbit, with $N_{\gamma}$ reflections at the boundaries is derived, yielding $$\begin{aligned}
\label{eq:sc_66}
\mathrm{Tr}K_{\gamma} = 4 f_{\gamma}\cos\left(\theta_{\gamma} + \frac{\pi}{2}N_{\gamma}\right)\cos\left(2 K \Lambda_{\gamma} + \vartheta_{\gamma} + \frac{\pi}{2}N_{\gamma}\right),\end{aligned}$$ if the total number of reflections on armchair edges, $N_{\mathrm{ac}}$, is even and $\mathrm{Tr}K_{\gamma} = 0$ otherwise. The prefactor is defined as , where $N_{\mathrm{zz}}$ denotes the number of reflections on zigzag edges. is the sum over all reflection angles along the orbit $\gamma$. denotes the distances between the Dirac points and the $\Gamma$ point of the Brillouin zone. is the sum over the distance between two subsequent reflections on armchair edges. Further denotes the sum over zz reflection angles $\vartheta_{i}$, where for reflection on A- and B-edges, respectively, and $s_{i}$ is the number of ac reflections occuring after the zz reflection $i$. One finds[@epub12143; @PhysRevB.84.205421] where $\gamma^{-1}$ denotes the time reversed partner of orbit $\gamma$,
\[sec:Main\_2\_trace\]Semiclassical approximation of the orbital magnetic susceptibility
----------------------------------------------------------------------------------------
In the authors showed how the semiclassic theory of integrable and non-integrable billiard systems with parabolic dispersion can be extended to include the effect of an homogeneous, constant magnetic field. Due to the formal similarity of the trace formulas of systems with parabolic dispersion, , and graphene, , the techniques used in can be readily transferred. We will focus on the low-field regime where the classical cyclotron radius is much larger than the linear system size, i.e. . In the following we will consider quantum dots with corresponding regular classical dynamics in the field-free case. A derivation of orbital magentic properties of cavities with chaotic underlying dyanmics can be derived correspondingly. Following , we treat the weak magnetic field perturbatively, such that the classical Hamiltonian of the system, $$\begin{aligned}
\label{eq:sc_67}
\mathcal{H} = \frac{\left[\bold{p} - e \bold{A}\left(\bold{q}\right)\right]^{2}}{2 m} + V\left(\bold{q}\right),\end{aligned}$$ can be decomposed into the unperturbed part, , and the small perturbation $ -\frac{1}{m}\bold{p}\cdot\bold{A}\left(\bold{q}\right)$. To leading perturbative order the action difference between an orbit in the perturbed and the unperturbed system reads n[@bohigas; @StephenC199660] $$\begin{aligned}
\label{eq:sc_70}
\delta S_{\gamma} \approx e \int_{\gamma}\mathrm{d}\bold{q}\cdot\bold{A}\left(\bold{q}\right) = e \bold{B}\cdot\bold{\mathcal{A}}_{\gamma},\end{aligned}$$ with $\bold{\mathcal{A}}_{\gamma}$ the directed, enclosed area of the unperturbed orbit $\gamma$. In it is moreover shown that in the presence of a weak magnetic field the trace formula (\[eq:sc\_64\]) for the field-free Schrödinger system is modified to $$\begin{aligned}
\label{eq:sc_71}
\begin{split}
d^{\mathrm{osc}}_{\mathrm{sc}}(E, B) =& \sum\limits_{\gamma}d^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E, B),\\
d^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E, B)\propto&\,\mathrm{Re}\left[D_{\gamma} {\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}S_{0, \gamma}}\times\mathcal{M}_{\gamma}(B)\right],
\end{split}\end{aligned}$$ with the field-dependent modulation factor$$\begin{aligned}
\label{eq:sc_72}
\mathcal{M}_{\gamma}(B) = \frac{1}{V_{g}}\!\int_{\mathbb{G}}\!\mathrm{d}\mu\!\left(g\right){\mathrm{e}}^{\frac{{\mathrm{i}}}{\hbar}\delta S_{\gamma(g)}}
= \frac{1}{V_{g}}\!\int_{\mathbb{G}}\!\mathrm{d}\mu\!\left(g\right){\mathrm{e}}^{{\mathrm{i}}\frac{2\pi}{\phi_{0}}\bold{B}\cdot\bold{\mathcal{A}}_{\gamma(g)}}.\end{aligned}$$ The index $g$ represents an element of the symmetry group $\mathbb{G}$ characterizing the degeneracy of orbits $\gamma(g)$ in one specific orbit family. Since $\mu(g)$ is the Haar measure[@Haar] of $\mathbb{G}$, the normalization factor can be understood as the volume of $\mathbb{G}$. Since $d^{\mathrm{osc}}_{\mathrm{sc}}$ contains all information of the orbital dynamics, including the influence of the $B$-field, we can adapt and derive the oscillatory part of the DOS for a regular graphene cavity in a weak magnetic field in semiclassical approximation: $$\begin{aligned}
\label{eq:sc_72b}
\begin{split}
\rho^{\mathrm{osc}}_{\mathrm{sc}}(E, B) =& \sum\limits_{\gamma}\rho^{\mathrm{osc}}_{\mathrm{sc},\gamma}(E, B),\\
\rho^{\mathrm{osc}}_{\mathrm{sc},\gamma}(E, B) \propto&\, d^{\mathrm{osc}}_{\mathrm{sc},\gamma}(E, B)\mathrm{Tr}K_{\gamma}.
\end{split}\end{aligned}$$ is applicable to both, systems that remain integrable in a weak magnetic field and systems which are no longer integrable due to the symmetry breaking caused by a weak magnetic field, e.g. a rectangular quantum dot considered in . The lengths of time-reversed partner orbits or families, $\gamma$ and $\gamma^{-1}$, (for $B\!=\!0$) are equal, but the directed, enclosed areas have opposite signs due to the propagation direction, i.e. and . The contribution of these orbit pairs to the DOS can be combined to $$\begin{aligned}
\label{eq:sc_72ba}
\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E, B) +& \rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma'}(E, B) = 2 \,\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E)\times\mathcal{C}_{\gamma}(B),\end{aligned}$$ where $\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E)$ is the contribution (\[eq:sc\_65\]) of the orbit family $\gamma$ to $\rho^{\mathrm{osc}}_{\mathrm{sc}}$ in the field-free system and $$\begin{aligned}
\label{eq:sc_72c}
\mathcal{C}_{\gamma}(B) = \frac{1}{V_{g}}\!\int_{\mathbb{G}}\!\mathrm{d}\mu\!\left(g\right)\cos\left(\frac{2\pi}{\phi_{0}}\bold{B}\cdot\bold{\mathcal{A}}_{\gamma(g)}\right).\end{aligned}$$ The field dependence of the DOS and therefore of related observables such as the magnetic susceptibility is governed by dephasing between time-reversed orbit families and affected by dephasing between different members of a given orbit family induced by the magnetic field. From definition (\[eq:m1\_28\]) of the grand potential one can deduce the semiclassical approximation of the oscillatory part[@Richter] $$\begin{aligned}
\label{eq:sc_73}
\Omega^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B) =& \int\limits_{-\infty}^{\infty}\mathrm{d}E\,\mathcal{N}^{\mathrm{osc}}_{\mathrm{sc}}(E, B) f'(E - \mu),\end{aligned}$$ where $\mathcal{N}^{\mathrm{osc}}_{\mathrm{sc}}$ is obtained from $\rho^{\mathrm{osc}}_{\mathrm{sc}}$ after integrating twice by parts. For the contribution of the orbit family $\gamma$ to the oscillatory DOS, $\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}$, one finds[@Richter] $$\begin{aligned}
\label{eq:sc_74}
\mathcal{N}^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E, B) = -\left(\frac{\hbar}{\mathrm{d}S_{\gamma}/\mathrm{d}E}\right)^{2} \rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(E, B).\end{aligned}$$ The energy integral (\[eq:sc\_73\]) is of the form of and solved as described in App. A of . Using and one eventually finds $$\begin{aligned}
\label{eq:sc_75}
\hspace{-0.4cm} \Omega^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B) \approx \sum\limits_{\gamma} \left(\frac{\hbar v_{F}}{\mathcal{L}_{\gamma}}\right)^{2}\!\!\!\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(\mu, B)
\mathrm{R}_{T}\left(\frac{\mathcal{L}\gamma}{\mathcal{L}_c}\right).\hspace{-0.3cm}\end{aligned}$$ At finite $T$ the sum converges due to the exponential suppression of orbit families with encoded in $\mathrm{R}_{T}$, . Taking twice the $B$-field derivative one finds the semiclassical, oscillatory contribution to the orbital susceptibility of a graphene nanostructure with underlying regular classical dynamics: $$\begin{aligned}
\label{eq:sc_76}
\begin{split}
\chi^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B) =& -\frac{\mu_{0}}{\mathcal{A}}
\sum\limits_{\gamma} \left(\frac{\hbar
v_{F}}{\mathcal{L}_{\gamma}}\right)^{2}\mathrm{R}_{T}\left(\frac{\mathcal{L}_{\gamma}}{\mathcal{L_c}}\right)\\
&\times f_{\gamma}\,\rho^{\mathrm{osc}}_{\mathrm{sc}, \gamma}(\mu)\frac{\partial^{2}}{\partial
B^{2}}\mathcal{C}_{\gamma}(B) \, .
\end{split}\end{aligned}$$ Here, the sum involves one propagation direction of orbit families $\gamma$. Time-reversed partners are considered by the factor $f_{\gamma}\!=\!2$. The magnetic phase factor $\mathcal{C}_{\gamma}$, , implies that only orbits contribute to $\chi^{\mathrm{osc}}_{\mathrm{sc}}$ that enclose a finite area in the field-free case, and hence self-retracing orbits ($f_{\gamma} \!=\! 1$) do not contribute. We note that the same formal expression (\[eq:sc\_76\]) holds true for Schrödinger-type systems and graphene, since the graphene-specific relevant information is implicitly contained in $\rho^{\mathrm{osc}}_{\mathrm{sc, \gamma}}$. In the following we compare these predictions for the orbital magnetic response with quantum mechanical results within the effective Dirac model () and full tight-binding calculations ().
\[sec:Main\_2\_disk\]Circular billiard with infinite-mass-type edges
--------------------------------------------------------------------
The first representative system we analyze is a disk-shaped graphene quantum dot with infinite-mass-type edges. Due to its rotational symmetry there is a separable quantum mechanical solution within the Dirac approximation even in the presence of a magnetic field. The resulting quantization condition reads[@BM; @PhysRevB.76.235404] $$\begin{aligned}
\label{eq:sc_d1}
J_{\bar{m}}\left(k_{\bar{m}n}R\right) = \tau J_{\bar{m} + 1}\left(k_{\bar{m}n}R\right) \, .\end{aligned}$$ Here, $\tau = \pm 1$ labels the two valleys of the graphene Brillouin zone, $R$ is the disk radius and $J_{v}(x)$ denotes the $v$-th order Bessel function of the first kind[@gradshtein]. The index $\bar{m} = m + \phi/\phi_{0}$ includes the magnetic flux $\phi$ and the azimuthal orbital angular momentum quantum number . The second quantum number $n \in \mathbb{Z}$ counts (for a given $\bar{m}$) the solutions $k_{\bar{m}n}$ to which are obtained numerically. Each energy level has a two fold spin degeneracy. Based on , one can calculate the orbital magnetic susceptibility quantum mechanically according to .
The semiclassical properties of the disk cavity with infinite-mass-type edges have already been considered (for $B=0$) in . In order to compute its magnetic properties within semiclassical approximation, we combine these results with results adapted from , where $\chi_{\mathrm{sc}}^{\mathrm{osc}}$ for the Schrödinger disk billiard was derived. For the disk geometry, one can characterize the periodic-orbit families by their winding number $w$ and their total number $v$ of reflections at the boundary (with $v \geq 2 w$). The sign of $w$ defines the direction of rotation. A few representative periodic-orbit families are depicted in for $w = 1, 2$, together with their lengths $\mathcal{L}_{w, v}$ and the enclosed areas $\mathcal{A}_{w, v}$ (green shaded). They can be calculated within basic geometry yielding[@Richter] $$\begin{aligned}
\label{eq:sc_d2}
\mathcal{L}_{w, v} &= 2 v R \sin\left(\left|\pi \frac{w}{v}\right|\right),\\
\label{eq:sc_d3}
\mathcal{A}_{w, v} &= \mathcal{A} \frac{v}{2 \pi} \sin\left(2\pi \frac{w}{v}\right),\end{aligned}$$ with area $\mathcal{A} = \pi R^{2}$. The trace over the pseudospin propagator for an orbit family characterized by the tupel $(w, v)$ can be calculated from and reads[@PhysRevB.84.075468; @epub12143] $$\begin{aligned}
\label{eq:sc_d4}
\mathrm{Tr}K_{w, v} = g\cos\left(v\, \theta_{w, v}\right)
\begin{cases}
(-1)^{v/2} & \mathrm{for\ even\ }v,\\
0 & \mathrm{for\ odd\ } v
\end{cases},\end{aligned}$$ with the reflection angle . Due to pseudospin interference only orbits with an odd number of reflections contribute to the DOS, in contrast to the corresponding Schrödinger system[@brack_sc; @Richter]. Therefore, the entire field-dependent, oscillatory contribution to the DOS reads $$\begin{aligned}
\label{eq:sc_d5}
\begin{split}
\rho_{\mathrm{sc}}^{\mathrm{osc}}(E, B) =& \frac{2}{\hbar v_{F}}\sqrt{\frac{k}{2 \pi}} \sum\limits_{w = 1}^{\infty}\sum\limits_{\substack{v \geq 2 w \\ \mathrm{even}}}^{\infty}
(-1)^{w + v/2} \frac{f_{w, v}}{v^{2}} \mathcal{L}_{w, v}^{3/2}\\
&\times \sin\left(k\mathcal{L}_{w, v} + \frac{3}{4}\pi\right) \mathcal{C}_{w, v}(B).
\end{split}\end{aligned}$$ Owing to the rotational symmetry, the $B$-field induced modulation of each contribution is only due to dephasing between time-reversed orbits such that the magnetic phase factor reads[@Richter] $$\begin{aligned}
\label{eq:sc_d6}
\mathcal{C}_{w, v}(B) = \frac{1}{2 \pi}\int\limits_{0}^{2 \pi}\mathrm{d}\varphi \cos\left({\frac{\mathcal{A}_{w, v}}{l_{B}^{2}}}\right) = \cos\left({\frac{\mathcal{A}_{w, v}}{l_{B}^{2}}}\right).\end{aligned}$$ Together with , one then finds for the semiclassical approximation of the oscillatory contribution to the orbital magnetic susceptibility (in terms of $\chi_{0}$, ): $$\begin{aligned}
\label{eq:sc_d7}
\begin{split}
\chi^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B) =&-\chi_{0}(B)\times \frac{8 \pi^{3/2}}{3\zeta\left(3/2\right)} \frac{R}{l_{B}} \sqrt{k_{F}R}\\
&\hspace{-1.5cm}\times \sum\limits_{w = 1}^{\infty}\sum\limits_{\substack{v \geq 2 w \\ \mathrm{even}}}^{\infty} \frac{(-1)^{w + v/2}}{v^{2}}\left(\frac{\mathcal{A}_{w,v}}{R^{2}}\right)^{2}\sqrt{\frac{R}{\mathcal{L}_{w,v}}} \\
&\hspace{-1.5cm}\times \sin\left(k_{F}\mathcal{L}_{w, v} + \frac{3}{4}\pi\right) \mathcal{C}_{w, v}(B) \mathrm{R}_{T}\left(\frac{L_{w,v}}{\hbar v_{F}}\right).
\end{split}\end{aligned}$$
Since bouncing-ball orbits $(w,2w)$ do not enclose a finite area in the weak field limit they are not considered in $\chi^{\mathrm{osc}}_{\mathrm{sc}}$, and we absorbed the factor $f_{w, v}=2$ into the overall prefactor. Expression (\[eq:sc\_d7\]) demonstrates that the confinement-induced magnetic response of an integrable geometry is parametrically larger (by a factor $\sqrt{k_{\rm F}R}$) than the bulk value $\chi_0$.
Panel a) of shows the length spectrum resulting from the Fourier transform of the quantum-mechanical result for $\chi^{\mathrm{osc}}(\mu)$ at $B=0$. One can clearly identify the peak positions with the lengths $\mathcal{L}_{w, v}$ of the shortest contributing orbits as expected from the semiclassical formula (\[eq:sc\_d7\]). Green arrows mark the lengths of those orbits that do not contribute due to destructive pseudospin interference according to . Apparently, as visible in , also bouncing-ball orbits yield a contribution to the quantum mechanical result $\chi^{\mathrm{osc}}$, even though, according to Eq. (\[eq:sc\_d6\]), their semiclassical contribution vanishes at weak fields if bending of the trajectories is not included. The temperature used in is equivalent to a short cut-off length of $\mathcal{L}_{c}\approx 1.5\,R$, implying that only the lowest harmonics contribute significantly to $\chi^{\mathrm{osc}}_{\mathrm{sc}}$. This may explain why the peak from the shortest orbits, the bouncing-ball orbits is comparable to the other peaks. The influence of this first peak causes small deviations between the semiclassical and the quantum mechanical result, as visible in . There, $\chi^{\mathrm{osc}}$ is normalized by $\sqrt{k_{F}R}$ and $$\begin{aligned}
\label{eq:sc_X}
X = (0.5\,\phi_{0}/\mathcal{A}) \chi_0 \approx-7.8\,R\cdot 10^{-5} \, .\end{aligned}$$ Due to the divergent character of $\chi_{0}$ for small values of $\phi$ (see ), the amplitude of the oscillations in $\chi^{\mathrm{osc}}$ appears to be smaller than the contribution from the filled valence band. Anyhow, one would not expect the quantum-mechanical and the semiclassical result to lie in perfect agreement with each other since the susceptibility as a second derivative is very sensitive to small deviations already on the level of the DOS. The length spectrum \[fig:disk\]a) shows an accumulation of contributing orbits in the vicinity of $2\pi\,w$. This clustering of orbit families can be identified with the so called ’whispering gallery’ modes, which yield a coherent contribution to $\rho^{\mathrm{osc}}_{\mathrm{sc}}$. Since these orbit families enclose nearly the whole disk area, i.e. $\mathcal{A}_{w, v} \approx \mathcal{A}\,w$, their contribution to $\chi^{\mathrm{osc}}_{\mathrm{sc}}$ converges as $(-1)^{v/2}/v^{2}$ for a fixed value of $w$ leading to an overall convergence of at finite temperatures[@Richter].
As has been done for corresponding systems of parabolic dispersion[@Richter] one can calculate the typical value of the oscillatory susceptibility contribution defined by the root mean square of $\chi^{\mathrm{osc}}_{\mathrm{sc}}$ with respect to energy[@Richter]: $$\begin{aligned}
\label{eq:sc_d8}
\bar{\chi}_{\mathrm{sc}}(\mu, B) = \sqrt{\langle\left[\chi^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B)\right]^{2}\rangle},\end{aligned}$$ with $$\begin{aligned}
\label{eq:sc_d9}
\hspace{-0.3cm}\langle\left[\chi^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B)\right]^{2}\rangle
=\frac{1}{\Delta k_{F} R}\hspace{-0.5cm}\int\limits_{k_{F}R}^{k_{F}R + \Delta k_{F} R}\hspace{-0.5cm}\mathrm{d}k'_{F}R \left[\chi^{\mathrm{osc}}_{\mathrm{sc}}(E'_{F}, B)\right]^{2}.
\hspace{-0.3cm}\end{aligned}$$ The energy interval is chosen classically negligible but quantum mechanically large, i.e. $k_{F} R \gg \Delta k_{F}R \gg 2 \pi$. As a consequence, semiclassical off-diagonal terms , where , vanish under integration in , whereas the diagonal terms yield a contribution of $1/2$. For a detailed discussion see . In the zero field limit simplifies to $$\begin{aligned}
\label{eq:sc_d10}
\begin{split}
\hspace{-0.4cm}\bar{\chi}_{\mathrm{sc}}(\mu, 0) =& -X\sqrt{k_{F}R}\times\frac{8\sqrt{\pi}}{3\zeta\left(\frac{3}{2}\right)} \\
&\times \left[\frac{1}{2} \hspace{-0.4cm}\sum\limits_{\substack{w \\ v > 2 w, \mathrm{even}}}\hspace{-0.4cm}
\frac{\mathrm{R}_{T}^{2}\left(\frac{\mathcal{L}_{w, v}}{\hbar v_{F}}\right)}{v^{4}}\frac{\left(\mathcal{A}_{w, v}/R^{2}\right)^{4}}{\mathcal{L}_{w, v}/R}
\right]^{1/2}
\end{split} \hspace{-0.4cm}\end{aligned}$$ in terms of $X$, Eq. (\[eq:sc\_X\]). Choosing a similar cut-off length as in , i.e. , yields $\bar{\chi}_{\mathrm{sc}}(\mu, 0) \approx $ $ -0.11\,X\,\sqrt{k_{F}R}$, marked as a horizontal line in Fig. \[fig:disk\](b). In contrast to that, a calculation[@Richter] yields for a circular quantum dot with parabolic dispersion , where the Landau susceptibility $\chi_{L}$, , corresponds to $\chi_{0}$ in graphene.
We additionally considered ring-shaped graphene billiards of various thickness with infinte-mass-type edges. As shown in this geometry can be quantized in Dirac approximation for arbitrary magnetic field strength yielding a condition similar to . The comparison of $\chi_{\mathrm{qm}}^{\mathrm{osc}}$ with $\chi_{\mathrm{sc}}^{\mathrm{osc}}$ does not yield convincing coincidence in that case due to additional diffraction effects at the inner disk. These effects are beyond the leading-order semiclassical expansion considered in this work.
\[sec:Main\_2\_rect\]Rectangular billiard with zigzag and armchair edges
------------------------------------------------------------------------
The second fundamental system we consider is a rectangular graphene quantum dot with zigzag edges in $x$- and armchair edges in $y$-direction similar as shown in . The side lengths are labeled as $\mathcal{L}_{\mathrm{zz}}$ and $\mathcal{L}_{\mathrm{ac}}$, respectively, such that . Similar to the comparable Schrödinger system, the Dirac equation for a rectangular graphene quantum dot cannot be solved analytically in the presence of a magnetic field. For this reason, we will calculate the eigenenergies numerically within tight-binding approximation to check the quality of the semiclassical prediction.
From it is clear that opposite zigzag edges are built from different sublattices and lead to an additional sign of the reflection angle at one of the zigzag edges, as mentioned in . The classical periodic paths in this system can be classified by the tuple of their primitive reflection numbers $m$ and $n$ on the edges and their number of repetitions $r$. The corresponding orbit has $M$ bounces at the armchair and $N$ bounces at the zigzag edges in total and closes after $r$ repetitions. shows as examples members of the family $(1, 1)$ and $(1, 2)$, respectively. The members of one orbit family can be transformed into each other via translation of the reflection point at the $x$-axis. Thus, all members of one family have the same path length[@brack_sc; @Richter] $$\begin{aligned}
\label{eq:sc_re1}
\mathcal{L}_{M, N} = 2 r \sqrt{\left(m \mathcal{L}_{\mathrm{zz}}\right)^{2} + \left(n \mathcal{L}_{\mathrm{ac}}\right)^{2}},\end{aligned}$$ and only the enclosed area depends on the translational group element $x_{0}$. From follows $$\begin{aligned}
\label{eq:sc_re2}
\hspace{-0.35cm}\mathcal{A}_{M, N}(x_{0}) = \begin{cases}\frac{2 r}{m} \mathcal{L}_{\mathrm{ac}} x_{0}\left(1 - \frac{x_{0}}{\mathcal{L}_{\mathrm{zz}}} n\right) & \mathrm{if\ }m\cdot n\mathrm{\ odd,}\\
0 & \mathrm{if\ }m\cdot n\mathrm{\ even}.
\end{cases}\end{aligned}$$ As visible in , the directed area does not vanish if $m\cdot n$ is odd. The flux-dependent dephasing factor reads $$\begin{aligned}
\label{eq:sc_re3}
\mathcal{C}_{M, N}(B) =& \frac{n}{\mathcal{L}_{\mathrm{zz}}} \int\limits_{0}^{\mathcal{L}_{\mathrm{zz}}/n}\mathrm{d}x_{0}\,\cos\left(\frac{\mathcal{A}_{M, N}(x_{0})}{l_{B}^{2}}\right)\\
\begin{split}
=& \frac{\sqrt{\pi/2}}{\sqrt{\phi_{M, N}}}\left[\cos\left(\phi_{M, N}\right)\mathrm{C}\left(\sqrt{\phi_{M, N}}\right)\right.\\
&\hspace{0.5cm}\left. \,\, + \sin\left(\phi_{M, N}\right)\mathrm{S}\left(\sqrt{\phi_{M, N}}\right)\right].
\end{split}\end{aligned}$$ The cosine Fresnel integral[@gradshtein] is defined analogous to $\mathrm{S}(x)$ in . The phase $$\begin{aligned}
\label{eq:sc_re4}
\phi_{M, N} = \frac{\mathcal{A}_{M, N}(\mathcal{L}_{\mathrm{zz}}/(2 n))}{l_{B}^{2}} = \pi \frac{r}{m n} \varphi\end{aligned}$$ corresponds to $2\pi$ times the flux through the area , which is enclosed by the time-reversed orbit partner with bounces at . It can be directly proven that the enclosed area of these two orbits of the $(M,N)$ orbit family is maximum and therefore the action () stays extremal only for these two paths. Corresponding to the Poincaré-Birkhoff theorem, these are the only members of the orbit family, which remain periodic in the presence of the perpendicular magnetic field[@Richter]. In contrast to rotational symmetric systems, the magnetic field factor is not only governed by the dephasing between time-reversed orbit twins but also due to dephasing of family members propagating in the same direction.
The trace over the pseudospin-propagator is \[, \] $$\begin{aligned}
\label{eq:sc_re5}
\begin{split}
\mathrm{Tr}K_{M, N} =& g(-1)^{r n}\cos\left(2 K \mathcal{L}_{\mathrm{zz}} r m- 2 r n \left|\theta_{\mathrm{zz}}\right|\right),
\end{split}\end{aligned}$$ where $K \!=\! 4 \pi/(3 a)$ is the distance between the $\Gamma$- and one of the $K$-points in the first Brillouin zone. The reflection angle $ |\theta_{\mathrm{zz}}| \!=\! \arctan(M \mathcal{L}_{\mathrm{zz}} / (N\mathcal{L}_{\mathrm{ac}})) $ appears in because the opposing zigzag edges are built from different sublattices (Fig. \[fig:rect\_ex\]). On a microscopic scale the distance between both armchair edges can only take values , , yielding[@epub12143; @PhysRevB.84.205421] $$\begin{aligned}
\label{eq:sc_re7}
K\mathcal{L}_{\mathrm{zz}} = \begin{cases} 0 \mathrm{\,mod\,}2\pi & \mathrm{if\ }q\mathrm{\,mod\,}3 = 0,\\
\pi/3 \mathrm{\,mod\,}2\pi & \mathrm{otherwise}.
\end{cases}\end{aligned}$$
Whenever the length of the zigzag edge is such that $q$ is a multiple of $3$, orbit families with , where $N$ is odd, are suppressed by the trace of the pseudospin-propagator and hence do not contribute to the DOS[@epub12143; @PhysRevB.84.205421]. Furthermore, when on a macroscopic scale, bouncing ball orbits with $(0, M)$ and $(N, 0)$ cancel each other exactly if $M$ and $N$ are odd[@epub12143; @PhysRevB.84.205421], respectively. Combining these considerations with the results for a rectangular Schrödinger system[@Richter; @brack_sc] we find the field-dependent expression $$\begin{aligned}
\label{eq:sc_re8}
\begin{split}
\hspace{-0.3cm}\rho^{\mathrm{osc}}_{\mathrm{sc}}(E, B) =& \frac{\mathcal{A}}{\hbar v_{F}} \sqrt{\frac{k}{2\pi^{3}}}
\sum\limits_{r = 1}^{\infty}\sum\limits_{\substack{m, n = 0 \\ m = 0 \vee n = 0}}^{\infty}\frac{f_{n, m}}{\sqrt{\mathcal{L}_{M, N}}}\\
&\times\cos\left(k\mathcal{L}_{M, N} - \frac{\pi}{4}\right)\mathrm{Tr}K_{M, N}\mathcal{C}_{M, N}(B). \hspace{-0.33cm}
\end{split}\end{aligned}$$ The factor $f_{M,N}=2$ whenever there exists a time-reversed version of the orbit family $(M, N)$ and for bouncing-ball orbits. In order to calculate $\chi^{\mathrm{osc}}_{\mathrm{sc}}$ we take the second derivative of the field factor, $$\begin{aligned}
\label{eq:sc_re9}
\mathcal{C}''_{M, N}(B) = -\left(\frac{2\pi \mathcal{A}_{M, N}}{\phi_{0}}\right)^{2} \frac{\sqrt{\pi/2}}{4} \times \tilde{\mathcal{C}}_{M,N}\left(\phi_{M, N}\right),\end{aligned}$$ with $\phi_{M, N}$ in and $$\begin{aligned}
\label{eq:sc_re10}
\hspace{-0.25cm}\begin{split}
\tilde{\mathcal{C}}_{M,N}\left(x\right) = &
\sqrt{\frac{2}{\pi}}\frac{3}{x^{2}}
-\frac{\mathrm{C}(\sqrt{x})}{x^{5/2}}\left[\left(3 - 4 x^2\right)\cos(x) + 4 x \sin(x)\right]\\
&
\qquad \quad - \frac{\mathrm{S}(\sqrt{x})}{x^{5/2}}\left[\left(3 - 4 x^2\right)\sin(x) - 4 x \cos(x)\right]
\end{split}\hspace{-0.25cm}\end{aligned}$$ In the zero field limit $\tilde{C}_{M, N}$ converges to the value . For $\chi^{\mathrm{osc}}_{\mathrm{sc}}$ we find, according to , $$\begin{aligned}
\label{eq:sc_re11}
\hspace{-1.5cm}\begin{split}
\chi^{\mathrm{osc}}_{\mathrm{sc}}(\mu, B) =& -\chi_{0}(B)\times \frac{\sqrt{2}2\pi}{3\zeta(3/2)}\left(\frac{\mathcal{L}_{\mathrm{zz}}}{\mathcal{L}_{\mathrm{ac}}}\right)^{2} \frac{\mathcal{L}_{\mathrm{ac}}}{l_{B}} \sqrt{k_{F} \mathcal{L}_{\mathrm{ac}}}\\
&\hspace{-1.5cm}\times\sum\limits_{r = 1}^{\infty}\sum\limits_{\substack{m, n = 1 \\ m\cdot n \mathrm{\,odd}}}^{\infty}\hspace{-0.15cm}\frac{\mathrm{Tr}K_{M, N}}{g}\left(\frac{\mathcal{A}_{M, N}}{\mathcal{A}}\right)^{2}\sqrt{\frac{\mathcal{L}_{\mathrm{ac}}}{\mathcal{L}_{M, N}}}\\
&\hspace{-1.5cm}\times\cos\left(k_{F}\mathcal{L}_{M, N} - \frac{\pi}{4}\right)\! \mathrm{R}_{T}\left(\!\frac{\mathcal{L}_{M, N}}{\hbar v_{F}}\!\right)\! \tilde{\mathcal{C}}_{M, N}\left(\!\phi_{M, N}\!\right).
\end{split}\hspace{-1.5cm}\end{aligned}$$ The factor $f_{M,N}=2$ for all contributing orbit families, is absorbed in the prefactor. The squared aspect ratio ${\cal L}_{\mathrm{zz}}/{\cal L}_{\mathrm{ac}}$ enters the prefactor yielding a strong dependence of the susceptibility on the geometry of the system.
Panels a) and c) of show the length spectra obtained after Fourier transform from $\chi^{\mathrm{osc}}$ calculated from the tight-binding eigenenergies of two rectangular graphene quantum dots with , and , , respectively. The green arrows mark the position of orbit families which are semiclassically predicted not to contribute to $\chi^{\mathrm{osc}}$. These length spectra are not as smooth as the one obtained for the graphene disk with infinite mass boundaries, , since the region of linear dispersion cannot be extended arbitrarily in tight-binding approximation. Still, one can clearly identify the peaks in and c) with the lengths of contributing orbits, such that the comparison of $\chi^{\mathrm{osc}}_{\mathrm{TB}}$, with $\chi^{\mathrm{osc}}_{\mathrm{sc}}$, , in and d) shows convincing agreement. In these cases the thermal energy is corresponding to the cut-off length . The normalization factor $X/\sqrt{k_{F}\mathcal{L}_{\mathrm{ac}}}$ is the same as the one chosen in , with $X$ defined by and $\mathcal{L}_{\mathrm{ac}} \approx R$, i.e. all parameters are similar to the disk. The amplitudes of the oscillation between para- and diamagnetic behavior of $\chi^{\mathrm{osc}}$ in the case of the rectangular quantum dots \[ and d)\] are similar to the amplitude of $\chi^{\mathrm{osc}}$ for the circular quantum dot \[\]. Though the agreement of the oscillation frequencies of $\chi^{\mathrm{osc}}_{\mathrm{TB}}$ and $\chi^{\mathrm{osc}}_{\mathrm{sc}}$ in and d) are convincing the tight-binding result in panel d) exhibits an additional modulation of the oscillations for which are not contained in the semiclassical approximation. In the corresponding energy range, the Dirac model and therefore the semiclassical approximation reaches the limit of validity[@epub12143] in describing the energy spectrum of graphene when $k\,a\lesssim 1$. Though the lengths $\mathcal{L}_{\mathrm{ac}, \mathrm{zz}}$ of the system considered in ($K\,\mathcal{L}_{\mathrm{zz}} = 0\,\mathrm{mod}\,2\pi$) are only one row of atoms shorter on each side than the system considered in ($K\,\mathcal{L}_{\mathrm{zz}} = \pi/3\,\mathrm{mod}\,2\pi$), the oscillation amplitude of $\chi^{\mathrm{osc}}$ differs by one order of magnitude. This is due to the suppression of orbit families with $(N\,\mathcal{L}_{\mathrm{ac}}/\mathcal{L}_{\mathrm{zz}}, N)$, where $N$ is odd, as noted above. Since the aspect ratio is not perfectly integer, those orbit families still yield a small contribution to $\rho^{\mathrm{osc}}$, and correspondingly to $\chi^{\mathrm{osc}}$, and appear in the length spectrum, .
\[sec:Conclusion\] Summary and Outlook
======================================
In this work we focused on the orbital magnetic properties of non-interacting ballistic bulk graphene and in particular on confined graphene-based systems with regular classical dynamics. To this end we considered the magnetic susceptibility $\chi$, calculated in the grand canonical ensemble, in the energy region of linear dispersion.
In the first part of this paper, we considered bulk graphene. There the orbital magnetic response distinctly depends on the particular energy scales involved, namely the different energy regimes associated with temperature, chemical potential and magnetic field that we considered with a comparitive look.
In a first step we derived the temperature-independent susceptibility contribution $\chi_0$ from the filled valence band, [*i.e.*]{} the graphene analogue to the Landau suceptibility $\chi_{L}$ of an ordinary two-dimensional electron gas. We found for $\chi_{0}$ the well-known diamagnetic $-B^{-1/2}$ behavior, assuming, in accordance with literature, the valence band to be linear and extended, so $\chi_{0}$ cannot be directly compared to realistic tight-binding calculations even for very large systems. Still, for finite temperatures we found the total orbital magnetic susceptibility $\chi = \chi_{0} + \chi_{T}$ to be regular in the limit $B \rightarrow 0$ for $T \neq 0$. We compared our analytic results for the temperature-dependent part $\chi_{T}$ of the susceptibility with the results from literature and to numerical tight-binding calculations. The latter were performed for finite nanostructures of mesoscopic dimensions, also in view of the confinement effects later addressed. Still, the magnetic response of these graphene cavities also exhibits bulk-like features, and we discussed initially the necessary conditions for comparision of our analytic bulk calculations with results for finite systems.
In the presence of a finite magnetic field the features of the susceptibility depend on the relative size of the associated Landau level spacing $\Delta_{LL}$, the chemical potential $\mu$ and the thermal energy $k_{\rm B}T$. If the latter is the smallest scale, we distinguish two regimes (see Fig. 1):
For $\mu > \Delta_{LL}$ we obtained the typical $\mu^{2}$- and $1/B$-equidistant, oscillatory behavior of $\chi_{T}$, similar to de Haas-van Alphen oscillations in two-dimensional electron gases. Though the corresponding numerically calculated magnetic response for the finite systems exhibits, as a signature of the confinement, a richer oscillatory structure in this regime, there is a clear coincidence of clustered peaks with the pattern of $\chi_{T}$ in the bulk system. This becomes even more obvious by raising the temperature in the numerics, since then the finite-size contributions are damped out and only Landau level signatures remain. The amplitudes of these de Haas-van Alphen-type oscillations in graphene are one order of magnitude larger than the diamagnetic $\chi_{0}$ for the considered parameters, implying that the total orbital magnetic susceptibility oscillates between para- and diamagnetic behaviour as a function of $\mu$ and $B$, respectively.
For $\mu < \Delta_{LL}$ the term $\chi_{T}$, and therefore $\chi$, is an exponentially decaying function of the magnetic field and diamagnetic. For field values high enough such that bulk effects dominate over finite-size signatures, the numerically calculated susceptibility of the quantum dots coincides very well with the analytic results.
If $k_{\rm B}T$ is larger than $\Delta_{LL}$, $\chi_{T}$ is a smooth function of temperature, chemical potential and magnetic field and shows paramagnetic behavior with values . Therefore, $\chi=\chi_0+\chi_{T}$ is diamagnetic and appears even to be independent of the magnetic field for arbitrary $\mu$. At the Dirac point ($\mu=0$), $\chi_T$, and correspondingly $\chi$, follow a Curie-type $T^{-1}$ power law which is confirmed by our numerical data for the (triangular) quantum dots at finite temperature. For $1/\beta \lesssim t$, deviations between $\chi_{T}$ for the bulk and the finite systems appear due to the increasing relevance of finite-size signatures in this limit. We also analytically confirmed the well-known $\delta(\mu)$ singularity [@PhysRev.104.666; @PhysRevB.20.4889; @PhysRevB.75.235333; @PhysRevB.76.113301; @PhysRevLett.102.177203; @JPSJ.80.114705; @1751-8121-44-27-275001; @PhysRevB.83.235409; @PhysRevB.80.075418] of $\chi$ at zero temperature.
Through the confirmation of the analytic results for extended graphene with numerical data of finite quantum dots, we could analyze the importance of bulk effects in finite system on the one hand and distinguish them from true confinement effects on the other hand. As one interesting aspect we found $\chi_{T}/\chi_{0}$ of the triangular quantum dot with zigzag edges to be smaller than $\chi_{T}/\chi_{0}$ for the armchair quantum dot with same parameters. This is due to the zigzag edge state and the lower average energy in that case. Moreover, especially in the energy range, where oscillations occur in $\chi_{T}$, the influence of the boundary is clearly observable.
In the second, major part of this work we then analyzed in detail such confinement effects. To this end we considered two representative geometries, a disk-shaped and a rectangular graphene cavity. We derived a generic analytic expression for the oscillatory part $\chi^{\mathrm{osc}}$ of the orbital magnetic susceptibility based on results \[\] for the susceptiblity of confined electron gases and working out generalizations to finite $B$-fields of semiclassical expressions for the field-free density of states for graphene cavities, . We demomstrated that graphene specific edge effects depending on the type of the boundaries enter the semiclassical expressions, and thereby orbital magnetism, through phases associated with the pseudospin propagator. This semiclassical approximation applies in particular to the low-field regime, where bulk contributions are suppressed and the energy spectrum (and correspondingly the orbital susceptibility) is governed by finite-size effects.
We found good agreement of our semiclassical approach with the quantum mechanical results for $\chi^{\mathrm{osc}}$ based on the calculation of the eigenergies for circular graphene quantum dots with infinite-mass type edges. The Fourier transform of $\chi^{\mathrm{osc}}_{\mathrm{qm}}$ with respect to the energy yielded a length spectrum with relatively sharp peaks reflecting the underlying classical orbit dynamics of this system. We showed that orbits with odd number of reflections are suppressed as it is predicted in our semiclassical approach due to destructive pseudospin interferences. This is distinctly different from the corresponding case of the electron gas system[@Richter], where all non self-retracing orbits yield a contribution to $\chi^{\mathrm{osc}}_{\mathrm{sc}}$. We found the typical value for $|\chi^{\mathrm{osc}}|$ to scale like $\sqrt{k_{F} R}$. Hence, similar as in Ref. \[\] $|\chi^{\mathrm{osc}}|$ can be larger than $X = \chi_{0}(0.5\,\phi_{0}/\mathcal{A})$. In contrast, the amplitudes of the $\chi^{\mathrm{osc}}$-oscillations show the same scaling behavior, but appear to be of the same order of magnitude than $X$. Similar agreement was found for rectangular-shaped graphene quantum dots, where we compared the semiclassical predictions with numerical tight-binding calculations. Depending on the length of the zigzag edges, the strength of the oscillatory modulations in $\chi$ were found to differ by one order of magnitude due to destructive pseudospin interferences.
We studied the magnetic response for individual systems, including the typical susceptibility, within the grand canonical formalism. To compute the average response of an ensemble of nanostructures, a canonical treatment starting from the free energy instead of the grand potential is required [@Imry-book]. Along the lines of [@PhysRevLett.74.383; @Richter], and with the semiclassical expressions for graphene derived here, it appears straight forward to compute the ensemble-averaged susceptibility.
A further interesting aspect concerns the role of disorder for orbital magnetism in graphene, both for the bulk and confined case. Again, previous work [@McCann; @McCann2] for the 2d Schrödinger case, covering the entire disorder range from clean to diffusive, could act as a guideline.
Our overall analysis demostrates pronounced confinement effects on orbital magnetism in graphene-based nanosystems that dominate the bulk response in wide parameter regimes. However, our approach is based on non-interacting models for graphene, as most of the works on orbital magnetism in graphene. An exception is Ref. where interaction effects are considered at $T=0$, however only to first order in the Coulomb repulsion. The physics of conventional two-dimensional electron systems shows that, while non-interacting terms are also crucial there, contributions from electron-electron interactions can usually not be disregarded. For instance, for the two-dimensional bulk Aslamazov and Larkin computed interaction corrections to the Landau susceptibility [@Aslamazov] (see also Ref. [@Ullmo1] for a semiclassical treatment). Moreover, this work demonstrated that higher-order diagrams are essential for an appropriate perturbative treatment of interaction effects, a treatment that is missing for graphene. In Ref. [@Ullmo2] it was furthermore shown that additional confinement-mediated interaction contributions to the susceptibility of 2d electron systems can be of the same order as those from the non-interacting model. To generalize such an analysis in terms of interaction effects for graphene is beyond the scope of the present work. Hence this interesting and challenging question is left for future research.
Acknowledgments
===============
We thank Inanc Adagideli and Jürgen Wurm for useful discussions and Viktor Krückl for help in numerical implementations. This work was funded by the [*Deutsche Forschungsgemeinschaft*]{} through GRK 1570: [*Electronic Properties of Carbon Based Nanostructures*]{}.
\[sec:App\_Int\_Fresnel\]Transformation of the Fresnel integral
===============================================================
Starting with the definition of the Fresnel integral[@gradshtein], , one finds after substituting and using the relation[@gradshtein] $$\begin{aligned}
\label{eq:ap_1}
\frac{1}{z^{\alpha}} = \frac{1}{\Gamma\left(\alpha\right)}\int\limits_{0}^{\infty}\mathrm{d}u\,\frac{{\mathrm{e}}^{-zu}}{u^{1-\alpha}},
\quad\mathrm{Re}\,z > 0, \mathrm{Re}\,\alpha > 0,\end{aligned}$$ where and , $$\begin{aligned}
\label{eq:ap_2}
\mathrm{S}\left(\left|x\right|\right) &= \frac{1}{\sqrt{2\pi}\Gamma\left(\frac{1}{2}\right)} \mathrm{Im}\left[
\int\limits_{0}^{\infty}\mathrm{d}u\, \frac{1}{\sqrt{u}}
\int\limits_{0}^{x^{2}}\mathrm{d}\tau\, {\mathrm{e}}^{-(u-{\mathrm{i}})\tau}\right]\\
\label{eq:ap_3}
&= \frac{1}{\sqrt{2}\pi}\mathrm{Im}\left[
\int\limits_{0}^{\infty}\mathrm{d}u\, \frac{1}{\sqrt{u}(u-{\mathrm{i}})}
-\int\limits_{0}^{\infty}\mathrm{d}u\, \frac{{\mathrm{e}}^{-(u-{\mathrm{i}})x^{2}}}{\sqrt{u}(u-{\mathrm{i}})}\right].\end{aligned}$$ The first term in yields $1/2$ and represents the smooth part of the Fresnel integral. Using[@gradshtein] $$\begin{aligned}
\label{eq:ap_4}
\mathrm{U}\left(1-\alpha; 1-\alpha; x\right) = \frac{x^{\alpha}}{\Gamma\left(1-\alpha\right)}
\int\limits_{0}^{\infty}\mathrm{d}t\,\frac{{\mathrm{e}}^{-t} t^{-\alpha}}{t+x}\end{aligned}$$ one finds ( and ) with $$\begin{aligned}
\label{eq:ap_5}
\tilde{\mathrm{S}}\left(x\right) = - \frac{1}{\sqrt{2\pi}}\mathrm{Im}
\left[{\mathrm{e}}^{{\mathrm{i}}\frac{\pi}{4}}{\mathrm{e}}^{{\mathrm{i}}x^{2}} \mathrm{U}\left(\frac{1}{2}; \frac{1}{2}; -{\mathrm{i}}x^{2}\right)\right].\end{aligned}$$
\[sec:App\_Int\_Omega\]Transformation of $\tilde{\Omega}_T$ for $\alpha, \gamma > 1$
====================================================================================
In order to calculate $\tilde{\Omega}_{T}$ as given in for $\alpha, \gamma > 1$ it is useful to apply the Taylor series representations of the logarithmic and exponential function yielding $$\begin{aligned}
\label{eq:ap_6}
\begin{split}
\tilde{\Omega}_{T} - \hat{\Omega}_{T} =& g\frac{\varphi}{\beta}
\sum_{s\pm 1}\sum\limits_{\substack{n=1 \\ m = 1}}^{\infty}\frac{(-1)^{m}}{m}{\mathrm{e}}^{s\frac{\gamma}{\alpha}\,m}\\
&\times\sum\limits_{k=0}^{\infty}
\frac{(-1)^{k}}{\Gamma\left(k+1\right)}( \sqrt{2}\gamma\, m)^{k}\,n^{\frac{k}{2}}.
\end{split}\end{aligned}$$ In the next step we interchange the order of summation[@PhysRevB.75.115123], which can be done without causing correction terms in this particular situation[@elizalde2012ten]. Computing the sum over the Landau index $n$ first yields[@gradshtein] , where $\zeta(z)$ is the Riemann zeta function. With use of[@gradshtein] $$\begin{aligned}
\label{eq:ap_7}
\zeta(z) = \frac{1}{\Gamma(z)}\int_{0}^{\infty}\mathrm{d}t\,\frac{t^{z-1}}{{\mathrm{e}}^{t}-1}\end{aligned}$$ transforms to $$\begin{aligned}
\label{eq:ap_8}
\begin{split}
\tilde{\Omega}_{T} - \hat{\Omega}_{T} =& \,g\frac{\varphi}{\beta}
\sum_{s\pm 1}\sum\limits_{m=1}^{\infty}\frac{(-1)^{m}}{m}{\mathrm{e}}^{s\frac{\gamma}{\alpha}\,m}\\
& \hspace*{-0.45cm}\times\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}}{\Gamma\left(k+1\right)\Gamma\left(-\frac{k}{2}\right)}
\int_{0}^{\infty}\!\!\! \mathrm{d}t\,\frac{\left(\frac{\sqrt{2}\gamma\, m}{\sqrt{t}}\right)^{k}}{t\left({\mathrm{e}}^{t}-1\right)}.
\end{split}\end{aligned}$$ We substitute such that the integral in can be approximated by $$\begin{aligned}
\label{eq:ap_9}
\int\limits_{0}^{\infty}\mathrm{d}y\,\frac{y^{-\frac{k}{2}-1}}{\exp\left(u\,y\right) - 1}
\stackrel{\gamma > 1}{\approx}& \int\limits_{0}^{\infty}\mathrm{d}y\,y^{-\frac{k}{2}-1}{\mathrm{e}}^{-u\,y} = \Gamma\left(-\frac{k}{2}\right) u^{\frac{k}{2}}.\end{aligned}$$ Calculating subsequently the sums over $k$ and $m$ in finally yields $$\begin{aligned}
\label{eq:ap_10}
\tilde{\Omega}_{T} - \hat{\Omega}_{T} \approx-g\frac{\varphi}{\beta} \sum\limits_{s\pm 1} \ln\left[1 + {\mathrm{e}}^{-\sqrt{2}\gamma + s\frac{\gamma}{\alpha}}\right].\end{aligned}$$
|
---
abstract: 'We consider, in the context of a 331 model with a single neutral right-handed singlet, the generation of lepton masses. At zeroth order two neutrinos and one charged lepton are massless, while the other leptons, two neutrinos and two charged leptons, are massive. However the charged ones are still mass degenerate. The massless fields get a mass through radiative corrections which also break the degeneracy in the charged leptons.'
address: |
$^a$ Instituto de Física Teórica\
Universidade Estadual Paulista\
Rua Pamplona, 145\
01405-900– São Paulo, SP\
$^b$ Instituto de Física da Universidade de São Paulo\
01498-970 C.P. 20516-São Paulo, SP\
Brazil
author:
- 'F. Pisano$^a$, V. Pleitez$^a$ and M.D. Tonasse$^b$'
title: ' Radiatively induced electron and electron-neutrino masses'
---
It is well known that in renormalizable theories, some masses or mass differences vanish at tree level if there are some symmetries in the theory which forbid them. This is maintained in higher order in perturbation theory. Or, if the respective higher order corrections are infinite the introduction of the counter term necessary to remove the infinity leaves these masses or mass differences as free parameters. Notwithstanding, in theories with spontaneously broken symmetry if a mass and mass counter term are forbidden by gauge structure, then higher order corrections are finite and calculable [@wggm]. In this spirit, many mechanisms for finding fermion masses as radiative corrections have been considered in the literature [@rev]. Here we will show how a mechanism of this kind can be implemented in the context of the recently proposed electroweak model based on the $SU(3)_L\otimes U(1)_N$ gauge symmetry [@pp; @pf].
In this model, leptons are treated democratically with the three generations transforming as $({\bf3},0)$ but with one quark generation (it does not matter which one) transforming differently from the other two. This condition arises because the model in order to be anomaly free must contain the same number of triplets as antitriplets. Hence, the number of generations is related to the number of quark colors.
In the minimal model the neutrinos remain massless since there is a global symmetry which prevents them to get a mass. This symmetry implies the conservation of the quantum number ${\cal F}=L+B$, where $L$ is the total lepton number $L=L_e+L_\mu+L_\tau$ and $B$ is the baryon number [@pt]. Here we will see that if we allow this symmetry to be explicitly broken and also adding a single right-handed neutrino singlet, one of the charged leptons and two neutrinos get mass through radiative corrections.
Let us introduce the following Higgs scalars, $\eta=(\eta^0,\eta^-_1,\eta^+_2)^T$, $\rho=(\rho^+,\rho^0,\rho^{++})^T$ and $\chi=(\chi^- ,\chi^{--}, \chi^0)^T$ which transform, under $SU(3)_L\otimes U(1)_N$ as $({\bf3},0),({\bf3},1)$ and $({\bf3},-1)$, respectively.
The leptonic triplets are $\psi_{aL}=(\nu''_a, l''_a, l''^c_a)^T\sim({\bf3},0)$, where the double primed fields denote weak eigenstates, $l''_a=e'',\mu'',\tau''$ and $\nu''_a=\nu''_e,\nu''_\mu,\nu''_\tau$.
The lepton mass term transforms as ${\bf3}\otimes{\bf3}={\bf3}^*_A\oplus{\bf6}_S$. Thus, we can introduce a triplet, like $\eta$, or a symmetric antisextet $S=({\bf6}^*_S,0)$. In the former case one of the charged leptons remains massless, and the other two are mass degenerate. For this reason it was chosen in Refs. [@pf; @fhpp] the latter one in order to obtain arbitrary mass for charged leptons.
Here we will not introduce the sextet $S$ but only the triplets $\eta,\rho$ and $\chi$; the respective VEV will be denoted by $v_\eta,v_\rho$ and $v_\chi$.
The more general $SU(3)\otimes U(1)$ gauge invariant renormalizable Higgs potential for the three triplets is $$\begin{aligned}
V(\eta,\rho,\chi)&=&\mu_1^2\eta^\dagger\eta+\mu^2_2\rho^\dagger\rho+
\mu^2_3\chi^\dagger\chi+
\lambda_1(\eta^\dagger\eta)^2 \nonumber
\\ & &\mbox{}+\lambda_2(\rho^\dagger\rho)^2
+\lambda_3(\chi^\dagger\chi)^2
+(\eta^\dagger\eta)\left[\lambda_4(\rho^\dagger\rho )\right.
\nonumber \\ & &\mbox{}\left.+\lambda_5(\chi^\dagger\chi)\right]
+\lambda_6(\rho^\dagger\rho)(\chi^\dagger\chi)+
\lambda_7(\rho^\dagger\eta)(\eta^\dagger\rho)\nonumber
\\ & &\mbox{} +
\lambda_8(\chi^\dagger\eta)(\eta^\dagger\chi)
+\lambda_9(\rho^\dagger\chi)(\chi^\dagger\rho)
\nonumber\\ & &\mbox{}+[\lambda_{10}(\eta^\dagger\chi)(\eta^\dagger\rho)
+\lambda'\epsilon^{ijk}\eta_i\rho_j\chi_k + H.c.].
\label{potential}\end{aligned}$$ As we said before let us define the [*lepto-baryon*]{} number ${\cal F}=L+B$, which is additively conserved. As usually $B(l,\nu_l)=0$ for any lepton $l,\nu_l$, and $L(q)=0$ for any quark $q$, ${\cal F}(l)={\cal F}(\nu_l)=+1$. In order to make ${\cal F}$ a conserved quantum number in the Yukawa sector, we also assign to the scalar fields the following values $
-{\cal F}(\chi^-)=-{\cal F}(\eta^+_2)={\cal F}(\rho^{++})=
-{\cal F}(\chi^{--})=+2$, and with all the other scalar fields carrying ${\cal F}=0$.
Notice that the ${\cal F}$-conservation forbids the quartic term $\lambda_{10}(\eta^\dagger\chi)(\eta^\dagger\rho)$ in Eq. (\[potential\]) [@laplata]. Hence, assuming that the $\lambda_{10}$ term does exist we are violating explicitly the ${\cal F}$ symmetry. We must stress that if we had introduced the scalar sextet and allow ${\cal F}$ to be broken, there will be additional terms involving the sextet and the triplets as well. In this case it is not possible to maintain neutrinos with calculable masses unless a fine tune is imposed [@pt].
The $\lambda_{10}$ term has interactions like $\rho^0\chi^0\eta^-_1\eta^+_2,\quad \eta^0\rho^0\eta^-_1\chi^+$, and we have mixing between $\eta^-_1$ and $\eta^+_2$, etc. In fact, the mass matrix in the singly charged scalars sector (in the $\eta^-_1,\rho^-,\eta^-_2,\chi^-$ basis) is $$v^2_\chi\left(
\begin{array}{cccc}
eba^{-1}-\lambda_7b^2\, &\, e-\lambda_7ab\, &\, \lambda_{10}b\, &\,
\lambda_{10}ab \\
e-\lambda_7ab\, &\, eab^{-1}-\lambda_7a^2\, &\, \lambda_{10}a\, &\,
\lambda_{10}a^2 \\
\lambda_{10}b \,&\, \lambda_{10}a\, &\, eba^{-1}-\lambda_8\, &\,
eb-\lambda_8a \\
\lambda_{10}ab \,&\, \lambda_{10}a^2\, &\, eb-\lambda_8a & eab-\lambda_8a^2
\end{array}
\right),
\label{oba2}$$ where $e=\lambda'/v_\chi$, $a=v_\eta/v_\chi$, $b=v_\rho/v_\chi$. Hence we have a mixing among all singly charged scalars. The spectrum from Eq. (\[oba2\]) will be given elsewhere but here we must stress that it has two Goldstone bosons. Notice that if there is no $\lambda_{10}$ term the mixing occurs between $\eta_1^-,\rho^-$ and between $\eta^-_2,\chi^-$.
Since right-handed neutrinos transforming as singlets under the gauge group do not contribute to the anomaly, their number is not constrained by the requirement of obtaining an anomaly free theory. Hence, we can introduce, as in the standard electroweak model, an arbitrary number of such fields. An interesting possibility is to introduce just a single neutral singlet [@cj].
The Yukawa interaction in the leptonic sector plus a Majorana mass term for the right-handed neutrino, is $${\cal
L}_{l\eta}=-\frac{i}{8}\sum_{a,b=e,\mu,\tau}
\epsilon^{ijk}F_{ab}\overline{(\psi_{Lai})^c}
\psi_{Lbj}\eta_k+\sum_a\hat{h}_a\overline{\psi_{aL}}\nu''_R\eta-\frac{1}{2}M
\overline{(\nu''_R)^c}\nu''_R
+H.c.
\label{yukawa}$$ All the arbitrary constants in Eq. (\[yukawa\]) may be taken real and positive. The Yukawa couplings $F_{ab}$ must be antisymmetric due to Fermi statistics. Explicitly we have $${\cal L}_{l\eta}=-i(v_\eta+\eta^0)\overline{l''_{Ra}}F_{ab}l''_{Lb}+
\frac{i}{2}\overline{l''_{Ra}}F_{ab}\nu''_{Lb}\eta^-_1
+\frac{i}{2}\overline{(\nu''^c)_{Ra}}F_{ab}l''_{Lb}\eta^+_2+H.c.,
\label{yukawa2}$$ where $$F_{ab}=\left(
\begin{array}{ccc}
0 \,&\, -f_{e\mu} \, &\, -f_{e\tau} \\
f_{e\mu}\, & \, 0 \, &\, -f_{\mu\tau} \\
f_{e\tau}\, & \, f_{\mu\tau}\, & \, 0 \\
\end{array}
\right).
\label{m1}$$ The mass spectrum of the charge leptons is $0,m,-m$. We can always define $e''$ as the state with zero mass. That is, we can choose a basis in which $f_{e\mu}=f_{e\tau}=0$. In this case we have $m=v_\eta
f^2_{\mu\tau}$. In this basis the matrix in (\[m1\]) can be diagonalized by an unitary matrix $$U=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \frac{1}{\sqrt2} & \frac{1}{\sqrt2} \\
0 & -\frac{i}{\sqrt2} & \frac{i}{\sqrt2}\\
\end{array}
\right).
\label{u}$$ Notice that one of the mass is negative. To get positive mass eigenvalues we let $m$ be positive and redefine the respective field with a $\gamma^5$ factor, i.e., $\tau'\to\gamma^5\tau'$. Because of this $\gamma^5$ the $CP$ of $\tau'$ is $-1$.
The neutrinos mass term is $${\cal
L}_\nu=-\sum_{a=e,\mu,\tau}
h_a\bar\nu'_{La}\nu'_R-\frac{1}{2}M\overline{(\nu'_R)^c} \nu'_R+H.c.,
\label{cecilia}$$ where $h_a=v_\eta\hat h_a$, and $\hat h_a$ are arbitrary dimensionless parameters. We can write the mass term as $-\frac{1}{2}\bar N''M^\nu N''^c$ with $N''=(\nu''_{eL},
\nu''_{\mu L},\nu''_{\tau L}, {\nu''_R}^c)^T$ and $$M^\nu=\left(
\begin{array}{cccc}
0\, & \, 0 \, & \, 0 \, &\, h_e \\
0\, & \, 0 \, & \, 0 \, & \, h_\mu \\
0 \, &\, 0 \, & \, 0 \, & \, h_\tau \\
h_e\, &\, h_\mu \, & \, h_\tau \, & \,M
\end{array}
\right)
\label{mnu}$$ We can diagonalize the neutrino mass matrix by making $N'=\Phi PN''$ with $P$ an orthogonal matrix, $$P=\left(
\begin{array}{cccc}
h_\mu/(A^2-h_\tau^2)^{\frac{1}{2}} & h_e/(A^2-h_\tau^2)^{\frac{1}{2}}
& 0 & 0 \\
h_eh_\mu h_\tau/(A^2-h_e^2)^{\frac{1}{2}}&
h_\tau h_\mu^2/(A^2-h_\tau^2)(A^2-h_e^2)^{\frac{1}{2}}
& -h_\mu/(A^2-h_e^2)^{\frac{1}{2}}& 0 \\
(h_em'_{\nu_P}-A^2)/D_1 &
h_\mu M/D_1 &
h_\tau M/D_1 & h_e M/D_1 \\
h_e(h_em'_{\nu_F}-A^2)/D_2 &
h_\mu(h_em'_{\nu_F}-A^2)/D_2 &
h_\tau(h_em'_{\nu_F}-A^2)/D_2 & -AM(h_em'_{\nu_F}-A^2)/D_2 \\
\end{array}
\right)
\label{nu1}$$ where $$D_1^2=(A^2-h_\tau^2)^2+M^2A^2+(A^2-h_\tau^2-h_em'_{\nu_P})
(h_\tau^2-h_em'_{\nu_P})$$ $$D_2^2=A(A^2-h_\tau^2)(A^2-h_\tau^2+M^2-2h_em'_{\nu_F})
+h_e^2({m'}_{\nu_F}^2+2h_\tau^2).$$ $N'=(\nu'_{1L},
\nu'_{2L},\nu'_{PL},\nu'_{FL})^T$, $A^2=h_e^2+h_\mu^2+h_\tau^2$ and $\Phi$ is a diagonal phase matrix $\Phi=diag(1,1,i,1)$. At this stage, the neutrino mass spectrum consists of two massless fields $\nu'_{1,2}$ and two Majorana massive $\nu'_{P,F}$ neutrinos [@cj] $$m'_{\nu_P}=\frac{1}{2}[(4A^2+M^2)^{\frac{1}{2}}-M],\;\;
m'_{\nu_F}=\frac{1}{2}[(4A^2+M^2)^{\frac{1}{2}}+M],
\label{fp}$$
Note that $m'_{\nu_F}$ is arbitrary and in fact could be heavier than the lepton-$\tau$. Constraints on the masses $m'_{\nu_P}$ and $m'_{\nu_F}$ coming from the measured $Z^0$ invisible width were considered in Ref. [@eovr].
In the primed basis for the charged leptons, Yukawa couplings with the scalars $\eta^-_1$ and $\eta^+_2$ can be written as $$\frac{i}{2}\overline{l'_{Ra}}\left(U F\right)_{ab}
\nu''_{Lb}\eta^-_1
+\frac{i}{2}\overline{(\nu''^c)_{Ra}}(FU^\dagger)_{ab}l'_{Lb}\eta^+_2+H.c.
\label{yukawac}$$ As there are four neutrinos but only three charged leptons, it is possible to extend the charged lepton column with a zero on the fourth row in such a way that in Eq. (\[yukawac\]) all matrices are $4\times 4$. In Eq. (\[yukawac\]) the neutrinos are still linear combinations of the mass eigenstates, $\nu''_{a L}=(-\Phi P^T)_{al}N'_{lL},\,l=1,2,3,4$. Let us denote $\Gamma= UF $. Notice that in Eq. (\[yukawac\]) the $F$ matrix is in the basis in which $f_{e\mu}=f_{e\tau}=0$ and for this reason the electron does not interact with any neutrino.
Due to the mixing of $\eta^-_1$ and $\eta^-_2$ and to the non-diagonal Yukawa couplings $\Gamma$, diagrams like the one showed in Fig. 1 exist and they are finite. Notice that the primed fields $(\mu',\tau')$ couple with the two massive neutrinos $\nu'_P,\nu'_F$, then the neutrino masses insertions are $m'_{\nu_P},m'_{\nu_F}$. Then, the diagrams in Fig. 1 induce a contribution to the mass matrix of the $\mu',\tau'$. In fact contributions from Fig. 1 have the following form $$\delta_{ab}\sim \lambda_{10}m'\alpha_l\beta_{l'}
\Gamma_{al}\Gamma_{l'b}\left(\frac{v_\rho v_\eta}{m_\eta^2}\right)
\ln\left(
\frac{m_{\eta_1}^2}{m_{\eta_2}^2}\right),\quad a,b=\mu,\tau
\label{lm}$$ where $m'$ means $m'_{\nu_P}$ or $m'_{\nu_F}$, $m_{\eta_1}^2,
m_{\eta_2}^2$ are typical masses in the scalar sector and $m^2_\eta$ is the greatest of them. The $\Gamma$’s are the couplings appearing in Eq. (\[yukawac\]) and $\alpha_l,\beta_{l'}$ denote the $\nu''$’s projections in the $\nu'_{P,F}$ components i.e., can be read off from $\nu''_{aL}=-(\Phi P^T)_{al}N_l,\;l=3,4$. As an example the $\nu'_F$ contribution in Fig. 1 induces the following mass matrix in the $\mu',\tau'$ basis $$m'\alpha_l\beta_{l'}
\Gamma_{al}\Gamma_{l'b}\sim\frac{1}{D_2^2}\left(
\begin{array}{cc}
mD_2^2+m'_{\nu_F}(h_em'_{\nu_F}-A^2)^2(h_\tau-h_\mu)^2 &
-im'_{\nu_F}(h_em'_{\nu_F}-A^2)^2(h_\tau^2-h_\mu^2) \\
im'_{\nu_F}(h_em'_{\nu_F}-A^2)^2(h_\tau^2-h_\mu^2)
&mD_2^2+m'_{\nu_F}(h_em'_{\nu_F}-A^2)^2(h_\tau+h_\mu)^2
\end{array}
\right).
\label{mutau}$$ Notice that the diagonal terms have the contributions of the tree level mass $m$. We see from Fig.1 that an arbitrary mass matrix arises for the charged leptons $\mu'$ and $\tau'$ at the 1-loop level breaking their mass degeneracy. Since $\nu'_F$ can be heavier than the tau lepton [@eovr] the mass difference $m_\tau-m_\mu$ may be fitted with reasonable values for the parameters present in the model. Notwithstanding, the electron is still massless.
Since at the tree level $\nu'_1$ and $\nu'_2$ are massless we may define linear combinations, say $\tilde\nu'_1$ and $\tilde\nu'_2$ in such a way that $\tilde\nu'_2$ does not couple to one of the charged leptons, say the electron. However, if one makes this choice in the vector current, $\tilde\nu'_2$ will still couple to the electron through the Yukawa interactions in Eq. (\[yukawac\]). This is due to the presence of the $F$ matrix. On the contrary if we choose that $\tilde\nu'_2$ does not couple to the electron in Eq. (\[yukawac\]) it will couple to it through the vector current.
At 1-loop level processes like those showed in Fig.2 are also possible. Here the mass insertions are $m$ i.e., the charged lepton mass at tree level. It implies also a symmetric matrix in addition to $M^\nu$ in Eq. (\[mnu\]). Hence, the massless neutrinos (at tree level) $\nu'_2$ acquire masses from 1-loop radiative corrections. In Fig. 2 we show the case of $\nu'_2\to\nu'_F$ mixing. However, $\nu'_1$ is still massless.
After these loop corrections have been taken into account we have the basis $(e'',\mu,\tau)$ and $(\nu'_1,\nu_2,\nu_P,\nu_F)$ with the respective masses $(0,m_\mu,m_\tau)$, and $(0,m_2,m_{\nu_P},m_{\nu_F})$.
In the model there are two singly charged vector bosons $W^\pm$ and $V^\pm$. Their interactions with the leptons are $\bar\nu_{lL}V_l\gamma^\mu
l_LW^+_\mu$ and $\bar l^c_LV^\dagger_l\gamma^\mu
\nu_{lL}V^+_\mu$ [@pp] respectively. At this stage since three neutrinos are already massive we have a general mixing among them i.e., $ V_l= P'^T\Phi P^TU^\dagger U'^\dagger$ is a $4\times 4$ matrix, $P'$ is the arbitrary unitary $3\times 3$ matrix which diagonalize the mass matrix of the three massive neutrinos after the contributions of Fig. 2 have been taken into account. On the other hand $U'$ has the same structure than $U$ in Eq. (\[u\]) but now with arbitrary elements. The matrices $U,U'$ and $P'$ are appropriately extended by adding zeros and with $U_{44}=U'_{44}=P'_{11}=1$, in order to write them as $4\times 4$ matrices and the charged lepton column written as $(e'',\mu,\tau,0)$. Hence, it is straightforward to convince ourselves that $V_l$ has the appropriate form to induce diagrams like that showed in Fig. 3. These diagrams which exchange one $W^-$ and one $V^+$ as in Ref. [@babu] are possible and they are responsible for the mass of the electron as is shown in Fig.3. In this figure, we show only the contribution involving $\nu_F$ and the tau lepton. This is a higher order contribution, however it is proportional to $m_\tau m_{\nu_F}^2$ and for this reason the electron mass can be of the appropriate size.
Let us back to the $\nu'_1$ neutrino. This particle is up to now massless, however notice that after the electron has got a mass there is a contribution similar to the Fig. 2 but now the electron in the internal line. This kind of processes mix all the four neutrinos and $\nu'_1$ acquire a mass.
We have shown that if one does not introduce the sextet $S$, it is possible to give the right mass to all leptons if at least a single right-handed neutrino and a ${\cal F}$-violating term in the scalar potential are added to the minimal model. It is possible, of course, the case with three right-handed neutrinos, but we will not treat this case here.
It is interesting that in a supersymmetric version of the model it is also possible to give mass to the charged leptons without introducing the sextet of scalars [@ema].
Hence, in this 331 model, the smallness of the masses of the electron, the lightest neutrinos and the mass difference between muon and tau arise naturally.
We are very grateful to Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for full financial support (M.D.T.) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for partial (V.P.) and full (F.P.) financial support. We also thank O.L.G. Peres for useful discussions.
S. Weinberg, Phys. Rev. Lett. [**29**]{}, 388(1972); H. Georgi and S.L. Glashow, Phys. Rev. D[**6**]{}, 2977(1972), D[**7**]{}, 2475(1973); R. N. Mohapatra, Phys. Rev. D[**9**]{}, 3461(1974). E. Ma, Phys. Rev. Lett. [**64**]{}, 2866(1990) and references therein. F. Pisano and V. Pleitez, Phys. Rev. D[**46**]{}, 410 (1992). P.H. Frampton, Phys. Rev. Lett. [**69**]{}, 2889(1992). V. Pleitez and M.D. Tonasse, Phys. Rev. D [**48**]{}, 5274(1993). R. Foot, O. F. Hernandez, F. Pisano and V. Pleitez, Phys. Rev. D[**47**]{}, 4158(1993). L. Epele, H. Fanchiotti, C. Garcia Canal and D. Gómez Dumm, CP violation in a $SU(3)\otimes U(1)$ electroweak model. We thank D.G. Dumm for sending us a copy of the manuscript. C. Jarlskog, Nucl. Phys. [**A518**]{}, 129(1990); Phys. Lett. [**B241**]{}, 579(1990). L. Wolfenstein, Phys. Lett. [**107B**]{}, 77(1981). C.O. Escobar, O.L.G. Peres, V. Pleitez and R. Zukanovich Funchal, Phys. Rev. D[**47**]{}, R1747(1993). K.S. Babu and E. Ma, Phys. Rev. Lett. [**61**]{}, 674(1988). T.V. Duong and E. Ma, Supersymmetric $SU(3)\otimes
U(1)$ model: Higgs structure at the electroweak energy level, preprint UCRHEP-T111, June 1993.
|
---
abstract: 'We define a diffeology on the Milnor classifying space of a diffeological group $G$, constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori.'
address:
- 'LAREMA, Université d’Angers, 2 Bd Lavoisier , 49045 Angers cedex 1, France and Lycée Jeanne d’Arc, 40 avenue de Grande Bretagne, 63000 Clermont-Ferrand, France'
- 'Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO, USA 80309.'
author:
- 'Jean-Pierre Magnot'
- Jordan Watts
title: 'The Diffeology of Milnor’s Classifying Space'
---
Introduction
============
Let $G$ be a topological group with a reasonable topology (Hausdorff, paracompact, and second-countable, say). Milnor [@milnor2] constructed a universal topological bundle $EG\to BG$ with structure group $G$ satisfying:
- for any principal $G$-bundle $E\to X$ over a space (again, assume Hausdorff, paracompact, and second-countable) there is a continuous “classifying map” $F\colon X\to BG$ for which $E$ is $G$-equivariantly homeomorphic to the pullback bundle $F^*EG,$
- any continuous map $F\colon X\to BG$ induces a bundle $F^*EG$ with $F$ a classifying map, and
- any two principal $G$-bundles are isomorphic if and only if their classifying maps are homotopic.
To push this classification into the realm of Lie groups and smoothness, one would need smooth structures on the spaces $EG$ and $BG$. Many approaches exist, which extend to infinite-dimensional groups (see, for example, [@mostow], [@KM Theorem 44.24]). The main point here is that $EG$ and $BG$ are not typically manifolds, and so a more general smooth structure is required.
In this paper, we take the approach of diffeology. The language is quite friendly, allowing one to differentiate and apply other analytical tools with ease to infinite-dimensional groups such as diffeomorphism groups, including those of non-compact manifolds, as well as projective limits of groups, including some groups which appear naturally in the ILB setting of Omori [@Om], and may not exhibit atlases. Moreover, we can include in this paper interesting groups which are not typically considered as topological groups. For example, irrational tori (see Example \[x:irrational torus\]) have trivial topologies (and hence have no atlas) and only constant smooth functions; however, they have rich diffeologies, which hence are ideal structures for studying the groups. Irrational tori appear in important applications, such as pre-quantum bundles on a manifold associated to non-integral closed 2-forms (see Subsection \[ss:irrational torus\]). Another benefit of using diffeology is that we can directly use the language to construct connection 1-forms on $EG$. Our main source for the preliminaries on diffeology is the book by Iglesias-Zemmour [@iglesias].
Our main results include Theorem \[t:BG\], which states that there is a natural bijection between isomorphism classes of so-called D-numerable principal $G$-bundles over a Hausdorff, second-countable, smoothly paracompact diffeological space $X$, and smooth homotopy classes of maps from $X$ to $BG$. This holds for any diffeological group $G$. As well, we prove Theorem \[t:iz-connection\], which states that if $G$ is a regular diffeological Lie group, and $X$ is Hausdorff and smoothly paracompact, then any D-numerable principal $G$-bundle admits a connection; in particular, such a bundle admits horizontal lifts of smooth curves. These theorems provide a method for constructing classifying spaces different, for example, to what Kriegl and Michor do in [@KM Theorem 44.24] with $G=\Diff(M)$ for $M$ a compact smooth manifold, where they show that the space of embeddings of $M$ into $\ell^2$ yields a classifying space for $G$.
Our framework is applied to a number of situations. We show that $EG$ is contractible (Proposition \[p:contractible\]), which allows us to study the homotopy of $BG$ (Proposition \[p:homotopy of BG\]). We also study smooth homotopies between groups, and how these are reflected in classifying spaces and principal bundles (Subsection \[ss:homotopies\]). We transfer the theory to general diffeological fibre bundles via their associated principal $G$-bundles (Corollary \[c:BG\]) and discuss horizontal lifts in this context (Proposition \[p:assoc lifts\]). We also transfer the theory to diffeological limits of groups, with an application to certain ILB principal bundles (Proposition \[p:ilb\]). We show that a short exact sequence of diffeological groups induces a long exact sequence of diffeological homotopy groups of classifying spaces (Proposition \[p:les\]), and apply this to a short exact sequence of pseudo-differential operators and Fourier integral operators (Example \[x:pdo\]). Finally, we apply this theory to irrational torus bundles, which are of interest in geometric quantisation [@weinstein], [@iglesias-bdles], [@iglesias Articles 8.40-8.42] and the integration of certain Lie algebroids [@crainic].
This paper is organised as follows. Section \[s:diffeology\] reviews necessary prerequisites on diffeological groups (including diffeological Lie groups and regular groups), internal tangent bundles, and diffeological fibre bundles. In Section \[s:classifying sp\] we construct the diffeological version of the Milnor classifying space, $EG\to BG$, and prove Theorem \[t:BG\]. In Section \[s:connection\], we introduce the theory of connections from the diffeological point-of-view, and prove Theorem \[t:iz-connection\]. Finally, in Section \[s:applications\], we have our applications.
A few open questions are inspired. The conditions on the D-topology (in particular, Hausdorff and second-countable conditions, and sometimes smooth paracompactness as well) seem out-of-place in the general theory of diffeology. Even though these conditions are satisfied in our examples, in some sense, topological conditions and arguments should be *replaced* with diffeological conditions and arguments. This leads to the first question.
Under what conditions is the D-topology of a diffeological space Hausdorff, second-countable, and smoothly paracompact? Can one weaken these conditions?
An answer to this would allow us to rephrase Theorem \[t:BG\] in a manner more natural to diffeology. A partial answer, for example, is to require that the smooth real-valued functions separate points: in this case, the weakest topology induced by the functions is Hausdorff, and this is a sub-topology of the D-topology. It follows that the D-topology is Hausdorff.
Another question is an obvious one:
Given a diffeological group $G$, if $E\to X$ is a principal $G$-bundle satisfying necessary mild conditions, and the total space $E$ is diffeologically contractible, is $E\to X$ **universal** in the sense that any principal $G$-bundle over a sufficiently nice diffeological space $Y$ has a unique (up to smooth homotopy) classifying map to $X$? (The answer is affirmative in the topological category.)
The theme of universality continues:
Let $G$ be a regular diffeological Lie group (Definition \[d:diffeol lie gp\]), and let $E\to X$ be a principal $G$-bundle in which the D-topology on $X$ satisfies mild conditions. Is every diffeological connection, or even every connection 1-form, the pullback of the diffeological connection (or connection 1-form) constructed on $EG\to BG$ in Theorem \[t:universal connection\]?
Some of these questions have been addressed in [@CW2]. As mentioned above, Kriegl and Michor give a classifying space for the group $\Diff(M)$ where $M$ is a compact smooth manifold, and also show that this classifying space has a universal connection. However, the proof uses the fact that every such $M$ has an isometric embedding into some Euclidean space, and we no longer have this advantage for general diffeological groups.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank Patrick Iglesias-Zemmour and the other organisers of the “Workshop on Diffeology, etc.”, held in Aix-en-Provence in June 2014, where the discussions that lead to this paper began. We would also like to thank Daniel Christensen and Enxin Wu, as well as the anonymous referee, for excellent comments and suggestions.
Preliminaries
=============
This section provides background on diffeological groups and fibre bundles. For a review of more basic properties of diffeological spaces, we refer to the book of Iglesias-Zemmour [@iglesias]. In particular, products, sub-objects, quotients, and underlying D-topologies of diffeological spaces, as well as homotopies of maps between them, are used throughout this paper.
Diffeological Groups and Internal Tangent Bundles
-------------------------------------------------
In this subsection we review diffeological groups and their actions. We then introduce the internal tangent bundle of a diffeological space with the goal of obtaining a Lie algebra for certain diffeological groups admitting an exponential map. For more details on the basics of diffeological groups, see [@iglesias Chapter 7]. For more on internal tangent bundles, see [@CW]. For more details on diffeological Lie groups, see [@leslie] and [@KM] (although the latter reference deals with infinite-dimensional groups, not diffeological ones).
A **diffeological group** is a group $G$ equipped with a diffeology such that the multiplication map $m\colon G\times G\to G$ and the inversion map $\operatorname{inv}\colon G\to G$ are smooth. A **diffeological group action** of $G$ on a diffeological space $X$ is a group action in which the map $G\times X\to X$ sending $(g,x)$ to $g\cdot x$ is smooth. Fixing $g\in G$, denote by $L_g$ left multiplication by $g$, and by $R_g$ right multiplication by $g$. Denote by $e$ the identity element of $G$.
In order to begin associating a Lie algebra to certain diffeological groups, we need to establish the theory of tangent spaces and bundles to a diffeological space.
Let $(X,\mathcal{D})$ be a diffeological space, and fix $x\in X$. Let $\mathbf{Plots_0}(\mathcal{D},x)$ be the category whose objects are plots $(p\colon U\to X)\in\mathcal{D}$ for which $U$ is connected, $0\in U$, and $p(0)=x$; and whose arrows from $p\colon U\to X$ to $q\colon V\to X$ are commutative triangles $$\xymatrix{
U \ar[rr]^{f} \ar[dr]_{p} & & V \ar[dl]^{q} \\
& X & \\
}$$ where $f\colon U\to V$ is smooth and $f(0)=0$. Let $F$ be the forgetful functor from $\mathbf{Plots}_0(\mathcal{D},x)$ to $\mathbf{Open}_0$, the category whose objects are connected open subsets of Euclidean spaces containing $0$ and whose arrows are smooth maps between these that fix $0$. $F$ sends a plot to its domain, and a commutative triangle as above to $f\colon U\to V$. Finally, let $\mathbf{Vect}$ be the category of vector spaces with linear maps between them, and let $T_0\colon \mathbf{Open}_0\to\mathbf{Vect}$ be the functor sending $U$ to $T_0U$ and $f\colon U\to V$ to $f_*|_0\colon T_0U\to T_0V$. Then define the **internal tangent space** $T_xX$ to be the colimit of the functor $T_0\circ F$. Denote by $TX$ the set $\bigsqcup_{x\in X}T_xX$.
In order to equip $TX$ with a suitable diffeology, we need a few more definitions.
Let $(X,\mathcal{D})$ be a diffeological space. For a plot $(p\colon U\to X)\in\mathbf{Plots}_0(\mathcal{D},x)$, denote by $p_*$ the map sending vectors in $T_0U$ to $T_{p(0)}X$ given by the definition of a colimit. Extend this to more general plots $p\colon U\to X$ in $\mathcal{D}$ as follows. Define $p_*\colon TU\to TX$ to be the map that sends $v\in T_uU$ to the element in $T_{p(u)}X$ given by $(p\circ\tr_u)_*((\tr_{-u})_*v)$, where $\tr_u$ is the translation in Euclidean space sending $U-u$ to $U$. If $c\colon\RR\to X$ is a smooth curve, then by $\frac{dc}{dt}$ we mean $c_*\left(\frac{d}{dt}\right)$ where $\frac{d}{dt}$ is the constant section $\RR\to T\RR\colon t\mapsto(t,1)$. We may also denote $p_*$ by $Tp$, emphasising the functoriality of $T$ (see Remark \[r:tangent bdle\]).
A **diffeological vector space** is a vector space $V$ over $\RR$ equipped with a diffeology such that addition $+\colon V\times V\to V$ and scalar multiplication $\cdot\colon\RR\times V\to V$ are smooth.
We use the terminology of [@pervova], but continue to follow [@CW].
A **(diffeological) vector pseudo-bundle** $\pi\colon E\to X$ is a pair of diffeological spaces $E$ and $X$ with a smooth surjection $\pi\colon E\to X$ between them such that for each $x\in X$ the fibre $\pi^{-1}(x)$ is a diffeological vector space. Moreover, fibrewise addition $+\colon E\times_X E\to E$, scalar multiplication $\cdot\colon\RR\times E\to E$, and the zero section $X\hookrightarrow E$ are required to be smooth.
Let $\pi\colon E\to X$ be a smooth map between diffeological spaces such that for each $x$, the fibre $\pi^{-1}(x)$ is a diffeological vector space. Then there is a smallest diffeology on $E$ that contains the original diffeology on $E$ and so that $\pi$ is a vector pseudo-bundle; see [@CW Proposition 4.6].
Define the **internal tangent bundle** of a diffeological space $X$ to be the set $TX:=\bigsqcup_{x\in X}T_xX$ equipped with the smallest diffeology $T\mathcal{D}$ such that
1. $TX$ is a vector pseudo-bundle,
2. for each plot $p\colon U\to X$, the induced map $Tp\colon TU\to TX$ is smooth.
$T$ is a functor, sending diffeological spaces $(X,\mathcal{D})$ to $(TX,T\mathcal{D})$. Consequently, we get the chain rule: given smooth maps $f\colon X\to Y$ and $g\colon Y\to Z$, we have that $T(g\circ f)=Tg\circ Tf$. Moreover, $T$ respects products: let $X$ and $Y$ be diffeological spaces. Then $T(X\times Y)\cong TX\times TY$ [@CW Proposition 4.13].
We now arrive at an important generalisation of what is very well known in standard Lie group theory.
Let $G$ be a diffeological group with identity element $e$. Then, $TG$ is isomorphic as a vector pseudo-bundle to $G\times T_eG$.
This is [@CW Theorem 4.15]. The diffeomorphism is given by sending $v\in T_gG$ to $(g,(L_{g^{-1}})_*v)$.
Diffeological groups admit adjoint actions of $G$ on $T_{e}G$.
Let $G$ be a diffeological group, and let $C$ be the (smooth) **conjugation action**: $C\colon G\times G\to G\colon (g,h)\mapsto ghg^{-1}$. Define the **adjoint action** of $G$ on $T_{e}G$ to be the (smooth) action $g\cdot\xi:=TC_g(\xi)$ for all $\xi\in T_eG$.
This does not immediately give us a well-defined infinitesimal adjoint action (*i.e.* Lie bracket) on $T_{e}G$. This issue is resolved by Leslie in [@leslie] by considering the following type of diffeological group.
A diffeological group $G$ is a **diffeological Lie group** if
1. for every $\xi\in T_eG$ there exists a smooth real-valued linear functional $\ell\colon T_eG\to\RR$ satisfying $\ell(\xi)\neq 0$,
2. the plots of $T_{e}G$ smoothly factor through a map $\varphi\colon V\to T_eG$, where $V$ is an open subset of a complete Hausdorff locally convex topological vector space.
Let $G$ be a diffeological Lie group. Then $T_{e}G$ is a Lie algebra under the infinitesimal adjoint action ([@leslie Theorem 1.14]). In this case, we denote $T_{e}G$ by $\g$.
Finally, in order to associate curves in $G$ with curves in $\g$, we need to introduce the notion of a *regular* Lie group, following [@leslie], [@KM Section 38], and [@Om]. This correspondence is important when obtaining horizontal lifts of fibre bundles from connection 1-forms later in the paper.
A diffeological Lie group $G$ is **regular** if there is a smooth map $$\exp\colon \CIN([0,1],\g)\to\CIN([0,1],G)$$ sending a smooth curve $\xi(t)$ to a smooth curve $g(t)$ such that $g(t)$ is the unique solution of the differential equation $$\begin{cases}
g(0)= e,\\
(R_{g(t)^{-1}})_*\frac{dg(t)}{dt}= \xi(t).
\end{cases}$$
Unless specified otherwise, we take the subset diffeology on $[0,1]\subseteq\RR$ throughout the paper.
Not all diffeological groups are regular. For example, the diffeological group $\Diff_+(0,1)$ see [@Ma2011] or [@Ma2015 Subsection 2.5.2].
If $G$ is a regular diffeological Lie group, then any $v\in\g$ is a germ of a smooth path $c(t)=\exp(tv)$. This fact is not guaranteed when $G$ is not regular, as in this case $v\in T_{e}G$ is only a sum of germs, and not necessarily a germ on its own. This explains the different notations and definitions found in [@Ma2013] for Frölicher Lie groups. However, when $G$ is regular and $\g$ complete, the different definitions coincide.
Diffeological Fibre Bundles
---------------------------
We now review diffeological fibre bundles and principal $G$-bundles; for more details, see [@iglesias]. Instead of using the original definition of a diffeological fibre bundle given in [@IgPhD], we take the equivalent definition below (see [@iglesias Article 8.9]). Similarly, we take an equivalent definition for a principal $G$-bundle (see [@iglesias Article 8.13]).
Let $\pi\colon E\to X$ be a smooth surjective map of diffeological spaces.
1. Define the **pullback** of $\pi\colon E\to X$ by a smooth map $f\colon Y\to X$ of diffeological spaces to be the set $$f^*E\colon =\{(y,e)\in Y\times E~|~f(y)=\pi(e)\}$$ equipped with the subset diffeology induced by the product. It comes with the two smooth maps $\tilde{f}\colon f^*E\to E$ and $\pi_f\colon f^*E\to Y$ induced by the projection maps, the latter of which is also surjective.
2. $\pi\colon E\to X$ is **trivial** with **fibre** $F$ if there is a diffeological space $F$ and a diffeomorphism $\varphi\colon E\to X\times F$ making the following diagram commute.
$$\xymatrix{
E \ar[rr]^{\varphi} \ar[dr]_{\pi} & & X\times F \ar[dl]^{\pr_1} \\
& X & \\
}$$
3. $\pi\colon E\to X$ is **locally trivial** with fibre $F$ if there exists an open cover $\{U_\alpha\}_{\alpha}$ in the D-topology on $X$ such that for each $\alpha$, the pullback of $\pi\colon E\to X$ to $U_\alpha$ via the inclusion map is trivial with fibre $F$. The collection $\{(U_\alpha,\varphi_\alpha)\}_\alpha$, where the diffeomorphism $\varphi_\alpha\colon E|_{U_\alpha}\to U_\alpha\times F$ is as above, make up a **local trivialisation** of $\pi$.
4. We say that $\pi\colon E\to X$ is a **diffeological fibration** or **diffeological fibre bundle** if for every plot $p\colon U\to X$, the pullback bundle $p^*E\to U$ is locally trivial.
5. Let $G$ be a diffeological group. We say that a diffeological fibre bundle $\pi\colon E\to X$ is a **principal $G$-bundle** if there is a smooth action of $G$ on $E$ for which every plot $p\colon U\to X$ induces a pullback bundle $p^*E\to U$ that is **locally equivariantly trivial**. That is, for any $u\in U$ there exists an open neighbourhood $V$ of $u$, a plot $q\colon V\to E$ satisfying $\pi\circ q=p|_V$, and an equivariant diffeomorphism $\psi\colon V\times G\to p|_V^*E$ sending $(v,g)$ to $(v,g\cdot q(v))$. Furthermore, a principal $G$-bundle itself is **locally equivariantly trivial** if it admits a local trivialisation $\{U_\alpha,\varphi_\alpha\}_\alpha$ in which each $\varphi_\alpha$ is $G$-equivariant.
6. We say that a diffeological fibre bundle is **weakly D-numerable** if there exists a local trivialisation $\{U_i\}_{i\in\NN}$ of $X$ and a pointwise-finite smooth partition of unity $\{\zeta_i\}_{i\in\NN}$ of $X$ such that $\zeta_i^{-1}((0,1])\subseteq U_i$ for each $i$. We say that a weakly D-numerable diffeological fibre bundle is **D-numerable** if the local trivialisation can be chosen to be locally finite and the smooth partition of unity subordinate to $\{U_i\}$; that is, $\supp(\zeta_i)\subseteq U_i$ for each $i\in\NN$.
The point of having weak D-numerability versus D-numerability is that *a priori*, as we will see below, the Milnor construction as a bundle is not necessarily D-numerable, only weakly D-numerable.
Diffeological fibre bundles pull back to diffeological fibre bundles. Moreover, trivial bundles pull back to trivial bundles. Consequently, (weakly) D-numerable bundles pull back to (weakly) D-numerable bundles. Similar statements hold for principal $G$-bundles.
Not every diffeological fibre bundle is locally trivial. The 2-torus modulo the irrational Kronecker flow is such an example [@iglesias Article 8.38].
A weakly D-numerable principal diffeological fibre bundle with Hausdorff, smoothly paracompact base is D-numerable. A locally trivial diffeological fibre bundle over a Hausdorff, second-countable, smoothly paracompact base is also D-numerable. We include the definition of smooth paracompactness below for completeness.
A diffeological space $X$ is **smoothly paracompact** if the D-topology of $X$ is paracompact (any open cover admits a locally finite open refinement), and any open cover of $X$ admits a *smooth* partition of unity subordinate to it.
Milnor’s Classifying Space
==========================
We begin this section with the construction of a classifying space of a diffeological group $G$, completely in the diffeological category (Proposition \[p:EG principal\]). Under certain topological constraints on a diffeological space $X$, we get the desired natural bijection between homotopy classes of diffeologically smooth maps $X\to BG$ and D-numerable principal $G$-bundles over $X$ (see Theorem \[t:BG\]).
The Milnor Construction
-----------------------
We construct Milnor’s classifying space in the diffeological category for a diffeological group $G$. This results in a principal $G$-bundle $\pi\colon EG\to BG$ that is at least weakly D-numerable.
Let $\{X_i\}_{i\in\NN}$ be a family of diffeological spaces. Define the **join** $\bigstar_{i\in\NN}X_i$ of this family as follows. Take the subset $S$ of the product $\left(\Pi_{i\in \NN}[0,1]\right)\times\left(\Pi_{i\in \NN}X_i\right)$ consisting of elements $(t_i,x_i)_{i\in \NN}$ in which only finitely many of the $t_i$ are non-zero, and $\sum_{i\in \NN}t_i=1$. Equip $S$ with the subset diffeology induced by the product diffeology. Let $\sim$ be the equivalence relation on $S$ given by: $(t_i,x_i)_{i\in \NN}\sim(t'_i,x'_i)_{i\in \NN}$ if
1. $t_i=t'_i$ for each $i\in \NN$, and
2. if $t_i=t'_i\neq 0$, then $x_i=x_i'$.
Then the join is the quotient $S/\!\sim$ equipped with the quotient diffeology. We denote elements of $\bigstar_{i\in \NN}X_i$ by $(t_ix_i)$.
Let $G$ be a diffeological group. Define $EG = \bigstar_{i \in \NN}G$. There is a natural smooth action of $G$ on $EG$ given by $h\cdot(t_ig_i)=(t_ig_ih^{-1})$ induced by the diagonal action of $G$ on $G^{\NN}$ and the trivial action on $[0,1]^{\NN}$. Denote the quotient $EG/G$ by $BG$ and elements of $BG$ by $[t_ig_i]$. This is the **(diffeological) Milnor classifying space** of $G$. Since we will make use of it, denote by $S_G$ the subset of $[0,1]^\NN\times G^\NN$ corresponding to the set $S$ in Definition \[d:join\].
The action of $G$ on $EG$ is smooth.
We want to show that the map $a_E\colon G\times EG\to EG$ sending $(h,(t_ig_i))$ to $(t_ig_ih^{-1})$ is smooth. Let $p\colon U\to G\times EG$ be a plot. It is enough to show that $a_E\circ p$ locally lifts to a plot of $S_G$ via the quotient map $\pi\colon S_G\to EG$. Let $\pr_i$ ($i=1,2$) be the natural projection maps on $G\times EG$. Then there exist an open cover $\{U_\alpha\}$ of $U$ and for each $\alpha$ a plot $q_\alpha\colon U_\alpha\to S_G$ such that $$\pr_2\circ p|_{U_\alpha}=\pi\circ q_\alpha.$$ Let $a\colon G\times S_G\to S_G$ be the smooth map sending $(h,(t_i,g_i))$ to $(t_i,g_ih^{-1})$. Then, $$a_E\circ p|_{U_\alpha}=\pi\circ a(q_\alpha,\pr_1\circ p|_{U_\alpha}),$$ where the right-hand side is a plot of $EG$.
$EG \to BG$ is a weakly D-numerable principal $G$-bundle.
Define for each $j\in\NN$ the function $s_j\colon EG\to[0,1]$ by $s_j(t_ig_i):=t_j$. These are diffeologically smooth, hence continuous with respect to the D-topology. For each $j\in\NN$ define the open set $$V_j:=s_j^{-1}(0,1]=\{(t_ig_i) \mid t_j > 0\}.$$ Since for any point $(t_ig_i)\in EG$ we have that $\sum t_i=1$, it follows that $\{V_j\}$ forms an open cover of $EG$. Since each $V_j$ is $G$-invariant, setting $U_j:= \pi(V_j)$ we get an open covering $\{U_j\}$ of $BG$.
Define for each $j\in \NN$ the map $\varphi_j\colon V_j \to G\times U_j$ by $$\varphi_j(t_ig_i) := (g_j,[t_ig_ig_j^{-1}]).$$ This map is well-defined, smooth, and $G$-equivariant where $G$ acts trivially on $BG$ and via $k\cdot g=gk^{-1}$ on itself. Moreover, $\varphi_j$ is invertible with smooth inverse: $$\varphi_j^{-1}(k,[t_ig_i]) = (t_ig_ik)$$ where we set $g_j=e$. Thus, $\{(U_j,\varphi_j)\}$ is a local trivialisation of $EG\to BG$. It follows that $EG\to BG$ is a locally trivial diffeological principal $G$-bundle.
Finally, for each $(t_ig_i)\in EG,$ $$\sum_{j \in \NN} s_j(t_ig_i) = \sum_{j \in \NN} t_j = 1,$$ and each $s_j$ is $G$-invariant and so descends to a smooth map $\zeta_j\colon BG\to[0,1]$ with $\zeta_j^{-1}(0,1]=U_j$. The collection $\{\zeta_j\}$ is a pointwise-finite smooth partition of unity on $BG$, and so we have shown that $\pi\colon EG\to BG$ is weakly D-numerable.
Classifying Bundles
-------------------
The main result concerning a classifying space of a diffeological group $G$ is that it classifies principal $G$-bundles, up to isomorphism. To establish this, we follow the topological presentation by tom Dieck (see [@tD Sections 14.3, 14.4]). While many of the proofs only require slight modifications to ensure smoothness, some such as the proof to Proposition \[p:homotopy\] requires the development of some diffeological theory related to the D-topology and homotopy, which appears in Lemmas \[l:open cover\], \[l:smoothly T4\], and \[l:homotopy\]. To make the classification result precise, we introduce the following notation.
Let $G$ be a diffeological group, and let $X$ be diffeological space. Denote by $\mathcal{B}_G(X)$ the set of isomorphism classes of D-numerable principal $G$-bundles over $X$, and denote by $[X,BG]$ the set of smooth homotopy classes of smooth maps $X\to BG$. Given another diffeological space $Y$ and a smooth map $\varphi\colon X\to Y$, define $\mathcal{B}(\varphi)$ to be the pullback $\varphi^*\colon \mathcal{B}_G(Y)\to\mathcal{B}_G(X)$, and $[\varphi,BG]$ to be the pullback $\varphi^*\colon [Y,BG]\to[X,BG]$.
The main theorem of this section is the following.
Let $\mathbf{Diffeol}_{\operatorname{HSP}}$ be the full subcategory of $\mathbf{Diffeol}$ consisting of Hausdorff, second-countable, smoothly paracompact diffeological spaces, and let $G$ be a diffeological group. Then, $\mathcal{B}_G(\cdot)$ and $[\cdot,BG]$ are naturally isomorphic functors from $\mathbf{Diffeol}_{\operatorname{HSP}}$ to $\mathbf{Set}$.
To prove this, we begin by constructing a correspondence $[X,BG]\to\mathcal{B}_G(X)$.
Let $X$ be a diffeological space, and let $G$ be a diffeological group. For any smooth map $F\colon X \to BG$ the pullback bundle $F^*EG$ is weakly D-numerable. Additionally, if $X$ is Hausdorff and smoothly paracompact, then $F^*EG$ is D-numerable.
Let $F\colon X\to BG$ be a smooth map. Let the collection $\{\zeta_j\colon BG\to[0,1]\}_{j\in\NN}$ be the partition of unity in the proof of Proposition \[p:EG principal\]. For each $j\in\NN$, define $\xi_j\colon X\to[0,1]$ by $\xi_j:=\zeta_j\circ F.$ Then $\{\xi_j\}$ is a pointwise-finite smooth partition of unity, and we have an open cover of $X$ given by open sets $W_j:=\xi_j^{-1}(0,1]$. We now show that $F^*EG$ is trivial over each $W_j$.
Recall $$F^*EG=\{(x,(t_ig_i))\in X\times EG\mid F(x)=[t_ig_i]\},$$ where for all $k\in G$ we have the smooth action $k\cdot(x,(t_ig_i))=(x,(t_ig_ik^{-1}))$. Define $\widetilde{F}\colon F^*EG\to EG$ to be the second projection map. Define $\psi_j\colon \pi^{-1}(W_j)\to G\times W_j$ by $$\psi_j(x,(t_ig_i)):=(g_j,x).$$ Then $\psi_j$ is well-defined, smooth, $G$-equivariant, and has a smooth inverse. It follows that $F^*EG$ is weakly D-numerable.
If $X$ is Hausdorff and smoothly paracompact, we can choose an appropriate open refinement of $\{W_j\}$ and smooth partition of unity subordinate to it (see, for example, [@munkres Lemma 41.6]). This completes the proof.
To ensure that the correspondence $[X,BG]\to\mathcal{B}_G(X)$ is well-defined, we must show that smoothly homotopic maps yield isomorphic bundles. This is Proposition \[p:homotopy\]. To prove this, we need to add another topological constraint, second-countability, and establish a series of lemmas. Also, we need to consider the D-topology of a product of diffeological spaces. Generally, the D-topology induced by the product diffeology contains the product topology. However, we have the following result of Christensen, Sinnamon, and Wu in [@CSW Lemma 4.1].
Let $X$ and $Y$ be diffeological spaces such that the D-topology of $Y$ is locally compact. Then the D-topology of the product $X\times Y$ is equal to the product topology.
Let $X$ be a Hausdorff, paracompact, second-countable diffeological space, and let $\pi\colon E\to X\times\RR$ be a locally trivial diffeological fibre bundle. Then for any $a<b$, there exists a countable locally finite open cover $\mathcal{U}$ of $X$ such that $\pi$ is trivial over $U\times[a,b]$ for any $U\in\mathcal{U}$. In the case of a principal bundle, the trivialisation can be chosen to be equivariant.
Since $\RR$ is locally compact, the D-topology on $X\times\RR$ is equal to the product topology by Lemma \[l:prod top\]. Fix a local trivialisation (equivariant in the case of a principal bundle) $\mathcal{V}$ of $\pi$, and fix $x\in X$. Since $[a,b]$ is compact there exist $k>0$, for each $i=1,\dots,k$ an open interval $(s_i,t_i)$ such that $\bigcup_i(s_i,t_i)\supseteq[a,b]$, and for each $i$ an open neighbourhood $U_i$ of $x$ such that $U_i\times(s_i,t_i)$ is contained in an element of $\mathcal{V}$.
Let $U=\bigcap_iU_i$. By [@iglesias Note 1 of Article 8.16 and Lemma 1 of Article 8.19], $\pi$ is trivial over $U\times[a,b]$. Now, take such an open neighbourhood $U$ for each $x\in X$; this is an open cover of $X$. We now take an appropriate refinement of this cover to obtain $\mathcal{U}$.
It is standard that given a normal topological space, one can find a continuous function that separates disjoint closed sets. We need a smooth version of this fact.
Let $X$ be a smoothly paracompact diffeological space, and let $A$ and $B$ be disjoint closed subsets of $X$. Then there exists a smooth function $f\colon X\to[0,1]$ such that $f|_A\equiv 0$ and $f|_B\equiv 1$.
Consider the open cover $\{X\smallsetminus A,X\smallsetminus B\}$. It admits a smooth partition of unity $\{\zeta_{X\smallsetminus A},\zeta_{X\smallsetminus B}\}$. Let $f=\zeta_{X\smallsetminus A}$.[^1]
Let $X$ be a Hausdorff, second-countable, smoothly paracompact diffeological space, and let $\pi\colon E\to X\times\RR$ be a locally trivial diffeological fibre bundle. Then $E|_{X\times\{0\}}$ is bundle-diffeomorphic to $E|_{X\times\{1\}}$. In the case of a principal bundle, the bundle-diffeomorphism can be chosen to be equivariant.
By Lemma \[l:open cover\], there exists a locally finite open cover $\{U_i\}_{i\in\NN}$ of $X$ such that $\pi$ is trivial over $U_i\times[0,1]$ for each $i$. Let $\{(U_i\times[0,1],\psi_i)\}$ be the corresponding local trivialisation (equivariant in the case of a principal bundle) for $\pi$ restricted over $X\times[0,1]$. Since $X$ is Hausdorff and paracompact, there is a locally finite open refinement $\{V_i\}_{i\in\NN}$ of $\{U_i\}$ such that $\overline{V_i}\subseteq U_i$ for each $i$. By Lemma \[l:smoothly T4\] there is a family of smooth maps $\{b_i\colon X\to[0,1]\}_{i\in\NN}$ such that $b_i|_{V_i}=1$ and $\supp(b_i)\subseteq U_i$ for each $i$.
Fix $i$, and denote by $F$ the fibre of $\pi$. Let $r_i\colon X\times[0,1]\to X\times[0,1]$ be the smooth map sending $(x,t)$ to $(x,t+(1-t)b_i(x))$ and let $R_i\colon E\to E$ be the map equal to the identity over the complement of $U_i\times[0,1]$, and such that for all $(x,t,a)\in U_i\times[0,1]\times F$, $$\psi_i\circ R_i\circ\psi_i^{-1}(x,t,a)=(x,t+(1-t)b_i(x),a).$$ The pair $(r_i,R_i)$ form a smooth bundle map over $X\times[0,1]$, whose restriction to $E|_{X\times\{1\}}$ is the identity map. Moreover, the restriction of $R_i$ to $E|_{X\times\{0\}}$ is a diffeomorphism onto its image. In the case of a principal bundle, this is equivariant.
Let $r$ be the composition of the maps $r_i$, taken in order: $r=\dots\circ r_2\circ r_1$. This is well-defined since $r_i(x)=x$ for all but finitely many $i$. Similarly, define $R$ to be the composition of all $R_i$. The pair $(r,R)$ is a smooth bundle map, whose restriction to $E|_{X\times\{1\}}$ is the identity map. Moreover, the restriction of $R$ to $E|_{X\times\{0\}}$ is a diffeomorphism onto $E|_{X\times\{1\}}$. Again, in the case of a principal bundle, this is also equivariant.
Let $X$ and $Y$ be diffeological spaces, and assume that $X$ is Hausdorff, second-countable, and smoothly paracompact. Let $f_i\colon X\to Y$ ($i=0,1$) be smooth maps with a smooth homotopy $H\colon X\times\RR\to Y$ between them, and let $\pi\colon E\to Y$ be a diffeological fibre bundle. If $H^*E\to X$ is locally trivial, then the pullback bundles $f_i^*E\to X$ ($i=0,1$) are bundle-diffeomorphic. In the case of a principal bundle, the bundle-diffeomorphism can be chosen to be equivariant.
Assume that $H^*E\to X$ is locally trivial. Then there is a bundle-diffeomorphism between $H^*E|_{X\times\{0\}}$ and $H^*E|_{X\times\{1\}}$ by Lemma \[l:homotopy\]. But these two restricted bundles are exactly bundle-diffeomorphic to $f_0^*E$ and $f_1^*E$, respectively.
Iglesias-Zemmour proves the above proposition for diffeological fibre bundles admitting a diffeological connection; see Definition \[d:iz-connection\], [@iglesias Article 8.32 and Article 8.34].
We now have a well-defined correspondence $[X,BG]\to\mathcal{B}_G(X)$. To construct an inverse correspondence, we must first define a “classifying map” $X\to BG$ for a principal $G$-bundle $E\to X$. In order to show that the classifying map is smooth, we require that $E$ be weakly D-numerable with a Hausdorff and smoothly paracompact base (in which case, $E$ is D-numerable).
If $\pi\colon E\to X$ is a weakly D-numerable principal $G$-bundle in which $X$ is Hausdorff and smoothly paracompact, then there is a smooth $G$-equivariant map $\widetilde{F}\colon E\to EG$ which descends to a smooth map $F\colon X\to BG$, called a **classifying map** of $\pi$, such that $E$ is isomorphic as a principal $G$-bundle to $F^*EG$.
Let $\pi\colon E\to X$ be a weakly D-numerable principal $G$-bundle in which $X$ is Hausdorff and smoothly paracompact (and hence $\pi$ is in fact D-numerable by Remark \[r:d-numerable\]), and fix a locally finite local trivialisation $\{(W_j,\psi_j)\}_{j\in\NN}$ and a smooth partition of unity $\{\xi_j\}_{j\in\NN}$ subordinate to $\{W_j\}$. Define a map $\widetilde{F}\colon E\to EG$ by $$\widetilde{F}(y):=\big((\xi_i\circ\pi(y))(\pr_2\circ\psi_i(y))\big).$$ Since $\xi_j$ has support contained in $W_j$, $\widetilde{F}$ is well-defined.
To show that $\widetilde{F}$ is smooth, fix a plot $p\colon U\to E$. It is enough to show that $\widetilde{F}\circ p$ locally lifts to a plot of $S_G$. Fix $u\in U$. Since $\{W_j\}$ is a locally finite open cover, there is an open neighbourhood $B$ of $p(u)$ such that $B$ intersects only finitely many of the open sets $\pi^{-1}(W_j)$; without loss of generality, assume that these are $W_1,\dots,W_k$. Next, of these, only $l\leq k$ contain $p(u)$. Again, without loss of generality, assume that these are $W_1,\dots,W_l$. Since $\{\xi_j\}$ is subordinate to $\{W_j\}$, we have that the closed set $\bigcup_{j=1}^{k-l}\supp(\xi_{l+j})$ is disjoint from $p(u)$, which itself is closed since $X$ is Hausdorff. Furthermore, since $X$ is also paracompact, it is normal, and so shrink $B$ so that it only intersects $W_{l+1},\dots,W_k$ outside of each $\supp(\xi_j)$ ($j=l+1,\dots,k$).
Let $p_B:=p|_{p^{-1}(B)}$ and let $\rho\colon S_G\to EG$ be the quotient map. Then, $\widetilde{F}\circ p_B=\rho\circ\sigma_B$ where $\sigma_B\colon p^{-1}(B)\to S_G$ is defined to be the smooth map $\sigma_B(u)=(t_i(u),g_i(u))$ where $g_i(u)=e$ if $i>l$, $g_i(u)=\pr_2(\psi_i(p_B(u)))$ if $i=1,\dots,l$, and $t_i(u)=\xi_i(\pi(p_B(u)))$ for each $i$. It follows that $\widetilde{F}$ is a smooth map into $EG$. It is also $G$-equivariant, and so descends to a smooth map $F\colon X\to BG$.
To show that $E$ and $F^*EG$ are isomorphic, define a map $\Phi\colon E\to F^*EG$ by $\Phi(y):=(\pi(y),\widetilde{F}(y))$. Then, $\Phi$ is a well-defined smooth bijection. Fixing $x\in X$, let $B$ be an open neighbourhood of $x$ intersecting only finitely many $W_j$. Then $\Phi^{-1}|_{\pi^{-1}(B)}(x,(t_ig_i))=\psi_j^{-1}(x,g_j)$ for any $j$ such that $x\in W_j$, which is smooth, and hence $\Phi$ is a diffeomorphism. $G$-equivariance of $\Phi$ follows from the $G$-equivariance of $\widetilde{F}$.
If $\pi\colon E\to X$ is a principal $G$-bundle that has a smooth classifying map $F\colon X\to BG$, then it follows that $F^*EG$ is weakly D-numerable, and so by definition of a classifying map, $\pi\colon E\to X$ is also weakly D-numerable.
To show that our inverse correspondence $\mathcal{B}_G(X)\to[X,BG]$ is well-defined, we need to show that isomorphic bundles yield smoothly homotopic classifying maps. This is a consequence of the following proposition.
Let $G$ be a diffeological group, $E\to X$ a principal $G$-bundle, and $f,h\colon E\to EG$ two $G$-equivariant smooth maps. Then, $f$ and $h$ are smoothly $G$-equivariantly homotopic, and so descend to smoothly homotopic maps $X\to BG$.
Denote the images of $f$ and $h$ by $$f(y)=\big(s_1(y)f_1(y),s_2(y)f_2(y),\dots\big) \text{ and } h(y)=\big(t_1(y)h_1(y),t_2(y)h_2(y),\dots\big)$$ where $f_i(y),h_i(y)\in G$ and $s_i(y),t_i(y)\in[0,1]$. We will construct an homotopy $H$ such that $H(y,0)=f(y)$ and $H(y,1)=h(y)$ for all $y\in E$. The construction will be a concatenation of homotopies that we construct now. Throughout these, we use a smooth function $b\colon\RR\to[0,1]$ such that $b(\tau)=0$ for all $\tau\leq \eps$, $b(\tau)=1$ for all $\tau\geq 1-\eps$, and $\frac{db}{d\tau}\geq 0$, for some fixed $\eps\in(0,1/2)$.
Let $F_1\colon E\times[0,1]\to EG$ be defined by $$\begin{aligned}
F_1(y,\tau):=&~\big(s_1(y)f_1(y),b(\tau)s_2(y)f_2(y),(1-b(\tau))s_2(y)f_2(y),\\
&~b(\tau)s_3(y)f_3(y),(1-b(\tau))s_3(y)f_3(y),\dots\big).\end{aligned}$$ Then $F_1$ is smooth, and satisfies $$\begin{gathered}
F_1(y,0)=\big(s_1(y)f_1(y),0,s_2(y)f_2(y),0,\dots\big) \text{ and }\\
F_1(y,1)=\big(s_1(y)f_1(y),s_2(y)f_2(y),0,s_3(y)f_3(y),0,\dots\big).\end{gathered}$$ Similarly, for each $n>1$, let $F_n\colon E\times[0,1]\to EG$ be defined by $$\begin{aligned}
F_n(y,\tau):=&~\big(s_1(y)f_1(y),\dots,s_n(y)f_n(y),b(\tau)s_{n+1}(y)f_{n+1}(y),\\
&~(1-b(\tau))s_{n+1}(y)f_{n+1}(y),b(\tau)s_{n+2}(y)f_{n+2}(y),\dots\big).\end{aligned}$$ Each $F_n$ is smooth, and $F_n(y,0)=F_{n-1}(y,1)$ for all $n>1$.
Define $F\colon E\times[0,\infty)\to EG$ by $F(y,\tau)=F_{\lfloor\tau+1\rfloor}(y,\tau-\lfloor\tau\rfloor),$ where $\lfloor\cdot\rfloor$ is the floor function. Since each $F_n$ is constant near $\tau=0$ and $\tau=1$ for fixed $y$, it follows that $F$ is smooth. Let $c\colon[0,1)\to[0,\infty)$ be an increasing diffeomorphism ($x\mapsto\tan(\frac{\pi}{2}x)$, say), and define $F'\colon E\times[0,1)\to EG$ as the smooth composition $F\circ(\id_E\times c)$. Define $F'(y,1):=f(y)$. This yields a smooth extension $F'\colon E\times[0,1]\to EG$.
Similarly, for each $n>0$, define smooth maps $S_n\colon E\times[0,1]\to EG$ by $$\begin{aligned}
S_n(y,\tau):=&~\big(t_1(y)h_1(y),\dots,t_{n-1}(y)h_{n-1}(y),b(\tau)t_n(y)h_n(y),\\
&~(1-b(\tau))t_n(y)h_n(y),b(\tau)t_{n+1}(y)h_{n+1}(y),\dots\big).\end{aligned}$$ We have $S_n(y,0)=S_{n-1}(y,1)$ for each $n>1$. Defining $S$ and $S'$ similar to $F$ and $F'$, by setting $S'(y,1)=h(y)$ we obtain a smooth extension $S'\colon E\times[0,1]\to EG$.
Finally, define $T\colon E\times[0,1]\to EG$ by $$\begin{aligned}
T(y,\tau):=&~\big((1-b(\tau))s_1(y)f_1(y),b(\tau)t_1(y)h_1(y),\\
&~(1-b(\tau))s_2(y)f_2(y),b(\tau)t_2(y)h_2(y),\dots\big).\end{aligned}$$ Then $T$ is smooth, $T(y,0)=F'(y,0)$, and $T(y,1)=S'(y,0)$.
We now define the smooth homotopy $H\colon E\times[0,1]\to EG$ between $f$ and $h$ by $$H(y,\tau):=\begin{cases}
F'(y,1-b(3\tau)) & \text{for $\tau\in[0,\frac{1}{3}]$,}\\
T(y,b(3\tau-1)) & \text{for $\tau\in[\frac{1}{3},\frac{2}{3}]$,}\\
S'(y,b(3\tau-2)) & \text{for $\tau\in[\frac{2}{3},1]$.}\\
\end{cases}$$ The $G$-equivariance of $H$ is clear. This completes the proof.
We are now ready to prove the main result of this section.
The fact that $\mathcal{B}_G(\cdot)$ and $[\cdot,BG]$ are functors is clear. Fix a diffeological space from $\mathbf{Diffeol}_{\operatorname{HSP}}$. It follows from Proposition \[p:EG classifying\] and Proposition \[p:homotopy\] that there is a map $\alpha$ from the set $[X,BG]$ to $\mathcal{B}_G(X)$. It follows from Proposition \[p:classifying maps\] and Proposition \[p:uniqueness\] that $\alpha$ has an inverse, and hence $\mathcal{B}_G(X)$ is in bijection with $[X,BG]$. Finally, this bijection is natural: given two diffeological spaces $X$ and $Y$ from $\mathbf{Diffeol}_{\operatorname{HSP}}$ and a smooth map $\varphi\colon X\to Y$, we have that $\mathcal{B}(\varphi)$ and $[\varphi,BG]$ commute with these bijections.
Diffeological Connections and Connection $1$-Forms
==================================================
In [@iglesias Article 8.32] Iglesias-Zemmour gives a definition of a connection on a principal $G$-bundle in terms of paths on the total space, generalising the classical notion for principal bundles with structure group a finite-dimensional Lie group. From this definition one obtains the usual properties that one expects from a connection (see Remark \[r:iz-connection\]). The purpose of this section is to prove Theorem \[t:iz-connection\]: that any principal $G$-bundle satisfying mild conditions admits one of these connections provided $G$ is a regular diffeological Lie group. We do this by way of constructing a connection $1$-form on the $G$-bundle $EG\to BG$, and showing that this induces a connection in the sense of Iglesias-Zemmour. Since connections pull back, we obtain our result.
Let $G$ be a diffeological group, and let $\pi\colon E\to X$ be a principal $G$-bundle. Denote by $\pathloc(E)$ the diffeological space of **local paths** $$\pathloc(E):=\{\gamma\in\CIN((a,b),E)\mid (a,b)\subseteq\RR\}$$ equipped with the subset diffeology induced by the standard functional diffeology *on a diffeology* (see [@iglesias Article 1.63]). We recall what this standard functional diffeology is: a parametrisation $p:U\to\mathcal{D}$ is a plot if and only if for each $u\in U$ and $v\in\dom(p(u))$ there exist open neighbourhoods $U'$ of $u$ and $V'$ of $v$ such that for each $u'\in U'$,
1. $V'\subseteq\dom(p(u'))$, and
2. the map $\Psi\colon U'\times V'\to X\colon(u',v')\mapsto p(u')(v')$ is in $\mathcal{D}$.
Denote by $\tbpath(E)$ the **tautological bundle of local paths**, equipped with the subset diffeology induced by $\pathloc(E)\times\RR$: $$\tbpath(E):=\{(\gamma,t)\in\pathloc(E)\times\RR\mid t\in\dom(\gamma)\}.$$ A **diffeological connection** is a smooth map $\theta\colon\!\tbpath(E)\to\pathloc(E)$ satisfying the following properties for any $(\gamma,t_0)\in\tbpath(E)$:
1. the domain of $\gamma$ equals the domain of $\theta(\gamma,t_0)$,
2. $\pi\circ\gamma=\pi\circ\theta(\gamma,t_0)$,
3. $\theta(\gamma,t_0)(t_0)=\gamma(t_0)$,
4. $\theta(g\cdot\gamma,t_0)=g\cdot\theta(\gamma,t_0)$ for all $g\in G$,
5. $\theta(\gamma\circ f,s)=\theta(\gamma,f(s))\circ f$ for any smooth map $f$ from an open subset of $\RR$ into $\dom(\gamma)$,
6. $\theta(\theta(\gamma,t_0),t_0)=\theta(\gamma,t_0)$.
Diffeological connections satisfy many of the usual properties that classical connections on a principal $G$-bundle (where $G$ is a finite-dimensional Lie group) enjoy; in particular, they admit unique horizontal lifts of paths into the base of a principal bundle [@iglesias Article 8.32], and they pull back by smooth maps [@iglesias Article 8.33].
We now state the main purpose of this section.
Let $G$ be a regular diffeological Lie group, and let $X$ be a Hausdorff smoothly paracompact diffeological space. Then any weakly D-numerable principal $G$-bundle $E\to X$ admits a diffeological connection. Consequently, for this diffeological connection, any smooth curve into $X$ has a unique horizontal lift to $E$.
To prove this, we begin by constructing a connection 1-form on $EG\to BG$.
Let $G$ be a diffeological group, and let $E\to X$ be a principal $G$-bundle. A **connection 1-form** on $E$ is a $G$-equivariant smooth fibrewise linear map $\omega\colon TE\to T_eG$ (with respect to the adjoint action on $T_{e}G$) such that for any $y\in E$ and $\xi\in T_eG$, we have $\omega(\xi_E|_y)=\xi$ where $$\xi_E|_y:=\frac{d}{dt}\Big|_{t=0}(g(t)\cdot y)$$ in which $g(t)$ is a smooth curve in $G$ such that $g(0)=e$ and $\dot{g}(0)=\xi$.
To make clear the connection between ordinary diffeological differential $1$-forms and smooth fibrewise linear maps as defined above, we present the following definition and proposition.
Denote by $\DVB$ the category of diffeological vector pseudo-bundles with smooth fibrewise linear maps between them. For a diffeological space $(X,\mathcal{D})$ denote by $\operatorname{Plots}(\mathcal{D})$ the category with objects plots in $\mathcal{D}$ and arrows commutative triangles $$\xymatrix{
U \ar[rr]^{f} \ar[dr]_{p} & & V \ar[dl]^{q} \\
& X & \\
}$$ in which $f$ is smooth. Let $V$ be a diffeological vector space. Denote $\Omega^1(X;V)$ to be the **$V$-valued differential $1$-forms on $X$**, defined to be the limit $$\underset{\operatorname{Plots}(\mathcal{D})}{\lim}\Hom_{\DVB}(T\circ F(\cdot),X\times V\to X).$$ Here, $X\times V\to X$ is the trivial $V$-bundle over $X$, $T$ is the tangent functor, and $F$ the forgetful functor sending plots to their domains and commutative triangles to the corresponding maps between Euclidean open sets.
Let $(X,\mathcal{D})$ be a diffeological space and $V$ a diffeological vector space. Then there is a natural identification between smooth fibrewise $\RR$-linear maps $TX\to V$ and $V$-valued $1$-forms $\Omega^1(X;V)$.[^2]
The set of smooth fibrewise $\RR$-linear maps $TX\to V$ is exactly $$\Hom_{\DVB}(TX,X\times V\to X).$$ But $TX$ is the colimit $\colim(T\circ F)$ in $\DVB$ over the category $\operatorname{Plots}(\mathcal{D})$ where $T$ and $F$ are as in Definition \[d:V-valued forms\]; see [@CW Theorem 4.17]. But then, $$\Hom_{\DVB}(TX,X\times V\to X)=\underset{\operatorname{Plots}(\mathcal{D})}{\lim}\Hom_{\DVB}(T\circ F(\cdot),X\times V\to X);$$ see, for example, [@awodey Corollary 5.29].
Ordinary differential $1$-forms on a diffeological space $X$ are, by definition, equal to $$\underset{\operatorname{Plots}(\mathcal{D})}{\lim}\Hom_{\DVB}(T\circ F(\cdot),X\times \RR\to X)$$
Returning to connection $1$-forms, the most basic connection $1$-form is the Maurer-Cartan form on a diffeological group.
Let $G$ be a diffeological group. Then $G$ has a **Maurer-Cartan form** $\alpha$; that is, a smooth fibrewise-linear map $\alpha\colon TG\to T_eG$ sending $v\in T_gG$ to $(L_{g^{-1}})_*v$. It is $G$-equivariant with respect to the adjoint action on $T_eG$ and the left action on $TG$ defined by $h\cdot v:= (R_{h^{-1}})_*v$.
This is immediate from the trivialisation of $TG$ given in Theorem \[t:group\].
Since $EG$ is constructed out of an infinite product of the group $G$, we can take the infinite sum of the Maurer-Cartan form, with coefficients $t_j$; since only finitely many of the $t_j$ are non-zero, the sum converges. The goal is to show that the result is smooth.
Let $G$ be a diffeological group. Then the principal $G$-bundle $EG\to BG$ admits a connection 1-form $\omega$.
Let $\alpha$ be the Maurer-Cartan form on $G$. Define $\widetilde{\omega}$ to be the smooth map $\widetilde{\omega}\colon TS_G\to T_eG$ given by $$\widetilde{\omega}|_{(g_i,t_i)} = \sum_{i \in \NN} t_i \pr_{g_i}^*\alpha$$ where $\pr_{g_i}\colon S_G\to G$ is the projection map onto the $i^\text{th}$ copy of $G$. Then $\widetilde{\omega}$ is a well-defined connection 1-form on the bundle $S_G\to S_G/G=:EG$.
We need to show that $\widetilde{\omega}$ descends to a form $\omega$ on $EG$. To do this, we use the following fact (see [@iglesias Article 6.38], noting that the proof goes through with our definition of $1$-form, independent of the fact that the forms in the proof are real-valued): if $\rho\colon S_G\to EG$ is the quotient map, then a form $\mu$ on $S_G$ is equal to $\rho^*\nu$ for some form $\nu$ on $EG$ if and only if for any two plots $p_1,p_2:U\to S_G$ satisfying $\rho\circ p_1=\rho\circ p_2$, we have $p_1^*\mu=p_2^*\mu$. Fix two such plots $p_1$ and $p_2$, and fix $u\in U$ and $v\in T_uU$. It is enough for us to show $$\labell{e:universal connection}
v\hook (p_1^*\widetilde{\omega}-p_2^*\widetilde{\omega})=0.$$
There are only finitely many $j\in\NN$ such that $\pr_{t_j}(p_1(u))\neq 0$ where $\pr_{t_j}$ is the $j^{\text{th}}$ projection from $S_G$ onto $[0,1]$. Hence there are only finitely many open sets $$\widetilde{V}_j=\{(g_i,t_i)\mid t_j\neq 0\}\subseteq S_G$$ containing $p_1(u)$. Thus, the intersection $\widetilde{V}$ of all such $\widetilde{V}_j$ is open, and its pre-image $W:=p_1^{-1}(\widetilde{V})\subseteq U$ is open. Moreover, it follows from $\rho\circ p_1=\rho\circ p_2$ that $W=p_2^{-1}(\widetilde{V})$.
Let $c\colon\RR\to U$ be a curve such that $c(0)=u$ and $\dot{c}(0)=v$. Then for all $\tau$ in $c^{-1}(W)$ we have $\pr_{g_j}\circ p_1\circ c(\tau)=\pr_{g_j}\circ p_2\circ c(\tau)$ for all $j$ such that $t_j\neq 0$. It follows that $$\alpha((\pr_{g_j})_*(p_1)_*v)=\alpha((\pr_{g_j})_*(p_2)_*v).$$ Equation follows.
We have the following corollary:
Let $G$ be a diffeological group. Then any principal $G$-bundle $E\to X$ admitting a smooth classifying map has a connection 1-form. In particular, if $E\to X$ is weakly D-numerable over a Hausdorff smoothly paracompact diffeological space $X$, then it admits a connection 1-form.
This follows from the existence of a $G$-equivariant map $E\to EG$ (Proposition \[p:classifying maps\]).
In order to continue, we need the following basic lemma about differentiating plots.
Let $(X,\mathcal{D})$ be a diffeological space, and let $T\colon\mathcal{D}\to T\mathcal{D}$ be the map sending a plot $p$ to its induced plot $Tp$. Then $T$ is a smooth map.
Fix a plot $p\colon U\to\mathcal{D}$ in the standard functional diffeology. Using the same notation as in Definition \[d:iz-connection\], by the definition of the diffeology on $TX$ and Remark \[r:tangent bdle\] the map $T\Psi\colon TU'\times TV'\to TX$ is a plot of $TX$. Restricting the first coordinate of $T\Psi$ to the zero-section of $TU'$ we get the smooth map $U'\times TV'\to TX\colon (u',w)\mapsto T\Psi|_{(u',v')}(0,w)$ where $v'$ is the foot-point of $w$. Since this is smooth, by the definition of the functional diffeology we have that the map $U'\to\CIN(TV',TX)$ sending $u'$ to $Tp(u')$ is a plot of $T\mathcal{D}$.
Now the main obstacle in proving Theorem \[t:iz-connection\] is to show that a connection 1-form on a principal $G$-bundle yields a diffeological connection; in particular, that one obtains a smooth map from the tautological bundle to the local paths of the total space. This is the content of the following proposition (*cf.* [@Ma2013 Section 2.1]).
Let $G$ be a regular diffeological Lie group, and let $\pi\colon E\to X$ be a principal $G$-bundle. Then a connection 1-form $\omega$ on $\pi$ induces a diffeological connection on $\pi$.
Fix a smooth curve $\gamma\colon(a,b)\to E$ and a point $t_0\in(a,b)$. Our first goal is to obtain a smooth curve $g\colon(a,b)\to G$ such that $g(t_0)=e$ and the smooth curve $t\mapsto g(t)\cdot\gamma(t)$ satisfies $$\omega\left(\frac{d(g\cdot \gamma)(t)}{dt}\right)=0.$$ We will denote $g(t)\cdot\gamma(t)$ by $\theta(\gamma,t_0)(t)$. By the chain rule (see Remark \[r:tangent bdle\]) the derivative of $\theta(\gamma,t_0)$ is $g\cdot\dot{\gamma}+\dot{g}\cdot\gamma$. Applying $\omega$ to this, we obtain the differential equation $$\omega(\dot{\gamma})(t)=-\Ad_{g^{-1}}\dot{g}|_{\gamma(t)}.$$
Choose $\eps>0$ so that $a<t_0-\eps$ and $t_0+\eps<b$. After composing with an appropriate translation and dilation, it follows from the regularity of $G$ that there is a smooth solution $[t_0-\eps,t_0+\eps]\to G$; in fact, uniqueness of solutions implies that applying this procedure to each $t_0\in(a,b)$ will yield a smooth curve $g\colon (a,b)\to G$ as required. We thus have proved the existence of $\theta\colon\!\tbpath(E)\to\pathloc(E)$.
To show that $\theta$ is smooth, note that the map $\gamma\mapsto\dot{\gamma}$ is smooth by Lemma \[l:diff of plots\], as well as $\omega$ and the exponential map in the definition of the regularity of $G$. Finally, the translations and dilations are smooth, and since $\theta$ is a composition of all of these things, smoothness follows.
It is an easy exercise to check that $\theta$ satisfies the six conditions in Definition \[d:iz-connection\]. This completes the proof.
By Corollary \[c:universal connection\] the principal $G$-bundle $\pi\colon E\to X$ has a connection 1-form. By Proposition \[p:iz-connection\] this induces a diffeological connection on $\pi$. In particular, we obtain horizontal lifts of smooth curves into $X$ (see Remark \[r:iz-connection\]).
We end this section with the following note. Let $G$ be a *regular* diffeological *Lie* group. By Proposition \[p:iz-connection\] we have a diffeological connection on any principal $G$-bundle. By Remark \[r:homotopy\], Proposition \[p:homotopy\] holds without any assumption on the topology of the base. Since Proposition \[p:uniqueness\] also has no conditions on the topology of the base, it follows that *if* you have two principal $G$-bundles (with no assumptions on them) $E\to X$ and $E'\to X$ *with classifying maps* $F\colon X\to BG$ and $F'\colon X\to BG$, then $E$ and $E'$ are isomorphic bundles if and only if $F$ and $F'$ are smoothly homotopic.
Applications
============
In this section, we apply the theory developed in the previous sections to various situations. Many of these are motivated by applications found in [@iglesias], [@KM], [@Ma2013], [@Ma2016], [@R], and [@tD].
Contractibility of $EG$ & Homotopy of $BG$
------------------------------------------
In this subsection, we show that $EG$ is smoothly contractible, which allows us to compute the (diffeological) homotopy groups of $BG$ in terms of those of $G$. Again, we look to the topological proof found in [@tD] for a template.
Let $G$ be a diffeological group. Then $EG$ is smoothly contractible.
Let $H_1\colon EG\times[0,1]\to EG$ be a smooth homotopy given by $$\begin{gathered}
H_1((t_ig_i),0)=(t_ig_i) \text{ and}\\
H_1((t_ig_i),1)=(t_1g_1,0,t_2g_2,t_3g_3,\dots). \end{gathered}$$ using the same notation as that in the proof of Proposition \[p:uniqueness\]. This exists by Proposition \[p:uniqueness\]. Let $b$ be the smooth function from the same proof and let $H_2\colon EG\times[0,1]\to EG$ be the smooth map $$H_2((t_ig_i),\tau)=\big((1-b(\tau))t_1g_1,b(\tau)e,(1-b(\tau))t_2g_2,(1-b(\tau))t_3g_3,\dots\big)$$
Concatenating $H_1$ and $H_2$ yields a smooth homotopy. Since $$H_2((t_ig_i),1)=\big(0,e,0,0,\dots\big),$$ which is constant on $EG$, it follows that $EG$ is smoothly contractible.
Let $G$ be a diffeological group. Then for each $k>0$, we have $\pi_k(BG)\cong\pi_{k-1}(G)$.
This is immediate from the fact that $EG\to BG$ is a principal $G$-bundle, Proposition \[p:contractible\], and the long exact sequence of (diffeological) homotopy groups; see [@iglesias Article 8.21].
Fix an irrational number $\alpha$. Let $G$ be the irrational torus $T_\alpha:=\RR/\ZZ^2$, where $\ZZ^2$ acts on $\RR$ by $(m,n)\cdot x=x+m+n\alpha$. By the long exact sequence of (diffeological) homotopy groups [@iglesias Article 8.21], it is immediate that $\pi_0(T_\alpha)=\{T_\alpha\}$, $\pi_1(T_\alpha)=\ZZ^2$, and $\pi_k(T_\alpha)=0$ for $k>1$. It follows from Proposition \[p:homotopy of BG\] that $\pi_0(BT_\alpha)=\{BT_\alpha\}$, $\pi_1(BT_\alpha)=0$, $\pi_2(BT_\alpha)=\ZZ^2$, and $\pi_k(BT_\alpha)=0$ for all $k>2$.
Smooth Homotopies of Groups and Bundles
---------------------------------------
Here we look at how smooth group homomorphisms between diffeological groups induce smooth maps between the classifying spaces, with applications to smooth strong deformation retractions.
Let $G$ and $H$ be diffeological groups, and let $\varphi\colon G\to H$ be a smooth map between them. Then $\varphi$ induces a smooth map $\widetilde{\Phi}\colon EG\to EH$ defined as $$\widetilde{\Phi}(t_ig_i)=(t_i\varphi(g_i)).$$ Moreover, if $\varphi$ is a smooth homomorphism, $\widetilde{\Phi}$ descends to a smooth map $\Phi\colon BG\to BH$.
The proof follows immediately from the definitions.
If $\varphi_\tau\colon G\to H$ is a smooth family of maps between diffeological groups $G$ and $H$, then a similar proof to the above yields a smooth family of maps $\widetilde{\Phi}_\tau\colon EG\to EH$ defined in the obvious way, and in the case of a smooth family of group homomorphisms, we obtain a smooth family of maps $\Phi_{\tau}\colon BG\to BH$.
Let $G$ be a diffeological group, and let $\Phi\colon G\times[0,1]\to G$ be a smooth strong deformation retraction of $G$ onto a subgroup $H$ such that for each $\tau\in[0,1]$, the map $\Phi(\cdot,\tau)\colon G\to G$ is a group homomorphism. Then, up to isomorphism, there is a smooth strong deformation retract of any weakly D-numerable principal $G$-bundle over a Hausdorff smoothly paracompact base to a principal $H$-bundle over the same base.
By Proposition \[p:maps between gps\] and Remark \[r:maps between gps\], we obtain a smooth strong deformation retract $\widetilde{\Phi}_\tau$ of $EG$ onto $EH$, which descends to a smooth strong deformation retract $\Phi_\tau$ of $BG$ onto $BH$. Let $E\to X$ be a weakly D-numerable principal $G$-bundle over a Hausdorff smoothly paracompact base $X$. By Proposition \[p:classifying maps\], there is a smooth map $\widetilde{F}\colon E\to EG$ which descends to a smooth classifying map $F\colon X\to BG$ for which $E$ is isomorphic as a principal $G$-bundle to $F^*EG$. Composing $F$ with $\Phi_\tau$, we obtain a smooth strong deformation retract of $F^*EG$ onto $(\Phi_0\circ F)^*EH$, where at any $\tau$ we have $$(\Phi_\tau\circ F)^*EG=\{(x,(t_ig_i))\mid \Phi_\tau\circ F(x)=[t_ig_i]\}.$$
Let $G=\Diff(\RR^n;0)$ be the diffeological group of diffeomorphisms of $\RR^n$ that fix the origin. Define $\Phi\colon G\times[0,1]\to G$ by $$\Phi(\varphi,\tau)=\begin{cases}
m_{1/\tau}\circ\varphi\circ m_\tau & \text{if $\tau\neq 0$,}\\
d\varphi|_0 & \text{if $\tau=0$,}\\
\end{cases}$$ where $m_\tau\colon\RR^n\to\RR^n$ is scalar multiplication by $\tau$ (which is smooth). By definition of the functional diffeology [@iglesias Article 1.57], it is an easy exercise to check that this is a smooth strong deformation retract of $G$ onto $\GL(n;\RR)$. Moreover, the chain rule shows that $\Phi(\cdot,\tau)$ is a group homomorphism for each $\tau$. It follows from Proposition \[p:maps between gps\] that $EG$ has a smooth strong deformation retract onto $E\GL(n;\RR)$, and this descends to a smooth strong deformation retract of $BG$ onto $B\GL(n;\RR)$. By Corollary \[c:maps between gps\] we have that any weakly D-numerable principal $G$-bundle over a Hausdorff smoothly paracompact base has a smooth strong deformation retract onto a principal $\GL(n;\RR)$-bundle over the same base.
Let $G=\Diff^+(\SS^2)$ be the diffeological group of orientation-preserving diffeomorphisms of $\SS^2$ (equipped with the functional diffeology). By the main result of [@LW2011] there is a smooth strong deformation retraction $\kappa\colon G\times[0,1]\to G$ onto $\SO(3)$. By Proposition \[p:maps between gps\] and Remark \[r:maps between gps\], $\kappa$ induces a smooth strong deformation retraction from $EG$ to $E(\SO(3))$. Unfortunately, the smooth strong deformation retraction in [@LW2011] does not give a group homomorphism from $G$ to $\SO(3)$, and so the deformation retraction does not descend to $BG$ and $B\SO(3)$.
Associated Fibre Bundles
------------------------
Let $G$ be a diffeological group, and let $\pi\colon E\to X$ be a principal $G$-bundle. Let $F$ be a diffeological space admitting a smooth action of $G$. Then we may construct the **associated bundle to $\pi$ with fibre $F$**, denoted $\widetilde{\pi}\colon \widetilde{E}\to X$, as follows: let $G$ act diagonally on $E\times F$; then define $\widetilde{E}$ as the quotient $E\times_G F:=(E\times F)/G$. Any diffeological fibre bundle with structure group $G$ can be constructed as an associated fibre bundle of some principal $G$-bundle. See [@iglesias Article 8.16] for more details. Note that if a fibre bundle is (weakly) D-numerable any associated principal $G$-bundle is (weakly) D-numerable as well. We have the following corollary to Theorem \[t:BG\]:
Let $\mathcal{B}^F_G(\cdot)$ be the functor from $\mathbf{Diffeol}_{\operatorname{HSP}}$ to $\mathbf{Set}$ sending an object $X$ to the set of isomorphism classes of locally trivial diffeological fibre bundles with fibre a fixed $G$-space $F$. Then there is a natural surjection from $[\cdot,BG]$ to $\mathcal{B}^F_G(\cdot)$.
Using the same notation as above, let $\theta$ be a diffeological connection on $E$ induced by a connection 1-form $\omega$ on $\pi\colon E\to X$ as in Proposition \[p:iz-connection\]. Now $\omega\oplus 0$ is a $T_{e}G$-valued 1-form on $E\times F$, and one can check using the fact that $\pi$ is D-numerable that $\omega$ descends to a $T_{e}G$-valued 1-form $\omega_{\widetilde{E}}$ on $\widetilde{E}$, which we call a **connection 1-form** on $\widetilde{E}$.
Let $\widetilde{\pi}\colon \widetilde{E}\to X$ be a diffeological fibre bundle, let $c\colon(a,b)\to X$ be a smooth curve, and fix $z\in\widetilde{E}$ and $t_0\in(a,b)$ such that $\widetilde{\pi}(z)=c(t_0)$. A curve $c_{\widetilde{E}}\colon(a,b)\to \widetilde{E}$ is a **horizontal lift** of $c$ through $z$ if
1. $\widetilde{\pi}\circ c_{\widetilde{E}}=c$,
2. $c_{\widetilde{E}}(t_0)=z$,
3. $\omega_{\widetilde{E}}\left(\frac{d}{dt}c_{\widetilde{E}}(t)\right)=0.$
Let $G$ be a regular diffeological Lie group, $F$ a diffeological space with a fixed smooth action of $G$, $X$ a Hausdorff smoothly paracompact diffeological space, and $\pi\colon E\to X$ a weakly D-numerable principal $G$-bundle. Equip $E$ with the connection 1-form $\omega$ as in Corollary \[c:universal connection\]. Then for any smooth curve $c\colon(a,b)\to X$, and any $z\in\widetilde{E}$, there is a horizontal lift $c_{\widetilde{E}}\colon(a,b)\to\widetilde{E}$ through $z$.
Take $c_{\widetilde{E}}(t):=\rho_{\widetilde{E}}(\theta(c,t_0)(t),z')$, where $\rho_{\widetilde{E}}\colon E\times F\to\widetilde{E}$ is the quotient map and $z'\in F$ such that $\rho_{\widetilde{E}}(\theta(c,t_0)(t_0),z')=z$. This definition is independent of the choice of $z'$. This satisfies all of properties required in Definition \[d:assoc lifts\].
We do not mention uniqueness of horizontal lifts in Proposition \[p:assoc lifts\]. This requires the uniqueness of solutions to the initial-value problem given by the differential equation in Item (\[i:assoc lifts de\]) with initial condition as given in Item (\[i:initial condition\]), both in Definition \[d:assoc lifts\]. It is not immediately clear under what conditions this would hold on a general diffeological space.
Let $\pi\colon \widetilde{E}\to X$ be a diffeological fibre bundle with fibre $F$ and structure group $G=\Diff(F)$. Define the set $\mathcal{FR}$ to be all commutative diagrams $$\xymatrix{
F \ar[r]^f \ar[d] & \widetilde{E} \ar[d]^{\pi} \\
\{*\} \ar[r] & X \\
}$$ where $f$ is a diffeomorphism onto a fibre of $\widetilde{E}$. Equip $\mathcal{FR}$ with the subset diffeology induced by $\CIN(F,\widetilde{E})$. One may think of this as a “non-linear frame bundle” of the bundle $\widetilde{E}$. From above, we have that $\mathcal{FR}\times_{\Diff(F)}F\cong \widetilde{E}$, and by Corollary \[c:BG\], we have that there is a natural surjection from smooth homotopy classes of maps $X\to B\Diff(F)$ to isomorphism classes of such bundles $\widetilde{E}$, provided that the bundles are locally trivial, and $X$ has Hausdorff, second-countable, and smoothly paracompact topology.
If $F$ is a smooth manifold, then $\Diff(F)$ is a regular diffeological Lie group [@KM Theorem 43.1], and so provided that $\widetilde{E}\to X$ is a locally trivial principal $\Diff(F)$-bundle, and the topology on $X$ is Hausdorff, second-countable, and smoothly paracompact, then by Proposition \[p:assoc lifts\] $\widetilde{E}$ has a diffeological connection, which in turn gives us horizontal lifts of curves.
Limit of Groups & ILB Principal Bundles
---------------------------------------
We consider in this subsection limits of diffeological group, and in particular infinite-dimensional groups. We rely heavily on [@Om] for terminology.
Fix a small category $J$, which we will think of as our “index category”. Let $\mathbf{DGroup}$ be the category of diffeological groups with smooth group homomorphisms between them, and let $F\colon J\to\mathbf{DGroup}$ be a functor. Denote by $G_j$ the image $F(j)$ for each object $j$ of $J$, and by $\varphi_f$ the image $F(f)$ for each arrow $f$ in $J$. Let $G=\lim F$ be the limit taken in the category of diffeological spaces; in particular, there is a smooth map $\varphi_j\colon G\to G_j$ for each object $j$ in $J$ such that if $f\colon j_1\to j_2$ is an arrow in $J$, then $\varphi_f\circ\varphi_{j_1}=\varphi_{j_2}.$
Let $F$ be the functor above, and define $G=\lim F$. Then $G$ is a diffeological group.
Recall that both the category of diffeological spaces and the category of groups are complete, and limits are constructed on the set-theoretical level: $$G=\left\{(g_j)\in\prod_{j\in J_0}G_j~\Biggr|~\forall(f\colon j_1\to j_2)\in J_1,~\varphi_f\circ\pr_{j_1}((g_j))=\pr_{j_2}((g_j))\right\}$$ where $J_0$ is the set of objects of $J$, $J_1$ the set of arrows, and $\pr_j$ is the $j^{\text{th}}$ projection onto $G_j$. The diffeology is given by the intersection of all pullback diffeologies on $G$ via the maps $\varphi_j:=\pr_j$. The group multiplication on $G$ is given by the coordinate-wise product: $(g_j)(h_j):=(g_jh_j)$. It follows that the maps $\varphi_j$ are group homomorphisms, from which it follows that multiplication and inversion are smooth maps $G\times G\to G$ and $G\to G$, respectively. Hence, $G$ is a diffeological group.
Let $G$ be the diffeological group constructed as a limit above. Then for every object $j$ of $J$, from Proposition \[p:maps between gps\] we obtain a commutative diagram:
$$\xymatrix{
EG \ar[r]^{\widetilde{\Phi}_j} \ar[d] & EG_j \ar[d] \\
BG \ar[r]_{\Phi_j} & BG_j. \\
}$$
Moreover, for any arrow $f\colon j_1\to j_2$ of $J$, we obtain a commutative diagram:
$$\xymatrix{
EG \ar[rr] \ar[dr] \ar[ddd] & & EG_{j_1} \ar[dl]^{\widetilde{\Phi}_{f}} \ar[ddd] \\
& EG_{j_2} \ar[d] & \\
& BG_{j_2} & \\
BG \ar[rr] \ar[ur] & & BG_{j_1} \ar[ul]_{\Phi_{f}} \\
}$$ where all of the maps are the induced ones via Proposition \[p:maps between gps\].
We now apply the theory developed to ILB-manifolds and ILB-bundles. Technically, our definition of an ILB-manifold below is only a special case of the definition given in [@Om Definition I.1.9]; however, it is sufficient for our needs. For this application to make sense, we must first note that infinite-differentiability in the Fréchet sense is equivalent to smoothness in the diffeological sense when we equip a Fréchet space/manifold with the diffeology comprising Fréchet infinitely-differentiable parametrisations. Moreover, Fréchet spaces form a full subcategory of diffeological spaces under this identification. See, for example, [@losik Theorem 3.1.1], [@KaWa]. Note that the limits used in the following definitions can be taken in either the Fréchet category or the diffeological category: in our cases below they coincide.
1. An **ILB-manifold** $M$ is the limit in the category of Fréchet manifolds of a family of Banach manifolds $\{M_n\}_{n \in \NN}$ in which there is a smooth and dense inclusion $M_{n+1}\hookrightarrow M_n$ for each $n$, and such that there exists a Banach atlas on $M_0$ that restricts to an atlas on $M_n$ for each $n$.
2. An **ILB-map** $f$ between two ILB-manifolds $M$ and $N$ is a smooth map $f\colon M\to N$ along with a family of smooth maps $\{f_n\colon M_n\to N_n\}$ such that $f$ and all $f_n$ commute with all inclusions maps $M\hookrightarrow M_{n+1}\hookrightarrow M_n$ and $N\hookrightarrow N_{n+1}\hookrightarrow N_n$.
3. An **ILB-principal bundle** is an ILB-map between two ILB-manifolds $(\pi\colon P\to M,~\pi_n\colon P_n\to M_n)$ such that for each $n$, the map $\pi_n\colon P_n\to M_n$ is a principal $G_n$-bundle where $G_n$ is a Banach Lie group.
4. An ILB-map $F$ between two ILB-principal bundles $\pi\colon P\to M$ and $\pi'\colon P'\to M'$ is an **ILB-bundle map** if $F_n\colon P_n\to P'_n$ is a ($G_n$-equivariant) bundle map for each $n$. Note that $F$ induces an ILB-map $M\to M'$. An ILB-bundle map is an **ILB-bundle isomorphism** if it has an inverse ILB-bundle map.
It follows from the above definitions that given an ILB-principal bundle $\pi\colon P\to M$, the structure groups $\{G_n\}$ of the principal bundles $\{\pi_n\colon P_n\to M_n\}$ satisfy: $G_{n+1}$ is smoothly and densely included in $G_n$ for each $n$. Hence $\pi\colon P\to M$ is a principal $G$-bundle where $G=\lim G_n$. An ILB-bundle map from $\pi\colon P\to M$ to $\pi'\colon P'\to M'$ is thus necessarily $G$-equivariant.
1. Given an ILB-principal bundle $\pi\colon P\to M$ with structure group $G$, an **ILB-classifying map** $F\colon M\to BG$ is a classifying map for $\pi$ such that there is a classifying map $F_n\colon M_n\to BG_n$ for each $n$, the following diagram commutes for all $n$,
$$\xymatrix{
P \ar[rrr] \ar[dr]_{\widetilde{F}} \ar[ddd] & & & P_n \ar[dl]^{\widetilde{F}_n} \ar[ddd] \\
& EG \ar[r]^{\widetilde{\Phi}_n} \ar[d] & EG_n \ar[d] & \\
& BG \ar[r]_{\Phi_n} & BG_n & \\
M \ar[rrr] \ar[ur]^{F} & & & M_n \ar[ul]_{F_n} \\
}$$ and these maps commute with the inclusion maps $EG\hookrightarrow EG_{n+1}\hookrightarrow EG_n$ and $BG\hookrightarrow BG_{n+1}\hookrightarrow BG_n$ as in Remark \[r:limit d-groups\].
2. An **ILB-homotopy** between two ILB-classifying maps $F,F'\colon M\to BG$ for ILB-principal bundles $P\to M$ and $P'\to M$ each with structure group $G$ is a family of smooth homotopies $H_n\colon M_n\times[0,1]\to BG_n$ such that $H_n(\cdot,0)=F_n$ and $H_n(\cdot,1)=F'_n$ for each $n$, and a smooth homotopy $H\colon M\times[0,1]\to BG$ such that $H(\cdot,0)=F$ and $H(\cdot,1)=F'$, and finally each $H_n$ and $H$ commute with the maps in Remark \[r:limit d-groups\].
Let $\pi\colon P\to M$ be an ILB-principal bundle, and assume that each $M_n$ is a Hilbert manifold (in this case, we say that $M$ is an **ILH-manifold**). Hilbert manifolds are smoothly paracompact [@KM Corollary 16.16], and (smooth) partitions of unity pull back. Hence, a smooth partition of unity subordinate to a local trivialisation of $\pi_0\colon P_0\to M_0$ induces similar partitions of unity for each $\pi_n\colon P_n\to M_n$, and on $\pi\colon P\to M$. Thus, in this case, each $\pi_n\colon P_n\to M_n$, as well as $\pi\colon P\to M$, is D-numerable.
Given an ILB-principal bundle $\pi\colon P\to M$ with ILB group $G$ and ILH base as above, by Proposition \[p:classifying maps\] there is a classifying map $F\colon M\to BG$ and so $\pi\colon P\to M$ is isomorphic to $F^*EG$. Moreover, for each $n$, there is a classifying map $F_n\colon M_n\to BG_n$ and so $P_n$ is isomorphic to $F_n^*EG_n$. It follows from Remark \[r:limit d-groups\] that we obtain an ILB-classifying map for $\pi\colon P\to M$. From Theorem \[t:BG\] and Remark \[r:limit d-groups\] we obtain the following proposition:
Let $\pi\colon P\to M$ be an ILB-principal $G$-bundle in which $M=\lim M_n$ is an ILH-manifold. Then $\pi$ has an ILB-classifying map. Moreover, smoothly homotopic ILB-classifying maps yield isomorphic ILB-principal bundles yield ILB-homotopic ILB-classifying maps.
Group Extensions
----------------
Fix a diffeological extension of diffeological groups (see [@iglesias Article 7.3]): $$\labell{e:ses}
1 \rightarrow G \rightarrow G' \rightarrow G'' \rightarrow 1;$$ By Proposition \[p:maps between gps\], we obtain the following commutative diagram of diffeological spaces.
$$\xymatrix{
EG \ar[r] \ar[d] & EG' \ar[r] \ar[d] & EG'' \ar[d] \\
BG \ar[r] & BG' \ar[r] & BG'' \\
}$$
Since diffeological extensions are examples of principal bundles, induces a long exact sequence of diffeological homotopy groups (see [@iglesias Article 8.21]). The following result thus is a consequence of Proposition \[p:homotopy of BG\].
Given a diffeological extension of diffeological groups as in , we obtain a long exact sequence $$\dots \to \pi_{n+1}(BG'') \to \pi_n(BG) \to \pi_n(BG') \to \pi_n(BG'') \to \pi_{n-1}(BG) \to \dots~.$$
Consider the rational numbers $\QQ$ as a diffeologically discrete subgroup of $\RR$. We have the short exact sequence of diffeological groups $$1 \to \QQ \to \RR \to \RR/\QQ \to 1.$$ From the above, we get a long exact sequence of homotopy groups: $$\dots \to \pi_{n+1}(B(\RR/\QQ)) \to \pi_n(B\QQ) \to \pi_n(B\RR) \to \pi_n(B(\RR/\QQ)) \to \pi_{n-1}(B\QQ) \to \dots~.$$
It follows from this long exact sequence and Proposition \[p:homotopy of BG\] that
$$\begin{gathered}
\pi_0(B(\RR/\QQ))=\{B(\RR/\QQ)\},\\
\pi_1(B(\RR/\QQ))\cong 1,\\
\pi_2(B(\RR/\QQ))\cong\QQ, \text{ and}\\
\pi_k(B(\RR/\QQ))\cong 1 \text{ for all $k\geq 3$}.\end{gathered}$$
Of course, we could have used other easier means of computing these, but the point of this example is to illustrate what one could do with Proposition \[p:les\].
Using short exact sequences and group extensions is very frequent in the study of infinite-dimensional Lie groups for constructing new groups. For example:
Consider the $\Diff(M)$-pseudo-differential operators $FIO_{\Diff}^{0,*}$ described in [@Ma2016]. Given a Hermitian bundle $E \rightarrow M$ over a closed manifold $M,$ the group $FIO_{\Diff}^{0,*}(M,E)$ is a group of Fourier integral operators acting on smooth sections of $E$, which can be seen as a central extension of the group of diffeomorphisms $\Diff(M)$ by the group of $0$-order invertible pseudo-differential operators acting on smooth sections of $E,$ with exact sequence: $$0 \rightarrow PDO^{0,*}(M,E) \rightarrow FIO_{\Diff}^{0,*}(M,E) \rightarrow \Diff(M) \rightarrow 0.$$ We obtain a long exact sequence of homotopy groups: $$\begin{gathered}
\dots \to \pi_{n+1}(B\Diff(M)) \to \pi_n\left(B(PDO^{0,*}(M,E))\right) \to \pi_n\left(B(FIO_{\Diff}^{0,*}(M,E))\right) \to \\
\pi_n(B\Diff(M)) \to \pi_{n-1}\left(B(PDO^{0,*}(M,E))\right) \to \dots~.\end{gathered}$$
Irrational Torus Bundles
------------------------
We now connect what we have done to the classical theory of Weil, in which circle (or complex line) bundles with connection over a fixed simply-connected manifold are classified by their curvatures, and conversely every integral 2-form on the manifold induces a circle bundle with connection whose curvature is that 2-form. This is extended to non-integral 2-forms in [@iglesias-bdles], in which the circle bundles are replaced with irrational torus bundles. See also the more general theory on diffeological spaces in [@iglesias].
Let $T_\alpha$ be the irrational torus, defined in Example \[x:irrational torus\]. It is a regular diffeological Lie group (indeed, the quotient map $\RR\to T_\alpha$ is the exponential map). Let $\omega$ be the connection 1-form on $ET_\alpha$ constructed in Theorem \[t:universal connection\], and let $\pi\colon ET_\alpha\to BT_\alpha$ be the projection map.
The 2-form $d\omega$ is basic; that is, there exists a unique $T_{e}G$-valued 2-form $\Omega$ on $BT_\alpha$ such that $\pi^*\Omega=d\omega$.
This statement in fact holds for any abelian diffeological group $G$. We prove this more general statement below.
We again use the following fact (see [@iglesias Article 6.38]): $d\omega$ is the pullback of a form $\Omega$ on $BG$ if and only if for any two plots $p_1,p_2:U\to EG$ satisfying $\pi\circ p_1=\pi\circ p_2$, we have $p_1^*d\omega=p_2^*d\omega$. Fix two such plots $p_1$ and $p_2$. Let $V_j\subseteq EG$ be the open set $$V_j=\{(t_ig_i)\mid t_j\neq 0\}.$$ We have $p_1^{-1}(V_j)=p_2^{-1}(V_j)=:V$. The projection $\pr_{g_j}\colon V_j\to G$ sending $(t_ig_i)$ to $g_j$ is well-defined on $V_j$, and so we have a smooth map $\gamma\colon V\to G$ sending $u$ to $\pr_{g_j}(p_2(u))^{-1}\pr_{g_j}(p_1(u))$. Thus, $\gamma(u)\cdot p_2(u)=p_1(u)$ for all $u\in V$.
Fix $u\in V$ and $v\in T_uV$. It follows from the chain rule (Remark \[r:tangent bdle\]) that $$(p_1)_*v\hook d\omega=(p_2)_*v\hook(\gamma(u)^*d\omega)+\eta\hook d\omega,$$ where $\eta=\frac{d}{dt}\Big|_{t=0}(\gamma(c(t))\cdot p_1(u))$ for some smooth curve $c\colon(-\eps,\eps)\to V$ such that $c(0)=u$ and $\dot{c}(0)=v$ ($\eps>0$). Since $\omega$ is invariant, we have $\gamma(u)^*d\omega=d\omega$.
Since $j$, $u$, and $v$ are arbitrary, to complete the proof, we only need to show that $\eta\hook d\omega=0$. Note that this is equal to the *contraction* of $d\omega$ by the map $g_{EG}\colon(-\eps,\eps)\to\Diff(EG)$ induced by the curve $g:=\gamma\circ c$ in $G$ (see [@iglesias Article 6.57]). By the Cartan-Lie formula ([@iglesias Article 6.72]), we have $$\labell{e:curvature1}
\eta\hook d\omega=\pounds_g(\omega)-d(\eta\hook\omega).$$ By definition of $\omega$ we know $\eta\hook\omega=\dot{g}(0)$, and so the right term of the right-hand side of vanishes. The left term of the right-hand side of vanishes since $\omega$ is invariant (see [@iglesias Article 6.55] for a definition of the Lie derivative).
Uniqueness of $\Omega$ follows from the fact that $\pi^*$ is injective on forms. This finishes the proof.
We refer to $\Omega$ in Lemma \[l:curvature\] as the **curvature form** of $\omega$. If $X$ is any Hausdorff smoothly paracompact diffeological space, and $F\colon X\to BT_\alpha$ a smooth function, we obtain the **curvature form** $F^*\Omega$ of the connection 1-form $\widetilde{F}^*\omega$ on $F^*EG$ by Proposition \[p:classifying maps\]. The 2-form $F^*\Omega$ satisfies $$\pr_1^*F^*\Omega=\widetilde{F}^*d\omega$$ where $\pr_1\colon F^*EG\to X$ is the pullback bundle.
Let $X$ be a connected diffeological space. In [@iglesias Article 8.40], Iglesias-Zemmour proves that every principal $T_\alpha$-bundle which can be equipped with a connection 1-form induces a unique class in $H^1(\paths(X),T_\alpha)$, its **characteristic class**. If additionally $X$ Hausdorff and smoothly paracompact, then it follows from Proposition \[p:classifying maps\] that *all* weakly D-numerable $T_\alpha$-bundles over $X$ induce a unique class in $H^1(\paths(X),T_\alpha)$.
In [@iglesias Article 8.42], Iglesias-Zemmour shows a converse for the simply-connected case: if $X$ is a simply-connected (and hence connected) diffeological space, and $\mu$ is a non-zero closed 2-form on $X$, then there is a principal $T_\alpha$-bundle on $X$ whose curvature is $\mu$, where $T_\alpha$ is the **torus of periods** of $\mu$. If $X$ is additionally Hausdorff and smoothly paracompact, and the $T_\alpha$-bundle constructed is weakly D-numerable, then $\mu=F^*\Omega$, where $F$ is the classifying map of the $T_\alpha$-bundle, and $\Omega$ is the curvature form on $BT_\alpha$.
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[^1]: Thanks to Dan Christensen for this proof, which is much simpler than the original.
[^2]: Proof suggested by Daniel Christensen via private communication.
|
---
abstract: |
In this paper we consider a number of natural decision problems involving $k$-regular sequences. Specifically, they arise from
- lower and upper bounds on growth rate; in particular boundedness,
- images,
- regularity (recognizability by a deterministic finite automaton) of preimages, and
- factors, such as squares and palindromes
of such sequences. We show that the decision problems are undecidable.
address:
- 'Fachbereich Mathematik, Paris Lodron University of Salzburg, Hellbrunnerstra[ß]{}e 34, 5020 Salzburg, Austria'
- ' School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada'
author:
- Daniel Krenn
- Jeffrey Shallit
title: 'Decidability and $k$-Regular Sequences'
---
Introduction
============
A sequence $(a(n))_{n \geq 0}$ over a finite alphabet is said to be [*$k$-automatic*]{}, for $k \geq 2$ an integer, if its $k$-kernel $$K_k({(a(n))_{n \geq 0}}) = \setm{ (a(k^e n + i))_{n \geq 0} }{ e \geq 0,\ 0 \leq i < k^e }$$ is of finite cardinality. There are many different equivalent definitions of this class of sequences [@Allouche-Shallit:1992:regular-sequences]. It is well-known that many questions about these sequences, such as the growth rate of $\sum_{0 \leq n < N} a(n)$, are decidable [@Cobham:1972].
The so-called [*$k$-regular sequences*]{} form a natural generalization of the automatic sequences. These are sequences $(a(n))_{n \geq 0}$ where the kernel $K_k ((a(n))_{n \geq 0}) $ is contained in a finitely generated module. Unlike the case of $k$-automatic sequences, it is known that some decision problems involving $k$-regular sequences are recursively unsolvable .
In this paper we examine a number of natural decision problems involving $k$-regular sequences, and show that they are undecidable.
Recursively solvable decision problems
--------------------------------------
A decision problem is one with a yes/no answer. To say that a decision problem is solvable means there exists an algorithm (or Turing machine) that unerringly solves it on all inputs. Throughout this paper we use the terms “recursively solvable”, “solvable”, and “decidable” interchangeably, and similarly for the terms “recursively unsolvable”, “unsolvable”, and “undecidable”.
Notation
--------
We let ${\mathbb{N}}_0$ denote the nonnegative integers (natural numbers) and ${\mathbb{N}}$ denote the positive integers.
For a word $z$ with symbols chosen from a finite set $D$, we let $\abs{z}$ denote its length and $\abs{z}_d$ the number of occurrences of the letter $d \in D$ in $z$.
For a fixed integer $k$ at least $2$, we consider base-$k$ representations with the usual digit set $D=\set{0,1,\dots,k-1}$. For a nonnegative integer $n$, we write $(n)_k$ for the standard $k$-ary representation of $n$, having no leading zeroes. The representation of $0$ is the empty word. Note that $(n)_k$ is a word over $D$ and that $\abs{(n)_k}=\floor{\log_k n}+1$ for $n > 0$.
Hilbert’s tenth problem
-----------------------
Showing that a certain decision problem is recursively unsolvable is usually carried out by reducing from another decision problem already known to be recursively unsolvable. One such problem is Hilbert’s tenth problem; see the result of Davis, Matijasevi[č]{}, Putnam, and Robinson [@Davis-Putnam-Robinson:1961:exp-diophantine-equations; @Matijasevic:1970:diophantiness-enumerable-sets].
The decision problem
> “Given a multivariate polynomial $p$ with integer coefficients, do there exist natural numbers $x_1$, $x_2$, …, $x_t$ such that $p(x_1,
> \dots, x_t) = 0$?”
is recursively unsolvable.
The analogous problem, where the $x_i$ need to be positive, is also recursively unsolvable. We will reduce from this problem quite frequently, namely in Theorems \[thm:unsolvable:Omega-n\], \[thm:unsolvable:preimage-dfa\], \[thm:unsolvable:image-N\], \[thm:unsolvable:image-Z\] and \[thm:unsolvable:value-twice\].
Representation of k-regular sequences
-------------------------------------
A $k$-regular sequence $(f(n))_{n \geq 0}$ can be finitely represented in a number of different ways, of which two are the most useful. First, a set of identities in terms of sequences in the $k$-kernel, where each identity represents a subsequence $(f(k^e n + i))_{n \geq 0}$ as a linear combination of subsequences in the $k$-kernel, and a set of initial values. Together it must be possible to compute $f(n)$ for all $n$ from this set of identities and initial values.
Second, a linear representation for $(f(n))_{n\geq 0}$, which consists of a row vector $v$, a column vector $w$, and $k$ square matrices $M_0, M_1, \ldots, M_{k-1}$ such that $$f(n) = v M_{n_{s-1}} \cdots M_{n_0} w,$$ for all $n$, where $(n)_k = n_{s-1} \cdots n_0$. Of course, the empty product of matrices is the identity matrix. See [@Allouche-Shallit:1992:regular-sequences Theorem 2.2].
For example, consider the $2$-regular sequence $(s_2(n))_{n\ge0}$, which counts the number of $1$’s in the binary representation of $n$. Then it is easy to see that $$\begin{aligned}
s_2 (0) &= 0 \\
s_2 (2n) &= s_2 (n) \\
s_2 (4n+1) &= s_2 (2n+1) \\
s_2 (4n+3) &= -s_2 (n) + 2 s_2 (2n+1)\end{aligned}$$ is an example of the former representation, and $$\begin{aligned}
v & = [ 0 \ \ 1 ] \\
M_0 &= \left[ \begin{array}{cc}
1 & 0 \\
0 & 1
\end{array} \right] \\
M_1 &= \left[ \begin{array}{cc}
0 & -1 \\
1 & 2
\end{array} \right] \\
w &= \left[ \begin{array}{c}
1 \\
0 \end{array} \right] .\end{aligned}$$
From now on, when we say an algorithm is “given” a $k$-regular sequence as input, we mean either one of these two representations. Note that we can transform between these two representations effectively, that is, with an algorithm [@Berstel-Reutenauer:2011:noncommutative-rational-series].
Some of our theorems involve algebraic numbers. When we say we are “given” an algebraic number $\alpha$, we mean we are given the minimal polynomial for $\alpha$, together with a rational interval that contains $\alpha$ and none of its conjugates. As is well-known ([@Froehlich-Shepherdson:1956:effective-proc-field-theory; @Griffor:1999:handbook-computability-theory]), we can effectively carry out arithmetic on algebraic numbers represented in this way.
Closure properties of k-regular sequences
-----------------------------------------
In this section we recall some closure properties of $k$-regular sequences: which operations on sequences preserve the property of $k$-regularity. For more details, see [@Allouche-Shallit:1992:regular-sequences]. It is important to note that not only do these operations preserve $k$-regularity; they also are [*effectively*]{} $k$-regular. Let $\circ$ be some operation on sequences. By the operation $\circ$ being “effectively $k$-regular”, we mean that there is an algorithm that, given some representation of $k$-regular sequences ${\bf a} = (a(n))_{n \geq 0}$ and ${\bf b} = (b(n))_{n \geq 0}$, computes a representation for ${\bf a} \circ {\bf b}$.
The class of $k$-regular sequences is closed under the following operations:
- sum, ${\bf a}+{\bf b} = (a(n) + b(n))_{n \geq 0}$;
- product, ${\bf a} {\bf b} = (a(n) \, b(n))_{n \geq 0}$;
- convolution, ${\bf a} \star {\bf b} =
(\sum_{0 \leq i \leq n} a(i) \, b(n-i) )_{n \geq 0}$;
- perfect shuffle, ${\bf a} \,\sha\, {\bf b} =
{\bf c} = (c(n))_{n \geq 0}$, where $c(2i) = a(i)$ and $c(2i+1) = b(i)$ for $i \geq 0$. The same is true for $t$-way perfect shuffle, where we combine $t$ sequences analogously.
For proofs, see [@Allouche-Shallit:1992:regular-sequences].
\[rem:poly-regular\] Let $p$ be a multivariate polynomial with integer coefficients, and suppose $d_1, \ldots, d_t\in\set{0,\dots,k-1}$. Then $$(\f{p}{\abs{z}_{d_1}, \abs{z}_{d_2}, \dots, \abs{z}_{d_t}})_{n \geq 0}$$ with $z = (n)_k$, is (effectively) a $k$-regular sequence in $n\in{\mathbb{N}}_0$ over ${\mathbb{Z}}$. This is true because the number of occurrences $\abs{z}_d$ of a digit $d$ in the standard $k$-ary expansion $z=(n)_k$ is a $k$-regular sequence by [@Allouche-Shallit:1992:regular-sequences Theorem 6.1] and, as above, $k$-regular sequences are closed under term-by-term addition and multiplication.
Growth of k-regular sequences
=============================
We use the standard notation for asymptotic growth of sequences. Let $(f(n))_{n\ge0}$ and $(g(n))_{n\ge0}$ be sequences. Then
- $f(n) \in O(g(n))$ means that there exist $n_0$, $c > 0$ such that $f(n) \leq c g(n)$ for all $n \geq n_0$, and
- $f(n) \in \Omega(g(n))$ means that there exist $n_0$, $c > 0$ such that $f(n) \geq c g(n)$ for all $n \geq n_0$.
For simplicity, we sometimes say that the sequence $(f(n))_{n\ge0}$ is in $O(g(n))$ or $\Omega(g(n))$.
In what follows, ${\mathbb{A}}$ denotes the set of real algebraic numbers.
Lower bounds {#sec:lower-bound}
------------
\[thm:unsolvable:Omega-n\] Let $k\geq2$ be an integer and ${\mathbb{S}}$ be a set with ${\mathbb{N}}\subseteq{\mathbb{S}}\subseteq{\mathbb{A}}$. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, is $f(n)$ in $\f{\Omega}{n}$?”
is recursively unsolvable.
We reduce from Hilbert’s tenth problem. For a given multivariate polynomial $p$ in $t$ variables over ${\mathbb{Z}}$, we choose $r\in{\mathbb{N}}$ such that $K=k^r \geq t+2$, and we construct the sequence $$f(n) \coloneqq
(n+1)
\bigl(\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t}\bigr)^2 \,
(\abs{z}_{t+1} + 1)$$ with $z = (n)_K$. The sequence $(f(n))_{n \geq 0}$ is $K$-regular (see Remark \[rem:poly-regular\]) and therefore $k$-regular as well [@Allouche-Shallit:1992:regular-sequences Theorem 2.9].
The following claim shows that the above indeed provides a reduction.
The sequence $(f(n))_{n \geq 0}$ is not in $\in \Omega(n)$ iff there exist nonnegative integers $x_1$, $x_2$, …, $x_t$ such that $p(x_1, \dots, x_t) = 0$.
To see this, note that $f(n) = 0$ iff at least one factor is zero, but the first and third factors defining it are never zero. Hence $f(n) = 0$ iff $\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t} = 0$. Moreover, note that if a zero of $(f(n))_{n\ge0}$ occurs once, it occurs infinitely often by the third factor of its construction.
Thus, if $p(x_1, \dots, x_t) = 0$ has at least one solution, then $f(n) = 0$ for infinitely many $n$, and $f(n)$ is not in $\Omega(n)$. Otherwise, if $p(x_1, \dots, x_t) = 0$ does not have any solution, then its absolute value is at least one, so the value of $f(n)$ is at least $n+1$, and hence $f(n)$ is in $\Omega(n)$. This contradiction completes the proof of the claim and consequently the proof of Theoren \[thm:unsolvable:Omega-n\].
Theorem \[thm:unsolvable:Omega-n\] can be extended to other growth rates as well.
\[cor:unsolvable:Omega\] Let $k\geq2$ be an integer. Suppose $\sigma$ is a real number and $\ell$ is a nonnegative integer, not both zero. Assume that $k^\sigma$ is an algebraic number. Let ${\mathbb{S}}$ be a ring with ${\mathbb{Z}}\subseteq{\mathbb{S}}\subseteq{\mathbb{A}}$ and containing $k^\sigma$. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, is $f(n)$ in $\f[\big]{\Omega}{n^\sigma (\log n)^\ell}$?”
is recursively unsolvable.
\[rem:growth-theta-sequence\] For a real number $\sigma$ and a nonnegative integer $\ell$ we construct a $k$-regular sequence $(h_{\sigma,\ell}(n))_{n \geq 0}$ with positive terms (except for the first few terms, which may be $0$) and with $$h_{\sigma,\ell}(n) \in \f[big]{\Theta}{n^\sigma (\log n)^\ell}.$$
Set $H_0=\cdots=H_{k-1}=J_{\ell+1}(k^\sigma)$, where $J_{\ell+1}(k^\sigma)$ is a Jordan block, of size $\ell + 1$, corresponding to the eigenvalue $k^\sigma$. We set $$h_{\sigma,\ell}(n) = e_1 \f{H}{n}\, e_{\ell+1},$$ where $H(n) = H_{n_0} \cdots H_{n_{\ell-1}}$ for $(n)_k = n_{\ell-1} \cdots n_0$, and the $e_i$ are the $i$th unit vectors. Therefore, the sequence $h_{\sigma,\ell}(n)$ is $k$-regular, as it is defined by a linear representation. Explicitly, we have $$h_{\sigma,\ell}(n) = \binom{s}{\ell} k^{(s-\ell)\sigma},$$ where $s=\floor{\log_k n}+1$. Thus, this sequence’s asymptotic behavior is $\f[big]{\Theta}{n^\sigma (\log n)^\ell}$. If $\ell=0$, then no term is $0$. If $\ell \not=0$, then only terms with $n \leq k^{\ell-1}$ are $0$.
The proof runs along the same lines as the proof of Theorem \[thm:unsolvable:Omega-n\]. For a given multivariate polynomial $p$ in $t$ variables over ${\mathbb{Z}}$, we instead choose $r\in{\mathbb{N}}$ such that $K=k^r \geq t+2$, and we define $$f(n) \coloneqq
(\f{h_{\sigma,\ell}}{n} + 1)
\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t} \,
(\abs{z}_{t+1} + 1)$$ with $z = (n)_K$ and $(\f{h_{\sigma,\ell}}{n})_{n\ge0}$ of Remark \[rem:growth-theta-sequence\]. Note that the factor $(\f{h_{\sigma,\ell}}{n} + 1)$ of $f(n)$ is increasing and always positive.
Upper bounds {#sec:upper-bound}
------------
Let $(h(n))_{n \geq 0}$ be a sequence. We say that a sequence $(M(n))_{n\ge0}$ of matrices with entries in a set ${\mathbb{S}}\subseteq{\mathbb{A}}$ is in $\Oh{h(n)}$, formally written as usual as $$M(n) \in \Oh{h(n)},$$ if each sequence of a fixed entry (fixed row and column) of the matrices is in $\Oh{h(n)}$. Rephrased, this means that the sequence of maximum norms of the matrices lies in $\Oh{h(n)}$. By the equivalence of norms, this is also true for any other norm. As in the one-dimensional case, we say that the sequence $(M(n))_{n\ge0}$ is *bounded* if it lies in $\Oh{1}$.
\[rem:growth:linear-deviation\] Let $\sigma\in{\mathbb{R}}$ and $\ell\in{\mathbb{N}}_0$, and set $h(n) = n^\sigma (\log
n)^\ell$. If we have $m(n) = cn + \tau(n)$ for some constant $c\neq0$ and some sequence $\tau(n)\in\oh{n}$, then $$\Oh{h(m)} = \Oh{h(n)}.$$ as $n\to\infty$. This follows from $$\begin{aligned}
h(m(n)) &= \bigl(cn + \tau(n)\bigr)^\sigma \bigl(\log (cn + \tau(n))\bigr)^\ell \\
&= c^\sigma h(n)
\Bigl(1 + \frac{\tau(n)}{cn}\Bigr)^\sigma
\biggl(1 + \frac{\log\bigl(c+\frac{\tau(n)}{n}\bigr)}{\log n}\biggr)^\ell \\
&= c^\sigma h(n) \bigl(1+\oh{1}\bigr).
\end{aligned}$$
\[thm:unsolvable:bounded\] Let $k\geq2$ be an integer and ${\mathbb{S}}$ be a ring with ${\mathbb{Q}}\subseteq{\mathbb{S}}\subseteq{\mathbb{A}}$. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, is $f(n)$ bounded?”
is recursively unsolvable.
The above problem is decidable for $k$-regular sequences that have a linear representation with integer matrices, as it is decidable whether products of integer matrices are bounded; see an algorithm of Mandel and Simon [@Mandel-Simon:1977:finite-matrix-semigroups] for matrices with nonnegative entries, and an algorithm of Jacob for general integer matrices.
\[lem:property:k-regular-products\] Let ${\mathbb{S}}$ be a ring. Let $P$ be a property of a sequence over ${\mathbb{S}}$, i.e., $P$ for each sequence over ${\mathbb{S}}$ is either true or false. Suppose we can extend property $P$ to sequences of matrices over ${\mathbb{S}}$ in one of the following ways: Property $P$ holds for a sequence of matrices iff $P$ holds for
1. \[itm:lem:property:k-regular-products:all\] all sequences
2. \[itm:lem:property:k-regular-products:any\] any sequence
consisting of a fixed entry (fixed row and column).
If $P$ is recursively solvable for $k$-regular sequences over ${\mathbb{S}}$ and for powers of one single square matrix, then $P$ is recursively solvable for products of matrices chosen from a finite set of square matrices over ${\mathbb{S}}$, all of the same dimension.
We will use this lemma in the proof of Theorem \[thm:unsolvable:bounded\] with the property $P$ being the boundedness of a sequence and in the proof of Theorem \[thm:unsolvable:poly\], where $P$ is true iff a sequence has polynomial growth.
We show (\[itm:lem:property:k-regular-products:all\]). Then (\[itm:lem:property:k-regular-products:any\]) follows by using the negation of property $P$.
Given square matrices $F_0$, …, $F_{k-1}$ over ${\mathbb{S}}$ all of the same dimension, we set $F(n) =
F_{n_0} \cdots F_{n_{s-1}}$ for $(n)_k = n_{s-1} \cdots n_0$, and we define $$f_{i,j}(n) = e_i F(n) e_j$$ where $e_i$ is the $i$th unit vector. Therefore, $(f_{i,j}(n))_{n\ge0}$ is the sequence of entries in row $i$ and column $j$ of $(F(n))_{n\ge0}$. All sequences $(f_{i,j}(n))_{n\ge0}$ are $k$-regular, as they are defined by a linear representation. Clearly all these sequences $(f_{i,j}(n))_{n\ge0}$ satisfy $P$ iff $F(n)$ satisfies $P$.
As the question of deciding property $P$ of a $k$-regular sequence is recursively solvable, we can decide $P$ for all $(f_{i,j}(n))_{n \geq 0}$, as there are only finitely many of them and therefore can decide $P$ for $(F(n))_{n\ge0}$. As we can decide $P$ for the sequence $(F_0^s)_{s\ge0}$ as well, we can decide $P$ for $$\setm[\big]{\f{F}{n}\, F_0^s}{\text{$n$, $s\in{\mathbb{N}}_0$}}.$$ As this set equals the set of all possible matrix products of $F_0,
\ldots, F_{k-1}$, the proof is complete.
We reduce from the question of boundedness of all products of matrices over the rationals, which is not recursively solvable; see Blondel and Tsitsiklis [@Blondel-Tsitsiklis:2000:boundedness-undecidable].
The reduction is provided by Lemma \[lem:property:k-regular-products\] with property $P$ being the boundedness of a sequence. Note that we can decide the boundedness of the powers of a matrix from knowledge of its Jordan decomposition.
\[cor:unsolvable:Oh\] Let $k\geq2$ be an integer, $\sigma$ a real number and $\ell$ a nonnegative integer. Assume that $k^\sigma$ is an algebraic number. Let ${\mathbb{S}}$ be a ring with ${\mathbb{Q}}\subseteq{\mathbb{S}}\subseteq{\mathbb{A}}$ and containing $k^\sigma$. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, is $f(n)$ in $\Oh{n^\sigma (\log n)^\ell}$?”
is recursively unsolvable.
We reduce from the decision problem stated in Theorem \[thm:unsolvable:bounded\]. For a $k$-regular sequence $(g(n))_{n\ge0}$, we construct $\f{f}{n} = \f{g}{n} \f{h_{\sigma,\ell}}{n}$ with $\f{h_{\sigma,\ell}}{n}$ as defined in Remark \[rem:growth-theta-sequence\].
Then, the $k$-regular sequence $(f(n))_{n \geq 0}$ is in $\Oh{n^\sigma (\log n)^\ell}$ iff $g(n)$ is in $\Oh{1}$, i.e., bounded. Therefore deciding if a $k$-regular sequence is in $\Oh{n^\sigma (\log n)^\ell}$ implies deciding the boundedness of a $k$-regular sequence, which contradicts Theorem \[thm:unsolvable:bounded\].
Let $\sigma\in{\mathbb{R}}$ and $\ell\in{\mathbb{N}}_0$. We say that a sequence $(f(n))_{n \geq 0}$ has [*exact growth*]{} $n^\sigma (\log n)^\ell$ if $$f(n) \in \Oh[\big]{n^\sigma (\log n)^\ell}$$ but for all $\sigma'\in{\mathbb{R}}$ and $\ell'\in{\mathbb{N}}_0$ with $(\sigma',\ell')$ lexicographically smaller than $(\sigma,\ell)$ we have $$f(n) \not\in \Oh[\big]{n^{\sigma'} (\log n)^{\ell'}}.$$
\[pro:growth-matrices-vs-sequence\] Let $(f(n))_{n \geq 0}$ be a $k$-regular sequence over a field ${\mathbb{S}}\subseteq {\mathbb{A}}$ with matrices $F_0, \dots,
F_{k-1}$ of a minimal $k$-linear representation, and set $F(n) =
F_{n_0} \cdots F_{n_{s-1}}$ for $(n)_k = n_{s-1} \cdots n_0$. Let $\sigma\in{\mathbb{R}}$ and $\ell\in{\mathbb{N}}_0$, and set $h(n) = n^\sigma (\log n)^\ell$. Then $$f(n) \in \Oh{h(n)}$$ if and only if $$F(n) \in \Oh{h(n)}.$$ In particular, both $f(n)$ and $F(n)$ have the same exact growth.
Let $\lambda$ and $\gamma$ be the vectors of our minimal representation, i.e., $f(n) = \lambda \f{F}{n} \gamma$ for all $n\in{\mathbb{N}}_0$. We start with the easy direction: As $f(n)$ is a finite linear combination of the entries in the matrix $F(n)$ and each of these entries is in $\Oh{h(n)}$, we have $f(n)$ is in $\Oh{h(n)}$.
Conversely, suppose $F(n)$ is not in $\Oh{h(n)}$. As there are only finitely many entries in each matrix $F(n)$, we can assume that one entry of $F(n)$ is not in $\Oh{n}$. Let $(g(n))_{n\ge0}$ denote the sequence of this fixed entry of the matrices.
For a finite sequence $P \colon N_P \to {\mathbb{S}}$ with $N_P\subseteq{\mathbb{N}}_0$ (i.e., $N_P$ is finite), we let $F(P)$ denote the finite linear combination $$F(P) \coloneqq \sum_{p\in N_P} \f{P}{p} \f{F}{p}.$$ Now, as our linear representation is minimal, there exist finite sequences $P$ and $Q$ with domains $N_P$ and $N_Q$ being subsets of ${\mathbb{N}}_0$ and codomain ${\mathbb{S}}$, and with $$\label{eq:growth-matrices-vs-sequence:sum}
g(n) = \lambda \f{F}{P} \f{F}{n} \f{F}{Q} \gamma$$ for all $n\in{\mathbb{N}}_0$; see [@Berstel-Reutenauer:2011:noncommutative-rational-series Corollary 2.3]. As $g(n)$ is not in $\Oh{h(n)}$, one of the finitely many summands $$\f{P}{p} \f{P}{q} \cdot \lambda \f{F}{p} \f{F}{n} \f{F}{q} \gamma$$ of , where $p \in N_P$ and $q \in N_Q$, is not in $\Oh{h(n)}$. Dividing this summand by $\f{P}{p} \f{P}{q}$ yields a subsequence of $(f(n))_{n \geq 0}$, namely $$\lambda \f{F}{p} \f{F}{n} \f{F}{q} \gamma
= \f{f}{m(n)}$$ with $m(n) = pk^{\abs{(n)_k}+\abs{(q)_k}} + nk^{\abs{(q)_k}} + q$ for all $n\in{\mathbb{N}}_0$. As $\abs{(n)_k}=\floor{\log_k n}+1$, we have $m(n)
= cn + o(n)$ for some constant $c$, and therefore, by Remark \[rem:growth:linear-deviation\], we obtain that the subsequence $\f{f}{m(n)}$ is not in $\Oh{h(n)} =
\Oh{h(m(n))}$. Thus the sequence $(f(n))_{n \geq 0}$ itself is not in $\Oh{h(n)}$.
Polynomial growth {#sec:polynomial-growth}
-----------------
The growth of a $k$-regular sequence is always at most polynomial. To be precise, for a $k$-regular sequence $(f(n))_{n \geq 0}$ with values in ${\mathbb{A}}$, there exists a real constant $\sigma\geq0$ such that $f(n)=\Oh{n^\sigma}$; see [@Allouche-Shallit:1992:regular-sequences Theorem 2.10].
\[thm:unsolvable:poly\] Let $k\geq2$ be an integer and ${\mathbb{S}}$ be a ring with ${\mathbb{Q}}\subseteq{\mathbb{S}}\subseteq{\mathbb{A}}$. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, does $f(n)$ have polynomial growth (or a smaller growth), i.e., is there an $\sigma > 0$ such that $f(n)$ is not in $\Oh{n^\sigma}$?”
is recursively unsolvable.
\[pro:jsr-growth\] Let ${\mathbb{S}}\subseteq{\mathbb{A}}$ be a ring, let $\rho\in{\mathbb{R}}$ be positive, let $F_0$, …, $F_{k-1}$ be square matrices over ${\mathbb{S}}$ all of the same dimension, and set $F(n) =
F_{n_0} \cdots F_{n_{s-1}}$ for $(n)_k =n_{s-1} \dots n_0$. Then the following two statements are equivalent:
1. The joint spectral radius of $F_0$, …, $F_{k-1}$ is $\rho$.
2. For all $\varepsilon>0$ we have $F(n) \in
\Oh[\big]{n^{(\log_k\rho)+\varepsilon}}$ and $F(n) \not\in \Oh[\big]{n^{(\log_k\rho)-\varepsilon}}$ as $n\to\infty$, and we have $F_0^s \in \Oh[\big]{(k^s)^{(\log_k\rho)+\varepsilon}}$ as $s\to\infty$.
In particular, the joint spectral radius $\rho$ is bounded by some positive $\rho'\in{\mathbb{R}}$, i.e., $\rho \leq \rho'$, iff for all $\varepsilon>0$ we have $F(n) \in \Oh[\big]{n^{(\log_k\rho')+\varepsilon}}$ as $n\to\infty$ and $F_0^s \in \Oh[\big]{(k^s)^{(\log_k\rho')+\varepsilon}}$ as $s\to\infty$.
If ${\mathbb{S}}$ is a field and the matrices $F_0$, …, $F_{k-1}$ are of a minimal representation of a $k$-regular sequence $(f(n))_{n \geq 0}$, then we may replace $F(n)$ by $f(n)$ in the statements of this proposition.
We will use the “in particular” part of the lemma with $\rho'=1$ in Theorem \[thm:unsolvable:poly\] to connect polynomial growth with the joint spectral radius $\rho$.
In this proof, we suppose that $s$ and $n$ are related by $s =
\floor{\log_k n} + 1$. Then, by Remark \[rem:growth:linear-deviation\], we have $\Oh{n^{\sigma}} = \Oh{k^{s\sigma}}$ as $n\to\infty$ for any $\sigma$.
We have that for any fixed real $\sigma$, $$\norm{F(n)} \in \Oh{n^{\sigma}}
= \Oh{k^{s\sigma}}$$ as $n\to\infty$ is equivalent to $$\label{eq:jsr-growth:max}
\max_{k^{s-1} \leq n < k^{s}} \norm{F(n)} \in \Oh{k^{s\sigma}}$$ as $s\to\infty$, because $s$ is the same for all $n$ within the given range of the argument of the maximum and the right-hand side $\Oh{k^{s\sigma}}$ only depends on $s$ (and not on $n$).
We set $$\rho_s \coloneqq
\max_{n_0,\dots,n_{s-1}\in\set{0,\dots,k-1}}
\norm{F_{n_0}\cdots F_{n_{s-1}}}^{1/s}$$ Then the bound together with $F_0^s \in \Oh{k^{s\sigma}}$ is equivalent to $$\label{eq:jsr-growth:rho-ell}
k^{s\log_k\rho_s} = \rho_s^s \in \Oh{k^{s\sigma}}$$ as $s\to\infty$, because there is a constant $c>0$ (only depending on the used norm) such that for all $n_0$, …, $n_{s-1}\in\set{0,\dots,k-1}$, there is either a largest index $j\in\set{1,\dots,s}$ with $n_{j-1}\neq0$ or $j=0$ and $$\norm{F_{n_0}\cdots F_{n_{s-1}}}
\leq c \norm{F_{n_0}\cdots F_{n_{j-1}}} \cdot \norm{F_0^{s-j}}
\in \Oh{k^{j\sigma}} \Oh{k^{(s-j)\sigma}}
= \Oh{k^{s\sigma}}.$$
Consequently, the bound is equivalent to the existence of an $S\in{\mathbb{N}}_0$ such that for all $s \geq S$, the inequality $\log_k\rho_s
\leq \sigma$ holds. So much for our preliminary considerations.
Now let $\varepsilon>0$. Then $F(n) \not\in
\Oh[\big]{n^{(\log_k\rho)-\varepsilon}}$, $F(n) \in
\Oh[\big]{n^{(\log_k\rho)+\varepsilon}}$ and $F_0^s \in
\Oh[\big]{(k^s)^{(\log_k\rho)+\varepsilon}}$ iff there is an $S\in{\mathbb{N}}_0$ such that for all $s \geq S$, the inequalities $$(\log_k\rho)-\varepsilon
< \log_k\rho_s
\leq (\log_k\rho)+\varepsilon$$ hold. But this is equivalent to $\log_k\rho = \lim_{s\to\infty}
\log_k\rho_s$ and therefore equivalent to $$\rho = \lim_{s\to\infty} \rho_s,$$ which completes the proof of the equivalence.
If $f(n)$ is as in the proposition, then by Proposition \[pro:growth-matrices-vs-sequence\] we have the equivalence of $$f(n) \in \Oh{n^{\sigma}}.$$ to $$\norm{F(n)} \in \Oh{n^{\sigma}}
= \Oh{k^{s\sigma}}$$ as $n\to\infty$ for any fixed real algebraic $\sigma$, so it is allowed to replace $F(n)$ by $f(n)$ in our statements.
We reduce from the question whether the joint spectral radius of a set of matrices over the rationals is bounded by $1$; see Blondel and Tsitsiklis [@Blondel-Tsitsiklis:2000:boundedness-undecidable].
By the “in particular”-part of Proposition \[pro:jsr-growth\] with $\rho'=1$, the joint spectral radius $\rho$ of $F_0$, …, $F_{k-1}$ being at most $1$ is equivalent to the condition that for all $\varepsilon=\sigma>0$ we have $F(n) \in
\Oh{n^\sigma}$ and $F_0^s \in \Oh{k^{s\sigma}}$. The property that $F(n)$ does not have polynomial growth is exactly that for all $\sigma>0$ we have $F(n) \in \Oh{n^\sigma}$.
Now, $F_0^s \in \Oh{k^{s\sigma}}$ can be decided by the Jordan decomposition of $F_0$. Therefore, the reduction of Lemma \[lem:property:k-regular-products\], where property $P$ is whether a sequence does not have polynomial growth, completes the proof.
Images and preimages
====================
By [@Allouche-Shallit:1992:regular-sequences Theorem 5.2], it is undecidable whether a given $k$-regular sequence $(f(n))_{n \geq 0}$ has a zero term, i.e., whether there exists an $n\in{\mathbb{N}}_0$ with $f(n)=0$.
Preimages
---------
In this section we use closure properties of regular languages without further comment. See, for example, .
\[thm:unsolvable:preimage-dfa\] Let $k\geq2$ be an integer and ${\mathbb{S}}\supseteq{\mathbb{N}}_0$ be a set. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$ and a number $q\in{\mathbb{S}}$, is the language associated with $f^{-1}(q)$ regular, i.e., can it be recognized by a deterministic finite automaton?”
is recursively unsolvable.
Above, the language associated with $f^{-1}(q)$ is $\setm{(n)_k}{f(n)=q}$, where $(n)_k$ is the standard $k$-ary expansion of $n$.
A result from [@Allouche-Shallit:1992:regular-sequences Theorem 5.3] states that there exists a $k$-regular sequence $(f(n))_{n\ge0}$ such that neither $\setm{(n)_k}{f(n)=0}$ nor $\setm{(n)_k}{f(n)\neq0}$ are context-free.
We can assume that $q=0$ by subtracting $q$ from the $k$-regular sequence. In order to prove the theorem, we reduce from Hilbert’s tenth problem.
For a given multivariate polynomial $p$ in $t$ variables over ${\mathbb{Z}}$, we choose $r\in{\mathbb{N}}$ such that $K=k^r \geq t+3$, and we construct $$\label{eq:preimage-dfa:f}
f(n) = \bigl(\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t}\bigr)^2
+ \bigl(\abs{z}_{t+1} - \abs{z}_{t+2}\bigr)^2,$$ where $z = (n)_K$. The sequence $(f(n))_{n \geq 0}$ is $K$-regular (see Remark \[rem:poly-regular\]) and therefore $k$-regular as well by [@Allouche-Shallit:1992:regular-sequences Theorem 2.9].
The following claim shows that the above indeed provides a reduction.
The set $f^{-1}(0)$ is not recognized by a deterministic finite automaton iff there exist nonnegative integers $x_1$, $x_2$, …, $x_t$ such that $p(x_1, \dots, x_t) = 0$.
If $p(x_1, \dots, x_t) = 0$ has no solution in ${\mathbb{N}}_0^t$, then $f(n)
\neq 0$ by its construction. Thus $f^{-1}(0)$ is the empty set and is accepted by a deterministic finite automaton.
Otherwise, suppose we have nonnegative integers $x_1$, …, $x_t$ with $p(x_1, \dots, x_t) = 0$, and suppose the language $L=\setm{(n)_K}{f(n)=0}$ is accepted by a deterministic finite automaton, i.e., $L$ is regular. We note that each $z=(n)_K \in L$ satisfies $\abs{z}_{t+1}=\abs{z}_{t+2}$ as $f(n)=0$ and this is equivalent to both squares in its definition being zero. Moreover, for each $s\in{\mathbb{N}}_0$, there is a $z\in L$ with $s=\abs{z}_{t+1}=\abs{z}_{t+2}$.
As $L$ is regular, so is $$L_1 = L \cap 1^{x_1} 2^{x_2} \cdots t^{x_t} (t+1)^+ (t+2)^+$$ where $d^+=\set{d,dd,ddd,\ldots}$ for a letter $d$. Furthermore, the language $$L_2 = L_1 / \set{1^{x_1}2^{x_2} \cdots t^{x_t} (t+1)}$$ is regular. This contradicts $$L_2 = \setm{(t+1)^{m-1} (t+2)^m}{m\geq1},$$ which is not regular; see [@Eilenberg:1974:automata-languages-machines-A Examples 2.8 and 5.2]. The proof of the claim is completed.
Now, if we can decide whether the language associated with $f^{-1}(q)$ is regular, then we can decide whether a solution of $p$ exists, and therefore decide Hilbert’s tenth problem. This completes the proof of Theorem \[thm:unsolvable:preimage-dfa\].
Images
------
\[thm:unsolvable:image-N\] Let $k\geq2$ be an integer and ${\mathbb{S}}\supseteq{\mathbb{N}}_0$ be a set. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, is $\setm{f(n)}{n \in {\mathbb{N}}_0} = {\mathbb{N}}_0$?”
is recursively unsolvable.
In order to prove the theorem, we again reduce from Hilbert’s tenth problem. For a given multivariate polynomial $p$ in $t$ variables over ${\mathbb{Z}}$, we choose $r\in{\mathbb{N}}$ such that $K=k^r \geq t$, and we construct $$f(n) =
\begin{cases}
n/2 + 1, & \text{if $n$ is even;} \\
\bigl(\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t}\bigr)^2, &
\text{if $n$ is odd},
\end{cases}$$ where $z = \bigl((n-1)/2\bigr)_K$. The sequence $(f(n))_{n \geq 0}$ is $K$-regular (Remark \[rem:poly-regular\] and [@Allouche-Shallit:1992:regular-sequences Theorem 2.7]) and therefore $k$-regular as well by [@Allouche-Shallit:1992:regular-sequences Theorem 2.9].
Once we have shown the following claim, we have a reduction to Hilbert’s tenth problem and therefore the proof of Theorem \[thm:unsolvable:image-N\] is completed.
The set $\setm{f(n)}{n \in {\mathbb{N}}_0}$ equals ${\mathbb{N}}_0$ iff there exist nonnegative integers $x_1$, $x_2$, …, $x_t$ such that $p(x_1, \dots, x_t) = 0$.
If $p(x_1, \dots, x_t) = 0$ has no solution in ${\mathbb{N}}_0^t$, then $f(n)
\neq 0$ by its construction. Thus $0$ is not in the set $\setm{f(n)}{n \in {\mathbb{N}}_0}$, so this set cannot be equal to ${\mathbb{N}}_0$.
Otherwise, suppose we have nonnegative integers $x_1, \ldots, x_t$ with $p(x_1, \dots, x_t) = 0$, then there exists an $n\in{\mathbb{N}}_0$ with $f(n)=0$. As $\setm{f(n)}{\text{$n\in{\mathbb{N}}_0$ is even}}$ already contains all the positive integers, all nonnegative integers appear as a value $f(n)$ somewhere.
\[thm:unsolvable:image-Z\] Let $k\geq2$ be an integer and ${\mathbb{S}}\supseteq{\mathbb{Z}}$ be a set. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, is $\setm{f(n)}{n \in {\mathbb{N}}_0} = {\mathbb{Z}}$?”
is recursively unsolvable.
The proof runs along the same lines as the proof of Theorem \[thm:unsolvable:image-N\], but for a given multivariate polynomial $p$ in $t$ variables over ${\mathbb{Z}}$, we choose $r\in{\mathbb{N}}$ such that $K=k^r \geq t$, and we construct $$f(n) =
\begin{cases}
n/3 + 1, & \text{if $n \equiv 0 \pmod 3$;} \\
-(n-1)/3 - 1, & \text{if $n \equiv 1 \pmod 3$;} \\
\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t}, &
\text{if $n \equiv 2 \pmod 3$,}
\end{cases}$$ where $z = \bigl((n-2)/3\bigr)_K$. Then the set $\setm{f(n)}{n \in {\mathbb{N}}_0}$ equals ${\mathbb{Z}}$ iff there exist nonnegative integers $x_1$, $x_2$, …, $x_t$ such that $p(x_1, \dots, x_t) = 0$.
We can extend the above to the question whether two $k$-regular sequences have the same image.
\[cor:unsolvable:image-Z\] Let $k\geq2$ be an integer and ${\mathbb{S}}\supseteq{\mathbb{N}}_0$ be a set. The decision problem
> “Given two $k$-regular sequences $(f(n))_{n \geq 0}$ and $(g(n))_{n \geq 0}$ over ${\mathbb{S}}$, do their images coincide, i.e., is $\setm{f(n)}{n \in {\mathbb{N}}_0} = \setm{g(n)}{n \in {\mathbb{N}}_0}$?”
is recursively unsolvable.
We reduce from the decision problem of Theorem \[thm:unsolvable:image-N\], so let $(f(n))_{n \geq 0}$ be a $k$-regular sequence over ${\mathbb{S}}$ and set $g(n)=n$. If we can decide whether these two sequences have the same image, then we decide whether $$\setm{f(n)}{n \in {\mathbb{N}}_0} = \setm{g(n)}{n \in {\mathbb{N}}_0} = {\mathbb{N}}_0,$$ which contradicts Theorem \[thm:unsolvable:image-N\].
\[thm:unsolvable:value-twice\] Let $k\geq2$ be an integer and ${\mathbb{S}}\supseteq{\mathbb{N}}_0$ be a set. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$ over ${\mathbb{S}}$, does $f(n)$ take the same value twice?”
is recursively unsolvable.
In order to prove the theorem, we reduce from Hilbert’s tenth problem. For a given multivariate polynomial $p$ in $t$ variables over ${\mathbb{Z}}$, we choose $r\in{\mathbb{N}}$ such that $K=k^r \geq t$, and we construct $$g(m) =
\bigl(\f{p}{\abs{z}_1, \abs{z}_2, \dots, \abs{z}_t}\bigr)^2$$ where $z = (m)_K$ and $$f(n) = \sum_{0 \leq m < n} g(m)$$ The sequence $(f(n))_{n \geq 0}$ is $K$-regular (Remark \[rem:poly-regular\] and [@Allouche-Shallit:1992:regular-sequences Theorem 3.1]) and therefore $k$-regular as well by [@Allouche-Shallit:1992:regular-sequences Theorem 2.9].
Once we have shown the following claim, we have a reduction to Hilbert’s tenth problem und therefore the proof of Theorem \[thm:unsolvable:value-twice\] is completed.
The sequence $(f(n))_{n \geq 0}$ takes the same value twice iff there exist nonnegative integers $x_1$, $x_2$, …, $x_t$ such that $p(x_1, \dots, x_t) = 0$.
If $p(x_1, \dots, x_t) = 0$ has no solution in ${\mathbb{N}}_0^t$, then $g(m)$ is strictly positive, and therefore $f(n)$ strictly increasing. So no value is taken twice.
Otherwise, suppose we have nonnegative integers $x_1$, …, $x_t$ with $p(x_1, \dots, x_t) = 0$, then there exists an $n\in{\mathbb{N}}_0$ with $g(n)=0$, and so $f(n) = f(n+1)$.
Squares and other alpha-powers
==============================
Given a sequence $(f(n))_{n \geq 0}$ and an integer $\alpha \ge 2$, an [*$\alpha$-power*]{} is a nonempty contiguous subsequence $(f(j))_{i \leq j < i+\alpha m}$ of length $\alpha $, for some $i$ and $m$, such that $f(i+t) = f(i+sm+t)$ for all $0 \leq s < \alpha$ and $0 \leq t < m$. We call a $2$-power a [*square*]{}. For example, the fractional part of the decimal representation of $e$ contains the square $18281828$.
A [*palindrome*]{} is a nonempty contiguous subsequence that reads the same forwards and backwards. A palindrome is nontrivial if it is of length $\geq 2$.
For automatic sequences, the presence of squares, higher powers, and nontrivial palindromes is decidable (see, e.g., ). We now show that, in contrast, the existence of these patterns is undecidable for $k$-regular sequences.
\[thm:undecide-squares\] Let $\alpha \ge 2$ be an integer. The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$, does $(f(n))_{n \geq 0}$ contain an $\alpha$-power?”
is recursively unsolvable.
We reduce from the problem of deciding whether a $k$-regular sequence has a $0$ term.
Given a $k$-regular sequence $(f(n))_{n \geq 0}$ for which we want to decide whether $f(n) = 0$ for some $n$, we can (effectively) transform it to the $k$-regular sequence $(g(n))_{n \geq 0}$ defined recursively by $g(0)=1$ and $$g(n) = g(n-1) + f(n-(\alpha-1))^2 \cdots f(n-2)^2 \, f(n-1)^2,
\quad\text{for $n\ge1$}.$$ (Note that we use the convention $f(-i)=1$ for $i\ge1$.) For squares, this simplifies to the explicit formula $g(n) = 1 + f(0)^2 + \cdots + f(n-1)^2$. Then $(g(n))_{n \geq 0}$ is (not necessarily strictly) increasing, so it contains an $\alpha$-power iff there exists $n \ge 0$ such that $g(n) = g(n+1) = \cdots = g(n+\alpha-1)$. But this occurs iff $f(n)$ = 0.
Using the same technique, we can prove following theorem for palindromes.
The decision problem
> “Given a $k$-regular sequence $(f(n))_{n \geq 0}$, does $(f(n))_{n \geq 0}$ contain a nontrivial palindrome?”
is recursively unsolvable.
The same proof given for squares above works unchanged.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
Jean-Paul Allouche and Jeffrey Shallit, [*The ring of $k$-regular sequences*](http://dx.doi.org/10.1016/0304-3975(92)90001-V), Theoret. Comput. Sci. **98** (1992), no. 2, 163–197.
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|
---
abstract: 'It has recently been suggested that, in the field, $\sim\!\!56\%$ of Sun-like stars ($0.8\,{\rm M}_{_\odot}\la M_\star\la 1.2\,{\rm M}_{_\odot}$) are single. We argue here that this suggestion may be incorrect, since it appears to be based on the multiplicity frequency of systems with Sun-like primaries, and therefore takes no account of Sun-like stars that are secondary (or higher-order) components in multiple systems. When these components are included in the reckoning, it seems likely that only $\sim\!46\%$ of Sun-like stars are single. This estimate is based on a model in which the system mass function has the form proposed by Chabrier, with a power-law Salpeter extension to high masses; there is a flat distribution of mass ratios; and the probability that a system of mass $M$ is a binary is $\,0.50 + 0.46\log_{_{10}}\!\left(M/{\rm M}_{_\odot}\right)\,$ for $\,0.08\,{\rm M}_{_\odot}\leq M\leq 12.5\,{\rm M}_{_\odot}$, $\,0\,$ for $\,M<0.08\,{\rm M}_{_\odot}$, and $\,1\,$ for $\,M>12.5\,{\rm M}_{_\odot}$. The constants in this last relation are chosen so that the model also reproduces the observed variation of multiplicity frequency with primary mass. However, the more qualitative conclusion, that a minority of Sun-like stars are single, holds up for virtually all reasonable values of the model parameters. Parenthetically, it is still likely that the majority of [*all*]{} stars in the field are single, but that is because most M Dwarfs probably are single.'
author:
- |
A. P. Whitworth[^1] and O. Lomax\
School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, Wales, UK
bibliography:
- 'antsrefs.bib'
title: 'Are the majority of Sun-like stars single?'
---
\[firstpage\]
Stars: formation, stars: low-mass, stars: mass function, stars: binaries.
Introduction
============
The multiplicity statistics of Sun-like stars in the field have recently been re-evaluated by @Raghetal2010, using a volume-limited sample of 454 stars within $25\,{\rm pc}$. They estimate that, for systems having a Sun-like primary, the multiplicity frequency is $m_{_{\rm S}}(M_{_1}\!=\!{\rm M}_{_\odot})\!=\!0.44\pm0.02$.[^2]
The multiplicity frequency for systems having a primary of mass $M_{_1}$ is defined as $$\begin{aligned}
m_{_{\rm S}}(M_{_1})&=&\frac{B+T+Q+...}{S+B+T+Q+...}\,,\end{aligned}$$ where $S$ is the number of single stars of mass $M_{_1}$, $B$ is the number of binary systems having a primary of mass $M_{_1}$, $T$ is the number of triple systems having a primary of mass $M_{_1}$, $Q$ is the number of quadruple systems having a primary of mass $M_{_1}$, and so on. Implicit in the above statement is the fact that in reality one must consider a finite interval of mass, in order to have meaningful statistics.
Thus, the @Raghetal2010 estimate of $m_{_{\rm S}}({\rm M}_{_\odot})$ indicates that the number of single Sun-like stars exceeds, by a factor of $1.28\pm0.08$, the number of Sun-like stars that are primaries in multiple systems. This is not quite the same as the inference made by @Raghetal2010, [*“the majority ... of solar-type stars are single”*]{}, and reiterated by @DuchKrau2013 [*“a slight majority of all field solar-type stars are actually single”*]{}, since it takes no account of Sun-like stars that are secondaries, tertiaries, etc. in multiple systems, only those that are primaries.
In the sequel we show that the majority of Sun-like stars may, in fact, be in multiples. To keep the analysis simple we consider only single and binary systems, so we are concerned with evaluating the number of Sun-like stars that are secondaries in binary systems. Consideration of higher-order multiple systems would only strengthen our conclusion by bringing into the reckoning additional components that might be Sun-like. Section \[SEC:MODEL\] presents the model of binary statistics that we use, the ranges in which we allow the model parameters to vary, and the basic analysis. Section \[SEC:RES\] presents the results, i.e. the fractions of Sun-like stars that are single, primary or secondary, and how these fractions depend on the parameter choices made. Section \[SEC:CONC\] summarises our conclusions.
Model parameters and analysis {#SEC:MODEL}
=============================
In order to model the binary statistics of Sun-like stars and estimate what fraction are secondaries in binaries, we need to specify the distribution of system masses, the fraction of systems that are binaries (as a function of [*system mass*]{}), and the distribution of mass ratios in binary systems.
We define the system mass to be $M$, and the masses of stars to be $M_i$, where $i\!=\!0$ corresponds to a single star (hereafter a [*single*]{}), $i\!=\!1$ corresponds to the primary in a binary (hereafter a [*primary*]{}), and $i\!=\!2$ to the secondary in a binary (hereafter a [*secondary*]{}). In addition, we introduce the corresponding logarithmic variables $$\begin{aligned}
\mu&=&\log_{_{10}}\!\left\{\!\frac{M}{{\rm M}_{_\odot}}\!\right\}\,,\hspace{1.0cm}{\rm (systems)}\,;\\
\mu_i&=&\log_{_{10}}\!\left\{\!\frac{M_i}{{\rm M}_{_\odot}}\!\right\}\,,\hspace{1.44cm} {\rm (stars)}\,.\end{aligned}$$ Since $$\begin{aligned}
M&=&{\rm M}_{_\odot}\,10^\mu\;\,=\;\,{\rm M}_{_\odot}\,{\rm e}^{\ell\mu}\,,\end{aligned}$$ where $\ell\equiv\ln(10)$, we have $$\begin{aligned}
\frac{dM}{d\mu}&=&\ell\,M\,.\end{aligned}$$ The mass ratio of a binary system is $q\!=\!M_{_2}/M_{_1}$.
The distribution of system masses
---------------------------------
We assume that low- and intermediate-mass systems ($\mu\!<\!\mu_{_{\rm C}}$, see below) have a log-normal mass distribution, as proposed by @Chabrier2005, $$\begin{aligned}
\label{EQN:SYSMASS_LN}
\frac{d{\cal N}}{d\mu}_{_{\rm LN}}&=&\frac{1}{(2\pi)^{1/2}\sigma_{_{\rm S}}}\,\exp\!\left\{-\,\frac{(\mu-\mu_{_{\rm S}})^2}{2\sigma_{_{\rm S}}^2}\right\}\,,\end{aligned}$$ where the subscript [ln]{} is for log-normal. @Chabrier2005 estimates that $\mu_{_{\rm S}}\!=\!-0.60$ and $\sigma_{_{\rm S}}\!=\!0.55$; the normalisation in @Chabrier2005 is different because he uses physical units. We assume that these values are accurate to $\pm 0.05$, i.e. $-0.65\leq\mu_{_{\rm S}}\leq -0.55$ and $0.50\leq\sigma_{_{\rm S}}\leq 0.60$.
For higher-mass systems, $M\!>\!M_{_{\rm C}}$, we adopt a power-law distribution of masses, with exponent $\alpha$, i.e. $$\begin{aligned}
\left.\frac{d{\cal N}}{dM}\right._{_{\rm PL}}&=&\frac{K}{\ell\,{\rm M}_{_\odot}}\,\left(\!\frac{M}{{\rm M}_{_\odot}}\!\right)^{\!-\alpha}\,,\end{aligned}$$ where the subscript [pl]{} is for power-law. It follows that $$\begin{aligned}
\label{EQN:SYSMASS_PL}
\left.\frac{d{\cal N}}{d\mu}\right._{_{\rm PL}}&=&K\,\exp\!\left\{-(\alpha-1)\ell\mu\right\}\,.\end{aligned}$$ The default exponent is $\alpha\!=\!2.35$, as first estimated by @Salpeter1955, and we assume this is accurate to $\pm 0.35$, i.e. $2.00\!\leq\!\alpha\!\leq\!2.70$.
We require that the join between the two distributions be smooth, i.e. at $\mu_{_{\rm C}}$ the distributions (Eqns. \[EQN:SYSMASS\_LN\] & \[EQN:SYSMASS\_PL\]) and their slopes should be equal, so $$\begin{aligned}
\label{EQN:muC}
\mu_{_{\rm C}}\!&\!=\!&\!\mu_{_{\rm S}}\,+\,(\alpha-1)\,\ell\,\sigma_{_{\rm S}}^2\,, \\
K\!&\!=\!&\!\frac{1}{(2\pi)^{1/2}\sigma_{_{\rm S}}}\,\exp\!\left\{(\alpha-1)\ell\mu_{_{\rm C}}\,-\,\frac{(\mu_{_{\rm C}}-\mu_{_{\rm S}})^2}{2\sigma_{_{\rm S}}^2}\right\}.\hspace{0.5cm}\end{aligned}$$ With the default values of the model parameters, the join occurs at $\mu_{_{\rm C}}\!\simeq\! 0.34$, corresponding to a system mass of $M_{_{\rm C}}\!\simeq\!2.2\,{\rm M}_{_\odot}$.
The fraction of systems that are binaries
-----------------------------------------
Since the distribution of mass ratios appears to vary slowly – if at all – with primary mass in the range $0.2\,{\rm M}_{_\odot}\!\la\!M_{_1}\!\la\!2\,{\rm M}_{_\odot}$ [@Jansetal2012; @Raghetal2010; @ReggMeye2013], the fraction of systems that are binaries must depend on system mass in a similar way to the dependence of multiplicity frequency on primary mass, but displaced to slightly higher masses. We therefore assume that the fraction of systems that are binaries is given by $$\begin{aligned}
\label{EQN:BINFRAC}
\beta(\mu)\!\!&\!\!=\!\!&\!\!\left\{\begin{array}{lr}
0\,,&\mu\leq-\,\beta_{_0}/\beta_{_1};\\
\beta_{_0}+\beta_{_1}\mu\,,&-\,\beta_{_0}/\beta_{_1}\!<\!\mu\!<\!(1-\beta_{_0})/\beta_{_1};\\
1\,,&\mu\geq(1-\beta_{_0})/\beta_{_1}.
\end{array}\right.\end{aligned}$$ The justification for adopting this functional form is given in §\[SEC:FITb\], where we also derive the default values of $\beta_{_0}$ and $\beta_{_1}$, and their ranges, by fitting the observed run of multiplicity frequency, $m_{_{\rm S}}$ against primary log-mass, $\mu_{_1}$ (see Fig. \[FIG:mfvsM1\]).
The distribution of mass ratios
-------------------------------
Estimates of mass ratios are sufficiently uncertain, and the samples subject to such severe selection effects, that the appropriate distribution is poorly constrained. A convention has emerged [e.g. @ReggMeye2013; @DuchKrau2013] that – at least in the first instance – the distribution should be fitted with a power-law (exponent $\gamma$), possibly with a minimum mass ratio ($q_{_{\rm MIN}}$). We therefore adopt the distribution $$\begin{aligned}
\label{EQN:MASSRATIO}
\frac{dP}{dq}&=&\left\{\begin{array}{ll}
0\,,&q<q_{_{\rm MIN}}\,;\\
(\gamma+1)\left(1-q_{_{\rm MIN}}^{(\gamma+1)}\right)^{-1}\,q^\gamma\,,\hspace{0.3cm}&q>q_{_{\rm MIN}}\,.\\
\end{array}\right.\end{aligned}$$ The default values are $q_{_{\rm MIN}}=0.00$ and $\gamma=0.00$, i.e. a flat distribution with no minimum [e.g. @Raghetal2010; @Jansetal2012].
Finite $q_{_{\rm MIN}}$ reflects the possibility that Sun-like primaries may eschew brown-dwarf or very low-mass secondaries (the so-called Brown Dwarf Desert). We therefore consider the range $0.00\leq q_{_{\rm MIN}}\leq 0.10$.
Regarding the range of $\gamma$, we adopt $-0.60<\gamma\leq 2.00$ [see @DuchKrau2013 and references therein].
Negative $\gamma$ means a preference for low-mass companions, and $\gamma\!=\!-0.60$ means that (with $q_{_{\rm MIN}}\!=\!0$) $\sim\!50\%$ of systems have $q\!<\!0.2$. This might be appropriate for intermediate-mass and/or wide systems. However, the observational evidence for a predominance of such low $q$ values may actually refer to low-mass tertiary components orbiting unresolved binaries.
Positive $\gamma$ means a preference for a companion of comparable mass , and $\gamma\!=\!2.00$ means that (with $q_{_{\rm MIN}}\!=\!0$) $\sim\!50\%$ of systems have $q\!>\!0.8$. However, the observational evidence for a predominance of such high $q$ values may reflect selection effects. It is mainly seen in systems at the extremes of primary mass, where the statistical limitations of the observational data are most severe, and our concern here is not with such extreme masses.
[Source]{} $m_{_{\rm S}}\!\left({\rm M}_{_\odot}\right)$ $\beta_{_0}$ $\beta_{_1}$ $b_{_\star}\!\left({\rm M}_{_\odot}\right)$ $f_{_\star}\!\left({\rm M}_{_\odot}\right)$
--------------- ----------------------------------------------- -------------- -------------- --------------------------------------------- ---------------------------------------------
@Raghetal2010 0.44 0.50 0.46 0.535 0.461
@DuquMayo1991 0.58 0.64 0.52 0.667 0.450
The distributions of singles, primaries and secondaries
-------------------------------------------------------
With these definitions, the mass distribution of singles is $$\begin{aligned}
\label{EQN:SINGLES}
\frac{d{\cal N}}{d\mu_{_0}}&=&\left(\frac{d{\cal N}}{d\mu}\right)_{_{\mu\!=\!\mu_{_0}}}\,\left\{1-\beta(\mu_{_0})\right\}\,;\end{aligned}$$ the mass distribution of primaries is $$\begin{aligned}
\nonumber
\frac{d{\cal N}}{d\mu_{_1}}&=&\int\limits_{\mu=\mu_{_1}}^{\mu=\mu_{_1}\!+\log_{_{10}}(2)}\,\frac{d{\cal N}}{d\mu}\,\beta(\mu)\\\nonumber
&&\hspace{2.0cm}\times\left(\frac{dP}{dq}\,\ell\,(q+1)\right)_{q=10^{(\mu-\mu_{_1})}-1}d\mu;\\\label{EQN:PRIMARIES}\end{aligned}$$ and the mass distribution of secondaries is $$\begin{aligned}
\nonumber
\frac{d{\cal N}}{d\mu_{_2}}&=&\int\limits_{\mu=\mu_{_2}\!+\log_{_{10}}(2)}^{\mu=\infty}\,\frac{d{\cal N}}{d\mu}\,\beta(\mu)\\\nonumber
&&\hspace{1.25cm}\times\left(\frac{dP}{dq}\,\ell\;q(q+1)\right)_{q=\left(10^{(\mu-\mu_{_2})}-1\right)^{-1}}d\mu.\\\label{EQN:SECONDARIES}\end{aligned}$$ The last terms in the integrands of Eqns. (\[EQN:PRIMARIES\]) and (\[EQN:SECONDARIES\]) represent, respectively, $|\partial P/\partial \mu|_{\mu_{_1}}$ and $|\partial P/\partial \mu|_{\mu_{_2}}$.
![The variation of multiplicity frequency, $m_{_{\rm S}}$, with primary log-mass, $\mu_{_1}\!\equiv\!\log_{_{10}}(M_{_1}/{\rm M}_{_\odot})$. The hatched acceptance boxes represent observational estimates from @Closetal2003, @BasrRein2006, @FiscMarc1992, @Jansetal2012, @DuquMayo1991, @Kouwetal2007, @Rizzetal2013, @Preietal1999, and @Masoetal1998; the width of each box represents the approximate range of masses considered, and the height of each box represents the range of $m_{_{\rm S}}$ deduced. Note that, for the higher mass ranges [i.e. @Kouwetal2007; @Rizzetal2013; @Preietal1999; @Masoetal1998], there are only lower limits on $m_{_{\rm S}}$. The lower (bolder) line is the best fit when we require the model to go through the @Raghetal2010 estimate of $m_{_{\rm S}}$ for systems with Sun-like primaries; this fit defines the default values of the parameters $\beta_{_0}$ and $\beta_{_1}$ (see Table \[TAB:PARAMS\]). The upper (feinter) line is the best fit when we require the model to go through the @DuquMayo1991 estimate of $m_{_{\rm S}}$ for systems with Sun-like primaries; evidently this line fits the other data more comfortably.[]{data-label="FIG:mfvsM1"}](BinFrac_mfvsM1.pdf){width="1.0\columnwidth"}
Statistics
----------
To obtain the binary statistics of stars having mass $M_\star$, we set $\mu_{_0}=\mu_{_1}=\mu_{_2}=\log_{_{10}}(M_\star/{\rm M}_{_\odot})$ in Eqns. (\[EQN:SINGLES\]), (\[EQN:PRIMARIES\]) and (\[EQN:SECONDARIES\]), and evaluate the integrals numerically. The multiplicity fraction for systems having primaries of this mass is then $$\begin{aligned}
m_{_{\rm S}}(M_\star)&=&\frac{d{\cal N}/d\mu_{_1}}{d{\cal N}/d\mu_{_0}+d{\cal N}/d\mu_{_1}}\,,\end{aligned}$$ but the fraction of such stars that are in binaries is $$\begin{aligned}
b_\star(M_\star)&=&\frac{d{\cal N}/d\mu_{_1}+d{\cal N}/d\mu_{_2}}{d{\cal N}/d\mu_{_0}+d{\cal N}/d\mu_{_1}+d{\cal N}/d\mu_{_2}}\,.\end{aligned}$$ We can also compute the ratio of stars of mass $M_\star$ that are secondaries, to stars of mass $M_\star$ that are primaries, $$\begin{aligned}
f_\star(M_\star)&=&\frac{d{\cal N}/d\mu_{_2}}{d{\cal N}/d\mu_{_1}}\hspace{0.8cm}\left(=\;\frac{\left(b_\star-m_{_{\rm S}}\right)}{m_{_{\rm S}}\left(1-b_\star\right)}\right)\,;\end{aligned}$$ this is the fractional increase in the number of stars of mass $M_\star$ in binaries that derives from taking account of secondaries. The subscripts $\star$ on $b$ and $f$ record that these are stellar properties.
Constraining $\beta_{_0}$ and $\beta_{_1}$ {#SEC:FITb}
------------------------------------------
$\stackrel{\mbox{\sc model}}{\mbox{\sc function}}$ $\stackrel{\mbox{\sc model}}{\mbox{\sc parameter},\,X}$ $\stackrel{\mbox{\sc default}}{\mbox{\sc value},\,X_{_{\rm O}}}$ $\stackrel{\mbox{\sc range}}{\Delta X}$ $\stackrel{dm_{_{\rm S}}}{\overline{dX}}$ $\stackrel{db_\star}{\overline{dX}}$ $\stackrel{df_\star}{\overline{dX}}$
---------------------------------------------------- --------------------------------------------------------- ------------------------------------------------------------------ ----------------------------------------- ------------------------------------------- -------------------------------------- --------------------------------------
[distribution]{} $\mu_{_{\rm S}}$ $-0.60$ $\pm 0.05$ 0.13 0.18 0.27
[of system]{} $\sigma_{_{\rm S}}$ $+0.55$ $\pm 0.05$ 0.32 0.44 0.75
[log-masses]{} $\alpha$ $+2.35$ $\pm 0.35$ 0.00 -0.02 -0.14
[binary]{} $\beta_{_0}$ $+0.50$ $\pm 0.05$ 0.90 0.90 -0.20
[fraction]{} $\beta_{_1}$ $+0.46$ $\pm 0.05$ 0.07 0.09 0.15
[distribution]{} $q_{_{\rm MIN}}$ $+0.00$ $+0.10,\,-0.00$ -0.07 0.03 0.59
[of mass ratios]{} $\gamma$ $+0.00$ $+2.00,\,-0.60$ -0.03 0.01 0.26
We identify the default values of $\beta_{_0}$ and $\beta_{_1}$ by requiring that the model (i) reproduce accurately the multiplicity frequency of Sun-like stars inferred from observation by @Raghetal2010, and (ii) reproduce as closely as possible the run of multiplicity frequency with primary mass derived from other observational studies by @Closetal2003, @BasrRein2006, @FiscMarc1992, @Jansetal2012, @DuquMayo1991, @Kouwetal2007, @Rizzetal2013, @Preietal1999 and @Masoetal1998. The resulting default values are $\beta_{_0}\!=\!0.50$ and $\beta_{_1}\!=\!0.46$, so the probability that a system of mass $M$ is a binary becomes $$\begin{aligned}
\beta(M)&=&\left\{\begin{array}{l}
0, \hspace{2.7cm} M\!<\!0.08\,{\rm M}_{_\odot};\\
0.50+0.46\log_{_{10}}\!\left(\!M/{\rm M}_{_\odot}\!\right)\!, \\
$\,$\hspace{1.5cm} 0.08\,{\rm M}_{_\odot}\!\leq\!M\!\leq\!12.5\,{\rm M}_{_\odot};\\
1, \hspace{2.7cm} M\!>\!12.5\,{\rm M}_{_\odot}.
\end{array}\right.\end{aligned}$$ The corresponding fit to the observational estimates is illustrated by the lower (bolder) line on Fig. \[FIG:mfvsM1\]. A brief discussion of this plot is appropriate.
The model equations and default parameters describing [*both*]{} the distribution of system masses (i.e. Chabrier log-normal at low and intermediate masses, with $\mu_{_{\rm S}}\!=\!-0.60$ and $\sigma_{_{\rm S}}\!=\!0.55$, plus Salpeter power law, with negative slope $\alpha\!=\!2.35$, at high masses), [*and*]{} the distribution of stellar mass ratios (flat, $\gamma\!=\!0.00$, with no minimum, $q_{_{\rm MIN}}\!=\!0.00$) appear to be the natural default choices; we will explore the consequences of varying the model parameters, both individually and collectively, in §\[SEC:RES\]. The justification for adopting Eqn. (\[EQN:BINFRAC\]) for the fraction of systems that are binaries as a function of system mass, and hence the choice of the model parameters $\beta_{_0}$ and $\beta_{_1}$, is more [*adhoc*]{}.
From the observational data presented in Fig. \[FIG:mfvsM1\], it appears that, for $\,-1\la\,\mu_{_1}\la 1$, $\,m_{_{\rm S}}$ increases approximately linearly with $\mu_{_1}$, and hence a linear relation between $\beta$ and $\mu$ (i.e. Eqn. \[EQN:BINFRAC\]) seems the simplest option to explore. However, it is also clear from Fig. \[FIG:mfvsM1\] that Eqn. (\[EQN:BINFRAC\]) gives a much better fit when the @DuquMayo1991 acceptance box is invoked for systems with Sun-like primaries (upper, feinter line) than whan the @Raghetal2010 acceptance box is invoked (lower, bolder line). Even if the @FiscMarc1992 and @DuquMayo1991 acceptance boxes are discounted, the @Raghetal2010 acceptance box is hard to reconcile with the others without introducing a more complex function $m_{_{\rm S}}(\mu_{_1})$, for which the slope has a distinct minimum around $\mu_{_1}\!\sim\!1$. Since – as far as we are aware – no physical explanantion for such a minimum has been advanced, we avoid this complication. It would certainly be convenient if, when in future the multiplicity frequencies for systems with non–Sun-like primaries are evaluated more accurately, the values are reduced (say by $\sim 20\%$) to bring them into line with @Raghetal2010, [*or*]{} if the @Raghetal2010 estimate is revised upwards.
In this context, it is important to note that Sun-like singles derive from systems with $\,\mu\!=\! 0$, Sun-like primaries from systems with $\,0\!<\!\mu\!\leq\log_{_{10}}(2)$, and Sun-like secondaries from systems with $\,\log_{_{10}}(2)\!<\!\mu\!<\!\infty$. In addition, with any sensible choice of the model parameters, the system mass function is falling quite rapdly with increasing $\mu$ for $\mu\!\geq\!0$. Thus the important range of applicability of Eqn. (\[EQN:BINFRAC\]) is $\,0\!\leq\!\mu\!\la\!0.6$.
At low $\,\mu\;(\mu\leq\! -\beta_{_0}/\beta_{_1}$, $M<0.08\,{\rm M}_{_\odot}$), $\beta$ has to be set to zero to avoid non-physical predictions, and this has the consequence that the multiplicity frequency falls to zero for $\mu_{_1}\!\la\!-1.3$ (i.e. $M_{_1}\la 0.05\,{\rm M}_{_\odot}$). The observed multiplicity frequency appears to be very low for such low-mass primaries (only one of the systems reported by @Closetal2003 has $M_{_1}\la 0.05\,{\rm M}_{_\odot}$), but it is probably not zero. We explore the effect of adopting model parameters that increase the multiplicity frequency of brown dwarfs and very low-mass stars in §\[SEC:RES\]. However, it is important to note that the value of $\beta$ at these low masses is irrelevant to the statistics of Sun-like stars, because a low-mass core, $M<{\rm M}_{_\odot}$, cannot spawn a Sun-like star.
At high $\,\mu\;(\mu\geq (1-\beta_{_0})/\beta_{_1}$, $M>12.5\,{\rm M}_{_\odot}$), $\beta$ has to be set to one to avoid non-physical consequences. In fact the acceptance boxes due to @Kouwetal2007, @Rizzetal2013, @Preietal1999 and @Masoetal1998 are all lower limits. At these high primary masses, the multiplicity frequency is an inadequate measure of multiplicity, because higher-order multiple systems become increasingly important at high primary mass, and the multiplicity frequency does not distinguish between higher-order multiples and binaries [@HubbWhit2005]. The pairing factor (number of orbits per system) or companion frequency (mean number of companions) would be more appropriate measures. In addition, the notion of high-mass [*field*]{} stars is fraught, because the highest-mass stars do not live long enough for their birth clusters to disolve completely into the field, and therefore many high-mass stars in the field are runaways, which have been ejected in violent $N$-body interactions, and therefore seldom, if ever, have companions. The multiplicity statistics for high-mass stars, which point to very high pairing factors and companion frequencies, tend actually to pertain to stars that are still intimately involved with their birth clusters.
Results {#SEC:RES}
=======
The default solution
--------------------
For Sun-like stars and the default parameter set, we obtain $m_{_{\rm S}}({\rm M}_{_\odot})\!=\!0.440$, $b_\star({\rm M}_{_\odot})\!=\!0.535$ and $f_\star({\rm M}_{_\odot})\!=\!0.42$. In other words, although for every $\sim\!56$ single Sun-like stars, there are only $\sim\!44$ Sun-like primaries in binary systems, there are also $\sim\!20$ Sun-like secondaries in binary systems, hence a total of $\sim\!66$ Sun-like stars in binaries.
The effect of varying the model parameters
------------------------------------------
Figs. \[FIG:Variance\] and \[FIG:MonteCarlo\] represent the $(m_{_{\rm S}}({\rm M}_{_\odot}),b_\star({\rm M}_{_\odot}))$-plane in the vicinity of the solution obtained with the default parameters (hereafter the default solution). The black dot at $m_{_{\rm S}}({\rm M}_{_\odot})\!=\!0.440$ and $b_\star({\rm M}_{_\odot})\!=\!0.535$ on Fig. \[FIG:Variance\] represents the default solution. The vertical dashed lines demark the limits on $m_{_{\rm S}}({\rm M}_{_\odot})$ obtained by @Raghetal2010, viz. $0.42\!<\!m_{_{\rm S}}({\rm M}_{_\odot})\!<\!0.46$. The horizontal dotted line is at $b_\star({\rm M}_{_\odot})\!=\!0.50$; for solutions below (above) this line more (less) than $50\%$ of Sun-like stars are single.
On Fig. \[FIG:Variance\], the lines passing through the default solution show how the solution changes if one of the model parameters is varied. The arrow at one end of a line indicates the direction in which the parameter increases, and the symbol for the parameter in question is given beside this arrow. The lines for $\mu_{_{\rm S}}$, $\sigma_{_{\rm S}}$ and $\beta_{_1}$ lie almost on top of one another; each of these lines extends approximately the same distance on either side of the default solution, and therefore the length can be inferred from the position of the corresponding arrow (the line for $\sigma_{_{\rm S}}$ is the longest, and that for $\beta_{_1}$ the shortest).
Over the range of system log-masses that contributes Sun-like stars, $0\!\leq\!\mu\!<\!\infty$, the mass function is decreasing with increasing $\mu$, i.e. $d{\cal N}/d\mu\!<\!0$. Increasing $\mu_{_{\rm S}}$ and/or increasing $\sigma_{_{\rm S}}$ and/or decreasing $\alpha$ increases $d{\cal N}/d\mu\!<\!0$, i.e. [*reduces*]{} the downward slope of the system mass function, so there are then more Sun-like primaries relative to singles (increased $m_{_{\rm S}}({\rm M}_{_\odot})$) and more Sun-like secondaries relative to primaries (increased $b_\star({\rm M}_{_\odot}))$. Conversely, decreasing $\mu_{_{\rm S}}$ and/or decreasing $\sigma_{_{\rm S}}$ and/or increasing $\alpha$ decreases $m_{_{\rm S}}({\rm M}_{_\odot})$ and $b_\star({\rm M}_{_\odot})$. Increasing $\alpha$ from its minimum value ($\alpha\!=\!2.00$) has an ever diminishing effect, because the switch from the log-normal system mass distribution to the power-law distribution occurs at ever increasing $\mu_{_{\rm C}}$ (see Eqn. \[EQN:muC\]); consequently the line for increasing $\alpha$ does not extend far beyond the default solution on Fig. \[FIG:Variance\], once $\alpha\!>\!2.35$.
Increasing $\beta_{_0}$ and/or $\beta_{_1}$ above the default values increases the number of primaries and secondaries, relative to singles, and hence increases both $m_{_{\rm S}}({\rm M}_{_\odot})$ and $b_\star({\rm M}_{_\odot})$; increasing $\beta_{_0}$ has a larger effect than increasing $\beta_{_1}$ because the binary statistics of Sun-like stars are dominated by systems with $0\!\leq\!\mu\!\la\!0.6$, i.e. relatively small $\mu$. Conversely, decreasing $\beta_{_0}$ and/or $\beta_{_1}$ below their default values decreases both $m_{_{\rm S}}({\rm M}_{_\odot})$ and $b_\star({\rm M}_{_\odot})$.
Increasing $q_{_{\rm MIN}}$ and/or $\gamma$ above the default values squeezes the distribution of systems involving Sun-like primaries towards the upper end of the available range ($0\!<\!\mu\!\leq\!\log_{_{10}}(2)$), i.e. [*down*]{} the falling system log-mass distribution, thereby reducing the number of Sun-like primaries. At the same time, it squeezes the distribution of systems involving Sun-like secondaries towards the lower end of the available range ($\log_{_{10}}(2)\!<\!\mu\!\leq\!\infty$), i.e. [*up*]{} the falling system log-mass distribution, thereby augmenting the number of Sun-like secondaries. Consequently $m_{_{\rm S}}({\rm M}_{_\odot})$ is reduced, but $b_\star({\rm M}_{_\odot})$ is augmented. $q_{_{\rm MIN}}$ cannot be reduced below the default value, but decreasing $\gamma$ below the default value augments $m_{_{\rm S}}({\rm M}_{_\odot})$ and reduces $b_\star({\rm M}_{_\odot})$.
Table \[TAB:PARAMS\] gives the parameter symbols; their default values and prescribed ranges; and the partial derivatives, $\partial m_{_{\rm S}}({\rm M}_{_\odot})/\partial X$, $\partial b_\star({\rm M}_{_\odot})/\partial X$ and $\partial f_\star({\rm M}_{_\odot}) /\partial X$, where $X$ is one of the model parameters (i.e. $X\equiv\mu_{_{\rm S}},\;\sigma_{_{\rm S}},\;\beta_{_0},\;\beta_{_1},\;q_{_{\rm MIN}},\;\gamma$), and all the derivatives are evaluated for the default parameter set.
Fig. \[FIG:MonteCarlo\] displays the distribution of solutions when the model parameters are varied simultaneously and randomly. This plot is produced by generating $\sim\!1.8\times 10^9$ different solutions ($\sim\!10^9$ of which fall within Fig. \[FIG:MonteCarlo\]). For each solution, the value of parameter $X$ is chosen by generating a random Gaussian deviate, ${\cal G}$, from a distribution with mean $0$ and standard deviation 1, and then putting $X\!=\!X_{_{\rm O}}\!+\!{\cal G}\,\Delta\! X$, where the values of $X_{_{\rm O}}$ and $\Delta X$ for each model parameter are given in Table \[TAB:PARAMS\] (Columns 3 and 4); with this procedure, $\sim 32\%$ of parameter values fall outside the range $X_{_{\rm O}}\pm\Delta\!X$.
If we consider all the solutions generated in this way (including those that fall outside Fig. \[FIG:MonteCarlo\]), $\sim 21\%$ of them give $f_\star\!<\!0.50$. However, the majority of these solutions involve very low values of $\beta_{_0}$, and therefore they also deliver $m_{_{\rm S}}({\rm M}_{_\odot})\!<\!0.42$ [i.e. below the range estimated by @Raghetal2010].
If we limit consideration to solutions that satisfy the constraints calculated by @Raghetal2010, i.e. $0.42\!\leq\!m_{_{\rm S}}({\rm M}_{_\odot})\!\leq\!0.46$ (including those that fall outside Fig. \[FIG:MonteCarlo\]), then $\sim 3\%$ of the allowed solutions give $f_\star\!<\!0.50$, because these solutions require both very low $\beta_{_0}$ and very low $\gamma$. We conclude that it is rather unlikely that the majority of Sun-like stars are single.
![The interdependence of the multiplicity frequency for systems having Sun-like primaries ($m_{_{\rm S}}({\rm M}_{_\odot})$; abscissa) and the fraction of Sun-like stars that are in binary systems ($b_\star({\rm M}_{_\odot})$; ordinate). The filled circle near the centre represents the combination obtained with the default parameters (see Table \[TAB:PARAMS\]). The lines track the changes in $m_{_{\rm S}}({\rm M}_{_\odot})$ and $b_\star({\rm M}_{_\odot})$ that occur when the model parameters are changed individually between the limits in Table \[TAB:PARAMS\] (for example, when $\mu_{_{\rm S}}$ varies from -0.65 to -0.55). In each case, an arrow at one end of the line indicates the direction in which the parameter increases, and the symbol for the parameter in question is always located beside this arrow. The lines for $\mu_{_{\rm S}}$, $\sigma_{_{\rm S}}$ and $\beta_{_1}$ are almost exactly on top of one another. Since each of these lines extends almost the same distance on either side of the default point, their lengths can be estimated from the locations of the corresponding arrows; the longest line is for $\sigma_{_{\rm S}}$ and the shortest for $\beta_{_1}$. The two vertical dashed lines mark the range of $m_{_{\rm S}}({\rm M}_{_\odot})$ deduced by @Raghetal2010. The horizontal dotted line is at $b_\star({\rm M}_{_\odot})\!=\!0.500$, so all solutions abve this line represent situations in which the majority of Sun-like stars are in multiple systems. We see that no individual parameter can reduce $b_\star({\rm M}_{_\odot})$ below this line without also taking $m_{_{\rm S}}({\rm M}_{_\odot})$ outside the range derived by @Raghetal2010.[]{data-label="FIG:Variance"}](BinFrac_Variance.pdf){width="1.0\columnwidth"}
![The likelihood of different combinations of $m_{_{\rm S}}({\rm M}_{_\odot})$ (the multiplicity frequency for systems having Sun-like primaries; abscissa) and $b_\star({\rm M}_{_\odot})$ (the fraction of Sun-like stars that are in binary systems; ordinate). Different random sets of model parameters have been generated by picking a value for each parameter from a Gaussian distribution having mean equal to the default value and standard deviation equal to the specified range; thus, for any parameter there is a $\sim 32\%$ chance that it falls outside the range $X_{_{\rm O}}\pm\Delta X$. The grey-scale and contours delineate the relative probability of different combinations of $m_{_{\rm S}}({\rm M}_{_\odot})$ and $b_\star({\rm M}_{_\odot})$ that fall within the bounds of the figure. The contours are at $15\%$, $30\%$, ... $90\%$ of the peak probability. If we consider all the combinations generated (including those that fall outside the bounds of the figure), there is a $\sim\!21\%$ chance that the majority of Sun-like stars are single. However, this result is dominated by parameter sets with very low $\beta_{_1}$ which also deliver $m_{_{\rm S}}({\rm M}_{_\odot})$ well below the minimum estimated by @Raghetal2010, i.e. the extension into the bottom lefthand corner of the figure. If we limit our consideration to the sets that deliver $m_{_{\rm S}}({\rm M}_{_\odot})$ in the range deduced by @Raghetal2010, the chance that the majority of Sun-like stars are single falls to $\sim\!3\%$.[]{data-label="FIG:MonteCarlo"}](BinFrac_MonteCarlo.pdf){width="1.0\columnwidth"}
Other masses
------------
If we accept the default parameters as a reasonable representation of the binary statistics of low- and intermediate-mass stars, we can estimate the fraction of such stars that are single. For example, for $M_\star\!\leq\!0.70\,{\rm M}_{_\odot}$, $\geq\!50\%$ of stars are single. Thus most M Dwarfs are single.
Conclusions {#SEC:CONC}
===========
We have developed a model to estimate the fraction of Sun-like stars that are secondaries in binary systems. Our model (which only considers singles and binary systems, and therefore gives a lower limit to the fraction of Sun-like stars that are in multiples) invokes a log-normal distribution of system masses [@Chabrier2005] with a power-law tail at high masses [@Salpeter1955]; a flat distribution of mass ratios [e.g. @Raghetal2010; @Jansetal2012; @ReggMeye2013]; and the probability that a system with mass $M$ is a binary, $$\beta\!=\!0.50+0.47\log_{_{10}}(M/{\rm M}_{_\odot})\,.$$ We find that, even if the multiplicity frequency is as low as the recent estimate of @Raghetal2010, $0.42\!\leq\!m_{_{\rm S}}({\rm M}_{_\odot})\!\leq\!0.46$, and even if we vary the model parameters significantly from their default values, when account is taken of secondaries (and higher-order components in mutiple systems), the majority of Sun-like stars are probably in multiple systems. Our model predicts that the fraction of stars that are in multiple systems is a monotonically increasing function of stellar mass, and that the majority of stars with mass $M_\star\!>\!0.7\,{\rm M}_{_\odot}$ are in multiple systems. Conversely, most M Dwarfs, and hence most stars overall, are single.
Acknowledgements {#acknowledgements .unnumbered}
================
APW and OL gratefully acknowledge the support of a consolidated grant (ST/K00926/1) from the UK STFC. We thank the referee, Mike Simon, for his useful comments.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: Here the subscript [s]{} records the fact that $m_{_{\rm S}}(M_{_1}\!=\!{\rm M}_{_\odot})$ is a system-property. $\,``(M_{_1}\!=\!{\rm M}_{_\odot})"\,$ is to emphasise that – for practical observational reasons – the convention is to estimate $m_{_{\rm S}}$ as a function of primary mass, $M_{_1}$, rather than system mass. We use $m_{_{\rm S}}$ for multiplicity frequency, and not $mf$ as is normal, because mathematical expressions involving a double symbol like $mf$ are inevitably confusing.
|
---
abstract: 'The increasing size of cosmological simulations has led to the need for new visualization techniques. We focus on Smoothed Particle Hydrodynamical (SPH) simulations run with the [GADGET]{} code and describe methods for visually accessing the entire simulation at full resolution. The simulation snapshots are rastered and processed on supercomputers into images that are ready to be accessed through a web interface (GigaPan). This allows any scientist with a web-browser to interactively explore simulation datasets in both in spatial and temporal dimensions, datasets which in their native format can be hundreds of terabytes in size or more. We present two examples, the first a static terapixel image of the MassiveBlack simulation, a [P-GADGET]{} SPH simulation with 65 billion particles, and the second an interactively zoomable animation of a different simulation with more than one thousand frames, each a gigapixel in size. Both are available for public access through the GigaPan web interface. We also make our imaging software publicly available.'
author:
- 'Yu Feng, Rupert A.C. Croft, Tiziana Di Matteo, Nishikanta Khandai, Randy Sargent, Illah Nourbakhsh, Paul Dille, Chris Bartley, Volker Springel, Anirban Jana, Jeffrey Gardner'
title: Terapixel imaging of cosmological simulations
---
Introduction
============
In the 40 years that $N$-body simulations have been used in Cosmology research, visualization has been the most indispensable tool. Physical processes have often been identified first and studied via images of simulations. A few examples are: formation of filamentary structures in the large-scale distribution of matter [@JENKINS; @COPLBERG; @MILLENIUM-I; @DOLAG], growth of feedback bubbles around quasars [@SIJACKI; @FEEDBACK]; cold flows of gas forming galaxies [@COLDFLOW-DEKELA; @KERES], and the evolution of ionization fronts during the re-ionization epoch [@BUBBLESHIN; @BUBBLEZAHN]. The size of current and upcoming Peta-scale simulation datasets can make such visual exploration to discover new physics technically challenging. Here we present techniques that can be used to display images at full resolution of datasets of hundreds of billions of particles in size.
Several implementations of visualization software for cosmological simulations already exist. IRFIT [@IFRIT] is a general purpose visualization suite that can deal with mesh based scalar, vector and tensor data, as well as particle based datasets as points. YT [@YT] is an analysis toolkit for mesh based simulations that also supports imaging. SPLASH [@SPLASH] is a visualization suite specialized for simulations that use smoothed particle hydrodynamics(SPH) techniques. Aside from the CPU based approaches mentioned above, [@GPUVIS] implemented a GPU based interactive visualization tool for SPH simulations.
The Millennium I & II simulations [@MILLENIUM-I; @MILLENIUM-II] have been used to test an interactive scalable rendering system developed by @FSW2009. Both SPLASH and the Millennium visualizer support high quality visualization of SPH data sets, while IRFIT treats SPH data as discrete points.
Continuing improvements in computing technology and algorithms are allowing SPH cosmological simulations to be run with ever increasing numbers of particles. Runs are now possible on scales which allow rare objects, such as quasars to form in a large simulation volume with uniform high resolution (see Section 2.1; and @DIMATTEO-PREP [@DEGRAF-PREP; @KHANDAI-PREP]). Being able to scan through a vast volume and seek out the tiny regions of space where most of the activity is occurring, while still keeping the large-scale structure in context necessitates special visualization capabilities. These should be able to show the largest scale information but at the same be interactively zoomable. However, as the size of the datasets quickly exceeds the capability of moderately sized in-house computer clusters, it becomes difficult to perform any interactive visualizations. For example, a single snapshot of the MassiveBlack simulation (Section 2.1) consists of 8192 files and is over 3 TB in size.
Even when a required large scale high resolution image has been rendered, actually exploring the data requires special tools. The GigaPan collaboration [^1] has essentially solved this problem in the context of viewing large images, with the GigaPan viewer enabling anyone connected to the Internet to zoom into and explore in real time images which would take hours to transfer in totality. The viewing technology has been primarily used to access large photographic panoramas, but is easily applicable to simulated datasets. A recent enhancement to deal with the time dimension, in the form of gigapixel frame interactive movies (GigaPan Time Machine[^2]) will turn out to give particularly novel and exciting results when applied to simulation visualization.
In this work we combine an off-line imaging technique together with GigaPan technology to implement an interactively accessible visual probe of large cosmological simulations. While GigaPan is an independent project (uploading and access to the GigaPan website is publicly available), we release our toolkit for the off-line visualization as [Gaepsi]{}[^3], a software package aimed specifically at [GADGET]{} [@GADGET; @GADGET2] SPH simulations.
The layout of our paper is as follows. In Section 2 we give a brief overview of the physical processes modeled in [GADGET]{}, as well as describing two [P-GADGET]{} simulations which we have visualized. In Section 3 we give details of the spatial domain remapping we employ to convert cubical simulation volumes into image slices. In Section 4, we describe the process of rasterizing an SPH density field, and in Section 5 the image rendering and layer compositing. In Section 6 we address the parallelism of our techniques and give measures of performance. In Section 7 we briefly describe the GigaPan and GigaPan Time Machine viewers and present examples screenshots from two visualizations (which are both accessible on the GigaPan websites).
Simulation
==========
Adaptive Mesh Refinement(AMR, e.g., @AMR) and Smoothed Particle Hydrodynamics (SPH, @SPH) are the two most used schemes for carrying out cosmological simulations. In this work we focus on the visualization of the baryonic matter in SPH simulations run with [P-GADGET]{} [@GADGET2].
[GADGET]{} is an SPH implementation, and [P-GADGET]{} is a version which has been developed specifically for petascale computing resources. It simultaneously follows the self-gravitation of a collision-less N-body system (dark matter) and gas dynamics (baryonic matter), as well as the formation of stars and super-massive black holes. Dark matter particles and gas particle positions and initial characteristics are set up in a comoving cube, and black hole and star particles are created according to sub-grid modeling [@STARFORMATION; @BLACKHOLE] Gas particles carry hydrodynamical properties, such as temperature, star formation rate, and neutral fraction.
Although our attention in this paper is limited to imaging properties the of gas, stars and black holes in [GADGET]{} simulations, similar techniques could be used to visualize the dark matter content. Also, the software we provide should be easily adaptable to the data formats of other SPH codes (e.g. [GASOLINE]{}, [@GASOLINE])
MassiveBlack
------------
The MassiveBlack simulation is the state-of-art SPH simulation of a $\mathsf{\Lambda CDM}$ universe [@DIMATTEO-PREP]. [P-GADGET]{} was used to evolve $2\times3200^{3}$ particles in a volume of side length $\unit[533]{h^{-1}Mpc}$ with a gravitational force resolution of $\unit[5]{h^{-1}Kpc}$. One snapshot of the simulation occupies 3 tera-bytes of disk space, and the simulation has been run so far to redshift $z=4.75$, creating a dataset of order $120$ TB. The fine resolution and large volume of the simulation permits one to usefully create extremely large images. The simulation was run on the high performance computing facility, Kraken, at the National Institute for Computational Sciences in full capability mode with 98,000 CPUs.
E5
--
To make a smooth animation of the evolution of the universe typically requires hundreds frames directly taken as snapshots of the simulation. The scale of the MassiveBlack run is too large for this purpose, so we ran a much smaller simulation (E5) with $2\times336^{3}$ particles in a $\unit[50]{h^{-1}Mpc}$ comoving box. The model was again a $\mathsf{\Lambda CDM}$ cosmology, and one snapshot was output per 10 million years, resulting in 1367 snapshots. This simulation ran on $256$ cores of the Warp cluster in the McWilliams Center for Cosmology at CMU.
Spatial Domain Remapping
========================
Spatial domain remapping can be used to transform the periodic cubic domain of a cosmological simulation to a patch whose shape is similar to the domain of a sky survey, while making sure that the local structures in the simulation are preserved [@REMAPPING; @REMAPPING2]. Another application is making a thin slice that includes the entire volume of the simulation. Our example will focus on the latter case.
A [GADGET]{} cosmological simulation is usually defined in the periodic domain of a cube. As a result, if we let $f(X=(x,y,z))$ be any position dependent property of the simulation, then $$f(X)=f((x+\mu L,y+\nu L,z+\sigma L)),$$ where $\mu,\nu,\sigma$ are integers. The structure corresponds to a simple cubic lattice with lattice constant $a=L$, the simulation box side-length. A bijective mapping from the cubic unit cell to a remapped domain corresponds to a choice of the primitive cell. Figure \[fig:TransformationPrimitive\] illustrates the situation in 2 dimensions.
![Transformation of the Primitive Cell\[fig:TransformationPrimitive\]. The cubical unit cell is shown using solid lines. The new primitive cell, generated by $\left[\protect\begin{smallmatrix} 1 &
1\protect\\ 1 & 2\protect\end{smallmatrix}\right]$ is shown with dash-dotted lines. The transformed domain is shown in gray. ](cell-transformation.png)
Whilst the original remapping algorithm by [@REMAPPING] results in the correct transformations being applied, it has two drawbacks: (i) the orthogonalization is invoked explicitly and (ii) the hit-testing for calculation of the shifting (see below) is against non-aligned cuboids. The second problem especially undermines the performance of the program. In this work we present a faster algorithm based on similar ideas, but which features a QR decomposition (which is widely available as a library routine), and hit-testing against an AABB (Axis Aligned Bounding Box).
First, the transformation of the primitive cell is given by a uni-modular integer matrix, $$M=\left(\begin{matrix}
M_{11} & M_{12} & M_{13}\\
M_{21} & M_{22} & M_{23}\\
M_{31} & M_{32} & M_{33}\end{matrix}\right),$$ where $M_{ij}$ are integers and the determinant of the matrix $|M|=1$. It is straight-forward to obtain such matrices via enumeration. [@REMAPPING] The $QR$ decomposition of $M$ is $$M=QR,$$ where $Q$ is an orthonormal matrix and $R$ is an upper-right triangular matrix. It is immediately apparent that (i) application of $Q$ yields rotation of the basis from the simulation domain to the transformed domain, the column vectors in matrix $Q^{T}$ being the lattice vectors in the transformed domain; (ii) the diagonal elements of $R$ are the dimensions of the remapped domain. For imaging it is desired that the thickness along the line of sight is significantly shorter than the extension in the other dimensions, thus we require $0<R_{33}\ll|R_{22}|<|R_{11}|$. Note that if a domain that is much longer in the line of sight direction is desired, for example to calculate long range correlations or to make a sky map of a whole simulation in projection, the choice should be $0<|R_{33}|<|R_{22}|\ll|R_{11}|$.
Next, for each sample position $X$, we solve the indefinite equation of integer cell number triads $I=(I_{1},I_{2},I_{3})^{T}$,
& =Q\^[T]{}X+aQ\^[T]{}I,
where $a$ is the box size, $\tilde{X}$ is the transformed sample position satisfying $\tilde{X}\in[0,R_{11})\times[0,R_{22})\times[0,R_{33})$. In practice, the domain of $\tilde{X}$ is enlarged by a small number $\epsilon$ to address numerical errors. Multiplying by $Q$ on the left and re-organizing the terms, we find $$I=\frac{Q\tilde{X}}{a}-\frac{X}{a}.$$ Notice that $Q\tilde{X}$ is the transformed sample position expressed in the original coordinate system, and is bounded by its AABB box. If we let $(Q\tilde{X}/a)_{i}\in[B_{i},T_{i}]$, where $B_{i}$ and $T_{i}$ are integers, and notice $\frac{X_{i}}{a}\in[0,1)$, the resulting bounds of $I$ are given by $$I_{i}\in[B_{i},T_{i}].$$ We then enumerate the range to find $\tilde{X}$.
When the remapping method is applied to the SPH particle positions, the transformations of the particles that are close to the edges give inexact results. The situation is similar to the boundary error in the original domain when the periodic boundary condition is not properly considered. Figure \[fig:Distortion\] illustrates the situation by showing all images of the particles that contribute to the imaging domain. We note that for the purpose of imaging, by choosing a $R_{33}$(the thickness in the thinner dimension) much larger than the typical smoothing length of the SPH particle, the errors are largely constrained to lie near the edge. These issues are part of general complications related to the use of a simulation slice for visualization. For example in an animation of the distribution of matter in a slice it is possible for objects to appear and disappear in the middle of the slice as they pass through it. These limitations should be borne in mind, and we leave 3D visualization techniques for future work.
The transformations used for the MassiveBlack and E5 simulations are listed in Table \[tab:Transformations\].
![Boundary effects and Smoothed Particles\[fig:Distortion\]. Four images of a particle intersecting the boundary are shown. The top-right image is contained in the transformed domain, but the other three are not. The contribution of the two bottom images is lost. By requiring the size of the transformed domain to be much larger than typical SPH smoothing lengths, most particles do not intersect a boundary of the domain and the error is contained near the edges.](distortion.png)
---------------------------------------------------------------------------------------------------
Simulation MassiveBlack E5
------------------------- ------------------------------------ ------------------------------------
Matrix $M$ $\left[\begin{smallmatrix} $\left[\tiny\begin{smallmatrix}
5 & 6 & 2\\ 3 & 1 & 0\\
3 & -7 & 3\\ 2 & 1 & 0\\
2 & 7 & 0\end{smallmatrix}\right]$ 0 & 2 & 1\end{smallmatrix}\right]$
$R_{11}$(h$^{-1}$Mpc) 3300 180
$R_{11}$ Pixels(Kilo) 810 36.57
$R_{22}$(h$^{-1}$Mpc) 5800 101
$R_{22}$ Pixels(Kilo) 1440 20.48
$R_{33}$(h$^{-1}$Mpc) 7.9 6.7
Resolution(h$^{-1}$Kpc) 4.2 4.9
---------------------------------------------------------------------------------------------------
: Transformations\[tab:Transformations\]
Rasterization
=============
In a simulation, many field variables are of interest in visualization.
- scalar fields : density $\rho$, temperature $T$, neutral fraction $x_{\text{HI}}$, star formation rate $\phi$;
- vector fields: velocity, gravitational force.
In an SPH simulation, a field variable as a function of spatial position is given by the interpolation of the particle properties. Rasterization converts the interpolated continuous field into raster pixels on a uniform grid. The kernel function of a particle at position $\mathbf{y}$ with smoothing length $h$ is defined as $$W(\mathbf{y},h)=\frac{8}{\pi h^{3}}\begin{cases}
1-6\left(\frac{y}{h}\right)^{2}+6 &
\left(\frac{y}{h}\right)^{3},0\le\frac{y}{h}\le\frac{1}{2}\\
2\left(1-\frac{y}{h}\right)^{3}, &
\frac{1}{2}<\frac{y}{h}\le1,\\ 0 &
\frac{y}{h}>1.\end{cases}$$ Following the usual prescriptions [e.g. @GADGET2; @SPLASH], the interpolated density field is taken as$$\rho(\mathbf{x})=\sum_{i}m_{i}W(\mathbf{x}-\mathbf{x}_{i},h_{i}),$$ where $m_{i}$, $\mathbf{x}_{i}$, $h_{i}$ are the mass, position, and smoothing length of the $i$th particle, respectively. The interpolation of a field variable, denoted by $A$, is given by $$A(\mathbf{x})=\sum_{i}\frac{A_{i}m_{i}W(\mathbf{x}-\mathbf{x}_{i},h_{i})}{\rho(\mathbf{x}_{i})}+O(h^{2}),$$ where $A_{i}$ is the corresponding field property carried by the $i$th particle. Note that the density field can be seen as a special case of the general formula.
Two types of pixel-wise mean for a field are calculated, the
1. volume-weighted mean of the density field,
|(P) & = =\_[P]{}d\^[3]{}()\
& =\_[i]{}m\_[i]{}\_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})\
& =\_[i]{}M\_[i]{}(P),
where $M_{i}(P)=m_{i}\int_{P}d^{3}\mathbf{x}\,
W(\mathbf{x}-\mathbf{x}_{i},h_{i})$ is the mass overlapping of the $i$th particle and the pixel, and the
2. mass-weighted mean of a field ($A$) [^4], $$\begin{aligned}
\bar{A}(P) & = & \frac{\int_{P}d^{3}\mathbf{x}\,
A(\mathbf{x})\rho(\mathbf{x})}{\int_{P}d^{3}\mathbf{x}\,\rho(\mathbf{x})}\\
& = &
\frac{\sum_{i}A_{i}M_{i}(P)}{M(P)}+O(h^{2}).\end{aligned}$$
To obtain a line of sight projection along the third axis, the pixels are chosen to extend along the third dimension, resulting a two dimensional final raster image. The calculation of the overlapping $M_{i}(P)$ in this circumstance is two dimensional. Both formulas require frequent calculation of the overlap between the kernel function and the pixels. An effective way to calculate the overlap is via a lookup table that is pre-calculated and hard coded in the program. Three levels of approximation are used in the calculation of the contribution of a particle to a pixel:
1. When a particle is much smaller than a pixel, the particle contributes to the pixel as a whole. No interpolation and lookup occurs.
2. When a particle and pixel are of similar size of (up to a few pixels in size), the contribution to each of the pixels are calculated by interpolating between the overlapping areas read from a lookup table.
3. When a particle is much larger than a pixel, the contribution to a pixel taken to be the center kernel value times the area of a pixel.
Note that Level 1 and 3 are significantly faster than Level 2 as they do not require interpolations.
The rasterization of the $z=4.75$ snapshot of MassiveBlack was run on the SGI UV Blacklight supercomputer at the Pittsburgh Supercomputing Center. Blacklight is a shared memory machine equipped with a large memory for holding the image and a fairly large number of cores enabling parallelism, making it the most favorable machine for the rasterization. The rasterization of the E5 simulation was run on local CMU machine Warp. The pixel dimensions of the raster images are also listed in Table \[tab:Transformations\]. The pixel scales have been chosen to be around the gravitational softening length of $\sim \unit[5]{h^{-1}Kpc}$ in these simulations in order to preserve as much information in the image as possible.
Image Rendering and Layer Compositing
=====================================
The rasterized SPH images are color-mapped into RGBA (red, green, blue and opacity) layers. Two modes of color-mapping are implemented, the simple mode and the intensity mode.
In the simple mode, the color of a pixel is directly obtained by looking up the normalized pixel value in a given color table. To address the large (several orders of magnitude) variation of the fields, the logarithm of the pixel value is used in place of the pixel value itself.
In the intensity mode, the color of a pixel is determined in the same way as done in the simple mode. However, the opacity is reduced by a factor $f_{m}$ that is determined by the logarithm of the total mass of the SPH fluid contained within the pixel. To be more specific, $$f_{m}=\begin{cases} 0 & \log M<a,\\ 1 & \log M>b,\\
\left[\frac{\log M-a}{b-a}\right]^{\gamma} &
\text{otherwise},\end{cases}$$ where $a$ and $b$ are the underexposure and overexposure parameters: any pixel that has a mass below $10^{a}$ is completely transparent, and any pixel that has a mass above $10^{b}$ is completely opaque.
The RGBA layers are stacked one on top of another to composite the final image. The compositing assumes an opaque black background. The formula to composite an opaque bottom layer $B$ with an overlay layer $T$ into the composite layer $C$ is [@PTDT1984] $$C=\alpha
F+(1-\alpha)T,$$ where $C$, $B$ and $T$ stand for the RGB pixel color triplets of the corresponding layer and $\alpha$ is the opacity value of the pixel in the overlay layer $T$. For example, if the background is red and the overlay color is green, with $\alpha=50\%$, the composite color is a $50\%$-dimmed yellow.
Point-like (non SPH) particles are rendered differently. Star particles are rendered as colored points, while black hole particles are rendered using circles, with the radius proportional to the logarithm of the mass. In our example images, the MassiveBlack simulation visualization used a fast rasterizer that does not support anti-aliasing, whilst the frames of E5 are rendered using matplotlib [@MATPLOTLIB] that does anti-aliasing.
The choice of the colors in the color map has to be made carefully to avoid confusing different quantities. We choose a color gradient which spans black, red, yellow and blue for the color map of the normalized gas density field. This color map is shown in Figure \[fig:Colormap-of-gas\]. Composited above the gas density field is the mass weighted average of the star formation rate field, shown in dark blue, and with completely transparency where the field vanishes. Additionally, we choose solid white pixels for the star particles. Blackholes are shown as green circles. In the E5 animation frames, the normalization of the gas density color map has been fixed so that the maximum and minimum values correspond to the extreme values of density in the last snapshot.
![Fiducial color-map for the gas density field\[fig:Colormap-of-gas\]. The colors span a darkened red through yellow to blue.](gascolormap.png){width="2in"}
Parallelism and Performance
===========================
The large simulations we are interested in visualizing have been run on large supercomputer facilities. In order to image them with sufficient resolution to be truly useful, the creation of images from the raw simulation data also needs significant computing resources. In this section we outline our algorithms for doing this and give measures of performance.
Rasterization in parallel
--------------------------
We have implemented two types of parallelism, which we shall refer to as “tiny” and “huge”, to make best use of shared memory architectures and distributed memory architectures, respectively. The tiny parallelism is implemented with OpenMP and takes advantage of the case when the image can be held within the memory of a single computing node. The parallelism is achieved by distributing the particles in batches to the threads within one computing node. The raster pixels are then color-mapped in serial, as is the drawing of the point-like particles. The tiny mode is especially useful for interactively probing smaller simulations.
The huge version of parallelism is implemented using the Message Passing Interface (MPI) libraries and is used when the image is larger than a single computing node or the computing resources within one node are insufficient to finish the rasterization in a timely manner. The imaging domain is divided into horizontal stripes, each of which is hosted by a computing node. When the snapshot is read into memory, only the particles that contribute to the pixels in a domain are scattered to the hosting node of the domain. Due to the growth of cosmic structure as we move to lower redshifts, some of the stripes inevitably have many more particles than others, introducing load imbalance. We define the load imbalance penalty $\eta$ as the ratio between the maximum and the average of the number of particles in a stripe. The computing nodes with fewer particles tend to finish sooner than those with more. The color-mapping and the drawing of point-like particles are also performed in parallel in the huge version of parallelism.
Performance
-----------
The time spent in domain remapping scales linearly with the total number of particles $N$,$$T_{\text{remap}}\sim
O(N).$$ The time spent in color-mapping scales linearly with the total number of pixels $P$,$$T_{\text{color}}\sim
O(P).$$ Both processes consume a very small fraction of the total computing cycles.
The rasterization consumes a much larger part of the computing resources and it is useful to analyze it in more detail. If we let $\bar{n}$ be the number of pixels overlapping a particle, then $\bar{n}=K^{-1}N^{-1}P$, where $K^{-1}$ is a constant related to the simulation. Now we let $t(n)$ be the time it takes to rasterize one particle, as a function of the number of pixels overlapping the particle. From the 3 levels of detail in the rasterization algorithm (Section 4), we have $$t(n)=\begin{cases} C_{1},
& n\ll1\\ C_{2}n & n\sim1\\ C_{3}n, & n\gg1\end{cases};$$ with $C_{2}\gg C_{1}\approx C_{3}$ . The effective pixel filling rate $R$ is defined as the total number of image pixels rasterized per unit time,$$\begin{gathered}
R=P[t(\bar{n)}N]^{-1}=\bar{n}K[t(\bar{n})]^{-1}\\
=\begin{cases} \bar{n}KC_{1}^{-1}, & \bar{n}\ll1\\
KC_{2}^{-1} & \bar{n}\sim1\\ KC_{3}^{-1}, &
\bar{n}\gg1\end{cases}.\end{gathered}$$
![MassiveBlack Simulation Rasterization Rate\[fig:RasterizationRate\]. We show the number of pixels fixed as a function of resolution (pixel scale). The rate peaks at $KC_{3}^{-1}$ at the high resolution limit and approaches $KC_{2}^{-1}$ as the resolution worsens. The $\bar{n}\ll1$ domain is not explored.](rastertime.png){width="2.7in"}
The rasterization time to taken to create images from a single snapshot of the MassiveBlack simulation (at redshift 4.75) at various resolutions is presented in Table \[tab:RasterizationTime\] and Figure \[fig:RasterizationRate\].
[>p[0.25in]{}>p[0.35in]{}>p[0.25in]{}>p[0.25in]{}>p[0.25in]{}>p[0.15in]{}>p[0.25in]{}]{} Pixels & Res\
($\unit{Kpc/px}$) & $\bar{n}$ & CPUs & Wall-time\
($\unit{hours}$) & $\eta$ & Rate\
($\unit{K/sec}$)[\
]{}5.6G & 58.4 & 80 & 128 & & 1.45 & 11.3[\
]{}22.5G & 29.2 & 330 & 256 & 3.17 & 1.66 & 13.4[\
]{}90G & 14.6 & 1300 & 512 & 3.35 & 1.57 & 24.0[\
]{}90G & 14.6 & 1300 & 512 & 3.06 & 1.39 & 23.3[\
]{}360G & 7.3 & 5300 & 512 & 7.65 & 1.39 & 37.2[\
]{}1160G & 4.2 & 16000 & 1344 & 10.1 & 1.56 & 37.2[\
]{}
The rasterization of the images were carried out on Blacklight at PSC. It is interesting to note that for the largest images, the disk I/O wall-time, limited by the I/O bandwidth of the machine, overwhelms the total computing wall-time. The performance of the I/O subsystem shall be an important factor in the selection of machines for data visualization at this scale.
Image and Animation Viewing
===========================
Once large images or animation frames have been created, viewing them presents a separate problem. We use the GigaPan technology for this, which enables someone with a web browser and Internet connection to access the simulation at high resolution. In this section we give a brief overview of the use of the GigaPan viewer for exploring large static images, as well as the recently developed GigaPan Time Machine viewer for gigapixel animations.
Gigapan
-------
Individual gigapixel-scale images are generally too large to be shared in easy ways; they are too large to attach to emails, and may take minutes or longer to transfer in their entirety over typical Internet connections. GigaPan addresses the problem of sharing and interactive viewing of large single images by streaming in real-time the portions of images actually needed by the viewer of the image, based on the viewers current area of focus inside the image. To support this real-time streaming, the image is divided up and rendered into small tiles of multiple resolutions. The viewer pulls in only the tiles needed for a given view. Many mapping programs (e.g., Google Maps) use the same technique.
We have uploaded an example terapixel image[^5] of the redshift $z=4.75$ snapshot of the MassiveBlack simulation to the GigaPan website, which is run as a publicly accessible resource for sharing and viewing large images and movies. The dimension of the image is $1440000\times810000$, and the finished image uncompressed occupies $\unit[3.58]{TB}$ of storage space. The compressed hierarchical data storage in Gigapan format is about 15% of the size, or $\unit[0.5]{TB}$. There is no fundamental limits to size, provided the data can be stored on the disk. It is possible to create directly the compressed tiles of a GigaPan, bypassing the uncompressed image as an intermediate step, and thus reducing the requirement on memory and disk storage. We leave this for future work.
On the viewer side, static GigaPan works well at different bandwidths; the interface remains responsive independent of bandwidth, but the imagery resolves more slowly as the bandwidth is reduced. is a recommended bandwidth for exploring with a $1024\times768$ window, but the system works well even when the bandwidth is lower.
An illustration of the screen output is shown in Figure \[fig:GigapanView\]. The reader is encouraged to visit the website to explore the image.
GigaPan Time Machine
--------------------
In order to make animations, one starts with the rendered images of each individual snapshot in time. These can be gigapixel in scale or more. In our example, using the E5 simulation (Section 2.2) we have 1367 images each with 0.75 gigapixels.
One approach to showing gigapixel imagery over time would be to modify the single-image GigaPan viewer to animate by switching between individual GigaPan tile-sets. However, this approach is expensive in bandwidth and CPU, leading to sluggish updates when moving through time.
To solve this problem, we created a gigapixel video streaming and viewing system called GigaPan Time Machine [@GIGAPAN], which allows the user to fluidly explore gigapixel-scale videos across both space and time. We solve the bandwidth and CPU problems using an approach similar to that used for individual GigaPan images: we divide the gigapixel-scale video spatially into many smaller videos. Different video tiles contain different locations of the overall video stream, at different levels of detail. Only the area currently being viewed on a client computer need be streamed to the client and decoded. As the user translates and zooms through different areas in the video, the viewer scales and translates the currently streaming video, and over time the viewer requests from the server different video tiles which more closely match the current viewing area. The viewing system is implemented in Javascript+HTML, and takes advantage of recent browser’s ability to display and control videos through the new HTML5 video tag.
The architecture of GigaPan Time Machine allows the content of all video tiles to be precomputed on the server; clients request these precomputed resources without additional CPU load on the server. This allows scaling to many simultaneously viewing clients, and allows standard caching protocols in the browser and within the network itself to improve the overall efficiency of the system. The minimum bandwidth requirement to view videos without stalling depends on the size of the viewer, the frame rate, and the compression ratios. The individual videos in “Evolution of the Universe” (the E5 simulation, see below for link) are currently encoded at with relatively low compression. The large video tiles require a continuous bandwidth of , and a burst bandwidth of .
We have uploaded an example animation[^6] of the E5 simulation, showing its evolution over the interval between redshift $z=200$ and $z=0$ with 1367 frames equally spaced in time by . Again, the reader is encouraged to visit the website to explore the image.
Conclusions
===========
We have presented a framework for generating and viewing large images and movies of the formation of structure in cosmological SPH simulations. This framework has been designed specifically to tackle the problems that occur with the largest datasets. In the generation of images, it includes parallel rasterization (for either shared and distributed memory) and adaptive pixel filling which leads to a well behaved filling rate at high resolution. For viewing images, the GigaPan viewers use hierarchical caching and cloud based storage to make even the largest of these datasets fully explorable at high resolution by anyone with an internet connection. We make our image making toolkit publicly available, and the GigaPan web resources are likewise publicly accessible.
This work was supported by NSF Awards OCI-0749212, AST-1009781, and the Moore foundation. The following computer resources were used in this research: , , . Development of this work has made extensive use of the Bruce and Astrid McWilliams eScience Video Facility at Carnegie Mellon University.
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[^1]: http://www.gigapan.org
[^2]: http://timemachine.gigapan.org
[^3]: http://web.phys.cmu.edu/\~yfeng1/gaepsi
[^4]: The second line is an approximation. For the numerator,
& \_[P]{}d\^[3]{} A()()=\
& \_[P]{}d\^[3]{}\_[i]{}\_[j]{}m\_[j]{}W(-\_[j]{},h\_[j]{})\
& =\_[i]{}\_[j]{}m\_[j]{}\_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})W(-\_[j]{},h\_[j]{}).
If we apply the mean value theorem to the integral and Taylor expand, we find that
& \_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})W(-\_[j]{},h\_[j]{})\
& =W(\_[ij]{}-\_[j]{},h\_[j]{})\_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})\
= &\
& \_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})\
& =W(\_[i]{}-\_[j]{},h\_[j]{})\_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})+O(h\^[2]{}).
Both $W'$ and $\mathbf{\xi}_{ij}-\mathbf{x}_{i}$ are bound by terms of $O(h_{j})$, so that the extra terms are all beyond $O(h^{2})$ (the last line). Noticing that $\rho_{i}=\sum_{j}m_{j}W(\mathbf{x}_{i}-\mathbf{x}_{j},h_{j})+O(h^{2})$, the numerator
& \_[P]{}d\^[3]{} A()()\
& =\_[i]{}A\_[i]{}m\_[i]{}\_[P]{}d\^[3]{} W(-\_[i]{},h\_[i]{})+O(h\^[2]{})\
& =\_[i]{}A\_[i]{}M\_[i]{}(P)+O(h\^[2]{}).
[^5]: image at http://gigapan.org/gigapans/76215/
[^6]: http://timemachine.gigapan.org/wiki/Evolution\_of\_the\_Universe
|
---
title: 'Randomization does not help much, comparability does '
---
[**Abstract**]{}
Following Fisher, it is widely believed that randomization “relieves the experimenter from the anxiety of considering innumerable causes by which the data may be disturbed.” In particular, it is said to control for known and unknown nuisance factors that may considerably challenge the validity of a result. Looking for quantitative advice, we study a number of straightforward, mathematically simple models. However, they all demonstrate that the optimism with respect to randomization is wishful thinking rather than based on fact. In small to medium-sized samples, random allocation of units to treatments typically [*yields*]{} a considerable imbalance between the groups, i.e., confounding due to randomization is the rule rather than the exception.
In the second part of this contribution, we extend the reasoning to a number of traditional arguments for and against randomization. This discussion is rather non-technical, and at times even “foundational” (Frequentist vs. Bayesian). However, its result turns out to be quite similar. While randomization’s contribution remains questionable, comparability contributes much to a compelling conclusion. Summing up, classical experimentation based on sound background theory and the systematic construction of exchangeable groups seems to be advisable.
[**Key Words**]{}. Randomization, Comparability, Confounding, Experimentation, Design of Experiments, Statistical Experiments
62K99
The Logic of the Experiment {#logic}
===========================
Randomization, the allocation of subjects to experimental conditions via a random procedure, was introduced by R.A. @fi35. Arguably, it has since become the most important statistical technique. In particular, statistical experiments are defined by the use of randomization [@ro02; @sh02], and many applied fields, such as [*evidence based medicine*]{}, draw a basic distinction between randomized and non-randomized evidence [e.g. the @ox09].
In order to explain randomization’s eminent role, one needs to refer to the logic of the experiment, largely based on J. S. Mill’s (1843: 225) [*method of difference*]{}:
> If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former: the circumstance in which alone the two instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon.
Therefore, if one compares two groups of subjects (Treatment $T$ versus Control $C$, say) and observes a substantial difference in the end (e.g. ${\bar X}_T > {\bar X}_C $), that difference must be due to the experimental manipulation - IF the groups were equivalent at the very beginning of the experiment. In other words, since the difference between treatment and control (i.e. the experimental manipulation) is the only perceivable reason that can explain the variation in the observations, it must be the cause of the observed effect (the difference in the end). The situation is quite altered, however, if the two groups already differed substantially at the beginning. Then, there are two possible explanations of an effect:
--------------------- -------------- -------------- --------------- ------------------ -------- --------
Start of Experiment $T$ $=$ $C$ $T$ $\neq$ $C$
Intervention Yes No Yes No
End of Experiment
(Observed Effect) ${\bar X}_T$ $>$ ${\bar X}_C $ ${\bar X}_T$ $>$ ${\bar
X}_C $
Conclusion Intervention OR Prior Difference
--------------------- -------------- -------------- --------------- ------------------ -------- --------
Comparability
=============
Thus, for the logic of the experiment, it is of paramount importance to ensure equivalence of the groups at the beginning of the experiment. The groups, or even the individuals involved, must not be systematically different; one has to compare like with like. Alas, in the social sciences exact equality of units, e.g. human individuals, cannot be maintained. Therefore one must settle for [*comparable*]{} subjects or groups $(T \approx C)$.
In practice, it is straightforward to define comparability with respect to the features or properties of the experimental units involved. In a typical experimental setup, statistical units (e.g. persons) are represented by their corresponding vectors of attributes (properties, variables) such as gender, body height, age, etc.:
Unit No. Gender Height Marital status Age Income $\ldots$ Attribute $j$
---------- ---------- ---------- ---------------- ---------- ---------- ---------- ---------------
1 f (1) 170cm married (1) 34 £50000 $\ldots$ $a_{1j}$
2 m (2) 188cm divorced (2) 44 £70000 $\ldots$ $a_{2j}$
3 m (2) 169cm single (0) 44 £35000 $\ldots$ $a_{3j}$
… … … … … … … …
$i$ $a_{i1}$ $a_{i2}$ $a_{i3}$ $a_{i4}$ $a_{i5}$ … $a_{ij}$
If the units are almost equal in as many properties as possible, they should be comparable, i.e., the remaining differences shouldn’t alter the experimental outcome substantially. However, since, in general, vectors have to be compared, there is not a single measure of similarity. Rather, there are quite a lot of measures available, depending on the kind of data at hand. An easily accessible and rather comprehensive overview may be found in
reference.wolfram.com/mathematica/guide/DistanceAndSimilarityMeasures.html
As an example, suppose a unit $i$ is represented by a binary vector ${\bf
a}_i=(a_{i1},\ldots,a_{im})$. The Hamming distance $d(\cdot,\cdot)$ between two such vectors is the number of positions at which the corresponding symbols are different. In other words, it is the minimum number of substitutions required to change one vector into the other. Let ${\bf a}_1=(0,0,1,0)$, ${\bf a}_2=(1,1,1,0)$, and ${\bf
a}_3=(1,1,1,1)$. Therefore $d({\bf a}_1,{\bf a}_2)=2$, $d({\bf a}_1,{\bf a}_3)=3$, $d({\bf a}_2,{\bf a}_3)=1$, and $d({\bf a}_i,{\bf a}_i)=0$. Having thus calculated a reasonable number for the “closeness” of two experimental units, one next has to consider what level of deviance from perfect equality may be tolerable.
Due to the reasons outlined above, coping with similarities is a tricky business. Typically many properties (covariates) are involved and conscious (subjective) judgement seems to be inevitable. That might at least partially explain why matching on the (rather objective) scalar valued propensity score, which is the probability of being assigned to $T$ given a set of observed properties, has become popular recently [@ru06].
An even more serious question concerns the fact that relevant factors may not have been recorded or might be totally unknown. In the worst case, similarity with respect to some known factors has been checked, but an unnoticed nuisance variable is responsible for the difference between the outcome in the two groups.
Moreover, comparability depends on the phenomenon studied. A clearly visible difference, such as gender, is likely to be important with respect to life expectancy, and can influence some physiological and psychological variables such as height or social behaviour, but it is independent of skin color or blood type. In other words, experimental units do not need to be twins in any respect; it suffices that they be similar with respect to the outcome variable under study.
Given a unique sample it is easy to think about a [*reference set*]{} of other samples that are alike in all relevant respects to the one observed. However, even Fisher failed to give these words a precise formal meaning [see @jo88]. Nowadays, epidemiologists use the term ‘unconfounded’ in order to communicate the same idea: “$[\ldots]$ the effect of treatment is unconfounded if the treated and untreated groups resemble each other in all relevant features” [@pe09 196]. Pearl shows that this problem is tightly linked to Simpson’s famous paradox [@si51] and devotes a major part of his book to the development of a formal calculus that is able to cope with it. Another major advance was the idea of [*exchangeability*]{}, a concept proposed by @fi74:
> $[\ldots]$ instead of judging whether two groups are similar, the investigator is instructed to imagine a hypothetical [*exchange*]{} of the two groups (the treated group becomes untreated, and vice versa) and then judge whether the observed data under the swap would be distinguishable from the actual data. [@pe09 196]
@ba93 gives some history on this idea and suggests the term ‘permutability’ instead, “which conveys the idea of replacing one thing by another similar thing.”
Experimental Techniques to Achieve Comparability {#compare}
================================================
There are a number of strategies to achieve comparability. Starting with the experimental units, it is straightforward to match similar individuals, i.e., to construct pairs of individuals that are alike in many (most) respects. @bo53 [583] says:
> You may match them individual for individual in respect to what seems to be their most important determinable and presumably relevant characteristics $[\ldots]$ You can match litter-mates in body-weight if your subjects are animals, and you can advertise for twins when your subjects are human.
An example could be
Gender Height Martial status Age Income Academic
------ -------- -------- ---------------- ----- -------- ----------
$T1$ 1 170 1 34 50000 1 (yes)
$C1$ 1 165 1 30 55000 1 (yes)
$T2$ 2 193 2 46 72000 0 (no)
$C2$ 2 188 2 44 70000 1 (yes)
… … … … … … …
Looking at the group level ($T$ and $C$), another obvious strategy is to balance all relevant variables when assigning units to groups. Many approaches of this kind are discussed in @se00 [140], minimization being the most prominent among them [@ta74; @tr98], as the latter authors explain:
> In our study of aspirin versus placebo $[\ldots]$ we chose age, sex, operating surgeon, number of coronary arteries affected, and left ventricular function. But in trials in other diseases those chosen might be tumour type, disease stage, joint mobility, pain score, or social class.
>
> At the point when it is decided that a patient is definitely to enter a trial, these factors are listed. The treatment allocation is then made, not purely by chance, but by determining in which group inclusion of the patient would minimise any differences in these factors. Thus, if group A has a higher average age and a disproportionate number of smokers, other things being equal, the next elderly smoker is likely to be allocated to group B. The allocation may rely on minimisation alone, or still involve chance but “with the dice loaded” in favour of the allocation which minimises the differences.
However, apart from being cumbersome and relying on the experimenter’s expertise (in particular in choosing and weighing the factors), these strategies are always open to the criticism that unknown nuisance variables may have had a substantial impact on the result. Therefore @fi35 [18f] advised strongly against treating every conceivable factor explicitly:
> $[\ldots]$ it would be impossible to present an exhaustive list of such possible differences appropriate to any kind of experiment, because the uncontrolled causes which may influence the result are always strictly innumerable $[\ldots]$ whatever degree of care and experimental skill is expended in equalising the conditions, other than the one under test, which are liable to affect the result, this equalisation must always be to a greater or less extent incomplete, and in many important practical cases will certainly be grossly defective.
Instead, he made the best of it and put forward his arguably most famous contribution:
> The full procedure of randomization \[is the method\] by which the validity of the test of significance may be guaranteed against corruption by the causes of disturbance which have not been eliminated $[\ldots]$ the random choice of the objects to be treated in different ways \[is\] a complete guarantee of the validity of the test of significance \[$\ldots$Randomization\] relieves the experimenter from the anxiety of considering and estimating the magnitude of the innumerable causes by which the data may be disturbed. [@fi35 19, 20, 44]
Consequently, “randomization controls for all possible confounders, known and unknown” became a common slogan. @tr98 explain: “The primary objective of randomisation is to ensure that all other factors that might influence the outcome will be equally represented in the two groups, leaving the treatment under test as the only dissimilarity.” In the same vein @be05 [9f] says:
> The idea of randomization is to overlay a sequence of units (subjects, or patients) onto a sequence of treatment conditions. If neither sequence can influence the other, then there should be no bias in the assignment of the treatments, and the comparison groups should be comparable.
In other words, the weight of evidence of a statistical experiment crucially depends on how well the tool of randomization achieves its ultimate goal: comparability. Non-comparable groups offer a straightforward alternative explanation, undermining the logic of the experiment. Thus the common term [*randomized evidence*]{} is a misnomer, if randomization fails to reliably yield comparable groups.
Randomization and Comparability {#randcomp}
===============================
Historically, Fisher’s idea proved to be a great success. @ha88 [427] states: “Indeed randomization is so commonplace that anyone untroubled by the fundamental principles of statistics must suppose that the practice is quite uncontroversial.” Nevertheless, quite a few scientists have argued that randomization does not work as advertised. They concluded:
1. @al85 [125]: “Randomised allocation in a clinical trial does not guarantee that the treatment groups are comparable with respect to baseline characteristics.”
2. @ur85 [266]: “It is a chilling thought that medical treatments that are worthless may have been endorsed, and valuable ones discarded, after randomized trials in which the treatment groups differed in ways that were known to be relevant to the disease under study, but where the strict rules of randomization were applied and no adjustment made.”
3. @tr98: “Indeed, if there are many possible prognostic factors there will almost certainly be differences between the groups despite the use of random allocation. In a small clinical trial a large treatment effect is being sought, but a large difference in one or more of the prognostic factors can occur purely by chance. In a large clinical trial a small treatment effect is being sought, but small but important differences between the groups in one or more of the prognostic factors can occur by chance. $[\ldots]$ At this point the primary objective of randomisation — exclusion of confounding factors — has failed.”
4. @gr99 [35]: $[\ldots]$ random imbalances may be severe, especially if the study size is small $[\ldots]$
5. @ro02 [21]: “The statement that randomization tends to balance covariates is at best imprecise; taken too literally, it is misleading $[\ldots]$ What is precisely true is that random assignment of treatments can produce some imbalances by chance, but common statistical methods, properly used, suffice to address the uncertainty introduced by these chance imbalances.”
6. @be05 [9]: “While it is certainly true that randomization is used for the purpose of ensuring comparability between or among comparison groups, we will see $[\ldots]$ that it is categorically not true that this goal is achieved.” \[See also his figure (p. 32) entitled: “Random Imbalance, No Selection Bias.”\]
7. @bo05 [94]: “Even with randomization the assumption of exchangeability can be violated.”
8. @ho06 [259]: “$[\ldots]$ [*the chief concern when designing a clinical trial should be to make it unlikely that the experimental groups differ on factors that are likely to affect the outcome*]{} $[\ldots]$ With this rule in mind, it is evident that a randomized allocation of subjects to treatments might sometimes be useful in clinical trials as a way of better balancing the experimental groups $[\ldots]$ [*But randomized allocation is not absolutely necessary; it is no*]{} sine qua non; [*it is not the only or even always the best way of constructing the treatment groups in a clinical trial*]{}” (emphasis in the original).
9. @wo07 [465]: “It is entirely possible that any particular randomization may have produced a division into experimental and control groups that is unbalanced with respect to ‘unknown’ factor $X [\ldots]$”
10. @au08 [2039]: “While randomization will, on average, balance covariates between treated and untreated subjects, it need not do so in any particular randomization.”
11. @ch12: “Despite randomization, imbalance in prognostic factors as a result of chance (chance imbalance) may still arise, and with small to moderate sample sizes such imbalance may be substantial.”
Moreover, quite early, statisticians - in particular of the Bayesian persuasion - put forward several rather diverse arguments against randomization [@sa62; @sa76; @ru78; @ba80; @li82; @ba88; @ka90]. The latter authors discuss the role of “randomized designs as methodological insurance against a ‘biased’ sample”. They conclude:
> What is it concerning randomization that makes such a judgement (of irrelevance of the allocation to the test outcomes) compelling for the reader? We can find none and suspect that randomization has little to do with whatever grounds there are for the belief that the allocation is irrelevant to the test results. [@ka90 335f]
Fortunately, it is not necessary at this point to delve into delicate philosophical matters or the rather violent Bayesian-Frequentist debate (however, see Section \[defense\]). Fairly elementary probabilistic arguments suffice to demonstrate that the above criticism hits its target: By its very nature a random mechanism provokes fluctuations in the composition of $T$ and $C$, making these groups (rather often) non-comparable. Therefore the subsequent argument has the advantage of being straightforward, mathematical, and not primarily “foundational”. Its flavour is Bayesian in the sense that we are comparing the [*actual groups*]{} produced by randomization which is the “posterior view” preferred by that school. At the same time its flavour is Frequentist, since we are focusing on the properties of a certain random [*procedure*]{} which is the “design view” preferred by this school.
There are not just two, but (at least) three, competing statistical philosophies: “In many ways the Bayesian and frequentist philosophies stand at opposite poles from each other, with Fisher’s ideas being somewhat of a compromise” [@ef98 98]. Since randomization is a Fisherian proposal, a neutral quantitative analysis of his approach seems to be appropriate, acceptable to all schools, and, in a sense, long overdue. Down-to-earth physicist @ja03 [xxii] explains:
> One can argue with philosophy; it is not easy to argue with a computer printout, which says to us: ‘Independently of all your philosophy, here are the facts of actual performance.’
To this end we resume our reasoning with a simple but striking example:
Greenland’s example {#greenlands-example .unnumbered}
-------------------
@gr90 [422] came up with “$[\ldots]$ the smallest possible controlled trial $[\ldots]$ to illustrate one thing randomization does [*not*]{} do: It does [*not*]{} prevent the epidemiologic bias known as confounding $[\ldots]$”(emphasis in the original; for a related example involving an interaction see p. ). That is, he flips a coin once in order to assign two patients to $T$ and $C$, respectively: If heads, the first patient is assigned to $T$, and the second to $C$; if tails, the first patient is assigned to $C$, and the second to $T$. Suppose ${\bar X}_T > {\bar X}_C $, what is the reason for the observed effect? Due to the experimental design, there are two alternatives: either the treatment condition differed from the control condition, or patient $P_1$ was not comparable to patient $P_2$.
-------------- ----- --------------- --------------------- -------------- ----- ---------------
$P_1$ $P_2$ Start of Experiment $P_2$ $P_1$
Yes No Intervention Yes No
${\bar X}_T$ $>$ ${\bar X}_C $ End of Experiment ${\bar X}_T$ $>$ ${\bar X}_C $
-------------- ----- --------------- --------------------- -------------- ----- ---------------
However, as each patient is only observed under either the treatment or control (the left hand side or the right hand side of the above table), one cannot distinguish between the patient’s and the treatment’s impact on the observed result. Therefore Greenland concludes:
> No matter what the outcome of randomization, the study will be completely confounded.
Suppose the patients are perfect twins with the exception of a single difference. Then Greenland’s example shows that randomization cannot even balance a [*single*]{} nuisance factor. To remedy the defect, it is straightforward to increase $n$. But how large an $n$ will assure comparability? Like many authors, Greenland states the basic result in a [*qualitative*]{} manner: “Using randomization, one can make the probability of severe confounding as small as one likes by increasing the size of the treatment cohorts” [@gr90 423]. However, no quantitative advice is given here or elsewhere. Thus it should be worthwhile studying a number of explicit and straightforward models, [*quantifying*]{} the effects of randomization.
Random Confounding {#models}
==================
Dichotomous factors {#dicho}
-------------------
Suppose there is a nuisance factor $X$ taking the value 1 if present and 0 if absent. One may think of $X$ as a genetic aberration, a medical condition, a psychological disposition or a social habit. Assume that the factor occurs with probability $p$ in a certain person (independent of anything else). Given this, $2n$ persons are randomized into two groups of equal size by a chance mechanism independent of $X$.
Let $S_1$ and $S_2$ count the number of persons with the trait in the first and the second group respectively. $S_1$ and $S_2$ are independent random variables, each having a binomial distribution with parameters $n$ and $p$. A natural way to measure the extent of imbalance between the groups is $D=S_1-S_2$. Obviously, $ED=0$ and $$\sigma^2(D)=\sigma^2(S_1)+
\sigma^2(-S_2)=2\sigma^2(S_1)=2np(1-p).$$
Iff $D =0$, the two groups are perfectly balanced with respect to factor $X$. In the worst case $|D|=n$, that is, in one group all units possess the characteristic, whereas it is completely absent in the other. For fixed $n$, let the two groups be comparable if $|D| \le n/i$ with some $i \in \{1,\ldots, n
\}$. Iff $i=1$, the groups will always be considered comparable. However, the larger $i$, the smaller the number of cases we classify as comparable. In general, $n/i$ defines a proportion of the range of $|D|$ that seems to be acceptable. Since $n/i$ is a positive number, and $S_1=S_2 \Leftrightarrow
|D|=0$, the set of comparable groups is never empty.
Given some constant $i(<n)$, the value $n/i$ grows at a linear rate in $n$, whereas $\sigma(D)=\sqrt{2np(1-p)}$ grows much more slowly. Due to continuity, there is a single point $n(i,k)$, where the line intersects with $k$ times the standard deviation of $D$. Beyond this point, i.e. for all $n\ge n(i,k)$, at least as many realizations of $|D|$ will be within the acceptable range $[0, n/i]$. Straightforward algebra gives, $$n_p(i,k) = 2p(1-p) i^2 k^2 .$$
A typical choice could be $i=10$ and $k=3$, which specifies the requirement that most samples be located within a rather tight acceptable range. In this case, one has to consider the functions $n/10$ and $3\sqrt{2p(1-p)n}$. These functions of $n$ are shown in the following graph:
Illustration: The linear function $n/10$, and $3\sqrt{2p(1-p)n}$ for $p=1/2, p=1/5, p=1/10,$ and $p=1/100$ (from above to below).
Thus, depending on $p$, the following numbers of subjects are needed per group (and twice this number altogether): $$\begin{array}{lll|l|l}
p & i & k & n_p(i,k) & 2 \cdot n_p(i,k) \\\hline
1/2 & 10 & 3 & 450 & 900 \\
1/5 & 10 & 3 & 288 & 576 \\
1/10 & 10 & 3 & 162 & 324 \\
1/100 & 10 & 3 & 18 & 36 \
\end{array}$$ Relaxing the criterion of comparability (i.e. a smaller value of $i$) decreases the number of subjects necessary: $$\begin{array}{lll|l}
p & i & k & n_p(i,k) \\\hline
1/2 & 5 & 3 & 113 \\
1/5 & 5 & 3 & 72 \\
1/10 & 5 & 3 & 41 \\
1/100 & 5 & 3 & 5 \
\end{array}$$ The same happens if one decreases the number of standard deviations $k$:$$\begin{array}{lll|l}
p & i & k & n_p(i,k) \\\hline
1/2 & 10 & 2 & 200 \\
1/5 & 10 & 2 & 128 \\
1/10 & 10 & 2 & 72 \\
1/100 & 10 & 2 & 8 \
\end{array}$$
This shows that randomization works, if the number of subjects ranges in the hundreds or if the probability $p$ is rather low. (By symmetry, the same conclusion holds if $p$ is close to one.) Otherwise there is hardly any guarantee that the two groups will be comparable. Rather, they will differ considerably due to random fluctuations.
The distribution of $D$ is well known [e.g. @jo05 142f]. For $d=-n,\ldots,n,$ $$P(D=d) = \sum_{\max(0,d) \le y \le \min(n,n+d)} \binom{n}{y} \binom{n}{y-d}
p^{2y-d} (1-p)^{2n-2y+d}$$
Therefore, it is also possible to compute the probability $q=q(i, n, p)$ that two groups, constructed by randomization, will be comparable. If $i=5$, i.e., if one fifth of the range of $|D|$ is judged to be comparable, we obtain: $$\begin{array}{ll|l}
p & n & q(i,n,p) \\\hline
1/2 & 5 & 0.66 \\
1/2 & 10 & 0.74 \\
1/2 & 25 & 0.88 \\
1/2 & 50 & 0.96 \
\end{array}
\hspace{5ex}
\begin{array}{ll|l}
p & n & q(i,n,p) \\\hline
1/10 & 5 & 0.898 \\
1/10 & 10 & 0.94 \\
1/10 & 25 & 0.98999 \\
1/10 & 50 & 0.999 \
\end{array}
\hspace{5ex}
\begin{array}{ll|l}
p & n & q(i,n,p) \\\hline
1/100 & 5 & 0.998 \\
1/100 & 10 & 0.9997 \\
1/100 & 25 & 0.999999 \\
1/100 & 50 & 1 \
\end{array}$$
Thus, it is rather difficult to control a factor that has a probability of about $1/2$ in the population. However, even if the probability of occurrence is only about $1/10$, one needs more than 25 people per group to have reasonable confidence that the factor has not produced a substantial imbalance.
The situation becomes worse if one takes more than one nuisance factor into account. Given $m$ independent binary factors, each of them occurring with probability $p$, the probability that the groups will be balanced with respect to all nuisance variables is $q^m$. Numerically, the above results yield: $$\begin{array}{lll|lll}
p & n & q & q^2 & q^5 & q^{10} \\\hline
1/2 & 5 & 0.66 & 0.43 & 0.12 & 0.015 \\
1/2 & 10 & 0.74 & 0.54 & 0.217 & 0.047 \\
1/2 & 25 & 0.88 & 0.78 & 0.53 & 0.28 \\
1/2 & 50 & 0.96 & 0.93 & 0.84 & 0.699 \\
\end{array}
\hspace{5ex}
\begin{array}{lll|lll}
p & n & q & q^2 & q^5 & q^{10} \\\hline
1/10 & 5 & 0.898 & 0.807 & 0.58 & 0.34 \\
1/10 & 10 & 0.94 & 0.88 & 0.74 & 0.54 \\
1/10 & 25 & 0.98999 & 0.98 & 0.95 & 0.90 \\
1/10 & 50 & 0.999 & 0.9989 & 0.997 & 0.995 \\
\end{array}$$$$\begin{array}{lll|lll}
p & n & q & q^2 & q^5 & q^{10} \\\hline
1/100 & 5 & 0.998 & 0.996 & 0.99 & 0.98 \\
1/100 & 10 & 0.9998 & 0.9996 & 0.999 & 0.9979 \\
1/100 & 25 & 0.9999998 & 0.9999995 & 0.999999 & 0.999998 \\
1/100 & 50 & 1 & 1 & 1 & 1\\
\end{array}$$
Accordingly, given $m$ independent binary factors, each occurring with probability $p_j$ (and corresponding $q_j=q(i,n,p_j)$), the probabilities closest to $1/2$ will dominate $1-q_1\cdots q_m$, which is the probability that the two groups are not comparable due to an imbalance in at least one variable. In a typical study with $2n=100$ persons, for example, it does not matter if there are one, two, five or even ten factors, if each of them occurs with probability of $1/100$. However, if some of the factors are rather common (e.g. $1/5 < p_j <4/5$), this changes considerably. In a smaller study with fewer than $2n=50$ participants, a few such factors suffice to increase the probability that the groups constructed by randomization won’t be comparable to 50%. With only a few units per group, one can be reasonably sure that some undetected, but rather common, nuisance factor(s) will make the groups non-comparable.
The situation deteriorates considerably if there are interactions between the variables that may yield convincing alternative explanations for an observed effect. It is possible that all factors considered in isolation are reasonably balanced (which is often checked in practice), but that a certain combination of them affects the observed treatment effect. For the purpose of illustration suppose four persons (being young or old, and male or female) are investigated:
\[interaktion\] T C
------------------- -----------
Old Man Old Woman
Young Woman Young Man
Although gender and (dichotomized) age are perfectly balanced between $T$ and $C$, the young woman has been allocated to the first group. Therefore a property of young women (e.g. pregnancy) may serve as an explanation for an observed effect, e.g. ${\bar X}_T > {\bar X}_C$.
Given $m$ factors, there are $m (m-1) /2$ possible interactions between just two of the factors, and $\binom{m}{\nu}$ possible interactions between $\nu$ of them. Thus, there is a high probability that some considerable imbalance occurs in at least one of these numerous interactions, in small groups in particular. For a striking early numerical study see @le80.
Detected or undetected, such imbalances provide excellent alternative explanations of an observed effect. Altogether our conclusion based on an explicit quantitative analysis coincides with the qualitative argument given by @sa62 [91]:
> Suppose we had, say, thirty fur-bearing animals of which some were junior and some senior, some black and some brown, some fat and some thin, some of one variety and some of another, some born wild and some in captivity, some sluggish and some energetic, and some long-haired and some short-haired. It might be hard to base a convincing assay of a pelt-conditioning vitamin on an experiment with these animals, for every subset of fifteen might well contain nearly all of the animals from one side or another of one of the important dichotomies $[\ldots]$
>
> Thus contrary to what I think I was taught, and certainly used to believe, it does not seem possible to base a meaningful experiment on a small heterogenous group.
In the light of this, one can only hope for some ‘benign’ dependence structure among the factors, i.e., a reasonable balance in one factor improving the balance in (some of) the others. Given such a tendency, a larger number of nuisance factors may be controlled, since it suffices to focus on only a few. Independent variables possess a ‘neutral’ dependence structure in that the balance in one factor does not influence the balance in others. Yet, there may be a ‘malign’ dependence structure, such that balancing one factor tends to actuate imbalances in others. We will make this argument more precise in Section \[intermediate\]. However, a concrete example will illustrate the idea: Given a benign dependence structure, catching one cow (balancing one factor) makes it easier to catch others. Therefore it is easy to lead a herd into an enclosure: Grab some of the animals by their horns (balance some of the factors) and the others will follow. However, in the case of a malign dependence structure the same procedure tends to stir up the animals, i.e., the more cows are caught (the more factors are being balanced), the less controllable the remaining herd becomes.
Ordered random variables
------------------------
In order to show that our conclusions do not depend on some specific model, let us next consider ordered random variables. To begin with, look at four units with ranks 1 to 4. If they are split into two groups of equal size, such that the best (1) and the worst (4) are in one group, and (2) and (3) are in the other, both groups have the same rank sum and are thus comparable. However, if the best and the second best constitute one group and the third and the fourth the other group, their rank sums (3 versus 7) differ by the maximum amount possible, and they do not seem to be comparable. If the units with ranks 1 and 3 are in the first group and the units with ranks 2 and 4 are in the second one, the difference in rank sums is $|6-4|=2$ and it seems to be a matter of personal judgement whether or not one thinks of them as comparable.
Given two groups, each having $n$ members, the total sum of ranks is $r=2n(2n+1)/2=n(2n+1)$. If, in total analogy to the last section, $S_1$ and $S_2$ are the sum of the ranks in the first and the second group, respectively, $S_2=r-S_1$. Therefore it suffices to consider $S_1$, which is the test statistic of Wilcoxon’s test. Again, a natural way to measure the extent of imbalance between the groups is $D=S_1-S_2=2S_1-r$. Like before $ED=0$ and because $\sigma^2(S_1)=
n^2(2n+1)/12$ we have $\sigma^2(D)=4\sigma^2(S_1)=n^2 (2n+1)/3$.
Moreover, $n(n+1)/2 \le S_j \le n(3n+1)/2$ $(j=1,2)$ yields $-n^2 \le D \le
n^2$. Thus, in this case, $n^2/i$ $(i \in \{1,\ldots,n^2 \})$ determines a proportion of the range of $|D|$ that may be used to define comparability. Given a fixed $i (<n^2)$, the quantity $n^2/i$ is growing at a quadratic rate in $n$, whereas $\sigma(D)=n\sqrt{(2n+1)/3}$ is growing at a slower pace. Like before, there is a single point $n(i,k)$, where $n^2/i = k \sigma(D)$. Straightforward algebra gives, $$n(i,k) = ik(ik+\sqrt{(ik)^2 +3})/3.$$ Again, we see that large numbers of observations are needed to ensure comparability: $$\begin{array}{ll|l}
i & k & n(i,k) \\\hline
10 & 3 & 501 \\
5 & 3 & 156 \\
10 & 2 & 267 \\
\end{array}$$
Again, it is possible to work with the distribution of $D$ explicitly. That is, given $i$ and $n$, one may calculate the probability $q=q(i, n)$ that two groups, constructed by randomization, are comparable. If $|D| \le n^2 /i$ is considered comparable, it is possible to obtain, using the function pwilcox() in R: $$\begin{array}{l|lllll}
& \multicolumn{5}{c}{n}\\
i & 5 & 10 & 25 & 50 & 100 \\\hline
3 & 0.58 & 0.78 & 0.96 & 0.996 & 1 \\
5 & 0.45 & 0.56 & 0.78 & 0.92 & 0.99 \\
10 & 0.16 & 0.32 & 0.45 & 0.61 & 0.78 \\
\end{array}$$
These results for ordered random variables are perfectly in line with the conclusions drawn from the binary model. Moreover, the same argument as before shows that the situation becomes (considerably) worse if several factors may influence the final result.
A continuous model
------------------
Finally, I consider a continuous model. Suppose there is just one factor $X \sim
N(\mu,\sigma)$. One may think of $X$ as a normally distributed personal ability, person $i$ having individual ability $x_i$. As before, assume that $2n$ persons are randomized into two groups of equal size by a chance mechanism independent of the persons’ abilities.
Suppose that also in this model $S_1$ and $S_2$ measure the total amount of ability in the first and the second group respectively. Obviously, $S_1$ and $S_2$ are independent random variables, each having a normal distribution $N(n\mu,\sqrt{n} \sigma)$. A straightforward way to measure the [*absolute*]{} extent of imbalance between the groups is
$$\label{gleich}
D=S_1-S_2=\sum_{\iota=1}^n X_{1,\iota}-\sum_{\iota=1}^n X_{2,\iota}=\sum_{\iota=1}^n
(X_{1,\iota}-X_{2,\iota}).$$
Due to independence, obviously $D \sim N(0,\sqrt{2n}\sigma)$.
Let the two groups be comparable if $|D| \le l \sigma$, i.e., if the difference between the abilities assembled in the two groups does not differ by more than $l$ standard deviations of the ability $X$ in a single unit. The larger $l$, the more cases are classified as comparable. For every fixed $l$, $l \sigma$ is a constant, whereas $\sigma(D)=\sqrt{2n} \sigma$ is growing slowly. Owing to continuity, there is yet another single point $n(l)$, where $l \sigma =
\sigma(D)=\sqrt{2n} \sigma$. Straightforward algebra gives, $$l \le \sqrt{2n} \Leftrightarrow 2 n \ge l^2 .$$ In particular, we have: $$\begin{array}{l|llllllllll}
l & 1 & 2 & 3 & 5 & 10 \\\hline
n & 1 & 2 & 5 & 13 & 50 \\
\end{array}$$ In other words, the two groups become non-comparable very quickly. It is almost impossible that two groups of 500 persons each, for example, could be close to one another with respect to total (absolute) ability.
However, one may doubt if this measure of non-comparability really makes sense. Given two teams with a hundred or more subjects, it does not seem to matter whether the total ability in the first one is within a few standard deviations of the other. Therefore it is reasonable to look at the [*relative*]{} advantage of group 1 with respect to group 2, i.e. $ Q = D/n$. Why divide by $n$ and not by some other function of $n$? First, due to equation (\[gleich\]), exactly $n$ comparisons $X_{1,\iota}-X_{2,\iota}$ have to be made. Second, since $$Q=\sum_{\iota =1}^n
X_{1,\iota}/n -\sum_{\iota =1}^n X_{2,\iota}/n = {\bar X}_T - {\bar X}_C,$$ $Q$ may be interpreted in a natural way, i.e., being the difference between the typical (mean) representative of group 1 (treatment) and the typical representative of group 2 (control). A straightforward calculation yields $Q \sim N(0,\sigma \sqrt{2/n})$.
Let the two groups be comparable if $|Q| \le l \sigma$. If one wants to be reasonably sure (three standard deviations of $Q$) that comparability holds, we have $ l\sigma \ge 3\sigma \sqrt{2/n} \Leftrightarrow n \ge 18/l^2 $. Thus, at least the following numbers of subjects are required per group: $$\begin{array}{l|llllll}
l & 5 & 2 & 1 & 1/2 & 1/4 & 1/8 \\\hline
n & 1 & 5 & 18 & 72 & 288 & 1152 \\
\end{array}$$ If one standard deviation is considered a large effect [@co88], three dozen subjects are needed to ensure that such an effect will not be produced by chance. To avoid a small difference between the groups due to randomization (one quarter of a standard deviation, say), the number of subjects needed goes into the hundreds. In general, if $k$ standard deviations of $Q$ are desired, we have, $$n
\ge 2 k^2 /l^2.$$ Thus, for $k=1,2$ and $5$, the following numbers of subjects $n_k$ are required in each group: $$\begin{array}{l|llllll}
l & 5 & 2 & 1 & 1/2 & 1/4 & 1/8 \\\hline
n_1 & 1 & 1 & 2 & 8 & 32 & 128 \\
n_2 & 1 & 2 & 8 & 32 & 128 & 512 \\
n_5 & 2 & 13 & 50 & 200 & 800 & 3200 \\
\end{array}$$ These are just the results for one factor. As before, the situation deteriorates considerably if one sets out to control several nuisance variables by means of randomization.
Intermediate Conclusions {#intermediate}
========================
The above models have deliberately been kept as simple as possible. Their results are straightforward and they agree: If $n$ is small, it is almost impossible to control for a trait that occurs frequently at the individual level, or for a larger number of confounders, via randomization. It is of paramount importance to understand that random fluctuations lead to considerable differences between small or medium-sized groups, making them very often non-comparable, thus undermining the basic logic of experimentation. That is, ‘blind’ randomization does not create equivalent groups, but rather [*provokes*]{} imbalances and subsequent artifacts: Even in larger samples one needs considerable luck to succeed in creating equivalent groups; $p$ close to 0 or 1, a small number of nuisance factors $m$, or a favourable dependence structure that balances all factors, including their relevant interactions, if only some crucial factors are to be balanced by chance.
“Had the trial not used random assignment, had it instead assigned patients one at a time to balance \[some\] covariates, then the balance might well have been better \[for those covariates\], but there would be no basis for expecting other unmeasured variables to be similarly balanced” [@ro02 21] seems to be the only argument left in favour of randomization. Since randomization treats known and unknown factors alike, it is quite an asset that one may thus infer from the observed to the unobserved without further assumptions. However, this argument backfires immediately since, for exactly the same reason, an imbalance in an observed variable cannot be judged as harmless. Quite the contrary: An observed imbalance [*hints at*]{} further undetectable imbalances in unobserved variables.
Moreover, treating known and unknown factors equivalently is cold comfort compared to the considerable amount of imbalance evoked by randomization. Fisher’s favourite method always comes with the cost that it introduces additional variability, whereas a systematic schema at least balances known factors. In subject areas haunted by heterogeneity it seems intuitively right to deliberately work in favour of comparability, and rather odd to introduce further variability.
Three types of dependence structures {#three-types-of-dependence-structures .unnumbered}
------------------------------------
In order to sharpen these arguments, let us look at an observed factor $X$, an unobserved factor $Y$, and their dependence structure in more detail. Without loss of generality let all functions $d(\cdot)$ be positive in the following. Having constructed two groups of equal size via randomization, suppose $d_R(X)={\bar X}_T
-{\bar X}_C >0$ is the observed difference between the groups with respect to variable $X$. Using a systematic scheme instead, i.e., distributing the units among $T$ and $C$ in a more balanced way, this may be reduced to $d_S(X)$. The crucial question is how such a manipulation affects $d_R(Y)$, the balance between the groups with respect to another variable.
A benign dependence structure may be characterized by $d_S(Y) < d_R(Y)$. In other words, the effort of balancing $X$ pays off, since the increased comparability in this variable carries over to $Y$. For example, given today’s occupational structures with women earning considerably less than men, balancing for gender should also even out differences in income. If balancing in $X$ has no effect on $Y$, $d_S(Y) \approx d_R(Y)$, no harm is done. For example, balancing for gender should not affect the distribution of blood type in the observed groups, since blood type is independent of gender. Only in the pathological case when increasing the balance in $X$ has the opposite effect on $Y$, one may face troubles. As an example, let there be four pairs $(x_1,y_1)=(1,4);
(x_2,y_2)=(2,2);(x_3,y_3)=(3,1)$; and $(x_4,y_4)=(4,3)$. Putting units 1 and 4 in one group, and units 2 and 3 in another, yields a perfect balance in the first variable, but the worst imbalance possible in the second.
However, suppose $d(\cdot) < c$ where the constant (threshold) $c$ defines comparability. Then, in the randomized case, the groups are comparable if both $d_R(X)$ and $d_R(Y)$ are smaller than $c$. By construction, $d_S(X)\le d_R(X)<c$, i.e., the systematically composed groups are also comparable with respect to $X$. Given a malign dependence structure, $d_S(Y)$ increases. Yet $d_S(Y)<c$ may still hold, since, in this case, the “safety margin” $c-d_R(Y)$ may prevent the systematically constructed groups from becoming non-comparable with respect to property $Y$. In large samples, $c-d_R(\cdot)$ is considerable for both variables. Therefore, in most cases, consciously constructed samples will (still) be comparable. Moreover, the whole argument easily extends to more than two factors.
In a nutshell, endeavouring to balance relevant variables pays off. A conscious balancing schema equates known factors better than chance and may have some positive effect on related, but unknown, variables. If the balancing schema has no effect on an unknown factor, the latter is treated as if randomization were interfering - i.e. in a completely nonsystematic, ‘neutral’ way. Only if there is a (very) malign dependence structure, when systematically balancing some variable provokes (considerable) “collateral damage”, might randomization be preferable.
This is where sample size comes in. In realistic situations with many unknown nuisance factors, randomization only works if $n$ is (really) large. Yet if $n$ is large, so are the “safety margins” in the variables, and even an unfortunate dependence structure won’t do any harm. If $n$ is smaller, the above models show that systematic efforts, rather than randomization, may yield comparability. Given a small number of units, both approaches only have a chance of succeeding if there are hardly any unknown nuisance factors, or if there is a benign dependence structure, i.e., if a balance in some variable (no matter how achieved) has a positive effect on others. In particular, if the number of relevant nuisance factors and interactions is small, it pays to isolate and control for a handful of obviously influential variables, which is a crucial ingredient of experimentation in the classical natural sciences. Our overall conclusion may thus be summarized in the following table:
Dependence structure X (observed) Y (unobserved) Preferable procedure
---------------------- ------------------- ------------------------- ------------------------
Benign $d_S(X) < d_R(X)$ $d_S(Y) < d_R(Y)$ Systematic allocation
Neutral $d_S(X) < d_R(X)$ $d_S(Y) \approx d_R(Y)$ Systematic allocation
Malign $d_S(X) < d_R(X)$ $d_S(Y) >d_R(Y)$ Rather systematic than
random allocation
The Frequentist Position {#defense}
========================
There is yet another important, some would say outstanding, defense of randomization that we have omitted so far. According to this point of view the major “$[\ldots]$ function of randomization is to generate the sample space and hence provide the basis for estimates of error and tests of significance $[\ldots]$” [@co76 419]. In a statistical experiment one controls the random mechanism, thus the experimenter knows the sample space and the distribution in question. This constructed and therefore “valid” framework keeps nuisance variables at bay and sound reasoning within the framework leads to correct results. Someone following this train of thought could therefore state - and several referees of this contribution indeed did so - that the above models underline the rather well-known fact that randomization can have difficulties in constructing similar groups (achieving exchangeablility/comparability, balancing covariates), in particular if $n$ is small. However, this goal is quite subordinate to the major goal of establishing a known distribution on which sound statistical conclusions can be based.
The latter view has been proposed and defended by Frequentist statisticians, in particular Fisher and Neyman. It once dominated the field of statistics and still has a stronghold in certain quarters, in particular medical statistics where [*randomized*]{} controlled trials have been the gold standard. In this section, we focus on the basic Frequentist viewpoint and some its major criticisms. Given this, several perspectives on randomization will be the thrust of the next section. Section \[cause\] then provides the link to causation and Section \[history\] gives a wider, historical perspective.
Traditionally, criticism of the Frequentist line of argument in general, and randomization in particular, has come from the Bayesian school of statistics. While Frequentist statistics is much concerned with the way data is collected, focusing on the design of experiments, the corresponding sample space and sampling distribution, Bayesian statistics is rather concerned with the data actually obtained. Its focus is on learning from the(se) data - in particular with the help of Bayes’ theorem - and the parameter space.
In a sense, both viewpoints are perfectly natural and not contradicting each other, so it may seem futile to decide which emphasis is more appropriate. However, the example of randomization shows that this cannot be the final word: For the pre-data view, randomization is essential, it constitutes the difference between a real statistical experiment and any kind of quasi-experiment. For the post-data view, however, randomization adds nothing to the information at hand, and is ancilliary or just a nuisance.
Consistently, @fr08a [238]\[free33\], an outstanding Frequentist, says a bit more generally that there are “[*two styles of inference*]{}.
- Randomization [*provides*]{} a [*known*]{} distribution for the assignment variables; statistical inferences are based on this distribution.
- Modeling [*assumes*]{} a distribution for the latent variables;\[latent4\] statistical inferences are based on that assumption.” (Emphasis added)
Yet @ja03, an outstanding Bayesian, devoting a section (16.4) of his book to pre- and post-data considerations, states (p. 500): “As we have stressed repeatedly, virtually all real problems of scientific inference are concerned with post-data questions.”
The crucial and rather fundamental issue therefore becomes how far-reaching the conclusions of each of these styles of inference are. To pin down the differences, @ba99 [192] gives a nice and important example:
> On the one hand, the procedure of using the sample mean (or some other measure) to estimate $\mu$ could be assessed in terms of how well we expect it to behave; that is, in the light of different possible sets of data that might be encountered. It will have some average characteristics that express the precision we initially expect, i.e. before we take our data $[\ldots]$
>
> The alternative concept of final precision aims to express the precision of an inference in the specific situation we are studying. Thus, if we actually take our sample and find ${\bar x}=29.8,$ how are we to answer the question ‘how close is $29.8$ to $\mu$’? This is a most pertinent question to ask - some might claim that it is the supreme consideration.
Now, within the Frequentist framework, the answer is rather disappointing. All we know is that “the interval $[\ldots]$ covers the true value of $\mu$ with frequency $95\%$ in a long series\[longrun8\] of independent repetitions $[\ldots]$”. Within the Bayesian framework, Barnett’s question can be answered, since it assumes a prior distribution on the parameter space and uses ${\bar x}=29.8$ to calculate a posterior distribution about $\mu$.
Because of its crucial dependence on the process generating the data the following phenomenon is also inevitable in the Frequentist framework: Suppose a scientist applies a standard I.Q. test and finds a value of 160 in a certain person. Since the distribution of I.Q. values is known, one can easily give a confidence interval around the observed value. However, suppose “on the day the score ${\bar x= 160}$ was reported, our test-grading machine was malfunctioning. Any score ${\bar x}$ below 100 was reported as 100. The machine functioned perfectly for scores ${\bar x}$ above 100” [@ef78 236f]. Although the observed value is well above the area where recording errors occured, this new bit of information on the process of data generation alters the confidence interval. Efron concludes: “it is disturbing that any change at all is necessary. \[We received\] no new information about the score actually reported, or about I.Q.’s in general. It only concerned something bad that might have happened but didn’t.” However, he adds: “Bayesian methods are free from this defect; the inferences they produce depend only on the data value $\bar x$ actually observed, since Bayesian averages $[\ldots]$ are conditional on the observed $\bar x$.”
It is also quite typical that the Frequentist school needs to reframe straightforward questions. Instead of answering them directly, it considers a similar situation or creates a suitable concept within its own frame of mind. In Barnett’s example an orthodox statistician would almost surely bring up the prominent notion of unbiased estimation. In Frequentist terms an estimator is a function of the data and some estimator $U$ of $\mu$ is called unbiased if $EU=\mu$. How, then, can @pe09 [332] complain that “$[\ldots]$ one would be extremely hard pressed to find a statistics textbook $[\ldots]$ containing a mathematical proof that randomization indeed produces unbiased estimates of the quantities we wish estimated $[\ldots]$”?
The reason is not hard to find: To him, “unbiased” means that all kinds of systematic error are excluded. Quite similarly, the centre for evidence-based medicine at the university of Oxford defines “Bias: Any tendency to influence the results of a trial (or their interpretation) other than the experimental intervention.” \[Footnote: See www.cebm.net/glossary.\] Of course, such a claim is much more difficult to prove than $E{\bar X}=\mu$. It is quite telling that Pearl needs a whole section (6) to illuminate these issues within his elaborated formal framework of causal graphs, while @ja03 has to denote a section (17.2) to the well-known defects of the rather crude classical concept, championed by Neyman. \[Footnote: For example, if $EV=\sigma$, i.e. if $V$ is an unbiased estimator of $\sigma$ in the restricted traditional sense, $V^2$ is a biased estimator of $\sigma^2$ (and vice versa). Therefore, early on, @fi73 [146] wished that these considerations would “have eliminated such criteria as the estimate should be ‘unbiased’ $[\ldots]$”\]
He further explains that “orthodoxians put such exaggerated emphasis on bias” due to a “a psychosematic trap of their own making. When we call the quantity $[EU-\mu]$ the ‘bias’, that makes it sound like something awfully reprehensible, which we must get rid of at all costs $[\ldots]$ Frequentist statisticians adopted the simple device of inventing virtuous-sounding terms (like unbiased, efficient, uniformly most powerful, admissible, robust) to describe their own procedures $[\ldots]$” [@ja03 508, 514]
Similarly, “validity” seems to be exactly what we need. However, randomization only guarantees that a test of significance is valid in a rather narrow, technical sense (for more details see the next section). Overlooking the fact that here, “valid” is associated with a restricted meaning, we are deceiving ourselves. Even more so, since randomisation provokes imbalances, thus evoking alternative explanations that threaten internal validity. Jaynes (ibid.) concludes: “This is just the price one pays for choosing a technical terminology that carries an emotional load, implying value judgements; orthodoxy falls constantly into this tactical error $[\ldots]$ Today these emotionally loaded terms are only retarding progress and doing a disservice to science.”
Hence, despite a “valid” framework and mathematically sound conclusions a (purely) Frequentist train of thought may easily miss its target or might even go astray. After decades of Frequentist - Bayesian comparisons like the above, it has become obvious that in many important situations the numerical results of Frequentist and Bayesian arguments (almost) coincide. However, the two approaches are conceptually completely different, and it also has become apparent that simple calculations within the sampling framework lead to reasonable answers to post-data questions only because of “lucky” coincidences (e.g. the existence of sufficient statistics for the normal distribution). Of course, in general, such symmetries do not exist, and pre-data results cannot be transferred to post-data situations. In particular, purely Frequentist arguments fail if the sampling distribution does not belong to the “exponential family”, if there are several nuisance parameters, if there is important prior information, or if the number of parameters is much larger than the number of observations $(p \gg n)$.
It is also no coincidence, but sheer necessity, that a narrow formal line of argument needs to be supplemented with much intuition and heuristics. So, on the one hand, an orthodox author may claim that “randomization, instrumental variables, and so forth have [*clear*]{} statistical definitions”; yet, on the other hand, he has to concede at once that “there is a long tradition of [*informal*]{} - but systematic and successful - causal inference in the medical sciences” (see @pe09 [387], my emphasis).
Random allocation
=================
In the Frequentist vein
> randomization in design $[\ldots]$ is supposed to provide the grounds for [*replacing*]{} uncertainty about the possible effects of nuisance factors with a probability statement about error” (@se79 [214], my emphasis).
In the light of the above it is straightforward to ask if this “reframing strategy” hits its target. Of course, it should come as no surprise that many, if not most, Bayesians have questioned this. For example, towards the end of his article @ba80 [582] writes quite categorically: “The randomization exercise cannot generate any information on its own. The outcome of the exercise is an ancillary statistic. Fisher advised us to hold the ancillary statistic fixed, did he not?” Basing our inferences on the distribution randomization creates seems to be the very reverse.
More recently, philosophers closer to the Bayesian persuation have gained ground [@ho06], and @wo07 explained “why there is no cause to randomize.” Yet even by the 1970s, members of the classical school noted that, upon using randomization and the distribution it entails, we are dealing with “the simplest hypothesis, that our treatment $[\ldots]$ has absolutely no effect in any instance”, and that “under this [*very tight*]{} hypothesis this calculation is obviously logically sound” [@br78 my emphasis]. Here is a similar, contemporary criticism from an outstanding scientist:
> $[\ldots]$ even under ideal conditions, unaided randomization cannot answer some very basic questions such as what fraction of a population benefits from a program $[\ldots]$ Randomization is not an effective procedure for identifying median gains, or the distribution of gains, under general conditions $[\ldots]$ By focusing exclusively on mean outcomes, the statistical literature converts a metaphor for outcome selection - randomization - into an ideal $[\ldots]$ [@he05 146, 21, emphasis in the original].
In more general terms he (pp. 48, 145, 86) complains:
> The absence of explicit models is a prominent feature of the statistical treatment effect literature “$[\ldots]$ a large statistical community implicitly appeal to a variety of conventions rather than presenting rigorous models and assumptions $[\ldots]$ Statistical causal models, in their current state, are not fully articulated models. Crucial assumptions about sources of randomness are kept implicit.
As for the sources of randomness, one should at least distinguish between natural variation and artificially introduced variability. A straightforward question then surely is, how inferences based on the “man-made” portion bear on the “natural” part. To this end, @ba80 [579ff] compares a scientist, following the logic we described in Section 1 and a statistician who counts on randomization. Let us eavesdrop on their conversation:
> [*Statistician:*]{} Observe that the randomization test argument does not depend on any probabilistic assumptions. The randomization probabilities are fully understood and completely under control.
>
> [*Scientist:*]{} I do not understand the relevance of the randomization probabilities $[\ldots]$ It is relevant to know that the 30 animals have been paired into 15 homogeneous blocks. The manner of my labeling the two animals in the $i$th block $[\ldots]$ does not seem to be of much relevance. The number $m$ of treatment allocations of the type $(t,c)$ seems to be of no consequence at all. $[\ldots]$ How can the level of significance depend so largely on such an irrelevant data characteristic as $m$? $[\ldots]$ I have not been asked on all the background information that I have on the problem $[\ldots]$ I am amazed to find that a statistical analysis of my data can be made without reference to these relevant bits of information.
>
> [*Statistician:*]{} You are trying to make a joke out of an excellent statistical method of proven value $[\ldots]$ Your criticisms are based on an extreme example and then on a misunderstanding of the very nature of the tests of significance. Tests of significance do not lead to probabilities of hypotheses $[\ldots]$ The randomization analysis of data is so simple, so free of unnecessary assumptions that I fail to understand how anyone can raise any objection against the method.
>
> [*Scientist:*]{} As a scientist I have been trained to put as much control into the experimental setup as I am capable of, to balance out nuisance factors as far as possible $[\ldots]$ I worked very hard on the project of striking a perfect balance between the treatment and the control groups.
>
> [*Statistician:*]{} Now the reference consists of only \[two points\] and the significance level \[works out to be $1/2$ or $1$\]. Your data is not significant at all. Had I known about this before, I would not have touched your data with a long pole.
>
> [*Scientist*]{} (utterly flabbergasted): But my experiment was better planned than a fully randomized experiment, was it not? With my group control (in addition to the usual local control) I made it much harder for $T=\sum t_i -\sum c_i$ to be large in the absence of any treatment difference.
>
> [*Statistician:*]{} My good man, you must realize that your experiment is no good $[\ldots]$ It appears that you do not have a clear understanding of the role of randomization in statistical experiments.
Not quite surprisingly, it turns out that the scientist and the statistician are talking past each other. While the foremost goal of the scientist is to make the groups comparable, the statistician focuses on the randomization distribution. Moreover, the scientist asks repeatedly to include important information, but with his inquiry falling on deaf ears, he disputes this statistician’s analysis altogether.
Frequentists say that the crucial role of randomization, stated right at the beginning of this discussion (see the last section), is to provide a known distribution. But is this really so? If the result of a random allocation is extreme (e.g. all women are assigned to T, and all men to C), everybody - Fisher included - seems to be prepared to dismiss this realization: “It should in fairness be mentioned that, when randomization leads to a bad-looking experiment or sample, Fisher said that the experimenter should, with discretion and judgement, put the sample aside and draw another” [@sa76 464].
The latter concession isn’t just a minor inconvenience that may be fixed by a “gentlemen’s agreement”. First, an informal correction is wide open to personal capriciousness: (already) “bad-looking” to you may be (still) “fine-looking” to me. Second, what’s the randomization distribution actually used when dismissing some samples? A vague selection procedure will inevitably lead to a badly defined distribution. Third, why reject certain samples at all? If the crucial feature of randomization is to provide a distribution (on which all further “valid” inference is based), one should not give away this advantage unhesitantly. At the very least, it is inconsistent to praise the argument of the known framework in theoretical work, and to turn a blind eye to it in practice.
As a matter of fact, in applications, the exact permutation distribution created by some particular randomization process plays a rather subordinate role. Much more frequently, randomization is used as a rationale for common statistical procedures. Here is one of these heuristics: Randomization guarantees independence and if many small uncorrelated (and also often unknown) factors contribute to the distribution of some observable variable $X$, this distribution should be normal - at least approximately, if $n$ is not too small. Therefore, in a statistical experiment, it seems to be justified to compare ${\bar X}_T$ and ${\bar X}_C$, using these means and the sample variance as estimators of their corresponding population parameters - which is nothing but a verbal description of the t-test [@go08]. Thus, this test’s rationale is supported by randomization. However, it may be noted that Student’s famous test was introduced much earlier and worked quite well without randomization’s assistance.
Let us look at this from a different angle. A statistical test - or any analytical procedure for that matter - is an algorithm, transferring some numerical input into a certain output which, in the simplest case, is just a single number. From a Frequentist point of view, there are two very different kinds of input: experimental and non-experimental data. However, from a look at the data at hand one cannot tell if they stem from a proper statistical experiment or not. \[Footnote: That’s the main reason why @pe09 classifies randomization as a causal and not as a statistical concept.\] The formal test does not distinguish either: Given the same data it yields exactly the same output. The crucial difference lies in the interpretation of the numerical result. Since randomization treated all variables (known and unknown) alike, the analytical procedure “catches” them all and their effect shows up in the output. For example, a confidence interval, so the story goes, gives a quantitative estimate of all of the variables’ impact. One can thus numerically assess how strong this influence is, and has, in a sense, achieved explicit quantitative control. In particular, if the combined influence of all nuisance factors (Seidenfeld’s “probability statement about error‘’’) is numerically small, one may safely conclude that a substantial difference between $T$ and $C$ must be due to the experimental intervention. In a nutshell, owing to randomization, a statistical experiment gives a “valid” result in the sense that it allows for far-reaching, in particular causal, conclusions. Seen this way, randomization is sufficient for a causal conclusion, and some are convinced that it is also necessary (Holland’s “no causation without manipulation”, see Section \[history\]).
Note, however, that the crucial part of the above argument is informal in a rather principled way. It is suspicious that in an experimental, as well as in a similar non-experimental situation, the formal machinery, i.e. the data at hand, the explicit analytical procedure (e.g. a t-test), and the final numerical result may be identical. It is just the narrative prior to the data that makes such a tremendous difference in the end. Since verbal, non-mathematical arguments have a certain power of persuasion which is certainly weaker than a crisp formal derivation or a strict mathematical proof, it seems to be no coincidence that opinion on this matter has remained divided. Followers of Fisher believed in his intuition and trusted randomization, critics did not. And since, sociologically speaking, the Frequentist school dominated the field for decades, so did randomization.
Today, informal reasoning is a bit out of fashion. First, at least in the natural sciences, mathematical arguments are more important than verbal considerations. Typically, the thrust of an argument consists of formulas and their implications, with words of explanation surrounding the formal core. Second, we have learnt that seemingly very convincing verbal arguments can be wrong. In particular, increased formal precision has often corrected intuition and “obvious” time-honored conclusions. Hence, the nucleus of most contemporary theories has become mathematical. Causality is no exception to that rule. First, in the last twenty years or so causal networks and causal calculus have formalized this field. Second, as was to be expected, this increased precision straightforwardly demonstrated that certain “reasonable” conventions do not work as expected (see @pe09, in particular Chapter 6 and p. 341). It is not necessary to delve deeply into these matters, since it suffices to look at Savage’s example again. He claims qualitatively, and the above models corroborate this quantitatively, that no matter how one splits a heterogenous group into two, the latter groups will always be systematically different. Randomization does not help: If you assign randomly and detect a large effect in the end, your experimental intervention [*or*]{} the initial difference between $T$ and $C$ may have caused it. All “valid” inferential statistics cannot exclude the straightforward second explanation; it’s the initial comparability of the groups that is decisive for a sound causal conclusion.
The phrase “if $n$ is not too small” is also a verbal argument, implicitly appealing to the central limit theorem. The same with groups created by random assignment: Owing to the latter theorem they tend to become similar. The informal assurance, affirming that this happens fast, ranks among the most prominent conventions of traditional statistics. However, explicit numerical models, e.g. those presented in this contribution, underline that our intuition needs to be corrected. Precise formal arguments - in particular rather straightforward calculations - show that fluctuations cannot be dismissed easily, even if $n$ is large.
Apart from the rather explicit rhetoric of a “valid framework”, there is also always the implicit logic of the experiment around. Thus, although the received theory emphasizes that “actual balance has nothing to do with validity of statistical inference; it is an issue of efficiency only” [@se94 227]; in practice, comparability has turned out to be crucial. Many, if not most, of those praising randomization hurry to mention that it promotes similar groups (see numerous quotations in this article). Nowadays, only a small minority is basing its inferences on the known permutation distribution created by the process of randomization; but an overwhelming majority is checking for comparability. Reviewers of experimental studies routinely request that authors provide randomization checks, that is, statistical tests designed to substantiate the equivalence of $T$ and $C$. At least, in almost every article a list of covariates - with their groupwise means and standard errors - can be found. \[Footnote: It is also not much of a surprise that the popular “propensity score” is defined as the coarsest [*balancing*]{} score [@ro83].\]
In a nutshell, hardly anybody follows “pure Frequentist logic”. In a strict sense, there is no logic at all, rather a certain kind of mathematical reasoning plus - since the formal framework is restricted to sampling - a fairly large set of conventions; rigid “pure” arguments being readily complemented by applied “flexibility.” (The latter consisting of time-honored informal reasoning and shibboleth, but also outright concessions.) Consequently, one finds a broad range of verbal arguments why randomization should be employed:
> In my view Fisher regarded randomization as being essential in all experiments in the same way that he regarded it as being essential in telepathic and psychophysical experiments: the estimate of error followed exactly from the richness of the randomization. [@se94 220]
>
> As I see it, the purpose of randomization in the design of agricultural field experiments was to help ensure the validity of normal-theory analysis. [@hi80 583]
>
> Randomization tends to produce study groups comparable with respect to known and unknown risk factors, removes investigator bias in the allocation of participants, and guarantees that statistical tests will have valid significance levels. [@fr98 61]
>
> The first property of randomization is that it promotes comparability among the study groups $[\ldots]$ The second property is that the act of randomization provides a probabilistic basis for an inference from the observed results when considered in reference to all possible results. [@ro02a 7]
Cause and effect {#cause}
================
The core question of a statistical experiment, in particular a clinical trial, is thus: Does the treatment under consideration work? More specifically: Did the experimental intervention (e.g. a new drug) cause an observed effect (e.g. that the patients in group $T$ lived longer than those in $C$)? The answer will be a straightforward yes, if alternative explanations can be excluded, i.e. if any competing cause can be barred.
In the above chapters we considered two lines of reasoning, the first one going back to J.S. Mill, the second one originating from R.A. Fisher’s work. In a sense, they are not competitors. However, at the end of the day, they both should be judged according to their contribution in establishing the desired causal connection.
The logic we have exposed in Section 1 is simple and strong: If the groups are comparable at the beginning and if conscious experimentation guarantees that the intervention remains the only systematic difference between the groups, then this intervention must be the reason for a finally observed difference between the groups.
However, many statisticians count on randomization. They give a variety of reasons why this technique should be used (for a sample see the end of the last section). When they refer to Mill’s logic, it is comparability that matters. Alas, as many have noted qualitatively (see Section \[randcomp\]), and as the above models demonstrate quantitatively, randomization often fails in this respect. Therefore, at least in theoretical discussions, traditional statisticians refer to the “known-distribution argument” which, as we have seen in the last section, is also rather short-legged. Finally, there is the “little-assumption argument:”
> A new eye drug was tested against an old one on 10 subjects. The drugs were randomly assigned to both eyes of each person. In all cases the new drug performed better than the old drug. The P-value from the observed data is $2^{-10}=0.001$, showing that what we observe is not likely due to chance alone, or that it is very likely the new drug is better than the old one $[\ldots]$ Such simplicity is difficult to beat. Given that a physical randomization\[rand1\] was actually used, very little extra assumption is needed to produce a valid conclusion. [@pa01 13]\[paw2\]
Since “there’s no such thing as a free lunch” (Tukey, see @da08 [195]), we should become suspicious upon reading such textbook examples. A narrow, restricted framework is only able to support weak conclusions. Therefore, upon reaching a strong conclusion, one should immediately think about implicit, hidden assumptions. Be reminded of @he05\[heck10\] who says (pp. 139, 155, his emphasis):
> Structural models do not ‘make strong assumptions.’ They make explicit the assumptions required to identify parameters\[param50\] in any particular problem. The treatment effect literature does not make fewer assumptions; it is much less explicit about its assumptions $[\ldots]$ The assumptions to justify randomization (no randomization bias, no contamination or crossover effects $[\ldots]$) are [*different*]{} and not weaker or stronger than the assumptions \[econometric models use\].
>
> \[Footnote: In a table, @he05 [87] compares econometric and statistical causal models. Not surprisingly, the “range of questions answered” by the latter is just “one focused treatment effect.”\]
A second look at Pawitan’s example thus reveals that it is the hidden assumption of comparability that carries much of the burden of evidence. It is no coincidence that an eye drug was tested. Suppose, one had tested a liver drug instead: The same numerical result would be almost as convincing if such a drug had been applied to twins or (very) similar persons. However, if the liver drug had been administered to a heterogenous set of persons or if the new drug had been given to a different biological species (mice instead of men, say), exactly the same formal result would not be convincing at all; since, rather obviously, a certain discrepancy a priori may cause a remarkable difference a posteriori.
Comparability keeps this crucial alternative explanation at bay, not randomization; and it is our endeavour to achieve similar groups. Remember that for exactly the same reason Savage (in Section \[dicho\]) came to the conclusion that “it does not seem possible to base a meaningful experiment on a small heterogenous group.” @wa93 [52, 70, emphasis in the original] adds:
> In statistics, the purpose of randomization is to achieve homogeneity in the sample units $[\ldots]$ it should be spelled out that stability and homogeneity are the foundation of the statistical solution, [*not*]{} the other way around. For instance, in a clinical trial, applications of a randomized study to new patients rely on both the stability and homogeneity assumptions of our biological systems.
Given this point of view, it turns out that minimization (see Section \[compare\]) is not just some supplementary technique to improve efficiency. Rather, it is a straightforward and elaborate device to enhance comparability, i.e. to consciously construct similar groups. \[Footnote: For the influence of unknown factors see Section \[intermediate\].\]
A broader perspective {#history}
=====================
In the 20th century, R.A. Fisher (1890-1962) was the most prominent statistician. However, while his early work on mathematical statistics is highly respected in all quarters, hardly anybody relies on his later ideas, in particular fiducial inference [@sa76]. Randomization lies in between, and, quite fittingly, public opinion on this formal technique has been divided. Like the Bayesians, @fi73 was looking for a general inductive logic. However, “Fisher’s main tactic was to logically reduce a given inference problem $[\ldots]$ to a simple form where everyone should agree that the answer is obvious $[\ldots]$ Fisher’s inductive logic might be called a theory of types, in which problems are reduced to a small catalogue of obvious situations [@ef98 97].”
Some fifty years after his death it can safely be said that this approach did not succeed. It turned out that a few remarkable concepts, united in a rather narrow mathematical framework, and augmented by a set of conventions could not reduce the complexity of the real world to a few prototype examples. Leaving out a crucial piece of probability theory, Fisher’s strategy was able to give a collection of ad hoc devices, working on many special occasions; but as a whole “Frequentist theory is shot full\[shotfull\] of contradictions $[\ldots]$” [@ef01]. @ba88 and others who doggedly tried to rectify Fisher’s ideas, soon became Bayesians. @ja03 [494ff] explains why:
> $[\ldots]$ Jeffreys\[jeff30\] was able to bypass Fisher’s\[fis182\] calculations and derive those parameter\[param65\] estimates in a few lines of the most elementary algebra $[\ldots]$ Fisher’s\[fis183\] difficult calculations calling for all that space intuition $[\ldots]$ were quite unnecessary for the actual conduct of inference. $[\ldots]$ Harold @je39\[jeff31\] was able to derive all the same results far more easily, by direct use of probability theory as logic,\[indlogik2\] and this automatically yielded additional information about the range of validity of the results\[valgen\] and how to generalize them, that Fisher\[fis184\] never did obtain.
What may thus be said about randomization? How does Fisher’s construction of a know distribution perform relative to the classical logic of experimentation?
First, most scientists check for comparability, i.e. they follow Mill’s argument. Or, as Frequentist statisticians put it: they all seem to have misunderstood randomization.
Second, within statistics, Bayesians have always disputed the orthodox point of view, and during the last decades the latter has become less popular.
Third, even within orthodox statistics, there is no consensus about how to analyze randomized data. Nobody, not even Fisher, relies on “pure randomization,” in particular the distribution it generates. It is quite striking that only a minority, perfectly in line with the received position, advices not to adjust data at all. @fr08b [180f, 191] argues thus:
> Regression adjustments are often made to experimental data. Since randomization does not justify the models, almost anything can happen $[\ldots]$ The reason for the breakdown is not hard to find: randomization does not justify the assumptions behind the OLS model $[\ldots]$ The simulations,\[simu2\] like the analytic results, indicate a wide range of possible behavior. For instance, adjustment may help or hurt.
>
> \[Footnote: For similar comments see @fr08a, @pe09 [340], and @bo14 [250ff].\]
Yet a majority, like @ro02 [@ro02a; @sh02], or @tu00, opts for an “adjustment of treatment effect for covariates in clinical trials.” The latter authors explain why (p. 511):
> Covariates that affect the outcome of a disease are often incorporated into the design and analysis of clinical trials. This serves two main purposes: 1. To improve the credibility of the trial results by demonstrating that any observed treatment effect is not accounted for by an imbalance in patient characteristics, and 2. To improve statistical efficiency.
\[Footnote: Notice that, again, the authors readily refer to balance, but not to the “known-distribution argument”.\]
How strong is the evidence produced by a randomized controlled trial (RCT)? Vis-à-vis the rather anecdotal and qualitative research that preceded today’s RCTs, the latter surely constituted real progress. Strict design and standardized analysis has raised the level and has fostered consensus among researchers. However, many classical experiments in the natural sciences are deterministic and have an even better reputation. If in doubt, physicists do not randomize, but replicate. @fi36 [58, my emphasis] gave similar advice:
> \[…\] no one doubts, in practice, that the probability of being led to an erroneous conclusion by the chances of sampling only, can, [*by repetition*]{} $[\ldots]$ of the sample, be made so small that the reality of the difference must be regarded as convincingly demonstrated.
Alas, a large number of important biomedical findings (RCTs included) failed this examination and turned out to be non-replicable [@io05; @pr11; @be12], so that the @nih13 was forced to launch the “Replication of Key Clinical Trials Initiative” \[Footnote: Also see @na13\]. The same with experimental psychology which has relied on (small) randomized trials for decades. Now, it is lamenting a “replicability crisis” that has proved to be so severe that an unprecedented “reproducibility project” needed to be launched [@ca12; @pa12]. \[Footnote: For further similar initiatives see http://validation.scienceexchange.com\]
The logic of Section \[logic\] offers a straightforward explanation to this unpleasant state of affairs. Given (almost) equal initial conditions, the same boundary conditions thereafter, and a well-defined experimental intervention, an effect once observed must reoccur. That’s how classical experiments work, which reliably and thus repeatedly hit their target. During the experiment, a controlled environment keeps disturbing factors at bay. Thus, if an effect cannot be replicated, the constructional flaw should be looked for at the very beginning of the endeavour. At this point, it is conspicuous that today’s studies do not focus explicitly on the crucial idea of comparability. With other issues - possibly rather irrelevant or even misleading - being at least as important, initial imbalances are the rule and not the exception. At the very least, with randomization, the starting point of researcher 2, trying to repeat the result of researcher 1, will always differ from the latter’s point of origin. Therefore, if an effect cannot be replicated, this may well be due to the additional variability introduced by randomization, yielding unequal initial conditions, and “drowning” the interesting phenomenon in a sea of random fluctuation. Chance has a Janus face. The idea that many (the more the better), small and rather uncorrelated random influences sum up to a “mild” distribution originated in the 19th century, culminating in the famous central limit theorem on which much of classical statistics is built. However, this was not the end of the story. Studying complex systems, physicists soon encountered “wild” distributions, in particular power laws [@so06 104]. It is well within this newer set of ideas that a single random event may have a major impact that cannot be neglected (e.g. the energy released by a particularly strong earthquake or the area devastated by a single large flood). Quite fittingly, Gumbel said (see @er08 [339]):
> It seems that rivers know the theory. It only remains to convince the engineers of the validity of this analysis.
The fact that the process of randomization can produce a major fluctuation, i.e. a pronounced imbalance in a covariate (and thereby between $T$ and $C$), exerting a tremendous influence on the final result of an RCT is in line with this more recent portrait of chance.
Randomization has tremendous prestige in orthodox statistics, downgrading all designs without a random element to quasi-experiments. One even distinguishes thoroughly between truly random allocation and “haphazard assignment, that is, a procedure that is not formally random but has no obvious bias” [@sh02 302]. Honoring thus the classical philosophical distinction between “deterministic” and “random‘’’, one readily neglects the fact that modern dynamical systems theory sees a continuum of increasing complexity between perfect (deterministic) order and “randomness \[which\] can be thought of as an extreme form of chaos” [@el90 340]. With the technique, or rather dogma, of randomization at its heart, Fisher’s conception of experiments could even develop into a “cult of the single study” [@ne99 262]\[nel5\], and catch phrases highlighting randomization’s outstanding role - e.g. “no causation without manipulation” [@ho86] - became increasingly popular. However, this determined point of view has also blocked progress and innovative solutions have been developed elsewhere.
In particular, the story of causation may be told as a “tale of statistical agony” [@pe09]. Econometrist @he05 [5, 147]\[heck22\] adds:
> Blind empiricism unguided by a theoretical framework for interpreting facts leads nowhere $[\ldots]$ Holland claims that there can be no causal effect of gender on earnings. Why? Because we cannot randomly assign gender. This confused statement conflates the act of definition of the causal effect $[\ldots]$ with empirical difficulties in estimating it $[\ldots]$ This type of reasoning is prevalent in statistics $[\ldots]$ Since randomization is used to define the parameters of interest, this practice sometimes leads to the confusion that randomization is the only way - or at least the best way - to identify causal parameters from real data.
Heckman earned a nobel prize for his contributions in 2000. Epidemiology, following the @us64, but not @fi59, made its way to causal graphs. On the one hand, this formalization straightforwardly gave crucial concepts a sound basis, on the other hand quite a few received “reasonable” practices turned out to be dubious [@pe09].
In a more positive vein, computer scientist @ri07 has shown how Fisher’s finest ideas may be reformulated and extended within a modern, fine-tuned mathematical framework [@li08]. In Rissannen’s work one finds a logically sound and general unifying theory of hypothesis testing, estimation and modeling; yet there is no link to randomization. Li and Vitányi even include a chapter (5.5) on “nonprobabilistic statistics”. In this contemporary theory the crucial concept turns out to be Kolmogorov complexity, allowing to express the idea that a regular sequence $\bf r$ (e.g. “1,0” repeated 13 times) is less complex than a sequence like ${\bf s}=(1,0,0,1,0,0,1,1,0,1,1,0,0,0,1,1,1,0,1,1,0,1,0,0,0,0)$ in a mathematically strict way. \[Footnote: $\bf s$ can be found in @li08 [48].\] It also turns out that stochastic processes typically produce complex sequences. However, contrary to the fundamental distinction (deterministic vs. random) mentioned above, given a certain sequence like $\bf s$, it is not possible to tell whether the process that generated $\bf s$ was systematic or not. $\bf s$ could be the output of a “(pseudo-)random number generator”, i.e. a deterministic algorithm designed to produce chaotic output, or of a truly random mechanism. \[Footnote: Whatever that is. For example, strictly speaking, coin tossing - being a part of classical physics - is not.\]
Hence, interpreting $\bf s$ as a particular assignment of units to groups ($1 \rightarrow T$, and $0 \rightarrow C$, say), the above fundamental distinction between “‘haphazard’’ and “random” assignment processes seems to be largely overdrawn, some might even question if it is relevant at all. However, it is difficult to deny that the (non-)regularity of the concrete mapping matters. Just compare $\bf r$ to $\bf s$: Since $\bf r$ invites a straightforward alternative explanation, most people would prefer $\bf s$. In today’s terminology, Fisher could have been looking for maximally complex sequences, i.e. allocations without any regularity. In his time, a simple stochastic process typically yielding an “irregular” sequence was a straightforward and convenient solution to this problem.
All in all, Fisher’s idea of randomization is still alive. However, at large, it looks more like a remarkable solitaire from a heroic past than like the indispensable key to statistic’s future. At first sight, its career has been remarkable, but, make no mistake, it is a sign of crisis if a technique attains the status of a doctrine. This gives those questioning it a hard time. Like all heretics in history, they have received an unfair amount of criticism (just look up the reactions to @ho06, 1st ed. 1986). Not too long ago, it sufficed to combine the evidence of several studies in a systematic way to be met with scorn and derision. (See, for example, @ey78 on meta-analysis.)
Conclusion: Good experimental practice
======================================
In the face of all the issues we have discussed, Fisher’s claim that “\[randomization\] relieves the experimenter from the anxiety of considering and estimating the magnitude of the innumerable causes by which the data may be disturbed” seems to be close to wishful thinking. In particular, quantitative arguments and formal theories show that quite the contrary is true. Since random assignment is [*no*]{} philosopher’s stone, almost effortlessly lifting experimental procedures in the medical and social sciences to the level of classical experiments in the natural sciences, it has lulled many researchers into a false sense of security, thereby degenerating the most successful information science [@ef01] into a mindless ritual [@gi04]. Why has it taken so long to address the question explicitly? A number of reasons spring to mind:
- Randomization considerably reduces the operating expense of experimentation and makes the analysis of data simple, producing results that seem to be far-reaching. Although this sounds too good to be true, we are rather inclined to accept such a “favourable” procedure, in particular, when it is put forward by an authority. To admit that there is no such thing as a free lunch, that one has to work hard in order to get (closer to) comparability, is unpleasant news that we tend to avoid.
- Randomization addresses the fundamental problem of controlling unknown nuisance factors. In particular, it is impartial and does not favour or discriminate against any particular variable. This seems to outweigh the obvious disadvantage of randomization, i.e., known factors are less balanced than they would be if one used some non-randomized procedure, such as minimization.
- Randomization fits well into the dominant Frequentist framework. Moreover, as the fundamental difference between random quantities and their realizations is rather blurred in traditional statistics, one easily confuses the two. Yet, as the above models and quotations show, the difference between process and realization is crucial: Although a random allocation mechanism is independent of any other variable, its effect, a particular allocation of units to groups, can be considerably out of balance.
With respect to the last point it is often said that “randomization equates the groups on expectation” (e.g. @mo07 [40, 82], @sh02 [250]). The latter authors explain: “In any given experiment, observed pretest means will differ due to luck $[\ldots]$ But we can expect that participants will be equal over conditions in the long run over many randomized experiments.”
What is wrong with this argument? First, it conflates the [*single experiment*]{} being conducted and analyzed with a [*hypothetical series*]{} of experiments. Second, it therefore downplays the fundamental problem pointed out in Section \[logic\] and emphasized by many authors (see Section \[randcomp\]): In each and every experiment there is a convincing alternative explanation if the groups differ from the outset. Third, these substantial objections do not just vanish “in the long run”. Rather, if each study can be seriously challenged, the whole of the evidence may remain rather shaky. Fourthly, although it is certainly correct that replicated, randomized experiments align the set of all those treated with the set of all those not treated (“in expectation”), this simply does not imply that such a symmetry property also holds in the single experiment. Typically, quite the reverse is true (see Section \[models\]). Finally, the argument praises randomization where credit ought to be given to replication.
Coming back to the main theme of this contribution, it may be said that chance in the guise of randomization by and large supports comparability. However, since the former is blind with respect to concrete factors and relevant interactions that may be present, it needs a large number of experimental units to do so. The intuition behind this result is easy to grasp: Without knowledge of the subject-matter, randomization has to protect against every conceivable nuisance factor. Such a kind of unsystematic protection is provided by number and builds up slowly. Thus, a huge number of randomly allocated subjects is needed to shield against a moderate number of potential confounders. And, of course, no finite procedure such as the flip of a coin is able to control for an [*infinite*]{} number of nuisance variables.
Therefore, it seems much more advisable to use background knowledge in order to minimize the difference between groups with respect to known factors or specific threats to experimental validity. As of today, minimization operationalizes this idea best. At the end of such a conscious construction process, randomization finds its proper place. Only if no reliable context information exists is unrestricted randomization the method of choice. It must be clear, however, that it is a weak guard against confounding, yet the only one available in such inconvenient situations.
All in all, the above analysis strongly recommends traditional experimentation, thoroughly selecting, balancing and controlling factors and subjects with respect to known relevant variables, thereby using broader context information - i.e., substantial scientific knowledge. I agree with @pe03 [76f]\[pen\] who says:
> $[\ldots]$ it is the existence of sound background theory which is crucial for the success of science. It is the framework against which observations are made, it allows strict definition of the items involved, it is the source of information about possible relevant variables and allows for the identification of homogeneous reference classes that ensure regularity and, hence, reliable causal inference.
Cumulative science is the result of a successful series of such experiments - each of them focusing on the crucial ingredients, like precise research questions, convincing operationalizations, explicit control, quantitative measures of effect, and - last but not least - comparability.
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---
abstract: 'Peer production projects such as Wikipedia or open-source software development allow volunteers to collectively create knowledge-based products. The inclusive nature of such projects poses difficult challenges for ensuring trustworthiness and combating vandalism. Prior studies in the area deal with descriptive aspects of peer production, failing to capture the idea that while contributors collaborate, they also compete for status in the community and for imposing their views on the product. In this paper, we investigate collaborative authoring in Wikipedia, where contributors append and overwrite previous contributions to a page. We assume that a contributor’s goal is to maximize ownership of content sections, such that content owned (i.e. originated) by her survived the most recent revision of the page. We model contributors’ interactions to increase their content ownership as a non-cooperative game, where a player’s utility is associated with content owned and cost is a function of effort expended. Our results capture several real-life aspects of contributors interactions within peer-production projects. Namely, we show that at the Nash equilibrium there is an inverse relationship between the effort required to make a contribution and the survival of a contributor’s content. In other words, majority of the content that survives is necessarily contributed by experts who expend relatively less effort than non-experts. An empirical analysis of Wikipedia articles provides support for our model’s predictions. Implications for research and practice are discussed in the context of trustworthy collaboration as well as vandalism.'
author:
- 'S. Anand$^*$, O. Arazy$^\dagger$, N. B. Mandayam$^*$ and O. Nov$^\ddagger$'
bibliography:
- 'refs.bib'
nocite:
- '[@teece]'
- '[@martin]'
- '[@peng]'
- '[@winter]'
title: '[A Game Theoretic Analysis of Collaboration in Wikipedia]{}'
---
[*Index Terms-*]{}[Peer production, Wikipedia, collaboration, non-cooperative game, trustworthy collaboration, vandalism]{}
Introduction {#sec:intro}
============
Recent years have seen the emergence of a web-based peer-production model for collaborative work, whereby large numbers of individuals co-create knowledge-based goods, such as Wikipedia, and open source software [@novkuk], [@oferodedMIS], [@wolf], [@benkler], [@MSmalla], [@LeeOS], [@hippel]. Increasingly, individuals, companies, government agencies and other organizations rely on peer-produced products, stressing on the need to ensure trust worthiness of collaboration (e.g., deter vandalism) as well as the quality of end products.
Our focus in this study is Wikipedia, probably the most prominent example of peer-production. Wikipedia has become one of the most popular information sources on the web, and the quality of Wikipedia articles has been the topic of recent public debates. Wikipedia is based on wiki technology. Wiki is a web-based collaborative authoring tool that allows contributors to add new content, append existing content, or delete and overwrite prior contributions. Wikis track the history of revisions – similarly to version control systems used in software development – allowing users to revert a wiki page to an earlier revision [@leuf], [@wagner2004], [@wagner2007].
Peer production projects face a key tension between inclusiveness and quality assurance. While such projects need to draw in a large group of contributors in order to leverage “the wisdom of the crowd,” there is also requirement for accountability, security, and quality control [@forte], [@kittur], [@towne]. Quality assurance measures are necessary not only in cases of vandalism; conflicts between contributors could also result from competition. For example, contributors to Wikipedia may wrestle to impose their own viewpoints on an article – especially for controversial topics – or attempt to dominate subsets of the peer-produced product. Another example is when contributors seeking status within the community compete to make the largest contribution, and in the process overwrite others’ previous contributions. The result of such competitions is often “edit wars” where articles are changed back-and-forth between the same contributors.
Prior studies investigating an individual’s motivation for contributing content to Wikipedia have identified a large number of motives driving participation[@oferodedMIS],[@krogh], including motives that are competitive in nature, such as ego, reputation enhancement, and the expression of one’s opinions [@krogh], [@MISQuarterly]. However, studies investigating individuals did not consider the competitive dynamics emerging from motives such as reputation. Research into group interactions at Wikipedia, have tended to emphasize the collaborative (rather than competitive nature of interactions) [@oferodedMIS]. Other studies investigated threats to security and trustworthiness resulting from malicious attacks (i.e. vandalism)[@krogh] and the organizational mechanisms used by Wikipedia to combat such attacks [@MISQuarterly]; yet these studies do not consider threats resulting from benevolent contributors. A relevant strand of the literature has looked at conflicts of opinions between contributors [@oferjasist], [@oferodedMIS]. However, the focus is on the result of these conflicts on content quality rather than the competitive mechanisms driving them. In summary, while peer-production projects, and in particular Wikipedia, have attracted significant attention within the research community, to the best of our knowledge, the competitive dynamics have not been investigated.
In order to better understand collaboration in Wikipedia and capture the competitive nature of interactions, we turn to game theory. Our underlying assumption is that a contributor’s goal is to maximize ownership of content sections, such that content “owned” (i.e. originated) by that user survived the most recent revision of the page. We model contributors’ interactions to increase their content ownership, as a non-cooperative game. A contributor’s motivation for trying to maximize her ownership within a certain topical page could be based on the need to express one’s views or to increase her status in the community; and competition could be the result of battles between opposing viewpoints (e.g. vandals and those seeking to ensure trustworthiness of content) or consequences of power struggles. The utility of each contributor in the non-cooperative game is the ownership in the page, defined as the fraction of content owned by the contributor in the page. Each contributor suffers a cost of contribution which is the effort expended towards making the contribution. The objective is then to determine the optimal strategies, i. e., the optimal number of contributions made by each contributor, so that her *net utility* is maximized. Here, the net utility is the difference between the utility (a measure of the ownership) and the cost (a measure of the effort expended). The optimal strategies are determined by determining the Nash equilibrium of the non-cooperative game that models the interactions between the contributors. We determine the conditions under which the Nash equilibrium of the game can be achieved and find its implications on the contributors’ expertise levels on the topic. We report of an empirical analysis of Wikipedia that validates the model’s predictions. The key results brought forth by our analysis include
- The ownership of contributors increases with the decreasing levels of effort expended by the contributor on the topic.
- Contributors expending equal amount of effort end up with equal ownership.
The rest of the paper is organized as follows. The non-cooperative game that models the interactions between contributors is described in Section \[sec:ncgame\]. We then use Wikipedia data to validate the modeling in Section \[sec:results\]. We then discuss in Section \[sec:twcvand\] the relevance of our analysis and modeling to trust worthy collaboration and vandalism. Conclusions are drawn in Section \[sec:concl\] along with pointers to future directions.
User Contribution as a Non-Cooperative Game {#sec:ncgame}
===========================================
We model the interactions of the $N$ content contributors to a page (i.e., users) as a non-cooperative game. The strategy set for each contributor is the amount and type of contribution she makes. Table \[tbl:notation\] describes the notations used in our analysis. and their descriptions.
Notation/Variable Description
------------------- -------------------------------------------------------
$N$ Number of users or content contributors
$x_i$ The amount of content owned by the $i^{th}$ user
$\beta_i$ Effort expended by user $i$ to make unit contribution
$u_i$ The fractional ownership held by the $i^{th}$ user
$n_i$ Net utility of contributor $i$
$\mathbf{1}$ The all-one vector
$\mathbf{I}$ The identity matrix
: \[tbl:notation\] Variables used in the analysis in this paper.
Let $x_i$ represent the content owned by the $i^{th}$ user in the current version of the page. We define the utility, $u_i$, as the fraction of content owned by the $i^{th}$ contributor, and is given by $$\begin{aligned}
\label{eqn:utilityi}
u_i= \frac{x_i}{\sum_{j=1}^N x_j}.\end{aligned}$$ The objective of contributor $i$ is to determine the optimal $x_i$ so that $u_i$ is maximum.
It is observed from (\[eqn:utilityi\]) that the optimal $x_i$ that maximizes $u_i$ is $x_i=\infty$. This is because the utility function is an increasing function of $x_i$. Intuitively, this result occurs because every time the $i^{th}$ user makes a contribution, his/her ownership increases. However this results in reduction in the ownership of other contributors, to counter which, they attempt to make additional contributions (by increasing their respective $x_k$’s). This, in turn, reduces the ownership of contributor $i$, thereby causing him/her to further increase $x_i$ to increase ownership. This process continues ad infinitum resulting in $x_i\rightarrow\infty$, $\forall$ $i$. This degenerate scenario can be mitigated as follows.
Suppose the $i^{th}$ contributor expends an effort, $\beta_i$, to make a unit contribution. For instance, $\beta_i$ can be the cost incurred by the $i^{th}$ user, in terms of time and effort spent in learning the topic and in posting content on a Wiki page. Therefore, the $i^{th}$ contributor expends a total effort $\beta_ix_i$, to achieve $x_i$ amount of content ownership in the page. The net utility experienced by the $i^{th}$ contributor, $n_i$, can be written as the difference between utility of contributor $i$, given by (\[eqn:utilityi\]) and the total effort expended by contributor $i$, i. e., $$\begin{aligned}
\label{eqn:netutilityi}
n_i=u_i-\beta_ix_i=
\frac{x_i}{\sum_{j=1}^N x_j}
-\beta_ix_i.\end{aligned}$$ It is observed that the net utility obtained by the $i^{th}$ contributor not only depends on the strategy of the $i^{th}$ contributor ([*i.e.*]{}, $x_i$), but also on the strategies of all the other contributors ([*i.e.*]{}, $x_j$, $j\neq i$). This results in the non-cooperative game of complete information [@neumannmorgen] between the contributors. The optimal $x_i$, $\forall$ $i$ (termed as $x_i^*$), which is determined by maximizing $n_i$ in (\[eqn:netutilityi\]), is then the Nash equilibrium of the non-cooperative game where no contributor can make a unilateral change.
Applying the first order necessary condition to (\[eqn:netutilityi\]), $x_i^*$ is obtained as the solution to $$\begin{aligned}
\label{eqn:firstderivative}
\begin{array}{cc}
\left .
\frac{\partial n_i}{\partial x_i}
\right |_{x_i=x_i^*}=
\frac{\sum_{k = 1\atop k \neq i}^{N} x_k^*}
{\left (\sum_{j = 1}^{N} x_j^*\right )^2} - \beta_i=0,
& \forall i
\end{array}\end{aligned}$$ subject to the constraints $x_i^*\geq 0$, $\forall$ $i$. From (\[eqn:firstderivative\]), we obtain $\frac{\partial^2 n_i}{\partial x_i^2} = - \frac{2
\sum_{k = 1\atop k \neq i}^{N} x_k}
{\left (\sum_{j = 1}^{N} x_j\right )^3} < 0$, $\forall$ $i$, when $x_i\geq 0$. Thus, $n_i$ is a concave function of $x_i$ and $x_i^*$, which solves (\[eqn:firstderivative\]) subject to $x_i^* \geq 0$, $\forall$ $i$, is a local as well as a global maximum point. In other words, according to [@nash], *the non-cooperative game has a unique Nash equilibrium*, $\mathbf{x}^*=
\left [
\begin{array}{cccc}
x_1^* & x_2^* & \cdots & x_N^*
\end{array}
\right ]^T$, obtained by numerically solving the system of $N$ non-linear equations specified by (\[eqn:firstderivative\]). However, to study the effect of the effort levels ($\beta_i$’s) on the strategies of the contributors, it is desirable to obtain an expression that relates the vectors, $\mathbf{x}^*$, $\mathbf{x}=\left [x_i\right ]_{1\leq i\leq N}$ and $\mbox{\boldmath $\beta$}=\left [\beta_i\right ]_{1\leq i\leq N}$.
Re-writing (\[eqn:firstderivative\]), $$\begin{aligned}
\label{eqn:omegai}
\begin{array}{cc}
\left (\sum_{j=1}^N x_j^*\right )^2
-\alpha_i\sum_{j=1\atop j\neq i}^Nx_j^*=0,
& \forall N,
\end{array}\end{aligned}$$ where $\alpha_i{\stackrel{\triangle}{=}}1/\beta_i$. Eqn. (\[eqn:omegai\] ) can be written as $$\begin{aligned}
\label{eqn:modifiedmatrixform}
\left (\mathbf{x}^*\right )^T
\mathbf{1}\mathbf{1}^T
\mathbf{x}^*\mathbf{1}-
\mathbf{D}_{\alpha}\left (\mathbf{1}\mathbf{1}^T-\mathbf{I}\right )
\mathbf{x}^*=\mathbf{0},\end{aligned}$$ where $(.)^T$ represents the transpose of a vector or a matrix, $\mathbf{D}_\alpha$ is the diagonal matrix $\mathbf{diag}\left (\alpha_1,
\alpha_2,\cdots,\alpha_N\right )$, $\mathbf{1}$ is the column vector in which all entries are one, $\mathbf{0}$ is the column vector in which all entries are zero and $\mathbf{I}$ is the identity matrix.
It can be easily verified the vectors, $\mathbf{y}_1=\left [
\begin{array}{cccccc}
\frac{1}{\sqrt{N}} &
\frac{1}{\sqrt{N}} &
\frac{1}{\sqrt{N}} &
\frac{1}{\sqrt{N}} &
\cdots &
\frac{1}{\sqrt{N}}
\end{array}
\right ]^T$ and for $j=2$, $3$, $\cdots$, $N$, $\mathbf{y}_j=
\left [
\begin{array}{cccccc}
y_{1j} & y_{2j} & y_{3j} & \cdots & y_{(N-1)j} & y_{Nj}
\end{array}
\right ]^T$, where $$\begin{aligned}
\label{eqn:yj}
y_{kj}=\left \{
\begin{array}{cc}
\frac{1}{\sqrt{j(j-1)}} & k<j\\
-\frac{j-1}{\sqrt{j(j-1)}} & k=j\\
0 & k>j,
\end{array}
\right .\end{aligned}$$ form a set of orthonormal eigen vectors to the matrix, $\mathbf{1}\mathbf{1}^T$. The eigen value corresponding to $\mathbf{y}_1$ is $N$ and those corresponding to $\mathbf{y}_2,\cdots,\mathbf{y}_N$ are $0$s. Let $\mathbf{P}=\left [
\mathbf{y}_1\vert\mathbf{y}_2\vert\cdots\vert\mathbf{y}_N\right ]$. Then, $\mathbf{P}$ is an orthogonal matrix and by orthogonality transformation, $$\begin{aligned}
\label{eqn:orthogonality}
\mathbf{P}^T\mathbf{1}\mathbf{1}^T\mathbf{P}=\mathbf{D}=
\mbox{diag}\left (N, 0, 0, \cdots, 0\right ).\end{aligned}$$ Let $\mathbf{z}=\left [
\begin{array}{cccccc}
z_1 & z_2 & z_3 & \cdots & z_{N-1} & z_N
\end{array}
\right ]^T$. Since the eigen vectors of a matrix form a basis for the $N-$dimensional sub-space [@meyerbook], the vector, $\mathbf{x}^*$, can be written as $\mathbf{x}^*=\mathbf{P}\mathbf{z}$. A similar expression has been solved in [@techreportpricing] in the context of pricing in wireless networks and we outline here the key steps to determine the optimal $\mathbf{x}^*$.
- Using $\mathbf{x}^*=\mathbf{P}\mathbf{z}$ in (\[eqn:modifiedmatrixform\]) and (\[eqn:orthogonality\]), we obtain $$\begin{aligned}
\label{eqn:nonlineZ}
\mathbf{z}^T\mathbf{D}\mathbf{z}\mathbf{1}
-\mathbf{D}_\alpha\left (\mathbf{1}\mathbf{1}^T-\mathbf{I}\right)
\mathbf{P}\mathbf{z}=\mathbf{0}.\end{aligned}$$
- The above is a set of non-linear equations in $\mathbf{z}$, in which the $k^{th}$ equation depends on $z_1$ and $z_j$, $k\leq j\leq N$. Solving the non-linear equations by backward substitution [@meyerbook], $z_k$, $2\leq k\leq N$ can be written in terms of $z_1$ as $$\begin{aligned}
\label{eqn:xkx1}
\frac{z_k}{\sqrt{k(k-1)}}=
\frac{Nz_1^2}{k(k-1)}\left [
\frac{k}{\alpha_k}+\sum_{j=k+1}^N\frac{1}{\alpha_j}\right ]
-\frac{z_1}{\sqrt{N}}\frac{N(N-1)}{k(k-1)}.\end{aligned}$$
- Using (\[eqn:xkx1\]) to replace all $z_k$’s in terms of $z_1$ in the set of non-linear equations in (\[eqn:nonlineZ\]), $z_1$ can be obtained as $$\begin{aligned}
\label{eqn:expz1}
z_1=\frac{N-1}{\sqrt{N}}
\frac{1}{G},\end{aligned}$$ where $$\begin{aligned}
\label{eqn:defnG}
G{\stackrel{\triangle}{=}}\sum_{j=1}^N\frac{1}{\alpha_j}.\end{aligned}$$
- Combining (\[eqn:xkx1\]) and (\[eqn:expz1\]), $$\begin{aligned}
\label{eqn:finalexpressionxk}
\begin{array}{cc}
\frac{z_k}{\sqrt{k(k-1)}}=
\frac{(N-1)^2}{k(k-1)}G^{-1}
\left [G^{-1}\left (\frac{k}{\alpha_k}+
\sum_{j=k+1}^N\frac{1}{\alpha_j}\right )-1\right ] &
2\leq k\leq N.
\end{array}\end{aligned}$$
- Using the facts $\mathbf{x}^*=\mathbf{P}\mathbf{z}$, and $\alpha_i=\frac{1}{\beta_i}$ in (\[eqn:expz1\]) and (\[eqn:finalexpressionxk\]), the unique Nash equilibrium of the non-cooperative game can be obtained as $$\begin{aligned}
\label{eqn:bistar}
x_i^*=\frac{\sum_{j=1}^N\beta_j-(N-1)\beta_i}
{\left (\sum_{j=1}^N\beta_j\right )^2}.\end{aligned}$$
Note that the unique Nash equilibrium $\mathbf{x}^*$, is feasible, [*i.e.*]{}, $x_i^* >0$, $\forall$ $i$ if and only if $$\begin{aligned}
\label{eqn:conditionnash}
(N-1)\beta_i< \sum_{j=1}^N\beta_j.\end{aligned}$$ The utility (ownership) of contributor $i$ at the Nash equilibrium, $u_i^*$, can then be obtained from (\[eqn:utilityi\]) and (\[eqn:bistar\]) as, $$\begin{aligned}
\label{eqn:uistar}
u_i^*=\left [1-\left ( \frac{(N-1)\beta_i}{\sum_{j=1}^N\beta_j}\right )\right ]^+,\end{aligned}$$ where $x^+=\max(x,0)$. It is observed that the ownership $u_i^*$ is non-zero if and only if (\[eqn:conditionnash\]) is satisfied, i.e., if the Nash equilibrium is feasible. The condition in (\[eqn:conditionnash\]) and the expression in (\[eqn:uistar\]) have the following interesting implications.
- From (\[eqn:uistar\]), the ownership of contributors depend on the $\beta_j$ of [*all the contributors*]{}. This is intuitively correct in a peer production project like Wikipedia because contributions are made by multiple users and the ownership held by a user will depend on the effort of all the users that worked together in making contributions to the page.
- The expression in (\[eqn:uistar\]) indicates that contributors who expend smaller effort have larger ownership and those who expend larger effort have low ownership, i.e., the fractional content ownership is a decreasing function of the effort expended.
- Asymptotically, i.e., as the number of contributors, $N$, becomes large, the ownership, $u_i^*$ in (\[eqn:uistar\]), can be written as $$\begin{aligned}
\label{eqn:uistarassym}
u_i*=\left ( 1-\frac{\beta_i}{E[\mbox{\boldmath $\beta$}]}\right )^+,\end{aligned}$$ where $E[\mbox{\boldmath $\beta$}]{\stackrel{\triangle}{=}}\frac{1}{N}\sum_{j=1}^N\beta_j$, is the [*average effort*]{} of all the users that make contributions to the page. From (\[eqn:uistarassym\]), only those contributors for whom $\beta_i<E[\mbox{\boldmath $\beta$}]$, i.e., only those contributors whose effort is below the average effort expended in posting content to a page, end up with non-zero ownership. In other words, given the effort involved in making a contribution, and the ease in which others can overwrite one’s contributions, only those who expend less effort in making their contributions than the average effort required, end up with non-zero ownership.
Empirical Validation with Data {#sec:results}
==============================
While the non-cooperative game theoretic models developed in Section \[sec:ncgame\] are based on intuitive notions of ownership and effort, it is necessary to validate these with real data from contributions to Wikipedia articles. We require a set of Wikipedia articles with data on: (a) the content “owned” by contributors at each revision (which can be analogous to the utility, $u_i^*$ in (\[eqn:uistar\]) and (b) the cumulative effort exerted by each contributor (including all of his/her contributions) up to each revision, which can represent the effort, $\beta_i$, used in the expressions in (\[eqn:finalexpressionxk\]) and (\[eqn:uistar\]). We use the data set from Arazy *et al* [@oferjasist], who explored automated techniques for estimated Wikipedia contributors’ relative contributions. The data set in [@oferjasist] includes nine articles randomly selected from English Wikipedia. Each article was created over an average period of 3.5 years. Section \[subsec:JASISTdata\] presents the details of the data set in [@oferjasist] and Section \[subsec:verification\] provides a validation of the same against the models developed in Section \[sec:ncgame\].
Extracting data from Wikipedia Articles {#subsec:JASISTdata}
---------------------------------------
----------------------------------- -------------- --------------- ---------- ------- ---------
Article title Start Date End Date Duration Edits Unique
(MM/DD/YYYY) (MM/DD/YYYY) (years) Editors
Aikodo [@aikido] $11/29/2001$ $06/13/2004$ 2.5 72 62
Angel [@angel] $11/30/2001$ $12/09/2005$ 4.0 341 277
Baryon [@baryon] $02/25/2002$ $ 08/25/2005$ 3.5 73 62
Board Game [@boardgame] $11/04/2001$ $12/30/2004$ 3.2 220 155
Buckminster Fuller [@buckminster] $12/13/2001$ $07/14/2004$ 2.6 65 55
Center for Disease $10/16/2001$ $03/05/2006$ 4.4 65 58
Control and Prevention [@center]
Classical Mechanics [@classical] $06/06/2002$ $08/13/2006$ 4.2 202 165
Dartmouth College [@dartmouth] $10/01/2001$ $08/26/2004$ 2.9 70 55
Erin Brockovich [@erin] $09/24/2001$ $02/02/2006$ 4.4 59 54
Total 31.7 1167 943
Average 3.5 129.7 104.8
----------------------------------- -------------- --------------- ---------- ------- ---------
The content “owned” by contributors at the end date of each article period was calculated using the method described in [@oferjasist]. A sentence was employed as the unit of analysis, and each full sentence was initially owned by the contributor who added it. As content on a wiki page evolves, a contributor may lose a sentence when more than 50% of that sentence was deleted or revised. A contributor making a major revision to a sentence can take ownership of that sentence. The algorithm tracks the evolution of content, recording the number of sentences owned by each contributor at each revision, until the study’s end date. The effort exerted by each contributor was, too, based on the method and data set described in [@oferjasist]. Two research assistants worked independently to analyze every “edit” made to the 9 articles in the sample set and record: contributor’s ID; the type of each “edit” to the wiki page (the categories used included: add new content, improve navigation, delete content, proofread, and add hyperlink); the scope of each edit (on a 5-point scale, from minor to major). For example, a particular edit might be categorized as major addition of new content. The two assessors reviewed the “History” section of articles (where Wikipedia keeps a log of all changes to a page), comparing subsequent versions. Once the assessors completed their independent work, and inter-rater agreement levels were calculated (yielding very high levels of agreement), the average of the two assessors was used in the analysis. Finally, the above data set was used to obtain the following parameters on each Wikipedia article listed in Table \[tab:oferdata\]:
- The number of exclusive contributors/users ($N$)
- The total effort expended by the $i^{th}$ user ($1\leq i\leq N$), $s_i$
- The number of edits made by the $i^{th}$ user ($1\leq i\leq N$), $e_i$
- The number of sentences owned by the $i^{th}$ user ($1\leq i\leq N$), $p_i$.
The following subsection provides a detailed explanation on how we use these parameters to verify the game theoretic analysis described in Section \[sec:ncgame\].
Numerical verification of the analysis {#subsec:verification}
--------------------------------------
Using the set of parameters obtained from the pages in Table \[tab:oferdata\], listed in Section \[subsec:JASISTdata\], we compute the effort expended by user $i$ for unit contribution, $\beta_i$, as $$\begin{aligned}
\label{eqn:betai}
\beta_i=\frac{s_i}{e_i}.\end{aligned}$$ Using the $\beta_i$’s thus obtained, we use the expression in (\[eqn:uistar\]) to determine the estimated fractional ownership on the Wikipedia page, that will be held by each contributor. We compare this with the fraction $\frac{p_i}{\sum_{j=1}^N p_j}$
Figs. \[fig:aikido\], \[fig:board\] and \[fig:erin\] show the comparison between the ownership obtained according to the game theoretic analysis described in Section \[sec:ncgame\] and that given by the data set in [@oferjasist] for the Wikipedia pages, “Aikido”, “Board Game” and “Erin Brockovich”, respectively. For this first analysis, we anonymized the data set, indexing the users in the decreasing order of $\beta_i$’s. We find that the patterns in the empirical data and that of the game-theoretic model closely match one another. In particular, the empirical data validates the following predictions made by the game theoretic model in Section \[sec:ncgame\][^1].
1. **Equivalence classes:**
1. Let the users be classified into equivalence classes according to their fractional ownership, i.e., all users having equal fractional ownership in the Wikipedia page belongs to the same equivalence class. It is observed that each page has five to six equivalence classes. For instance, Aikido, has five equivalence classes (Fig. \[fig:aikido\]) and Board game (Fig. \[fig:board\]) and Erin Brockovich (Fig. \[fig:erin\]), have six equivalence classes each. *Note that the number of equivalence classes obtained from the data is the same as that predicted by the game theoretic analysis described in Section \[sec:ncgame\]*.
2. From (\[eqn:uistar\]), $u_i^*=u_j^*$ if and only if $\beta_i^*=\beta_j^*$. This indicates that the distribution of the data into number of equivalence classes applies not only to fractional ownership, but also to the effort expended by users. In other words each Wiki page is expected to have five to six categories of contributors/users. A more detailed analysis of the distribution suggests that the majority of users fall into the equivalence middle classes, while the classes on the extreme representing very low and very high levels of effort (and content ownership) comprise of relatively few users. [*While the above can be inferred from the data alone, the game theoretic analysis provides a mathematical framework that validates this observation.*]{}
2. **Non-zero ownership:** It is observed from (\[eqn:uistar\]) that $u_i^*=0$ if and only if the condition in (\[eqn:conditionnash\]) is violated. The number of users in our sample data with zero ownership matches the number predicted by the game-theoretic model thus providing validation for the condition (14) (at least for the Wikipedia pages included in our analysis). [*Again, it is observed that the relation between the number of users with zero ownership and their corresponding $\beta_i$’s could have been inferred from the data alone, the game theoretic analysis presented in Section \[sec:ncgame\] provides a mathematical framework to model this phenomenon.*]{}
After establishing that the general trend (i.e. anonymized data) for the empirical data and the model’s predictions match one another, we perform a more detailed analysis where we pay attention to users’ identities. That is, we organize both data sets, namely the fractional ownership data taken directly from [@oferjasist] and the ownership values our model in Section \[sec:ncgame\] predicted, for each user. We then calculate the correlation between the two data sets, using the Pearson’s correlation coefficient [@wolf]. The result of the analysis for the nine articles in our data set is presented in Fig. \[fig:pearson\]. As could be seen from the figure, correlation coefficients range between 0.47 and 0.88, representing moderate-high correlation. When combining the entire data from the nine articles into a single data set, the Pearson correlation was 0.65 (with a $p-$value, $p\approx 0.04$). Therefore, we now proceed to verify if the discrepancies in the values of the ownership obtained by the game theoretic analysis and that obtained from the data can be offset by establishing a linear fit that maps the set of values obtained by analysis to the ones obtained from the data.
Let $\mathbf{a}{\stackrel{\triangle}{=}}\left [
\begin{array}{ccccc}
a_1 & a_2 & a_3 & \cdots & a_N
\end{array}
\right ]$ represent the ownership of the contributors obtained by the game theoretic analysis and let $\mathbf{d}{\stackrel{\triangle}{=}}\left [
\begin{array}{ccccc}
d_1 & d_2 & d_3 & \cdots & d_N
\end{array}
\right ]$ represent the ownership of the contributors obtained from the data as described in [@oferjasist]. For each page, we fit a function $$\begin{aligned}
\label{eqn:linfit}
\begin{array}{cc}
\hat{d}_i=\rho a_i+\delta & 1\leq i\leq N,
\end{array}\end{aligned}$$ where the parameters $\rho$ and $\delta$ are obtained by the method of least squares [@meyerbook]. We use the values for the pages “Aikido”, “Angel”, “Baryon” and “Board Game” as the training data to obtain $\rho$ and $\delta$. We then use the values of $\rho$ and $\delta$ thus obtained to determine $\hat{d}_i$ for the other five pages. We then compare $\hat{d}_i$ and $d_i$ and compute the estimation error for each page. Fig. \[fig:error\] shows the estimation error for the remaining five pages. It is observed that the error is between 7-9%. The error for the training set of data was found to be around 5%. *This indicates that the game theoretic analysis presented in Section \[sec:ncgame\] models the contributor interactions in peer production projects like Wikipedia accurately upto a linear scaling factor*.
Trustworthy Collaboration and Vandalism {#sec:twcvand}
=======================================
An important insight provided by our non-cooperative game model (and validated by our empirical analysis) is that only contributors with below-average effort levels are able to maintain fractional ownership on wiki pages. That is, by and large only the edits made by contributors who exert little effort survive the collaborative authoring process. In Section \[sec:intro\], we referred to two key concerns that are associated with trustworthy collaboration in peer-production projects: (a) a risk that non-experts will contribute content of low quality, and (b) a threat that malicious participants would vandalize Wikipedia pages. In spite of these serious concerns, the content on Wikipedia articles is generally of high quality and Wikipedia maintains the status as one of the most reliable sources of information on the web [@kittur]. How then, does Wikipedia maintain high-quality content in the face of threats of low-quality or malicious contributions? Our results can have important implications for investigation of trustworthy collaboration on Wikipedia (and more broadly, in peer-production projects). In the sections that follow, we provide two interpretations of our results that help explain how the threats highlighted above are mitigated.
1. **Trustworthiness/Quality of Wikipedia pages:** The first interpretation of the model and its empirical validation involves the concern of non-expert, low quality contributions eroding the trustworthiness of peer-produced product. This interpretation suggests that low effort is associated with greater likelihood of content survival due to a skill advantage: contributors who are experts in their field of contribution expend less effort, and their contributions are of higher quality [@anthony]. Thus, the effort associated with contribution is inversely related with its quality and consequently with its likelihood of survival of subsequent editing.
2. **Vandalism:** The second interpretation concerns the danger of vandalism activities reducing the trustworthiness of the peer-produced products. Since the underlying Wiki mechanisms allow any editor to easily revert the edits of other contributors, the effort involved in vandalistic edits is higher than the effort of reverting such edits. Thus, high effort is associated with vandalism and relatively lower effort is linked to correction of vandalism. The game theoretic analysis presented in Section \[sec:ncgame\] predicts that the contributions made by users expending large effort do not survive the edit process and end up with zero ownership. Therefore, most vandalistic edits would not survive over time, as also observed in [@kitturconflict], [@suhchi2007], [@stvtwi2008].
In summary, following on the intuition observed in [@kitturconflict], [@suhchi2007], [@stvtwi2008], we modeled competition between players as a non-cooperative game, where a player’s utility is associated with surviving fractional content owned, and cost is a function of effort exerted. Broader design implications emerging from this interpretation include the need to make version control mechanisms not only highly usable, but also highly open and egalitarian, and accessible to participants in a peer-production process. In addition, these insights suggest the importance of concurrent use of other quality control mechanisms, including user-designated alerts (where users are notified when changes are made to an article, or other part of the collaboratively-created product); watch lists (where users can track certain articles); and IP or user blocking in cases where repeated attacks from the same source are deemed to be acts of vandalism. The combination of these mechanisms make three important contributions to the trustworthiness of peer-production projects: first, their existence deter potential vandals; second, they reduce the costs of identifying and responding quickly to attacks; and third, they enable users to easily revert the consequences of vandalism .
Conclusion {#sec:concl}
==========
To better understand the success of peer production, we developed a non-cooperative game theoretic model of the creation of Wikipedia articles. The utility of a contributor was her relative ownership of the peer-produced product that survived a large number of iterations of collaborative editing. The work presented here contributes to better understanding of the trustworthiness of peer-production by
- Solving the game and demonstrating the conditions under which a Nash equilibrium exists, showing that asymptotically only users with below average effort would maintain ownership
- Empirically validating the model, demonstrating that only users with below average effort would maintain ownership, as well as showing editors’ equivalence classes
- Offering interpretations and implications for research on trustworthy peer-production (in terms of expertise and vandalism).
To the best of our knowledge, this is the first modeling of user interactions on Wikipedia as a non-cooperative game. Our analysis points to the benefits of deploying multiple mechanisms which afford the combination of large-scale and low-effort quality control as way to ensure the trustworthiness of products created through web-based peer-production. Further research is needed to analyze the effectiveness of each of these mechanisms, and to address other aspects of peer production through game theoretic analysis.
[Acknowledgment]{}
This work was supported in part by a National Academies Keck Futures Initiative (NAKFI) grant.
[^1]: These trends were observed not only for the three articles shown in Figs. \[fig:aikido\]-\[fig:erin\] but also for all the nine articles listed in Table \[tab:oferdata\]. We show results for three articles here due to lack of space.
|
---
abstract: 'The bioinformatical methods to detect lateral gene transfer events are mainly based on functional coding DNA characteristics. In this paper, we propose the use of DNA traits not depending on protein coding requirements. We introduce several semilocal variables that depend on DNA primary sequence and that reflect thermodynamic as well as physico-chemical magnitudes that are able to tell apart the genome of different organisms. After combining these variables in a neural classificator, we obtain results whose power of resolution go as far as to detect the exchange of genomic material between bacteria that are phylogenetically close.'
author:
- |
C. Calderón$^1$, L. Delaye$^2$, V. Mireles$^{3}$ and P. Miramontes$^{1,4,*}$[[^1]]{}\
1. Facultad de Ciencias, Universidad Nacional Autónoma de México, México 04510 DF, México\
2. Departamento de Ingeniería Genética, Centro de Investigación y Estudios Avanzados, Irapuato, Guanajuato, 36821, México.\
3. International Max Planck Research School for Computational Biology and Scientific Computing. Freie Universität Berlin, D-14195 Berlin, Germany\
4. Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, México 04510 DF, México
title: Detecting lateral genetic material transfer
---
Introduction
============
There is a general agreement that horizontal gene transfer (HGT) is important in genome evolution. To which degree is still a matter of debate. The discussion oscillates between two extreme positions: first, the idea that the rate of transfer and its impact are of such a magnitude as to be the “essence of phylogeny” -at least for Prokaryotic organisms [@Doolittle01] and, on the other hand, the researchers who opine that the role of HGT in evolution has been overestimated [@Kurland] and that, while a matter of interest, it is not relevant when compared to other causes of genomic evolution like paralog duplication and secondary gene looses. Most probably, the real weight of HGT in evolution is posed somewhere between these extremes. Independently of the outcome of the discussion, there is a general agreement that it is relevant to detect events of HGT.
There are several proposed methods to detect HGT. They can be classified into four categories: deviant composition, anomalous phylogenetic distribution, abnormal sequence similarity and incongruent phylogenetic trees [@Eisen; @Ragan; @Philippe].
Deviant composition methods are based on the different phenotypic characteristics among divergent genomes. They are mainly focused on bias in GC contents or codon usage [@Mrazek] and bias in the nucleotide composition in the third and first codon position [@Lawrence]. Deviant genes might exist for reasons other than HGT, and only recently transferred genes would be detected by this method [@Eisen; @Hooper]. Also, this group of methods normally does not detect transferred genes from phenotypically similar genomes.
Anomalous phylogenetic distribution is based on the identification of homologous genes shared by genomes in disjunct phylogenetic lineages and its absence in close relatives (in one or both lineages). However, polyphyletic gene looses and rapid sequence divergence can mislead the identification of HGT [@Eisen].
Abnormal sequence similarity is based on the assumption of overall similarity as a measure of phylogenetic relatedness. Usually, BLAST searches (or other similar algorithms) are used to detect sequences in one genome more similar to sequences in divergent genomes than those sequences found in phylogenetic closer genomes (the phylogenetic relationships between genomes are set prior the analysis according to some other criterion like rRNA phylogeny). While these methods work fast, they are not fully reliable because the similarity between a gene in two different species can be explained by a number of phenomenons besides HGT. For instance, evolutionary rate variation can lead to misleading results in the identification of HGT genes, both as false positives and false negatives [@Eisen].
Phylogenetic analysis is often considered to be perhaps the best way to investigate the occurrence of HGT because it remains the only one to reliably infer historical events from gene sequences [@Eisen]; Accordingly, incongruent phylogenetic trees between different families of genes will be caused by HGT; however conflicting phylogenies can be a result of either artifacts of phylogenetic reconstruction, HGT or unrecognized paralogy [@Zhaxybayeva].
In this study, we propose a method that does not depend on DNA functional traits and due to this reason is no longer correct to say that it detects HGT because it might also detect transfer of non-coding DNA (ncDNA). From now on, we will refer to horizontal genetic material transfer (HGMT). We use eight variables that can be measured over a set of windows covering whole genomes and combine them with an Artificial Neural Network (NN). The field of using DNA measurables other than those derived from protein coding requirements have been largely ignored. Up to our knowledge, there are no methods based exclusively on structural DNA traits to detect neither HGT or HGMT events.
Methods
=======
Our approach was to take a pair of DNA sequences –genomes in case of prokaryotes, chromosomes for eukaryotes-, the first one is the *donor* genome and the second one the *acceptor*. The data were taken from Genbank release 24.
To calculate the variables that characterize locally the DNA sequences, a window is placed over the chromosomes (Figure \[fig:01\]). The window can slide over the sequences or can be put randomly (see below).
![A window is placed over DNA sequence. On every position, some primary DNA sequence variables $(x_1, x_2,\dots, x_n$) are calculated. For the NN prediction stage, this window slides along the DNA sequence. (see text)[]{data-label="fig:01"}](figure1.eps)
We used a “classic NN approach” –a backpropragation multilayer perceptron (MLP), very similar to the model reported by Uberbacher [@Uberbacher] (Figure \[fig:02\])
![The sensors are mixed in a Multilayer Perceptron. The output can be zero or one depending on to which class the DNA sequence belongs.The MLP in the figure does not necessarily have the architecture used in the study.[]{data-label="fig:02"}](figure2.eps)
The novelty and main contribution of this paper is the set of measurables we use to evaluate a DNA sequence. Following the nomenclature of Uberbacher we will call them [*sensors*]{}.
For the training stage of the MLP A window of fixed length (300bp unless otherwise stated) was placed repeatedly over both genomes at random independent positions and every time eight sensors were evaluated over the subsequence in the window and were used feed and train a MLP with binary output; $'0'$ corresponding to acceptor DNA sequences and $'1'$ to donor ones. For the prediction stage the window was allowed to slide along the acceptor sequence and a plot of its position against the outcome was obtained.
Definition of the sensors
-------------------------
We worked with a total of eight sensors divided in three groups. The first one includes traditional measures of DNA variance, the second reflects the DNA local correlations structure and the third one is a measure of the DNA spatial structure according to the dinucleotide distribution:
1. This group comprises CG and CpG contents. There is a number of publicactions reporting the bias of these measures among different organisms. CpG is well known to tell appart prokaryal and eukaryal lineages [@Shimizu]. Even if the underlying reasons are still unclear [@Wang], it gives a good first clue.
2. In 1995, we proposed an index of DNA heterogeneity that disclosed different styles of genomic structural organization [@Miramontes]. Given a DNA sequence, it is translated into the three possible binary sequences using the groupings purine-pyrimidine ($YR$), weak-strong ($WS$) and amino-keto ($MK$).
the following index is calculated over each binary derivative
$$d=\frac{N_{00}N_{11}-N_{10}N_{01}}{N_0N_1}$$
Where $N_{ij}$ stands for the number of $i$ bases followed by the $j$ base, where $i$ and $j$ are zero or one. The phenomenology behind this index is discussed in [@Miramontes].
3. In 1992 the group of R.E. Dickerson [@Quintana01] reported the variability in the DNA structural angles depending on the dinucleotide steps. Their results can be summarized in the Table \[Tab:01\] (page 345 of the cited reference)
--- --- --- --- ---
A C G T
A L I L I
C V L V L
G H H L I
T V H V L
--- --- --- --- ---
[Matrix of angular variability according to one dimer step along the DNA sequence]{}
H stands for high twist profile steps for two stacked consecutive base pairs along the double helix. They are characterized for having high twist, positive cup and negative roll angels. The parameters L, I, and V are the Low, Intermediate and Variable twist.
Results
=======
One of the open problems in designing a MLP for pattern detection is to set the number of neurons in the hidden layer (the number of neurons in the output layer is determined by the number of classes to classify). In order to find out the best suited to our interests, we ran several configurations and tested the resulting output with an artificial problem: To detect a fragment of *E. coli* (donor) inserted *in silico* in a mouse (*Mus musculus*) chromosome (acceptor). To this end, a set of 20000 fragments of length 300 of both genomes was picked up randomly to train the MLP. The network configuration 8-5-1 yielded the results shown in Figure \[fig:03\]. With a 300bp sliding window and an overlap of 30bp the response of the MLP is steadily $'0'$ while the sliding window travel across the acceptor chromosome and then jumps to $'1'$ when it enters the donor insertion and goes back to $'0'$ for the rest of the acceptor sequence.
![*Mus musculus* chromosome 1 (acceptor) in the horizontal axis with a sequence of *E. coli* (donor) inserted in the middle. The ordinate can only be $'0'$ or $'1'$ depending whether the output of the MLP classify the sequence as an acceptor or as a donor. The horizontal scale is arbitrary.[]{data-label="fig:03"}](figure3.eps)
Once the MLP was well tuned, we carried out three case studies:
1. To detect a prokaryal insertion in a prokaryal genome. In this case we selected, for no particular reasons, the genome of *Archeoglobus fulgidus* as the acceptor genome and *Pseudomonas aeruginosa* as the donor species in a second [*in silico* ]{} experiment. Figure \[fig:04\] shows that the results are good enough to encourage one step further.
![*Archeoglobus fulgidus* (acceptor) in the horizontal axis with a sequence of *Pseudomonas aeruginosa* (donor) inserted in the middle. The ordinate can only be $'0'$ or $'1'$ depending whether the output of the MLP classify the sequence as an acceptor or as a donor.[]{data-label="fig:04"}](figure4.eps)
2. Next attempt is the detection of a real horizontal gene transfer already reported in the literature [@Kroll]. It is a gene (Cu-Zn-superoxide dismutase) of *Haemophylus ducreyi* inserted in the *Neisseria mengiditis* genome. Figure \[fig:05\] shows a clear detection.
![*Neisseria mengiditis* (acceptor) in the horizontal axis with a sequence of *Haemophylus ducreyi* (donor) inserted in the middle. The ordinate can only be $'0'$ or $'1'$ depending whether the output of the MLP classify the sequence as an acceptor or as a donor. The horizontal thick line emphasizes the region of insertion.[]{data-label="fig:05"}](figure5.eps)
3. Last example is a case of organelle to nuclear genome transfer. Figure \[fig:06\] shows chromosome 2 of [*Arabidopsis thaliana*]{} as the acceptor sequence for its own mitochondrion. As the horizontal scale unit is 30bp, the inserted fragment goes from 107,653 to 116,985 which correspond to the nucleotides from approximately 3,230,000 to 3,510,000 there is then is a clear inserted fragment of the order of 270kb which coincides with the unexpected case of an organelle to nuclear transfer event [@Xiaoying]. This case is important for our claims because approximately more than sixty percent of the *Arabidopsis thaliana*’s mitochondrial genome is not translated into aminoacids [@Unseld]. Notwithstanding, our method clearly detects the transfer.
![[*Arabidopsis thaliana*]{} (acceptor) in the horizontal axis with a sequence of its own mitochondrial genome (donor) inserted in the middle. The horizontal scale unit is 30bp[]{data-label="fig:06"}](figure6.eps)
Discussion
==========
In this paper we show the feasibility of using variables not related to the DNA function as measurables that can be used to detect horizontal exchange of genetic material between different species. The results are very good and encourage the further development of this line of research. It is a matter of future work to build on a complete set of variables with the minimum size but having the maximum power of resolution.
This paper also contributes to clarify the biology behind the phenomenon of horizontal gene transfer. For instance, the degree of amelioration of horizontally transferred genes is somehow related to the accuracy to which different methods detects xenologous genes. i,e., the sequencing of the chromosome 2 from *Arabidopsis thaliana*, has revealed a large and unexpected organellar-to-nuclear genetic material transfer event of the mitochondrial genome (20). The sequence in the nucleus is 99% identical to the mitochondrial genome, suggesting that the transfer event is very recent. Therefore, amelioration has been negligible, and the MLP clearly detects the transferred DNA. On the other hand, as shown in Figure \[fig:05\], the method presented here, effectively identifies DNA that has been transferred from *Haemophilus sp*. (a Gamma-proteobacteria) to *Neisseria meningitidis*, (a member of the Beta-proteobacteria subdivision) [@Kroll]. However, the high degree of spreading of the points in Figure \[fig:05\] suggest that accumulated mutations since the horizontal gene transfer event, have likely ameliorated sodC in *N. meningitidis*. There are 136 nucleotide differences between *H. ducreyi* sodC and the homologous gene from *N. meningitidis*. If we assume equal rates of substitutions among the two sequences, then an approximate of 68 new mutations have accumulated since the horizontal transfer event in each sequences (out of 561 nucleotides in *N. meningitidis* sodC). This is a substantial amount of change if we compare to the number of differences among 16S rRNA in the two species (83 differences accumulated in each gene, out of 1544 nucleotides in *N. meningitidis* 16S rRNA) and if we take into account that the transfer event must have happened after the divergence of the two species. The extent to which the NN can go in detection when amelioration occurs will be reported elsewhere.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research was partially supported by PAPIIT-UNAM grant IN115908. The investigation is part of Citlali’s B.Sc. dissertation in Computer Science at UNAM.
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[^1]: Corresponding author. e-mail:` [email protected]`
|
---
abstract: 'In this paper, we use the $2$-decent method to find a series of odd non-congruent numbers $\equiv1\pmod 8$ whose prime factors are $\equiv1\pmod4$ such that the congruent elliptic curves have second lowest Selmer groups, which includes Li and Tian’s result [@LT] as special cases.'
address: 'Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China'
author:
- Yi Ouyang and Shenxing Zhang
title: |
On non-congruent numbers\
with $1$ modulo $4$ prime factors
---
[^1]
Introduction
============
The congruent number problem is about when a positive integer can be the area of a rational right triangle. A positive integer $n$ is a non-congruent number is equivalent to that the congruent elliptic curve $$E:=E^{(n)}: y^2=x^3-n^2x$$ has Mordell-Weil rank zero. In [@Fe1] and [@Fe2], Feng obtained several series of non-congruent numbers for $E^{(n)}$ with the lowest Selmer groups. In [@LT], Li and Tian obtained a series of non-congruent numbers whose prime factors are $\equiv1
\pmod 8$ such that $E^{(n)}$ has second lowest Selmer groups. The essential tool of the above results is the $2$-descend method of elliptic curves. In this paper, we will use this method to get a series of odd non-congruent numbers whose prime factors are $\equiv1
\pmod 4$ such that $E^{(n)}$ has second lowest Selmer groups, which includes Li and Tian’s result as special cases.
Suppose $n$ is a square-free integer such that $n=p_1\cdots
p_k\equiv1\pmod 8$ and primes $p_i\equiv 1\pmod 4$, then by quadratic reciprocity law $\left(\frac{p_i}{p_j}\right)=\left(\frac{p_j}{p_i}\right)$.
Suppose $n=p_1\cdots p_k\equiv1\pmod 8$ and $p_i\equiv 1\pmod 4$. The graph $G(n):=(V,A)$ associated to $n$ is a simple undirected graph with vertex set $V:=\{\textrm{prime}\ p\mid n\}$ and edge set $A:=\{\overline{p q}: \left(\frac{p}{q}\right)=-1\}$.
Recall for a simple undirected graph $G=(V,A)$, a partition $V=V_0\cup V_1$ is called *even* if for any $v\in V_i$ ($ i=0,1$), $\#\{v{\rightarrow}V_{1-i}\}$ is even. $G$ is called an *odd graph* if the only even partition is the trivial partition $V=\emptyset\cup V$. Then our main result is:
\[maintheo\] Suppose $n=p_1\cdots p_k\equiv1\pmod 8$ and $p_i\equiv 1\pmod 4$. If the graph $G(n)$ is odd and $\delta(n)$ (as given by ) is $1$, then for the congruent elliptic curve $E=E^{(n)}$, $$\operatorname{\mathrm rank}_{\mathbb Z}(E({\mathbb Q}))=0\ \text{and}\ {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2^\infty]\cong ({\mathbb Z}/2{\mathbb Z})^2.$$ As a consequence, $n$ is a non-congruent number.
The following Corollary is Li and Tian’s result, cf. [@LT]:
\[coro:lt\] Suppose $n=p_1\cdots p_k$ and $p_i\equiv 1\pmod 8$. If the graph $G(n)$ is odd and the Jacobi symbol $\left(\frac{1+\sqrt{-1}}{n}\right)=-1$, then for $E=E^{(n)}$, $$\operatorname{\mathrm rank}_{\mathbb Z}(E({\mathbb Q}))=0\ \text{and}\ {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2^\infty]\cong ({\mathbb Z}/2{\mathbb Z})^2.$$ As a consequence, $n$ is a non-congruent number.
Acknowledgement. {#acknowledgement. .unnumbered}
----------------
This paper was prepared when the authors were visiting the Academy of Mathematics and Systems Science and the Morningside Center of Mathematics of Chinese Academy of Sciences, and was grew out of a project proposed by Professor Ye Tian to the second author. We would like to thank Professor Ye Tian for his vision, insistence and generous hospitality. We also would like to thank Jie Shu and Jinbang Yang for many helpful discussions.
Review of $2$-descent method. {#sec:descent}
=============================
In this section, we recall the $2$-descent method of computing the Selmer groups of elliptic curves. This section follows [@LT] pp 232-233, also cf. [@BSD] §5 and [@Si1] X.4.
For an isogeny $\varphi: E{\rightarrow}E'$ of elliptic curves defined over a number field $K$, one has the following fundamental exact sequence $$\label{eq:fun} 0{\rightarrow}E'(K)/\varphi E(K){\rightarrow}S^{(\varphi)}(E/K){\rightarrow}{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/K)[\varphi]{\rightarrow}0.$$ Moreover, if $\psi: E'{\rightarrow}E$ is another isogeny, for the composition $\psi\circ\varphi: E{\rightarrow}E$, then the following diagram of exact sequences commutes (cf. [@XZ] p 5): $$\label{eqn1}\xymatrix{
&0\ar@{.>}^{\iota_1}[d] &0\ar@{.>}^{\iota_2}[d] &0 \ar[d]& \\
0\ar[r]&E'(K)/\varphi E(K) \ar[r]\ar[d]^{\psi} & S^{(\varphi)}(E/K) \ar[r]\ar[d]&{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/K)[\varphi] \ar[r]\ar[d]&0\\
0\ar[r]&E(K)/\psi\varphi E(K)\ar[r]\ar[d] & S^{(\psi\varphi)}(E/K)\ar[r]\ar[d]&{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/K)[\psi\varphi]\ar[r]\ar[d]&0\\
0\ar[r]&E(K)/\psi E'(K) \ar[r]\ar[d] & S^{(\psi)}(E'/K) \ar[r] &{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/K)[\psi] \ar[r] &0\\
&0 & & &
}$$ Now suppose $n$ is a fixed odd positive square-free integer, $K={\mathbb Q}$, and $E/{\mathbb Q}$, $E'/{\mathbb Q}$, $\varphi$, $\psi=\varphi^{\vee}$ are given by $$E=E^{(n)}: y^2=x^3-n^2x,\quad E'=\widehat{E^{(n)}}: y^2=x^3+4n^2x,$$ $$\varphi: E{\rightarrow}E',\ (x,y)\mapsto(\frac{y^2}{x^2},\frac{y(x^2+n^2)}{x^2}),$$ $$\psi: E'{\rightarrow}E,\ (x,y)\mapsto(\frac{y^2}{4x^2},\frac{y(x^2-4n^2)}{8x^2}).$$ Then $\varphi\psi=[2], \psi\varphi=[2]$. In this case $\iota_1$ and $\iota_2$ are exact. Let $\tilde S^{(\psi)}(E'/{\mathbb Q})$ denote the image of $S^{(\psi\varphi)}(E/{\mathbb Q})$ in $S^{(\psi)}(E'/{\mathbb Q})$. Then $$\# {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[\varphi]=\frac{\#S^{(\varphi)}(E/{\mathbb Q})}{\#E'({\mathbb Q})/\varphi E({\mathbb Q})},\quad
\# {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[\psi]=\frac{\#S^{(\psi)}(E'/{\mathbb Q})}{\#E({\mathbb Q})/\psi E'({\mathbb Q})},$$ and $$\# {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2]=\frac{\#S^{(\varphi)}(E/{\mathbb Q})\cdot \#\tilde S^{(\psi)}(E'/{\mathbb Q})}{\#E'({\mathbb Q})/\varphi E({\mathbb Q})\cdot \#E({\mathbb Q})/\psi E'({\mathbb Q})}.$$ Similarly, $$\label{eq:4} \# {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[2]=\frac{\#S^{(\psi)}(E'/{\mathbb Q})\cdot \#\tilde S^{(\varphi)}(E/{\mathbb Q})}{\#E({\mathbb Q})/\psi E'({\mathbb Q})\cdot \#E'({\mathbb Q})/\varphi E({\mathbb Q})}.$$
The $2$-descent method to compute the Selmer groups $S^{(\varphi)}(E/{\mathbb Q})$ and $S^{(\psi)}(E'/{\mathbb Q})$ is as follows (cf. [@Si1] for general elliptic curves). Let $$S=\{\textrm{prime factors of}\ 2n\}\cup\{\infty\},$$ $${\mathbb Q}(S,2)=\{b\in{\mathbb Q}^\times/{{\mathbb Q}^{\times 2}}: 2\mid \operatorname{\mathrm ord}_p(b),\forall p\not\in S\}.$$ Note that ${\mathbb Q}(S,2)$ is represented by factors of $2n$ and we identify these two sets. By the exact sequence $$0{\rightarrow}E'({\mathbb Q})/\varphi E({\mathbb Q}){\stackrel{i}{\to}} {\mathbb Q}(S,2){\stackrel{j}{\to}} WC(E/{\mathbb Q})[\varphi],$$ where $$\begin{split} & i:\ (x,y)\mapsto x, \ O\mapsto 1,\ (0,0)\mapsto 4n^2,\qquad
j: d\mapsto\{C_d/{\mathbb Q}\}\end{split}$$ and $C_d/{\mathbb Q}$ is the homogeneous space for $E/{\mathbb Q}$ defined by the equation $$\label{eq:Cd} C_d: dw^2=d^2+4n^2z^4,$$ the $\varphi$-Selmer group $S^{(\varphi)}(E/{\mathbb Q})$ is then $$S^{(\varphi)}(E/{\mathbb Q})\cong\{d\in {\mathbb Q}(S,2) : C_d({\mathbb Q}_p)\neq\emptyset,\ \forall p\in S\}.$$ Similarly, suppose $$\label{eq:Cd1} C'_d: dw^2=d^2- n^2z^4.$$ The $\psi$-Selmer group $S^{(\psi)}(E'/{\mathbb Q})$ is then $$S^{(\psi)}(E'/{\mathbb Q})\cong\{d\in {\mathbb Q}(S,2) : C'_d({\mathbb Q}_p)\neq\emptyset,\ \forall p\in S\}.$$
The method to compute $\tilde S^{(\varphi)}(E/{\mathbb Q})$ follows from [@BSD] §5, Lemma 10:
\[lemma:descent\] Let $d\in S^{(\varphi)}(E/{\mathbb Q})$. Suppose $(\sigma, \tau,\mu)$ is a nonzero integer solution of $d\sigma^2=d^2
\tau^2+4n^2\mu^2$. Let ${\mathcal{M}}_b$ be the curve corresponding to $b\in{\mathbb Q}^\times/{{\mathbb Q}^\times}^2$ given by $$\label{eq:mb} {\mathcal{M}}_b:\ dw^2=d^2t^4+4n^2z^4,\ \ d\sigma w-d^2\tau t^2-4n^2\mu z^2=bu^2.$$ Then $d\in\tilde S^{(\varphi)}(E/{\mathbb Q})$ if and only if there exists $b\in{\mathbb Q}(S,2)$ such that ${\mathcal{M}}_b$ is locally solvable everywhere.
Note that the existence of $\sigma, \tau,\mu$ follows from Hasse-Minkowski theorem (cf. [@Se]).
Local computation
=================
We need a modification of the Legendre symbol. For $x\in {\mathbb Q}_p$ or $\in {\mathbb Q}$ such that $\operatorname{\mathrm ord}_p(x)$ is even, we set $$\label{eq:legendre} \left (\frac{x}{p}\right):=\left (\frac{xp^{-\operatorname{\mathrm ord}_p(x)}}{p}\right).$$ Thus $(\frac{\ }{p})$ defines a homomorphism from $\{x\in
{\mathbb Q}^\times/{\mathbb Q}^{\times 2} : \operatorname{\mathrm ord}_p(x)\ \textrm{is even}\}$ to $\{\pm
1\}$.
Computation of Selmer groups
----------------------------
In this subsection, we will find the conditions when $C_d$ or $C_d'$ is locally solvable. We will not give details since one only need to consider the valuations and quadratic residue.
\[lem:oddphi\] $d\in S^{(\varphi)}(E/{\mathbb Q})$ if and only if $d$ satisfies
1. $d>0$ has no prime factor $p \equiv3\pmod 4$;
2. $\left(\frac{n/d}{p}\right)=1$ for all odd $p\mid d$;
3. $\left(\frac{d}{p}\right)=1$ for all odd $p\mid (2n/d)$;
4. if $2\mid d$, $n\equiv\pm1\pmod8$.
In this case $C_d: dw^2=d^2t^4+4n^2z^4$. It is obvious that $C_d({\mathbb R})\neq\emptyset\Leftrightarrow d>0$. Assume $d>0$.
\(i) If $2\nmid d\mid n$, then $C_d: w^2=d(t^4+4(n/d)^2z^4)$.
- $p=2$. $C_d({\mathbb Q}_2)\neq\emptyset{\Longleftrightarrow}d\equiv1\pmod 4$.
- $p\mid d$. $C_d({\mathbb Q}_p)\neq\emptyset{\Longleftrightarrow}\left(\frac{n/d}{p}\right)=1$ and $p\equiv1\pmod 4$.
- $p\nmid d$. $C_d({\mathbb Q}_p)\neq\emptyset{\Longleftrightarrow}\left(\frac{d}{p}\right)=1$.
\(ii) If $2\mid d\mid 2n$, then $C_d: w^2=d(t^4+(2n/d)^2z^4)$.
- $p=2$. $C_d({\mathbb Q}_2)\neq\emptyset{\Longleftrightarrow}d\equiv2\pmod 8,\ n\equiv\pm1\pmod 8$.
- $2\neq p\mid d$. $C_d({\mathbb Q}_p)\neq\emptyset{\Longleftrightarrow}\left(\frac{n/d}{p}\right)=1$ and $p\equiv1\pmod 4$.
- $p\nmid d$. $C_d({\mathbb Q}_p)\neq\emptyset{\Longleftrightarrow}\left(\frac{d}{p}\right)=1$.
Combining (i) and (ii) follows the lemma.
\[lem:oddpsi\] $d\in S^{(\psi)}(E'/{\mathbb Q})$ if and only if $d$ satisfies
1. $d\equiv\pm1\pmod 8$ or $n/d \equiv \pm1 \pmod 8$
2. $\left(\frac{n/d}{p}\right)=1$ for all $p\mid d, p\equiv 1\pmod 4$;
3. $\left(\frac{d}{p}\right)=1$ for all $p\mid (n/d), p\equiv 1\pmod 4$.
In the case $C'_d: dw^2=d^2t^4-n^2z^4$.
\(i) If $2\mid d$, consider the $2$-valuation of each side, we see $C'_d({\mathbb Q}_2)=\emptyset$.
\(ii) If $2\nmid d\mid n$, then $C'_d: w^2=d(t^4-(n/d)^2z^4)$.
- $p=2$. $C'_d({\mathbb Q}_2)\neq\emptyset {\Longleftrightarrow}d\equiv\pm1\pmod 8$ or $n/d\equiv\pm1\pmod 8$.
- $p\mid d$. $C'_d({\mathbb Q}_p)\neq\emptyset{\Longleftrightarrow}\left(\frac{n/d}{p}\right)=1$ or $\left(\frac{-n/d}{p}\right)=1$.
- $p\nmid d$. $C'_d({\mathbb Q}_p)\neq\emptyset{\Longleftrightarrow}\left(\frac{d}{p}\right)=1$ or $\left(\frac{-d}{p}\right)=1$.
Combining (i) and (ii) follows the lemma.
Computation of the images of Selmer groups
------------------------------------------
Suppose $0<2d\in S^{(\varphi)}(E/{\mathbb Q})$, $d$ is odd with no $\equiv
3\pmod4$ prime factor, we want to find a necessary condition for $2d\in \tilde S^{(\varphi)}(E/{\mathbb Q})$. Write $2d=\tau^2+\mu^2$ and select the triple $(\sigma,\tau,\mu)$ in Lemma \[lemma:descent\] to be $(2n, n\tau/d, \mu)$. Then the defining equations of ${\mathcal{M}}_{4ndb}$ in can be written as $$\label{eq:mb1} w^2=2d(t^4+(n/d)^2 z^4),\quad w-\tau t^2-(n/d)\mu z^2 =b u^2.$$ By abuse of notations, we denote the above curve by ${\mathcal{M}}_b$. We use the notation $O(p^m)$ to denote a number with $p$-adic valuation $\geq m$.
0.3cm **The case $p\mid d$.** For $i_p
\equiv\tau/\mu\pmod{p{\mathbb Z}_p}$, $i_p\in{\mathbb Z}_p$ and $i_p ^2=-1$, then $$p\mid (\tau-i_p \mu),\quad p\nmid (\tau+i_p \mu).$$ It’s easy to see $v(t)=v(z)$, we may assume that $z=1,\ t^2\equiv
\pm \frac{i_p n}{d}\pmod p$, then ${\mathcal{M}}_b$ is given by $${\mathcal{M}}_b: w^2=2d(t^4+(n/d)^2),\quad
w-\tau t^2-(n/d)\mu=bu^2.$$
\(i) If $v(bu^2)=m\geq3$, then by $w^2=(\tau t^2+\frac{n\mu}{d}+O(p^m))^2=2d(t^4+\frac{n^2}{d^2})$, $$\Bigl(\mu t^2-\frac{n\tau}{d}\Bigr)^2=O(p^m).$$ Let $t^2=\frac{n\tau}{d\mu}+\beta$, where $v(\beta)=\alpha\geq\frac{m}{2}$, then $$\begin{split}
w^2=&2d\left((\frac{n}{d})^2+(\frac{n\tau}{d\mu})^2+2\frac{n\tau}{d\mu}\beta+\beta^2\right)\\
=&\frac{4n^2}{\mu^2}(1+\frac{\tau\mu}{n}\beta+\frac{d\mu^2}{2n^2}\beta^2),
\end{split}$$ Take the square root on both sides, then $$\begin{split}
w=&\pm\frac{2n}{\mu}\left(1+\frac{1}{2}(\frac{\tau\mu}{n}\beta+\frac{d\mu^2}{2n^2}\beta^2)-\frac{1}{8}(\frac{\tau\mu}{n}\beta)^2+O(p^{3\alpha-3})\right)\\
=&\pm\left(\frac{2n}{\mu}+\tau\beta+n\mu(\frac{\mu\beta}{2n})^2+O(p^{3\alpha-2})\right),\end{split}$$ but on the other hand, $$w =\tau t^2+\frac{n\mu}{d}+bu^2 =\frac{2n}{\mu}+\tau\beta+bu^2.$$ The sign must be positive and $$bu^2=n\mu(\frac{\mu\beta}{2n})^2+O(p^{3\alpha-2}),$$ thus $p\mid b$, $\left(\frac{b/p}{p}\right)=\left(\frac{n\mu/p}{p}\right)$, $\left(\frac{n/b}{p}\right)=\left(\frac{\mu}{p}\right)=\left(\frac{2\tau}{p}\right)$.
\(ii) If $v(bu^2)=m\leq2$ and $t^2\equiv \frac{i_p n}{d} \pmod p$, let $t^2=\frac{i_p n}{d} +p\alpha i_p $, then $$w^2=2d\cdot p\alpha i_p \cdot \Bigl( \frac{2i_p n}{d} +p\alpha i_p \Bigr)
=-4p^2\cdot\frac{n\alpha}{p}\Bigl(1+\frac{pd\alpha}{2n}\Bigr),$$ and $$\begin{split} w_1=& \frac{w}{p}=\pm 2i_p \sqrt{\frac{n\alpha}{p}}\Bigl(1+\frac{pd\alpha}{4n}+O(p^2)\Bigr),\\
bu^2 =&w-\tau t^2-\frac{n\mu}{d}\\
=&\pm 2pi_p \sqrt{\frac{n\alpha}{p}}\Bigl(1+\frac{pd\alpha}{4n}\Bigr)-\frac{i_p \tau n}{d}-\frac{n\mu}{d}-\tau\alpha i_p p+O(p^3)\\
=&-\frac{p^2 i_p \tau}{n}\Bigl(\sqrt{\frac{n\alpha}{p}}\mp\frac{n}{p\tau}\Bigr)^2-
\frac{ni_p }{2d\tau}(\tau-i_p \mu)^2 \pm 2p^2i_p \sqrt{\frac{n\alpha}{p}}\frac{d\alpha}{4n}+O(p^3).
\end{split}$$ If $v(bu^2)=2$, then $\sqrt{\frac{n\alpha}{p}}\equiv \pm
\frac{n}{p\tau}\pmod p$, and $$\begin{split}
bu^2=&-\frac{ni_p }{2d\tau}(\tau-i_p \mu)^2\pm2p^2i_p \sqrt{\frac{n\alpha}{p}}\frac{d\alpha}{4n}+O(p^3)\\
=&\frac{-ni_p (\tau-i_p \mu)^3(3\tau+i_p \mu)}{8d\tau^3}+O(p^3)\\
=&\frac{-ni_p (\tau-i_p \mu)^3}{2d\tau^2}+O(p^3)=O(p^3),
\end{split}$$ which is impossible! Thus $v(bu^2)=1$ and $p\mid b$, $$\left(\frac{b/p}{p}\right)=\left(\frac{-pi_p \tau/n}{p}\right)=\left(\frac{2p\tau/n}{p}\right),
\ \text{or}\ \left(\frac{n/b}{p}\right)=\left(\frac{2\tau}{p}\right).$$
\(iii) If $v(bu^2)=m\leq2$ and $t^2\equiv -i_p (n/d)\pmod p$, then $$\begin{split}
bu^2=&w-\tau t^2-(n/d)\mu
=(\tau i_p -\mu)n/d+O(p)\\
=&2i_p \tau n/d+O(p)
=(1+i_p )^2\cdot\frac{n}{d}\cdot \tau+O(p),
\end{split}$$ thus $p\nmid b$ and $\left(\frac{b}{p}\right)=\left(\frac{\tau}{p}\right)\left(\frac{n/d}{p}\right).$
Note that $2\tau\equiv\tau+\mu i_p\pmod p$ and $\left(\frac{2n/d}{p}\right)=1$, hence we have
\[lem:image1\] The curve ${\mathcal{M}}_b$ defined by is locally solvable at $p\mid d$ if and only if $$\textrm{either}\ \ p\mid b,\ \left(\frac{n/b}{p}\right)=\left(\frac{\tau+\mu i_p}{p}\right); \quad
\textrm{or}\ \ p\nmid b, \ \left(\frac{b}{p}\right)=\left(\frac{\tau+\mu i_p}{p}\right).$$
0.3cm **The case $p\mid \frac{n}{d}$.** In this case $t$ is a $p$-adic unit if and only if $w$ is so.
\(i) If $v(w)=v(t)=0$, then $w\equiv \pm\sqrt{2d}t^2\pmod p$ and $(\pm\sqrt{2d}-\tau)t^2\equiv bu^2\pmod p$. Since $(\sqrt{2d}-\tau)(\sqrt{2d}+\tau)=2d-\tau^2=\mu^2$ and $\sqrt{2d}\pm
\tau$ are co-prime, $\operatorname{\mathrm ord}_p(\sqrt{2d}-\tau)$ is even and $\left(\frac{\sqrt{2d}-\tau}{p}\right)$ is well defined. Then ${\mathcal{M}}_b$ is locally solvable if and only if $$p\nmid b, \left(\frac{2d}{p}\right)=1\ \textrm{and}\ \left(\frac{b}{p}\right)=\left(\frac{\sqrt{2d}-\tau}{p}\right).$$
\(ii) If $v(z)=0$ and $w=pw_1, t=pt_1$, then $w_1^2=2d(p^2t_1^2+(\frac{n}{pb})^2z^4)$, $w_1\equiv
\pm\sqrt{2d}\frac{n}{pd}z^2\pmod p$ and $bu^2/p\equiv
(\pm\sqrt{2d}-\mu)\frac{n}{pd}z^2\pmod p$. Thus ${\mathcal{M}}_b$ is locally solvable if and only if $$p\mid b, \left(\frac{2d}{p}\right)=1\ \textrm{and}
\ \left(\frac{n/(db)}{p}\right)=\left(\frac{\sqrt{2d}-\mu}{p}\right).$$
Note that $$2(\sqrt{2d}-\tau)(\sqrt{2d}-\mu)=(\tau+\mu-\sqrt{2d})^2\Rightarrow
\left(\frac{\sqrt{2d}-\mu}{p}\right)=\left(\frac{2(\sqrt{2d}-\tau)}{p}\right).$$
From now on, suppose $n=p_1\cdots p_k\equiv1\pmod 8$ and $p_i\equiv
1\pmod 4$. Pick $i_p\in {\mathbb Z}_p$ such that $i_p^2=-1$, then $$\sqrt{2d}-\tau
=-(\tau+\mu i_p)\cdot\frac{1}{2}\Bigl(1-\frac{\sqrt{2d}}{\tau+\mu i_p}\Bigr)^2.$$ Note that $\left(\frac{2d}{p}\right)=1$, we have
\[lem:image2\] ${\mathcal{M}}_b$ defined by is locally solvable at $p\mid \frac{n}{d}$ if and only if $$\begin{split}
p\mid b,&\quad \left(\frac{2d}{p}\right)=1\ \textrm{and}\ \left(\frac{n/b}{p}\right)=\left(\frac{\tau+\mu i_p}{p}\right)\left(\frac{2}{p}\right),\\
\textrm{or}\ p\nmid b,&\quad \left(\frac{2d}{p}\right)=1\ \textrm{and}\ \left(\frac{b}{p}\right)=\left(\frac{\tau+\mu i_p}{p}\right)\left(\frac{2}{p}\right).\\
\end{split}$$
By Lemmas \[lemma:descent\], \[lem:oddphi\], \[lem:image1\] and \[lem:image2\], and we have
\[prop:dn\] Suppose $n=p_1\cdots p_k\equiv1\pmod 8$ and $p_i\equiv
1\pmod 4$, then $2d\in S^{(\varphi)}(E/{\mathbb Q})$ if and only if $d>0$ and $\left(\frac{2n/d}{p}\right)=1$ for $p\mid d$, $\left(\frac{2d}{p}\right)=1$ for $p\mid \frac{n}{d}$. In this case $2d\in
\tilde{S}^{(\varphi)}(E/{\mathbb Q})$ only if there exists $b\in {\mathbb Q}(S,2)$ satisfying:
\(1) If $p\mid d$, $i_p\equiv \tau/\mu\pmod{p{\mathbb Z}_p}$, $i_p^2=-1$, $$p\mid b,\ \left(\frac{n/b}{p}\right)=\left(\frac{\tau+\mu i_p}{p}\right),\quad
\textrm{or}\quad p\nmid b,\quad \left(\frac{b}{p}\right)=\left(\frac{\tau+\mu i_p}{p}\right).$$
\(2) If $p\mid \frac{n}{d}$, $i_p^2=-1$, $$p\mid b,\ \left(\frac{n/b}{p}\right)=\left(\frac{2(\tau+\mu i_p)}{p}\right),\quad
\textrm{or}\quad p\nmid b,\quad \left(\frac{b}{p}\right)=\left(\frac{2(\tau+\mu i_p)}{p}\right).$$
Proof of the main result
========================
Some facts about graph theory.
------------------------------
We now recall some notations and results in graph theory, cf. [@Fe1; @Fe2].
\[matofgraph\] Let $G=(V,A)$ be a simple undirected graph. Suppose $\# V=k$. The *adjacency matrix* $M(G)=(a_{ij})$ of $G$ is the $k\times k$ matrix defined as $$a_{ij}:=\begin{cases}
0, &\ \textrm{if}\ \overline{v_i v_j}\not\in A;\\
1, &\ \textrm{if}\ \overline{v_i v_j}\in A.
\end{cases}$$ The *Laplace matrix* $L(G)$ of $G$ is defined as $$L(G)=\operatorname{\mathrm{diag}}\{d_1,\ldots,d_k\}-M(G)$$ where $d_i$ is the degree of $v_i$.
\[graphthm\] Let $G$ be a simple undirected graph and $L(G)$ its Laplace matrix.\
(1) The number of even partitions of $V$ is $2^{k-1-r}$, where $r=\operatorname{\mathrm rank}_{{\mathbb F}_2} L(G)$.\
(2) The graph $G$ is odd if and only if $r=k-1$.\
(3) If $G$ is odd, then the equations $$L(G)\left(\begin{smallmatrix}c_1\\ \vdots \\c_k\end{smallmatrix}\right)
=\left(\begin{smallmatrix}t_1\\ \vdots \\t_k\end{smallmatrix}\right)$$ has solutions if and only if $t_1+\cdots+t_k=0$.
The proof of the first two parts follows from [@Fe1]. We have a bijection $$\begin{split} {\mathbb F}_2^k/\{(0,\cdots,0),(1,\cdots,1)\} &\stackrel{\sim}{\longrightarrow} \{\textrm{partitions of $V$}\}\\
(c_1,\ldots,c_k)&\longmapsto (V_0, V_1)\end{split}$$ where $V_i=\{v_j: c_j=i\ (1\leq j\leq k)\},\ i\in\{0,1\}$.
Regard $L(G)=\operatorname{\mathrm{diag}}\{d_1,\ldots,d_k\}-(a_{ij})$ as a matrix over ${\mathbb F}_2$. If $$L(G)\left(\begin{smallmatrix}c_1\\ \vdots \\c_k\end{smallmatrix}\right)=\left(\begin{smallmatrix}b_1\\ \vdots \\b_k\end{smallmatrix}\right)\in{\mathbb F}_2^k,$$ then if $v_i\in V_t, t\in\{0,1\}$, $$\begin{split}
b_i&=d_ic_i+\sum_{j=1}^k a_{ij}c_j=\sum_{j=1}^k a_{ij}(c_i+c_j)\\
&=\sum\limits_{j=1}^k a_{ij}(t+c_j)=\sum\limits_{c_j=1-t}a_{ij}=\#\{v_i{\rightarrow}V_{1-t}\}\in{\mathbb F}_2.
\end{split}$$
\(1) The number of even partitions is $$\frac{1}{2}\#\left\{(c_1,\ldots,c_k)\in{\mathbb F}_2^n : L(G)\left(\begin{smallmatrix}c_1\\ \vdots \\c_k\end{smallmatrix}\right)
=\left(\begin{smallmatrix}0\\ \vdots \\0\end{smallmatrix}\right)\right\}=2^{k-1-r}.$$
\(2) follows from (1) easily.
\(3) Since $L$ is of rank $k-1$, the image space of $L$ is of dimensional $k-1$, but it lies in the hyperplane $x_1+\cdots+x_k=0$, thus they coincide and the result follows.
Graph $G(n)$ and Selmer groups of $E$ and $E'$.
-----------------------------------------------
From now on, we suppose
> $n=p_1\cdots p_k\equiv1\pmod 8$ and $p_i\equiv 1\pmod 4$.
Recall for an integer $a$ prime to $n$, the Jacobi symbol $\left(\frac{a}{n}\right)=\prod_{p\mid n} \left(\frac{a}{p}\right)$, which is extended to a multiplicative homomorphism from $\{a\in
{\mathbb Q}^{\times}/{\mathbb Q}^{\times 2} : \operatorname{\mathrm ord}_p(a)\ \text{even for } p\mid n\}$ to $\{\pm 1\}$. Set $$\left[\frac{a}{n}\right]:=\frac{1}{2}\Bigl(1-\left(\frac{a}{n}\right)\Bigr).$$ The symbol $[\frac{}{n}]$ is an additive homomorphism from $\{a\in {\mathbb Q}^{\times}/{\mathbb Q}^{\times 2} : \operatorname{\mathrm ord}_p(a)\ \text{even} \ \text{for} \
p\mid n\}$ to ${\mathbb F}_2$.
By definition, the adjacency matrix $M(G(n))$ has entries $a_{ij}= \left[\frac{p_i}{p_j}\right]$. For $0<d\mid n$, we denote by $\{d,\frac{n}{d}\}$ the partition $\{p:p\mid d\}\cup\{p:p\mid \frac{n}{d}\}$ of $G(n)$.
The following proposition is a translation of results in Lemma \[lem:oddphi\] and Lemma \[lem:oddpsi\]:
\[lem:6.3\] Given a factor $d$ of $n$.
\(1) For the Selmer group $S^{(\varphi)}(E/{\mathbb Q})$,
- $d\in S^{(\varphi)}(E/{\mathbb Q})$ if and only if $d>0$ and $\{d, n/d\}$ is an even partition of $G(n)$;
- Suppose $$c_i=\begin{cases} 1,\ &\text{if}\ p_i\mid d,\\ 0,\ &\text{if}\ p_i\mid \frac{n}{d}; \end{cases} \qquad
t_i=\left[\frac{2}{p_i}\right].$$ Then $2d\in S^{(\varphi)}(E/{\mathbb Q})$ if and only if $d>0$ and $$L(G)\left(\begin{smallmatrix}c_1\\ \vdots \\c_k\end{smallmatrix}\right)
=\left(\begin{smallmatrix}t_1\\ \vdots \\t_k\end{smallmatrix}\right).$$
\(2) For the Selmer group $S^{(\psi)}(E'/{\mathbb Q})$,
- $d\in S^{(\psi)}(E'/{\mathbb Q})$ if and only if $d\equiv\pm1\pmod 8$ and $\{d,n/d\}$ is an even partition of $G(n)$;
- $2d\notin S^{(\psi)}(E'/{\mathbb Q})$.
One only has to show (1-b), the rest is easy. For any $i$, let $[i]$ be the set of $j$ such that $p_i$ and $p_j$ are both prime divisors of $d$ or $n/d$. Then $$d_i c_i+\sum_{j\neq i} a_{ij} c_j=\sum_{j\neq i} a_{ij}(c_i+c_j)=\sum_{j\notin [i]} a_{ij}=\left[\frac{d}{p_i}\right]\ \text{or}\ \left[\frac{n/d}{p_i}\right].$$ Then (1-b) follows from Lemma \[lem:oddphi\].
Applying Theorem \[graphthm\](3) to Proposition \[lem:6.3\], then we have
\[cor:d\] If $G(n)$ is odd, there exists a unique factor $0<d<\sqrt{2n}$ of $n$ such that $$S^{(\varphi)}(E/{\mathbb Q})=\{1,2d,2n/d,n\}\cong {\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2{\mathbb Z},$$ and $$S^{(\psi)}(E'/{\mathbb Q})=\{\pm1, \pm n\}\cong {\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2{\mathbb Z}.$$
For the $d$ given in Corollary \[cor:d\], write $2d=\tau^2+\mu^2$. If $2d\in \tilde{S}^{(\varphi)}(E/{\mathbb Q})$, we suppose $b$ satisfies the condition that ${\mathcal{M}}_b$ defined by is locally solvable everywhere. Suppose $c'=(c'_1,\cdots, c'_k)^T$ and $t'=(t'_1,\cdots t'_k)^T$ are given by $$c'_j=\begin{cases} 1,\ &\text{if}\ p_j\mid b,\\ 0,\ &\text{if}\ p_j\nmid b; \end{cases} \qquad t'_j=\begin{cases} \Big[\dfrac{\tau+\mu i_{p_j}}{p_j}\Big],\ &\text{if}\ p_j\mid d,\\ \Big[\dfrac{2(\tau+\mu i_{p_j})}{p_j}\Big],\ &\text{if}\ p_j\mid \frac{n}{d}. \end{cases}$$ By Proposition \[prop:dn\], $L c'=t'$, i.e., $Lv=t'$ has a solution $v=c'$, which means that the summation of $t'_j$ must be zero in ${\mathbb F}_2$ by Theorem \[graphthm\](3).
\[defn:dn\] Suppose $n$ is given such that $G(n)$ is an odd graph. For the unique factor $d$ given in Corollary \[cor:d\], write $2d=\tau^2+\mu^2$ and $\frac{2n}{d}=\tau'^2+\mu'^2$, Let $i\in{\mathbb Z}/n{\mathbb Z}$ be defined by $$i\equiv\frac{\tau}{\mu}\pmod d,\quad i\equiv\frac{\tau'}{\mu'}\pmod{\frac{n}{d}}.$$ We define $$\label{eq:dn} \delta(n):=\left[\frac{\tau+\mu i}{n}\right]+ \left[\frac{2}{d}\right]
\in{\mathbb F}_2.$$
Then the following is a consequence of Proposition \[prop:dn\].
\[cor:image\] If $G(n)$ is odd and $\delta(n)=1$, then $$\tilde S^{(\varphi)}(E/{\mathbb Q})=\{1\}.$$
Let $\lambda^*$ be the ${\mathbb F}_2$-rank of $\tilde S^{(\varphi)}(E/{\mathbb Q})$, $\lambda$ be the ${\mathbb F}_2$-rank of $S^{(\varphi)}(E/{\mathbb Q})$, then $\lambda=2$. The existence of the Cassels’ skew-symmetric bilinear form on ${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}$ implies that the difference $\lambda-\lambda^*$ is even.
By the above analysis, $\delta(n)=\sum\limits_j t'_j\neq 0$, thus $2d\notin\tilde S^{(\varphi)}(E/{\mathbb Q})$, we have $\lambda^*<\lambda$, $\lambda^*=0$.
If we replace $d$ by $\frac{n}{d}$ in the definition, $\delta(n)$ is invariant. Indeed, $[\frac{2}{d}]=[\frac{2}{n/d}]$. For the other term, $$\left[\frac{\tau+\mu i}{n}\right]=\left[\frac{\tau+\mu i}{d}\right]+\left[\frac{\tau+\mu i'}{n/d}\right]$$ where $i\equiv \tau/\mu\pmod d$, $i'\equiv\tau'/\mu'\pmod{n/d}$. Let $u=(\tau\tau'-\mu\mu')/2$, $v=(\tau\mu'-\mu\tau')/2$, then $$\begin{split}
u+vi=&(\tau+\mu i)(\tau'+\mu' i)/2 \equiv \tau(\tau'+\mu'\cdot\frac{\tau}{\mu}) \\ \equiv& \tau\mu(\tau'\mu+\tau\mu')/\mu^2 \equiv(\tau+\mu)^2/\mu^2\cdot v/2\pmod d.
\end{split}$$ Similarly, $u+vi'\equiv (\tau'+\mu')^2/\mu'^2\cdot v/2\pmod{(n/d)}$. If we interchange $d$ and $n/d$, $\delta(n)$ will differ $$\begin{split}&\left[\frac{\tau+\mu i}{d}\right]+\left[\frac{\tau+\mu
i'}{n/d}\right]+\left[\frac{\tau'+\mu' i'}{n/d}\right]+\left[\frac{\tau'+\mu' i}{d}\right]\\
=&\left[\frac{2(u+vi)}{d}\right]+\left[\frac{2(u+vi')}{n/d}\right]=\bigg[\frac{v}{d}\bigg]
+\left[\frac{v}{n/d}\right]\\
=&\left[\frac{v}{n}\right]=\left[\frac{n}{v}\right]=0\in{\mathbb F}_2.
\end{split}$$ Thus $\delta(n)$ does not change, which implies that $\delta(n)$ does not depend on the choice of $d,\tau,\mu$ and only depend on $n$.
Proof of the main result.
-------------------------
We shall use the fundamental exact sequence and the commutative diagram in §\[sec:descent\] frequently.
Since $E({\mathbb Q})_{\operatorname{\mathrm tor}}\cap\psi E'({\mathbb Q})=\{O\}$ and $\# E({\mathbb Q})_{\operatorname{\mathrm tor}}=4$, $\# E({\mathbb Q})/\psi E'({\mathbb Q})\geq4$. Since $G(n)$ is odd, $\# S^{(\psi)}(E'/{\mathbb Q})=4$ and $\# E({\mathbb Q})/\psi E'({\mathbb Q})=4$, by , ${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[\psi]=0$. Apparently $\tilde S^{(\psi)}(E'/{\mathbb Q})\supseteq E({\mathbb Q})/\psi E'({\mathbb Q})$ and thus $\#\tilde S^{(\psi)}(E'/{\mathbb Q})=4$.
By Corollary \[cor:image\], $\tilde{S}^{(\varphi)}(E/{\mathbb Q})=\{1\}$, then $\# E'({\mathbb Q})/\varphi E({\mathbb Q})=1$. The facts $\# E({\mathbb Q})/\psi E'({\mathbb Q})=4$ and $E({\mathbb Q})_{\operatorname{\mathrm tor}}\cong ({\mathbb Z}/2{\mathbb Z})^2$ imply that $\#E({\mathbb Q})/2 E({\mathbb Q})=4$ and $$\operatorname{\mathrm rank}_{\mathbb Z}E({\mathbb Q})=\operatorname{\mathrm rank}_{\mathbb Z}E'({\mathbb Q})=0.$$ From ${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[\psi]=E'({\mathbb Q})/\varphi E({\mathbb Q})=0$, the diagram tells us that $${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2]\cong {{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[\varphi]\cong S^{(\varphi)}(E/{\mathbb Q}) \cong{\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z},$$ and tells us that $${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[2]\cong{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[\psi]\cong 0.$$ Hence ${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[2^\infty]=0$ and $
{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[2^k \psi]=0$. By the exact sequence $$0{\rightarrow}{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[\varphi]{\rightarrow}{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2^k]{\rightarrow}{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E'/{\mathbb Q})[2^{k-1} \psi],$$ we have for every $k\in {\mathbb N}_{+}$, $${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2^k]\cong{{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[\varphi]\cong({\mathbb Z}/2{\mathbb Z})^2,$$ and thus ${{\mbox{{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}}(E/{\mathbb Q})[2^\infty]\cong({\mathbb Z}/2{\mathbb Z})^2$.
In this case, $d=1$ and $\tau=\mu=1$, $\delta(n)=\Big[\frac{1+\sqrt{-1}}{n}\Big]$, thus the result follows.
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[^1]: Research partially supported by Project 11171317 from NSFC
|
---
abstract: 'The estimation of poroelastic material parameters based on ultrasound measurements is considered. The acoustical characterisation of poroelastic materials based on various measurements is typically carried out by minimising a cost functional of model residuals, such as the least squares functional. With a limited number of unknown parameters, least squares type approaches can provide both reliable parameter and error estimates. With an increasing number of parameters, both the least squares parameter estimates and, in particular, the error estimates often become unreliable. In this paper, the estimation of the material parameters of an inhomogeneous poroelastic (Biot) plate in the Bayesian framework for inverse problems is considered. Reflection and transmission measurements are performed and 11 poroelastic parameters, as well as 4 measurement setup-related nuisance parameters, are estimated. A Markov chain Monte Carlo algorithm is employed for the computational inference to assess the actual uncertainty of the estimated parameters. The results suggest that the proposed approach for poroelastic material characterisation can reveal the heterogeneities in the object, and yield reliable parameter and uncertainty estimates.'
author:
- 'Matti Niskanen$^{a,b,}$[^1], Aroune Duclos$^{b}$, Olivier Dazel$^{b}$, Jean-Philippe Groby$^{b}$, Jari Kaipio$^{c}$, Timo Lähivaara$^{a}$'
date: |
$^{a}$ Department of Applied Physics, University of Eastern Finland, Kuopio, Finland\
$^{b}$ Laboratoire d’Acoustique de l’Université du Mans, LAUM - UMR CNRS 6613, Le Mans, France\
$^{c}$ Department of Mathematics, University of Auckland, Auckland, New Zealand\
title: Estimating the material parameters of an inhomogeneous poroelastic plate from ultrasonic measurements in water
---
Introduction
============
Methods for characterising poroelastic media are needed in a wide range of applications, such as studying living bone tissue [@cowin1999bone], characterising seabed [@chotiros2017acoustics], geophysical exploration [@slatt2013stratigraphic], design of materials for noise treatment [@allard2009propagation; @cox2016acoustic], and industrial filtration [@espedal2007filtration]. The inverse problem of estimating the physical parameters of such media from acoustic measurements has received a lot of interest [@sebaa2006ultrasonic; @jocker2009ultrasonic; @buchanan2011recovery; @ogam2011non; @chazot2012acoustical; @verdiere2017inverse; @chen2018biot], in part because the acoustic measurements are non-destructive, and the measurement set up is simple compared to many direct measurements of the material properties. A concern that remains in using inverse methods for the characterisation is that currently the reproducibility seems to be poor even when estimating only the properties of rigid frame porous media, as shown in a recent study [@pompoli2017reproducible], or only the elastic properties, see [@bonfiglio2018reproducible]. Furthermore, usually a larger number of unknowns leads to larger uncertainty in the estimates.
Early approaches to the inverse problem were deterministic, i.e. concentrated on finding the best fit parameters by minimising a cost function, or by finding the analytical inverse mapping (if it exists) that connects data to parameters. With linear problems, and problems that can be reasonably linearised, deterministic approaches can produce reliable parameter and uncertainty estimates, and often with a relatively low computational cost. On the other hand, deterministic approaches do not provide a general way of treating errors in nonlinear problems or specifying prior information. Note that all characterisation methods in the two reproducibility studies [@pompoli2017reproducible; @bonfiglio2018reproducible] are deterministic. It has also been proposed [@chazot2012acoustical] to treat the inverse problem in the statistical (Bayesian) framework, where instead of trying to find a single value for the parameters, we are looking for their *posterior probability distribution* (ppd) [@kaipio2006statistical]. The ppd is constructed based on the measured data by assessing the uncertainty in the measurements, and by incorporating possible prior knowledge on the parameters. We can then calculate the most likely parameter values, as well as their credible interval estimates. Despite the interest in characterising poroelastic materials, only a few studies that consider Bayesian inversion exist so far. Bayesian inversion has been, however, widely used in characterisation of media that can be modelled as an equivalent fluid, such as rigid frame porous materials [@niskanen2017deterministic; @roncen2018bayesian], and seabed [@dettmer2007full; @dettmer2010trans; @dettmer2012trans; @dettmer2013transdimensional; @holland2013situ].
Bayesian inversion was used by Chazot *et al.* [@chazot2012acoustical] to study the characterisation of highly porous foam and fibre materials using measurements done in an impedance tube. To see the elasticity effects in the impedance tube, the materials need to be relatively soft, which means that the method is suitable for mostly foams and fibrous materials. Bonomo *et al.* [@bonomo2018comparison] used the Bayesian approach to compare three poroelastic models for sandy sediments and infer their parameters, using either compressional or shear wave speed data acquired from water saturated fine-grained silica sand. It was found that while the models could be used to explain most of the measurements, the parameters related to the elastic behaviour of the solid were mostly unidentifiable from the data. It was concluded that the Biot model is not directly suitable for modelling sandy materials since the unconsolidated granular medium does not form an effective solid frame. Niskanen *et al.* [@niskanen2019characterising] considered Bayesian inversion numerically in the case of poroelastic materials that have a solid porous frame such as limestone, trabecular bone, or porous ceramic. Simulation data for the inversion were acoustic reflection and transmission coefficients measured at low ultrasound frequencies. The study also considered errors related to the measurement setup, and showed that reliable parameter estimates can be found to all parameters in the Biot model, even in the presence of relatively high levels of noise. However, the study mainly considered additive white measurement noise whose characteristics are known, whereas with real measurements the noise term includes modelling errors that are often known approximately at best.
In this paper, we apply the inversion method in [@niskanen2019characterising] to characterise a porous ceramic plate using real measurements made in a water tank. Using focused ultrasound transducers, we measure the normal incidence acoustic reflection and transmission coefficients of the plate at more than 200 locations. These measurements are linked to the physical parameters using the Biot-Johnson model [@johnson1987theory], and a Global Matrix Method solution, which assumes plane wave propagation. Then, we use Bayes’ formula to construct the ppd, and compute the parameter estimates and their uncertainties with a Markov chain Monte Carlo sampling algorithm. We find that the plane wave model can produce model predictions that fit the actual measurements, and like in the numerical case [@niskanen2019characterising], data carry information on the variables in the sense that posterior variance is reduced compared to the prior variance. We also find that the proposed method can find the inhomogeneities in the specimen, for each parameter separately.
The rest of the paper is organised as follows. First we describe the measurement set-up in section \[sec:experiments\]. Next we discuss the modelling of the poroelastic wave propagation in section \[sec:forwardproblem\]. Then, in section \[sec:inverse\_problem\] we formulate the inverse problem and consider some uncertainties in the measurements. Results are presented and discussed in section \[sec:results\], followed by a conclusion in section \[sec:conclusion\].
Experiment and data processing {#sec:experiments}
==============================
We are interested in testing the inversion method on a porous material that is stiff, so that wave propagation in air cannot move its frame, and the material needs to be submerged in water to excite the solid phase. Porous ceramics usually fit such description and we therefore choose the porous ceramic QF-20, which is made of glass bonded silica and manufactured by Filtros Ltd. An approximately rectangular, parallel-faced, plate cut of the material is submerged in a water tank, and proper precaution is taken to ensure the pores are saturated with water.
Data needed for the inversion are the plate’s acoustic reflection and transmission coefficients, which in the frequency domain we denote by $\bm{R}$ and $\bm{T}$, respectively. The basic idea of the measurement is that a known wave pulse is sent towards the object, and the waves reflected back and transmitted through the material are recorded. These can be measured with ultrasound transducers in two measurement configurations, one for reflection and another for transmission. By comparing the incident and recorded waves we can estimate the desired acoustic responses. A schematic of the set up is shown in Fig. \[fig:watertank\_render\].
To keep the forward model simple we assume plane wave propagation and homogeneity of the material, as discussed in section \[sec:forwardproblem\]. Therefore, the obvious choice for the experiments is to measure the plane wave reflection and transmission coefficients. To realise conditions resembling plane waves, we could use very large transducers [@castaings2000inversion] or synthetic plane wave techniques [@jocker2007minimization]. However, these methods average the measurements over a large area, and small scale resolution of the object would be lost. In addition, if the object is inhomogeneous in the macroscopic sense, the forward model would be incompatible with measurements averaged over regions with differing physical properties. It would therefore be advantageous to measure as small a portion of the plate at a time as possible, and be able to locally approximate the plane wave condition. Being able to take multiple measurements from different places of the same object allows us not only to assess the homogeneity of the material, but also the validity of the inversion method. In this work, we use focused transducers for the reason that they have a small footprint on the plate, and that the waveform they send spreads less than the waveform from a flat transducer, approximating plane wave conditions better.
The measurement system
----------------------
![A schematic of the a) reflection, and b) transmission measurement setups. The dots on the plate denote the locations that were measured, and the shading is an indication of the -6 dB beam width.[]{data-label="fig:watertank_render"}](Figure1){width="0.50\linewidth"}
We first consider the frequency range of the measurements. In order to see the effects of the Biot waves (i.e. waves propagating both in the frame and fluid), the frequency needs to be below the regime where scattering effects dominate. For QF-20, based on conducting tests on the model fit and considering the validity the long-wavelength condition, we approximate the scattering regime to start from 400 kHz onwards. The front and back interfaces of the plate are smooth down to the pore scale so that we do not have separate scattering from roughness of the interfaces. The low frequency limit is in practice defined by the frequency response of the transducer, since the Biot model is valid down to audible frequencies. To produce low-frequency ultrasound we use a point-focused piezoelectric transducer (model Panametrics V389) as both the transmitter and the receiver. This broadband transducer has a diameter of 38 mm, focal length of 96 mm, and the central frequency at 500 kHz, which in water corresponds to the transducer diameter per wavelength ($D/\lambda$) of 12. A practical lower frequency limit for the transducer is 200 kHz, where it is still directive ($D/\lambda \approx 5$) and the pulse-echo signal is attenuated by 14 dB compared to the peak. The -6 dB beam diameter of a pulse-echo signal at the transducer’s focal point is 10 mm at 400 kHz [@fowler2012important].
The transducers are rigidly connected to a computer-controlled $XYZ$-positioning system. The largest face of the measured object is aligned along the $yz$-plane, and the transducers are angled at normal incidence to the same plane (see Fig. \[fig:watertank\_render\]). To maximise the signal strength we put each transducer at a distance of 105 mm from the nearest face of the measured object, where the object is in the transducer’s focal zone.
As the signal source we use a Sofranel 5072PR pulser/receiver, which can act both in a through-transmission and a pulse-echo mode. The source works by sending a -360 V spike excitation to the emitting transducer. The signal from the receiving transducer is then sent through the pulser/receiver, without any analogue filtering, to a Picoscope 5244B 16 bit AD-converter. The received signals show contributions from two waves with different velocities, which we can attribute to the fast and the slow longitudinal wave predicted by the Biot theory. This confirms that the measurements carry information on the Biot effects. Finally, the received signals are digitally averaged over 40 pulses, to reduce the effect of random electrical noise.
The computer positioning system is used to move the transducers and repeat the measurement procedure over a 90$\times$240 mm grid, with 10 mm spacing. This results in 216 measurement locations, represented by the black dots in Fig. \[fig:watertank\_render\]. The spacing between the measurement points is chosen small to achieve a high spatial resolution of the object, but not smaller than the -6 dB beam width to avoid making redundant measurements.
![Magnitude of the a) measured incident and reflected fields, b) calculated reflection coefficient, as a function of frequency.[]{data-label="fig:Wienerfilter"}](Figure2){width="0.50\linewidth"}
Spectral ratio technique
------------------------
Let us model a measurement in the reflection configuration, Fig. \[fig:watertank\_render\] a), as
$$\bm{y}_R(t) = \bm{x}_R(t)*\bm{h}_R(t) + \bm{n}(t) \:,$$
where $\bm{x}_R(t)$ is the incident wave sent by a transducer towards the plate, $\bm{y}_R(t)$ is a measurement of the wave that is reflected back (including multiple reflections from within the plate), $\bm{h}_R(t)$ is the plate’s impulse response, the operator $*$ denotes convolution, $\bm{n}(t)$ is random measurement noise, and $t$ denotes time. In the reflection configuration we use only one transducer that operates in a pulse-echo mode, so that it both sends and receives the acoustic waves.
The impulse response $\bm{h}_R(t)$ is related to the reflection coefficient of the plate by the Fourier transform $\bm{R}(\omega) = \int_{-\infty}^{\infty}\bm{h}_R(t)e^{-\I\omega t}dt$, where $\omega = 2\pi f$ and $f$ denotes frequency. In the presence of noise, deconvolution to find $\bm{R}(\omega)$ can be done for example using the Wiener filter [@chen1990effective]:
$$\label{eq:Wiener_filter_R}
\bm{R}(\omega) = \dfrac{\bm{Y}_R(\omega)\bm{X}_R^*(\omega)}{|\bm{X}_R(\omega)|^2 + q} \:,$$
where $\bm{Y}_R(\omega)$ and $\bm{X}_R(\omega)$ are the Fourier transforms of $\bm{y}_R(t)$ and $\bm{x}_R(t)$, respectively, $^*$ denotes the complex conjugate, and $q$ is the variance of the noise, sometimes also called the noise desensitising factor, that regularises the filter in frequencies where the signal-to-noise ratio is low.
In order to calculate the reflection coefficient accurately, we need to also measure the incident wave transmitted by the source, since the transducer response greatly affects the signal. If both $\bm{y}_R(t)$ and $\bm{x}_R(t)$ are measured using exactly the same system, the response of the measurement system mostly cancels out. The incident signal $\bm{x}_R(t)$ can be measured by pointing the transducer upwards to the air-water interface, which in theory gives a total reflection due to the high specific impedance difference between water and air. The 180 degree phase difference from the water-air reflection needs to be accounted for before further analysis. An example of the magnitude of the measured $\bm{Y}_R(\omega)$, $\bm{X}_R(\omega)$, and $\bm{R}(\omega)$, calculated using , is shown in Fig. \[fig:Wienerfilter\].
In transmission measurements, Fig. \[fig:watertank\_render\] b), we use two transducers operating in through-transmission mode, and a reference signal $\bm{x}_T(t)$ is recorded in the same configuration but with the plate removed. Otherwise the measurement is modelled in the same way as in the reflection case, and the transmission coefficient is obtained as
$$\label{eq:Wiener_filter_T}
\tilde{\bm{T}}(\omega) = \dfrac{\bm{Y}_T(\omega)\bm{X}_T^*(\omega)}{|\bm{X}_T(\omega)|^2 + q}e^{-\I k_f L} \:,$$
where $k_f = \omega/c_f$ is the wavenumber in water, $c_f$ speed of sound in water, and $L$ is the thickness of the plate. The exponential term accounts for the phase difference introduced when the object is removed from the signal path. Since the term includes the plate’s thickness, which we model as one of the unknown parameters and hence do not know prior to the inversion, what we actually measure and use in the inversion is $\bm{T}(\omega) := \tilde{\bm{T}}(\omega)e^{\I k_f L}$.
The forward problem {#sec:forwardproblem}
===================
In this section, let us briefly discuss the forward problem, i.e. how we describe wave motion in poroelastic media and how the theoretical reflection and transmission coefficients are obtained from a set of model parameters. A more comprehensive treatment of the related equations and solution methods is outside the scope of the current work, and can be found for example in [@gautier2011propagation; @niskanen2019characterising].
As is usually done, we model poroelastic media after the Biot theory [@biot1956theoryLow; @biot1956theoryHigh; @biot1962mechanics], where the material is seen to consist of two interlinked phases, a porous solid frame and a fluid saturating the pores. Our numerical model consists of three homogeneous and isotropic layers, where a poroelastic layer is sandwiched between two water layers extending to infinity. Waves are assumed to be propagating normally to the interfaces, making the problem effectively one-dimensional. When the physical properties of the fluid and poroelastic media are known, the theoretical plane wave transmission and reflection coefficients of the system can be computed by solving the Biot equations. This can be done, for example, by using the Global Matrix Method [@knopoff1964matrix; @lowe1995matrix]. The basic Biot model requires several input parameters: open porosity $\phi$, static viscous permeability $k_0$, geometric tortuosity $\alpha_\infty$, bulk modulus of the solid frame $K_b$, bulk modulus of the solid from which the frame is made of $K_s$, shear modulus of the frame $N$, density of the solid $\rho_s$, bulk modulus of the fluid $K_f$, density of the fluid $\rho_f$, and dynamic viscosity of the fluid $\eta$. We assume that the properties of the saturating fluid are known, and set $K_f = 2.19$ GPa, $\rho_f = 1000$ kg$\cdot$m$^{-3}$, and $\eta = 1.14\cdot 10^{-3}$ Pa$\cdot$s.
Several models are available to represent attenuation of waves propagating in a poroelastic medium. The two main types of attenuation are related to viscous losses due to movement of the fluid and to viscous losses in the solid frame. Attenuation in the fluid can be accounted for by the dynamic tortuosity model of Johnson *et al.* [@johnson1987theory], which introduces another parameter, viscous characteristic length $\Lambda$. A common way to represent losses in the solid is to give the elastic constants $K_b$, $K_s$, and $N$ a small imaginary part, as was done in [@niskanen2019characterising]. However, an attenuating model with constant real and imaginary parts can be shown to be weakly non-causal [@turgut1991investigation], and in reality the real and imaginary parts of the elastic moduli should be frequency dependent.
In this work, we adopt the Kjartansson model [@kjartansson1979constantq], which satisfies the causality requirement while only having two independent parameters. Dissipation in the solid can be quantified by the quality factor $Q(\omega)$, which is defined as $Q(\omega) = M_R(\omega)/M_I(\omega)$, the ratio of the real part to the imaginary part of a general elastic modulus $M = M_R + \I M_I$ [@bourbie1987acoustics]. In the Kjartansson model the quality factor is constant over the frequencies, and elastic moduli are represented as
$$M(\omega) = M_0(\I\omega/\omega_0)^{\frac{2}{\pi}\tan^{-1}(1/Q)},$$
where $\omega_0$ is an arbitrary reference frequency, $M_0$ is the value of the elastic modulus at $\omega_0$, and $Q^{-1}$ is the specific attenuation. Highly attenuating materials have a small $Q$, and vice versa. In our model, each elastic modulus is now represented by a reference value and a quality factor, i.e. we have $K_{b,0}$ and $Q_{K_b}$, $K_{s,0}$ and $Q_{K_s}$, as well as $N_0$ and $Q_N$.
The inverse problem {#sec:inverse_problem}
===================
The Bayesian approach for the present inverse problem was numerically studied in [@niskanen2019characterising]. In the following, we present an overview of the inversion method, and for the implementation details see [@niskanen2019characterising]. For general references on Bayesian inversion, see [@kaipio2006statistical; @calvetti2007introduction; @tarantola2005inverse].
The solution of an inverse problem in the Bayesian framework is the ppd, the probability density of the unknown parameters conditioned on the measured data. The ppd is proportional to the product of a likelihood and a prior probability density, as
$$\label{eq:Bayes}
\pi(\bm{\theta}|\bm{y}) \propto \pi(\bm{y}|\bm{\theta})\pi(\bm{\theta}),$$
where $\bm{y}$ denotes the measurement data, and $\bm{\theta}$ the unknowns. The likelihood $\pi(\bm{y}|\bm{\theta})$ includes the forward model that maps the parameters to the measurements, and information about the measurement noise and modelling uncertainty. The prior probability density $\pi(\bm{\theta})$ is straightforward to construct based on physical constraints and information obtained from other sources such as previous experiments.
Once the posterior has been derived, we can compute parameter estimates such as the maximum a posteriori (MAP) or the conditional mean (CM) estimate
$$\begin{aligned}
\bm{\theta}_{\mathrm{MAP}} & = \arg\max_{\bm{\theta}}\pi(\bm{\theta}|\bm{y}), \\
\bm{\theta}_{\mathrm{CM}} & = \mathbb{E}\{\bm{\theta}|\bm{y}\} = \int\bm{\theta}\pi(\bm{\theta}|\bm{y})d\bm{\theta}. \label{eq:CMestimate}\end{aligned}$$
We can also compute parameter uncertainty estimates, such as credible intervals. A 95 % credible interval $I_k(95) = [a_I,b_I]\subset\mathbb{R}$ for $\theta_k$ is defined as
$$\int_{a_I}^{b_I}\pi(\theta_k|\bm{y})d\theta_k = 0.95, \label{eq:95credIntval}$$
where $\pi(\theta_k|y) = \int_{\mathbb{R}^{n_\theta-1}}\pi(\theta_1,\dots,\theta_{n_\theta}|\bm{y})d\theta_1\cdots\theta_{k-1}\theta_{k+1}\cdots\theta_{n_\theta}$ is the marginal density of the $k$-th component of $\bm{\theta}$, and $n_\theta$ is the number of unknowns in the model. In the Bayesian framework, credible intervals can be directly interpreted as statements on the probabilities (i.e. uncertainties) of the parameter values.
Likelihood
----------
The data vector $\bm{y}\in\mathbb{C}^{2n_\omega}$, where $n_\omega$ is the number of frequencies, consists of the measured reflection and transmission coefficients
$$\bm{y} = [\bm{R}, \bm{T}],$$
where $\bm{R} = [R(\omega_1),\dots,R(\omega_{n_\omega})]$ and $\bm{T} = [T(\omega_1),\dots,T(\omega_{n_\omega})]$. We assume that the data include complex valued additive measurement noise $\bm{e}$, which is circularly symmetric, i.e. its mean is zero and the real and imaginary parts are independent and have equal variance. Then, with the usual assumption of mutual independence between $\bm{e}$ and $\bm{\theta}$, we can write the likelihood as
$$\label{eq:likelihood}
\pi(\bm{y}|\bm{\theta}) \propto \det(\bm{\Gamma}_{\bm{e}})^{-C}\exp\left\{-C{\left\|\bm{L}_{\bm{e}}(\bm{y} - f(\bm{\theta}))\right\|}^2\right\},$$
where $f(\bm{\theta}):\mathbb{R}^{n_\theta}\rightarrow\mathbb{C}^{2n_\omega}$ is the forward model solved at frequencies $\omega_1,\dots,\omega_{n_\omega}$, $n_\theta$ is the number of unknown parameters, $\bm{\Gamma}_{\bm{e}}$ is the covariance matrix of the measurement noise, and $\bm{L}_{\bm{e}}$ is a matrix square root of the inverse of the noise covariance, i.e. $\bm{L}_{\bm{e}}^T\bm{L}_{\bm{e}}^{} = \bm{\Gamma}_{\bm{e}}^{-1}$. Since our data are complex valued and circularly symmetric we have $C=1$, and Eq. corresponds to the complex normal distribution [@lapidoth2017foundation]. In the case of real valued data, we would have $C=1/2$.
We assume that the noise level of the measurements is unknown, but that it stays constant over the whole frequency range. Further, the measurements of $\bm{R}$ and $\bm{T}$ are not expected to be correlated with each other because they are carried out separately, and dissipation in the medium prevents us from using conservation principles that could relate them. The measurements may have differing noise levels and we can therefore write the diagonal measurement noise covariance matrix as
$$\bm{\Gamma}_{\bm{e}} = \begin{bmatrix}
\sigma_{e_R}^2 \bm{I} & 0 \\
0 & \sigma_{e_T}^2 \bm{I} \\
\end{bmatrix},$$
where $\sigma_{e_R}^2$ and $\sigma_{e_T}^2$ denote the noise variance in the reflection and transmission measurements, respectively, and $\bm{I}$ is the $n_\omega\times n_\omega$ identity matrix.
Errors related to the measurement
---------------------------------
The Bayesian approach makes it possible to take errors related to the measurement set-up into account. These include, for example, errors in the positioning of the sample and/or transducers. Kaczmarek *et al.* [@kaczmarek2015ultrasonic] studied the errors of sample positioning in ultrasonic reflectometry, and found that errors in the position of the sample influence mainly the phase of $\bm{R}$, while errors in the sample inclination mainly affect the magnitude of $\bm{R}$.
In our case, to measure the phase of $\bm{R}$ exactly, we would need to make sure that the distance from the sound source to the ceramic plate is precisely the same as the distance from the source to the air-water interface which is used to record the reference signal. However, there is always some uncertainty in measuring distances in an experimental setup. Moreover, in a scanning system the distance to the object can change if the object is not perfectly straight or aligned along the scanning axis, or if the positioning system itself flexes slightly while moving. We therefore take this possible discrepancy into account by multiplying the $\bm{R}$ given by the forward model with a distance correction term $\exp(-\I k_f\epsilon_R)$, where $\epsilon_R$ is the distance mismatch. Multiplying by this distance term we can match the phase of the reflection coefficient to the measured one, and thus remove the dependence on the exact sample position. As we will see in section \[sec:results\], this parameter is identified independently of the other parameters and improves the accuracy of the model.
To account for an incorrect sample inclination (i.e. when the sample is not normal to the incoming ultrasound field), we would need to model the actual finite-sized transducers and the ultrasonic field they produce, instead of the current plane wave approximation. Doing so would substantially increase the computational cost of the model, and render the current approach to the inverse problem infeasible. One possibility to take the effects of any inclination error into account is to use the approximation error approach [@kaipio2007statistical; @kaipio2013approximate], but this was not pursued in the current paper. We will discuss the validity of the normal incidence assumption in the current setup at the end of section \[sec:results\].
Other parameters related to the measurements are the noise levels $\sigma_{e_R}$ and $\sigma_{e_T}$. The noise level of a measurement is related to how accurately the unknown parameters can be estimated, and is therefore essential information. We estimate the noise levels simultaneously with the other parameters. Let us denote all the measurement uncertainty parameters by $\bm{\xi} = [L,\epsilon_R,\sigma_{e_R},\sigma_{e_T}]$.
Prior density {#ssec:prior}
-------------
----------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- --
**Parameter & & & & & & & & & & & $\sigma_{e_R}$, $\sigma_{e_T}$\
**Unit & & & & & & & & & -\
**Min & 0.001 & 1 & $-14$ & 0.01 & 0.01 & 0.01 & 1 & 1 & 10 & -3 & 0\
**Mean & 30 & 2 & $-12$ & 0.4 & 15 & 30 & 50 & 2500 & 25 & 0 & 0.1\
**Std & 20 & 1 & 2 & 0.1 & 10 & 20 & 50 & 1000 & 0.2 & 0.5 & 0.2\
**Max & 1000 & 10 & $-8$ & 1 & 400 & 400 & 1000 & 6000 & 30 & 3 & 1\
************
----------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- --
Since the properties of the fluid ($K_f, \rho_f, \eta$) are assumed to be known, we have 11 unknown material parameters in the model. Let us represent these by $\bm{\theta} = [\Lambda,\alpha_\infty,k_0,\phi,K_{b,0},Q_{K_b},K_{s,0},\\ Q_{K_s},N_0,Q_N, \rho_s]$. To denote all the unknown parameters, we define $\tilde{\bm{\theta}} = [\bm{\theta}, \bm{\xi}]$. We will use a prior that is a truncated normal distribution, i.e. $\tilde{\bm{\theta}} \sim \mathcal{N}(\tilde{\bm{\theta}}_*,\bm{\Gamma}_{\tilde{\bm{\theta}}}) \times B(\tilde{\bm{\theta}})$, where $\tilde{\bm{\theta}}_*$ denotes the mean, $\bm{\Gamma}_{\tilde{\bm{\theta}}}$ the covariance, and $B(\tilde{\bm{\theta}})$ is an indicator function:
$$\label{eq:indicatorfcn}
B(\tilde{\bm{\theta}}) =
\begin{cases}
1, & \text{if } \tilde{\theta_k} \in \text{physical bounds } \forall\: k,\\
0, & \text{otherwise.}
\end{cases}$$
The indicator function returns a probability one if all parameters are within the predefined bounds, and otherwise a probability of zero. The bounds are based on considering what values are physically possible, and are found in Table \[tab:prior\].
The prior mean is chosen based on available information from the manufacturer [@filtrosManual], other studies [@johnson1994probing], and our experience with similar materials. For example, the manufacturer reports typical data on the porosity, pore size, and permeability of QF-20. However, the prior knowledge is still limited, and we adjust the prior variance so that all expected parameter values have some probability. We do not assume correlations between the parameters a priori, so the matrix $\bm{\Gamma}_{\tilde{\bm{\theta}}}$ is diagonal. The prior mean and variance are also given in Table \[tab:prior\]. Permeability is expressed on a logarithmic scale since it can take values that span multiple orders of magnitude [@schon2015physical].
In addition to minimum and maximum values, the indicator function can also be used to impose other prior constraints. For example, from the physical point of view, we know that the frame of a porous material is either rigid (it does not move under pressure), limp (it does not resist movement at all), or something in between. This condition can be expressed as $0\le K_{b,0}\le K_{s,0}(1 - \phi)$ [@biot1957elastic], which restricts the possible values of $K_{b,0}$ and $K_{s,0}$. Other physical constraints we impose are that the Poisson’s ratio of the porous frame is non-negative, which can be expressed as $K_{b,0} \ge 2/3N_0$, and that attenuation in the frame is greater than attenuation in pure solid, $Q_{K_b}^{-1} > Q_{K_s}^{-1}$.
Sampling the posterior using MCMC
---------------------------------
Considering the form of the likelihood and the prior, the posterior can now be written out. For numerical reasons, it is preferable to work with the logarithm of the posterior, which reads
$$\label{eq:log_posterior}
\begin{split}
\log\pi(\tilde{\bm{\theta}}|\bm{y}) \propto &-\left\Vert \bm{L}_e(\bm{y} - f(\tilde{\bm{\theta}}))\right\Vert^2 -2n_{\omega}(\log\sigma_{e_R} + \log\sigma_{e_T}) \\
& - \frac{1}{2}\Vert \bm{L}_{\tilde{\bm{\theta}}}(\tilde{\bm{\theta}} - \tilde{\bm{\theta}}_*)\Vert^2 + \log B(\tilde{\bm{\theta}}),
\end{split}$$
where we have denoted $\bm{L}_{\tilde{\bm{\theta}}}^{}\bm{L}_{\tilde{\bm{\theta}}}^T = \bm{\Gamma}_{\tilde{\bm{\theta}}}^{-1}$.
In this work, we use the CM, Eq. , as the parameter point estimate, and the 95 % credible interval, Eq. , as the uncertainty estimate. Computing these estimates requires the solving of high-dimensional integrals, which are in practice often approximated with Markov chain Monte Carlo (MCMC) [@metropolis1953equation] methods. MCMC methods generate an ensemble of samples that are distributed according to the ppd, and given these samples it is easy to compute point and interval estimates for the parameters. For a general reference on MCMC methods, see e.g. Ref. [@brooks2011handbook].
MCMC methods are computationally heavy, and the efficiency of the sampler plays a big part on whether sampling methods are viable in a given problem. Here we use the sampler developed in [@niskanen2019characterising], which is based on an adaptive random walk Metropolis algorithm [@haario2001adaptive; @andrieu2008tutorial], with parallel tempering [@geyer1991markov; @earl2005parallel]. We use 10 temperatures, two of which are set to one and the rest are adapted during the MCMC run to optimise efficiency. The first 15,000 samples are removed as burn-in, and the sampler is run until an objective convergence criterion is fulfilled. Our stopping criterion is based on computing the Monte Carlo Standard Error [@flegal2008markov] for each parameter and ensuring that it is small enough compared to the posterior variance of the parameter. On average, the stopping criterion was reached in 35,000 samples after burn-in, and the multivariate effective sample size [@vats2019multivariate] was 850.
Characterisation results {#sec:results}
========================
Let us now present and discuss the results of the inversion. Due to the large number of measurement locations, we first focus on a few selected ones, shown in Fig. \[fig:example\_locations\], and then summarise all results by plotting the CM and uncertainty estimates of each parameter as two dimensional maps.
Individual locations
--------------------
{width="0.4\linewidth"}
The measured reflection and transmission coefficients vary significantly depending on where on the ceramic plate the measurement is taken. This already shows that the plate is likely not homogeneous in the macroscopic sense, and that, if accurate knowledge on the parameters is needed, one should avoid averaging measurements over several locations. Fig. \[fig:all\_modelfits\] demonstrates the variability in the measurements by showing the magnitude and phase of the measured $\bm{R}$ and $\bm{T}$ coefficients from three locations around the object. In addition, the figure shows the model prediction corresponding to the CM estimate, and indicates the range of the model predictions with shading that corresponds to one, two, and three standard deviations. Variance of the model predictions is mainly induced by the noise, which by our assumption is normally distributed. Therefore standard deviations describe the distribution of the model predictions well.
As Fig. \[fig:all\_modelfits\] shows, the frequency regions of almost zero reflection or transmission are different in all three locations, and the goodness of the model fit differs from location to location. In locations 1 and 3, the model underestimates the transmission coefficient amplitude at low frequencies, whereas in location 2 the model fit to the transmission coefficient is excellent over the whole frequency range. The goodness of the model fit can be evaluated from the width of the model prediction standard deviations. What should be noted, however, is how well the Biot model fits to the measurements, despite the variability in the measurement data and simplifications (such as the plane wave assumption) in the forward model.
-- -- -- -- -------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- --
**Ref. [@johnson1994probing]& **Ref. [@filtrosManual]\
$\Lambda$ ($\mu$m) & 47.7 & **66.9** & 90.2 & 36.8 & **53.7** & 70.1 & 64.7 & **89.2** & 113.0 & 19.0 & 84\
$\alpha_\infty$ & 2.13 &**2.20** & 2.25 & 2.05 & **2.13** & 2.17 & 2.02 & **2.10** & 2.16 & 1.89 &\
$\log_{10}k_0$ (m$^2$) & -11.3 &**-10.2** & -8.5 & -11.4 & **-10.2** & -8.6 & -11.0 & **-9.9** & -8.4 & -7.77 & -10.7 – -10.4\
$\phi$ & 0.33 &**0.34** & 0.36 & 0.32 & **0.34** & 0.35 & 0.36 & **0.38** & 0.40 & 0.402 & 0.35 – 0.45\
$K_{b,0}$ (GPa) & 12.0 &**13.2** & 14.5 & 9.8 & **10.5** & 11.4 & 9.3 & **10.6** & 12.0 & 9.47 &\
$Q_{K_b}$ & 11.3 &**14.3** & 18.0 & 6.7 & **7.7** & 8.8 & 10.1 & **12.5** & 15.3 & &\
$K_{s,0}$ (GPa) & 18.7 &**21.8** & 26.5 & 15.3 & **17.0** & 19.2 & 15.7 & **19.3** & 24.0 & 36.6 &\
$Q_{K_s}$ & 45.3 &**105.1** & 171.8 & 28.2 & **88.8** & 155.1 & 21.3 & **75.8** & 142.1 & &\
$N_0$ (GPa) & 17.3 &**18.7** & 20.1 & 14.1 & **15.0** & 15.9 & 12.7 & **14.4** & 15.9 & 7.63 &\
$Q_N$ & 71.3 &**134.4** & 203.8 & 58.5 & **117.6** & 177.8 & 58.7 & **115.1** & 177.3 & &\
$\rho_s$ (kg$\cdot$m$^{-3}$) & 2824 &**3053** & 3296 & 2613 & **2733** & 2855 & 2880 & **3133** & 3398 & 2760 &\
****
-- -- -- -- -------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- --
{width="\linewidth"}
Let us now consider to what degree does the variability in the measurements show as variability of the parameter estimates. Although porous materials are inherently heterogeneous, it is expected that measurements of the same material should lead to similar parameter estimates. The computed CM and 95 % CI estimates for locations 1–3 are shown in Table \[tab:inversion\_results\]. The table also shows data from the manufacturer [@filtrosManual], and some results of other characterisation processes and some textbook values reported by Johnson *et al.* [@johnson1994probing], shown here only as an illustration of the order of magnitude some parameters may take. These are not expected to coincide exactly to the currently estimated parameters, due to heterogeneities in the material, different frequency ranges, and uncertainties associated with the direct characterisation methods. In addition, Johnson *et al.* do not consider attenuation in the solid. We can see from Table \[tab:inversion\_results\] that the estimated parameter values do vary with location, but in most cases the variability is within the 95 % uncertainty bounds.
{width="\linewidth"}
Sampling the ppd using MCMC gives us access to the marginal and joint marginal posterior densities, which are helpful in assessing the identifiability of each model parameter. Identifiability, however, is not a straightforward concept to define, and depends on both the model and data. One way to think of identifiability is in terms of posterior variance, and how much does incorporating the data reduce the posterior variance compared to the prior variance. A visual representation of posterior and prior variances is found in Fig. \[fig:loc1\_1D\], where the marginal posterior distributions of measurement location 1 are drawn over the marginal prior distributions. According to the criteria of posterior variance reduction, the parameters $\alpha_\infty, \phi, K_{b,0}, Q_{K_b}, K_{s,0}, N_0$, and $\rho_s$ are the best identified, while the posterior of the rest of the model parameters is more similar to the prior. However, no parameters have a marginal posterior exactly like the prior, which shows that the data carry some information on all parameters. The reasonably low uncertainties of many parameters show that the measurement data, which include contributions from both the fast and the slow wave, contain a lot of information.
{width="\linewidth"}
Inspecting the joint densities can reveal other type of identifiability issues, namely strong correlations between parameters. Fig. \[fig:loc1\_2D\] shows an example of the joint marginal posterior distributions of the model parameters. Again, the results are from measurement location 1, but the findings are representative of the whole measurement set. Because all the measurements are at normal incidence where the shear wave is not generated, we can expect that the frame bulk modulus $K_{b,0}$ and the shear modulus $N_0$ are correlated and not necessarily identifiable separately [@niskanen2019characterising]. Fig. \[fig:loc1\_2D\] shows that these parameters do exhibit some correlation, but there is also a sharp cut in their joint marginal posterior. This cut is the result of adding to the prior the requirement that the Poisson’s ratio has to be positive, and the line can be seen to occur at $K_{b,0} = 2/3N_0$. The strongest correlations are between $\rho_s$ and $N_0$, as well as between $\rho_s$ and $K_{b,0}$. Also the solid bulk modulus $K_{s,0}$ exhibits correlation with several parameters.
Spatial variability of the parameter estimates
----------------------------------------------
{width="\linewidth"}
Let us now examine the results from all the measured locations together. To reveal possible spatial structures or variations in the measured plate, we plot the inversion results as a two dimensional map. The first and third columns of Fig. \[fig:map\_CMandCI\] show the CM model parameter estimates, and the second and fourth columns show the uncertainty related to each CM estimate as the width of the 95 % CI. Each subplot consists of a $9\times24$ grid of pixels, and the value of each pixel corresponds either to the CM or CI computed based on the measurement made in that location.
Figure \[fig:map\_CMandCI\] shows that the parameter estimates vary smoothly from point to point. Because the inversion is carried out individually at each point, with no connection to the neighbouring measurements, we can conclude that the smoothness is found in the measurement data. Furthermore, if the spatial changes of all parameter estimates were similar in smoothness to the one of $\alpha_\infty$, for example, we could argue that perhaps the effective measurement area of one transducer is much larger than the -6 dB diameter of 10 mm, and the smoothness is the result of low measurement resolution. The scale of changes in $\alpha_\infty$ would point to a effective measurement diameter of about 10 cm. However, parameters such as $\phi$ and $\rho_s$ vary smoothly as well, but in a much smaller scale. This points toward a conclusion that we do have a resolution of 10–20 mm, in line with the geometrical analysis in section \[sec:experiments\], and that the large smooth areas of some parameters are a property of the object. In conclusion, results in Fig. \[fig:map\_CMandCI\] show that the measured object is inhomogeneous, and that there is a gradual change in the porous frame going from left to right. When bulk and shear moduli decrease, porosity increases, which is consistent with the effective medium theory of porous ceramics [@munro2001effective].
Another notable feature of Fig. \[fig:map\_CMandCI\] is a small spot in the middle-right side of the plate, where $\Lambda$ and $\alpha_\infty$ are clearly smaller than in the other locations, whereas $K_{s,0}$ is much larger than the average. In addition, the estimated parameter uncertainty at that location is several times higher than elsewhere for many parameters. This result points to an anomaly in the plate, and it is interesting that it can be seen in basically every parameter. We selected this anomaly as the example location 2, and the model fit and parameter estimates at this point were considered earlier.
Uncertainty parameters
----------------------
An integral part in carrying out the parameter estimation in the Bayesian framework is the ability to also account for the so called nuisance parameters, parameters that we are not interested in but nevertheless affect the result of the inversion.
![CM estimates of the measurement uncertainty parameters, and the widths of the related 95 % credible intervals. Units are the same as in Table \[tab:prior\].[]{data-label="fig:map_CMandCI_nuisance"}](Figure8){width="0.5\linewidth"}
Fig. \[fig:map\_CMandCI\_nuisance\] shows the CM and CI estimates of the measurement uncertainty parameters $\bm{\xi}$ considered in this paper. Let us first comment on the estimated measurement noise levels. These parameters include not only the random white noise component but also any model errors, which is why the estimated noise levels vary around the plate. For example, in location 2 the estimated transmission noise level is lower than at any other point, which is confirmed by the good model fit seen in Fig. \[fig:all\_modelfits\]. Apart from the small area around location 2, the noise levels do not change much. This shows that the Biot model fits equally well to measurements from all over the object.
We can also see that the measured plate does not have perfectly parallel faces, but its thickness varies up to almost a millimetre along the plate. The associated uncertainty interval is 0.1 mm on average, which shows that the data provide accurate information on the thickness. The uncertainties in the estimates of $L$ follow a pattern similar to the estimates of the transmission measurement noise level $\sigma_{e_T}$, which suggests that most of the information on thickness is found in the transmission coefficient. This can be explained by the fact that the phase of $\bm{T}$ is directly linked to $L$, as can be seen from , whereas the link between $\bm{R}$ and $L$ is more implicit and thickness can be correlated with other parameters. We can also rule out changes in thickness as the reason for the smooth change in some parameter estimates, since the way the thickness estimates change is different to the change in the Biot parameters. The varying thickness is another reason why we could not use synthetic plane wave techniques in the measurements. It also shows that using a constant value of thickness in the measurements would add modelling error.
The estimates of $\epsilon_R$ are also very accurate, since the maximum 95 % CI for the estimates is less than 0.06 mm. However, the distance from the reflection transducer to the plate changes by over a millimetre, and without accounting for this distance mismatch the model fit would not be nearly as good. Interestingly, we can even see the scanning pattern the measurement device has taken, and that when scanning from left to right the transducer has been slightly closer to the plate than when scanning in the opposite direction.
The estimated uncertainty parameters give detailed information on the inclination of the plate which we can use to assess the validity of the normal incidence assumption. First, changes in $\epsilon_R$ tell us about the orientation of the plate’s closest face with respect to the reference plane the measurement system moves in. Fig. \[fig:map\_CMandCI\_nuisance\] shows that if we start from the centre of the plate, moving horizontally the plate is approximately parallel to the reference plane, whereas vertically the plate is at a small angle, with the top leaning away. This can be compensated by tilting the transducers towards the normal of the plate. However, the plate tilt angle changes from the left hand side to the right, and some error is inevitably introduced. Based on the estimates of $\epsilon_R$, the difference in the vertical tilt angle of the plate between the left and right sides is $1.5$ degrees. We can also calculate the tilt angle that the changes in $L$ produce. As a conservative estimate, assuming that one side of the plate is flat, a 1 mm change in thickness over the distance from the thinnest to the thickest part corresponds to about $0.5$ degree tilt from the reference plane.
Numerical simulations show that the relative difference between measurements at normal and oblique incidences are less than $2$ % when the deviation is one degree, but typically between $3$ and $6$ % when the deviation is two degrees. Thus the assumption of normal incidence may induce some error into the inversion, but this error is small compared to the deviations between the actual measurements and the best model prediction (as in Fig. \[fig:all\_modelfits\]). Possible reasons for the observed of model discrepancy include depth-wise heterogeneities in the plate, and the plane wave approximation, where we are modelling the focused ultrasound field as a single plane wave. Investigating the possible bias this approximation introduces to the parameter estimates would be an interesting topic of future research.
Conclusion {#sec:conclusion}
==========
In this paper, we estimated the physical parameters of a poroelastic (Biot) object using only ultrasonic reflection and transmission measurements made in a water tank. We measured over 200 different points on the object to assess how the parameters change spatially. The inverse problem was solved in the Bayesian framework, which allowed us to account for measurement and model errors, and to quantify the uncertainty related to the parameter estimates. The parameter inference was carried out using a Markov chain Monte Carlo algorithm. We found that the computational model described the measurements well, and that the measured data carried information on every parameter in the Biot model. With the proposed method, we were able to identify spatial changes in the parameters along the object, and provide uncertainty estimates for all parameters.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work has been supported by the strategic funding of the University of Eastern Finland, by the Academy of Finland (Finnish Centre of Excellence of Inverse Modelling and Imaging, and project 321761), by the RFI Le Mans Acoustique (Pays de la Loire) Decimap project, and the Jenny and Antti Wihuri Foundation. This article is based upon work initiated under the support from COST Action DENORMS CA-15125, funded by COST (European Cooperation in Science and Technology).
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[^1]: Corresponding author. email: *[email protected]*
|
---
abstract: 'The sup-norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function-field version of this problem can be reduced to the geometric problem of finding the largest dimension of the $i$th stalk cohomology group of a given Hecke eigensheaf at any point. This problem, in turn, can be reduced to the intersection-theoretic problem of bounding the “polar multiplicities" of the characteristic cycle of the Hecke eigensheaf, which in known cases is the nilpotent cone of the moduli space of Higgs bundles. We solve this problem for newforms on $GL_2 (\mathbb A_{\mathbb F_q(t)})$ of squarefree level, leading to bounds on the sup-norm that are stronger than what is known in the analogous problem for newforms on $GL_2(\mathbb A_{\mathbb Q})$ (i.e. classical holomorphic and Maass modular forms.)'
address: |
Department of Mathematics\
Columbia University\
New York, NY 10027
author:
- Will Sawin
bibliography:
- 'references.bib'
title: 'A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on $GL_2(\mathbb F_q(T))$ with squarefree level'
---
Introduction
============
The sup-norm problem in the analytic number theory of automorphic forms studies the sup-norms of automorphic forms when viewed as functions on locally symmetric spaces or on adelic groups. In this paper, we prove upper bounds on the sup-norms of certain automorphic forms over function fields:
\[sup-norm-intro\] Let $ F = \mathbb F_q(t)$, let $N$ be a squarefree effective divisor on $\mathbb P^1$, and let $f: GL_2(\mathbb A_F) \to \mathbb C$ be a cuspidal newform of level $N$ with unitary central character. Assume that for each place $v$ in the support of $N$, the restriction of the central character of $f$ to $\mathbb F_q^\times \subset F_v^\times$ is trivial. Then $$||f||_{\infty} = O \left( \left(\frac{ 2 \sqrt{q} +2}{ \sqrt{ 2 \sqrt{q}+ 1}}\right)^{ \deg N} \right)$$ if $f$ is Whittaker normalized and $$||f||_{\infty} = O \left( \left(\frac{ 2 (1+ q^{-1/2} ) }{\sqrt{ 2 \sqrt{q}+ 1}}\right)^{ \deg N} \log(\deg N)^{3/2} \right)$$ if $f$ is $L^2$-normalized.
We leave the definitions of most of these terms until Subsection \[ss-fourier\], while assuring readers familiar with classical modular forms or automorphic forms over number fields that the definitions are very similar to the corresponding definitions in those settings. (In the adelic language, they can be written almost identically).
The condition on the central character is satisfied automatically if $N$ is prime or the central character is trivial.
In the analogous case of classical or Maass modular forms of squarefree level $N$, @HarcosTemplier proved bounds of the form $|N|^{\epsilon -1/6}$ in the $L^2$ normalization, using the purely analytic amplification method. This is the best known result in that setting, though it has been generalized, with a bound depending subtly on the factorization of $N$ and the central character, to the non-squarefree case by @SahaPowerful2. We expect that similar bounds can be attained by the same method in the function field case. More specifically, because we can view $q^{\deg N}$ as the norm $|N|$ of $N$, we expect that these methods will prove bounds of the form $O \left( (q^{\deg N })^{\epsilon -1/6} \right)$ for all $\epsilon>0$. Theorem \[sup-norm-intro\] is better than this expected bound as long as $$\frac{ 2 (1+ 1/\sqrt{q}) }{ q\sqrt{ 2 \sqrt{q}+ 1}}< q^{-1/6}$$ which occurs for $q>134$. As $q$ goes to $\infty$, our bound will approach $|N|^{-1/4}$. This exponent occurs has occurred several times in the sup norm theory of newforms: It matches the lower bound @TemplierLowerBounds proved in the case where $N$ is square and the conductor of the central character is $N$, as well as the upper bound of [@SahaPowerful2] in the limit where $N$ becomes more powerful while the conductor of the central character divides $\sqrt{N}$, and the upper bound of @Comtat in the same case as [@TemplierLowerBounds]. It is possible that our method could be generalized to hold regardless of the level, which would partially explain why a bound of the form $|N|^{\epsilon -1/4}$ appears.
An exponent below $5/24< 1/4$ appears, for division algebras, in the recent work of [@HuSaha] and it is likely possible to obtain a similar statement, in particular one with exponent $<1/4$, over $GL_2$ (for highly ramified level, with the conductor of the central character not too large), by the same method. Doing this well by our method would require further geometric ideas.
Before we explain the proof, we observe that the proof can be viewed as proceeding via a geometric problem that may be of independent interest. For this reason, we introduce the geometric problem, and a natural approach to it, first. We then explain how to modify it to produce our method for the sup-norm problem.
From a purely analytic perspective, the most interesting feature of the proof might be the way that the Theorem \[sup-norm-intro\] follows from a series of bounds (Lemmas \[cusp-newform-bound\], \[Atkin-Lehner-bound\], and \[local-squarefree-bound\]) for the value of the form $f$ at a point, that depend in an intricate way on the geometry of the point (specifically, its distance to cusps and “virtual cusps"). In the amplification method, by contrast, the bound at a given point depends on some lattice point counts. We do not know to what extent these are related to virtual cusps. Because of the nature of our proof, how this local bound varies from point to point has some meaning for the geometry of the “modular curve", being directly related to the characteristic cycle of this space, which is independent of the choice of modular form. This geometric perspective suggests that it may be fruitful for some purposes to reformulate the sup-norm problem as studying $\sup_f |f(x)|$ for points $x$ instead of $\sup_x |f(x)|$ for eigenforms $f$.
Geometric Langlands and the general sup-norm problem
----------------------------------------------------
Let $C$ be a smooth proper geometrically connected curve over a field $k$, $G$ an algebraic group over $k$, and $\Bun_G$ the moduli space of $G$-bundles on $C$. Let $\mathcal F$ be a Hecke eigensheaf on $\Bun_G$, i.e. an irreducible perverse sheaf satisfying the conditions [@BeilinsonDrinfeld 5.4.2] studied in the geometric Langlands program. We can ask the following interrelated set of questions about $\mathcal F$:
1. For a point $x \in \Bun_G(k)$, how large is the stalk dimension $\dim \mathcal H^i(\mathcal F)_x$ for each integer $i$? In particular, for which $i$ does the stalk cohomology vanish?
2. Assume $\mathcal F$ is a pure perverse sheaf. (Either using the weights of Frobenius over a finite field or an abstract weight filtration as in the theory of mixed Hodge modules). For a point $x \in \Bun_G(k)$, how large is the sum over $i$ of the dimension of the weight $w$ graded piece of $H^i(\mathcal F)_x$ for each integer $w$? In particular, for which $w$ does the weight $w$ part vanish?
3. Assume $k$ is a finite field $\mathbb F_q$. How large is $\sum_i (-1)^i \operatorname{tr} (\operatorname{Frob}_q, \mathcal H^i(\mathcal F)_x)$?
Bounds for question (1) imply corresponding bounds for question (2) because the $i$th stalk cohomology of a perverse sheaf pure of weight $w$ is mixed of weight $\leq w+i$. Bounds for question (2) imply bounds for question (3) by the definition of weights of Frobenius.
The general sup-norm problem in analytic number theory asks for the maximum value, or the maximum value on some region, taken by a (cuspidal) Hecke eigenform. For $\mathcal F$ a Hecke eigensheaf, $x \mapsto \sum_i (-1)^i \operatorname{tr} (\operatorname{Frob}_q, \mathcal H^i(\mathcal F)_x)$ is a Hecke eigenform, so question (3) is a special case of the function field version of the classical sup-norm problem. It is equivalent to the full sup-norm problem in cases where we know every cuspidal Hecke eigenform comes from a Hecke eigensheaf, as in the case of $GL_n$ by combining the main results of [@Lafforgue] and [@FrenkelGaitsgoryVillonen].
Question (1) also may have interest as a purely geometric problem.
Massey proved bounds for the dimensions of the stalk cohomology groups of a perverse sheaf in terms of the “polar multiplicities" of its characteristic cycle. In [@mypaper] we generalized these from characteristic zero to characteristic $p$. Because $\mathcal F$ is perverse, we can use the appropriate results to give bounds for these three questions if we can calculate its characteristic cycle. The characteristic cycle lies in the cotangent bundle of $\Bun_G$, which is also the moduli space of Higgs bundles.
The characteristic cycle of the Hecke eigensheaves was first studied by Laumon. Laumon predicted that, in the $GL_n$ case, the characteristic cycle should be contained in the nilpotent cone of the moduli space of Higgs bundles [@Laumon Conjecture 6.3.1]. (He stated this only over characteristic zero, but now that the characteristic cycle is known to exist in characteristic $p$ [@saito1], we can extend the conjecture there as well.) This was by analogy to Lusztig’s theory of character sheaves. He verified this in the case of $GL_2$ using Drinfeld’s explicit construction [@Laumon (5.5)].
Beilinson and Drinfeld constructed Hecke eigensheaves, in the form of $D$-modules, associated to special local systems known as opers. They calculated the characteristic cycle of their Hecke eigensheaves especially, and found that it was equal to the nilpotent cone. More precisely, it is equal as a cycle to the zero fiber of the Hitchin fibration [@BeilinsonDrinfeld Proposition 5.1.2(ii)].[^1].
It is reasonable to expect that this formula for the characteristic cycle holds for eigensheaves on $\Bun_G$ in arbitrary characteristic, at least when the centralizer of the associated local system is the center of the Langlands dual group. Whenever this is proved, we can bound the stalk cohomology by the polar multiplicities of the zero fiber of the Hitchin fibration. In particular we have the following vanishing result:
\[automorphic-cohomology-vanishing\] Let $\mathcal F$ be a perverse sheaf on $\Bun_G$ whose characteristic cycle is contained in the nilpotent cone of the moduli space of Higgs bundles. Then for a $G$-bundle $\alpha$ on $X$, $\mathcal H^i(\mathcal F)_{\alpha} $ vanishes for $$i > \dim \{ v \in H^0 ( X, \operatorname{ad} (\alpha) \otimes K_X) | v \textrm { nilpotent} \} - (g-1) \dim G.$$
Proposition \[automorphic-cohomology-vanishing\] applies to the category of perverse sheaves with characteristic cycle contained in the nilpotent cone of the moduli space of Higgs bundles, which is also studied in the Betti geometric Langlands program.
We now explain how the proof of Theorem \[sup-norm-intro\] differs from this setup.
It is not clear how to calculate the polar multiplicities exactly in this level of generality, which would be required to get numerical bounds for the sup-norm problem. The main difficulty in computing the polar multiplicities for a general $C$ and $G$ is the potentially complicated geometric structure of the moduli of nilpotent Higgs fields on a given vector bundle. To make this as simple as possible, we have chosen to work with $C = \mathbb P^1$ and $G= GL_2$. Because the canonical bundle is negative, all Higgs fields must preserve the Harder-Narasimhan filtration, and all such fields are nilpotent, so here the moduli of nilpotent fields for a given vector bundle is simply a vector space. This allows us to calculate the polar multiplicities of the nilptotent cone in this setting.
Because there are no cusp forms of level $1$ on $\mathbb P^1$, we have chosen to work with nontrivial level structure. Working with newforms, there is an appropriate analogue of the nilpotent cone, which is not much more complicated - the polar multiplicities are just sums of the polar multiplicities from the unramified case.
Finally, we do not work with the Hecke eigensheaf, but rather with the Whittaker model of it. Recall that the key difficulty in the construction of Hecke eigensheaves studied by @Drinfeld, @Laumon, and @FrenkelGaitsgoryVillonen is the descent of an explicit perverse sheaf from some covering $\Bun_n'$ of $\Bun_n$ to $\Bun_n$. With regards to applications to the sup-norm problem, it is no loss to work on $\Bun_n'$, because the numerical function we are computing is the same in each case. This carries two advantages. First, we can avoid the descent step, and therefore work in greater generality than [@Drinfeld]. Second, the polar multiplicities are often smaller on this covering than the base, and so we get better bounds this way.
Because we are working on a covering, $\Bun_G$, the moduli of Higgs bundles, and the nilpotent cone almost never appear explicitly in our proof, replaced by the moduli space of extensions of two fixed line bundles, its cotangent bundle, and a certain explicit cycle in that cotangent bundle. Despite this, the fundamental idea is the same.
We calculate the polar multiplicities precisely, obtaining a bound for each point in $\Bun_2'$ over our chosen point of $\Bun_2$. To get the best possible bound we choose the optimal point of $\Bun_2'$. Roughly speaking, our bound consists of contributions from different cusps, that grow larger as we get closer to the cusps, but we do not have to count the cusp we are performing a Whittaker expansion around. To get a good bound, we need to perform a Whittaker expansion around whichever cusp the point is closest to. Because we have only done our geometric calculations for the expansion at the standard cusp, we transform an arbitrary cusp into the standard cusp using Atkin-Lehner operators, which requires $N$ to be squarefree and leads to our condition on the central character.
Notation for automorphic forms on $GL_2$ {#ss-fourier}
----------------------------------------
Let us now explain the notation, and basic theory, needed to understand the statement of Theorem \[sup-norm-intro\].
Let $C$ be a smooth projective geometrically irreducible curve over $\mathbb F_q$ and let $F= \mathbb F_q(C)$ its field of rational functions.
We can identify the set of places of $F$ with the set of closed points $|X|$ of $X$. For $v$ a place of $F$, let $F_v$ be the completion of $F$ at $v$, $\mathcal O_{F_v}$ the ring of integers of $F_v$, and $\pi_v$ a uniformizer of $\mathcal O_{F_v}$. Let $\mathbb A_F = \prod'_v F_v$ be the adeles of $F$.
A divisor on $X$ is a finite $\mathbb Z$-linear combination of closed points of $X$. The degree of the divisor is the corresponding $\mathbb Z$-linear combinations of the degrees of its points. For an adele $a$, we define the divisor $\div a = \sum_v v(a)[v]$, and $\deg a = \deg \div a$. Using this convention, adeles contained in $\prod_v \mathcal O_{F_v}$ have divisors that are effective and degrees that are nonnegative.
Fix $N$ an effective divisor on $C$. Write $N = \sum_{v \in |N|} c_v [v]$ for some set $|N|$ of places $v$ of $X$ and some positive integer multiplicities $c_v$. We say $N$ is squarefree if the multiplicities $c_v$ are all at most $1$. We always take $c_v=0$ if $v$ is not in the support of $N$.
We will always use $v$ to refer to places of $C$, or closed points, and $x$ to refer to points of $C(\overline{\mathbb F}_q)$. So we will also write $N= \sum_{x \in \Sing} c_x [x]$, where $c_x$ is the multiplicity of the $\overline{\mathbb F}_q$-point $x$ (which equals the multiplicity of the closed point $v$ that $x$ lies over.)
An *automorphic form of level $N$* on $GL_2(\mathbb A_F)$ is a function $f: GL_2(\mathbb A_F) \to \mathbb C$ which is left invariant under $GL_2(F)$ and right invariant under $$\Gamma_1(N) = \prod_{v \in |X| }\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2 ( \mathcal O_{F_v} ) \mid b \equiv 0\mod \pi_v^{c_v} , d \equiv 1 \mod \pi_v^{c_v} \right \}.$$
We say that $f$ is *cuspidal* if $\int_{z \in \mathbb A_F/F} f\left( \begin{pmatrix} 1& z \\ 0 & 1 \end{pmatrix} {\mathbf g} \right) =0$ for all $ {\mathbf g} \in GL_2(\mathbb A_F)$.
We say $f$ is a *Hecke eigenform* if it is an eigenfunction of the two standard Hecke operators at each place $v$ of $X$ not in the support of $N$.
We say $f$ is a *newform* if it is a Hecke eigenform and the same set of Hecke eigenvalues is not shared by any Hecke eigenform of level $N'<N$.
For any eigenform, there is a unique character $\eta: (\mathbb A_F^\times / F^\times) \to \mathbb C^\times$ such that $f ( t^{-1} {\mathbf g} ) = \eta(t) f({\mathbf g})$ for any scalar $t \in \mathbb A_F^\times$. We call $\eta$ the central character of $f$.
Fix a meromorphic $1$-form $\omega_0$ on $X$ and a character $\psi_0: \mathbb F_q \to \mathbb C^\times$. Define a character $\psi: \mathbb A_F / F \to \mathbb C^\times$ by $\psi(z) = \psi_0( \langle z, \omega_0\rangle)$ where $\langle, \rangle$ is the residue pairing.
For $\mathcal F$ a middle-extension $\overline{\mathbb Q}_\ell$-sheaf on $X$, let $r_{\mathcal F}(D)$ be the unique function from effective divisors $D$ to $\overline{\mathbb Q}_\ell$ satisfying $r_{\mathcal F}(D_1 + D_2) = r_{\mathcal F}(D_1) r_{\mathcal F}(D_2)$ if $D_1$ and $D_2$ are relatively prime and, for $v$ a closed point of $X$, $$\sum_{n=0}^\infty r_{\mathcal F} (n[v]) u^n = \frac{1}{ \det ( 1 - u \operatorname{Frob}_{ |\kappa_v|}, \mathcal F_v)}$$ where $\kappa_v$ is the residue field of $\mathcal O_{F_v}$ and $\mathcal F_v$ is the stalk of $\mathcal F$ at some geometric point lying over $v$.
The following Whittaker expansion is essentially due to @Drinfeld [(4)].
For any newform $f$ of level $N$ whose central character has finite order, there exists $\mathcal F$ an irreducible middle extension sheaf of rank two on $X$, pure of weight $0$, of conductor $N$, and $C_f\in \mathbb C$ such that
$$f \left( \begin{pmatrix} a & bz \\ 0 & b \end{pmatrix} \right) =C_f q^{- \frac{ \deg (\omega_0 a/b) }{2} } \eta(b)^{-1} \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} (w \omega_0 a/b ) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div}( w \omega_0 a/b ))$$ for all $a,b,z \in \mathbb A_F$.
$\mathcal F$ and $C_f$ are unique with this property.
In the future, we will use the notation $\mathcal F$ and $C_f$ for the unique $\mathcal F$ and $C_f$ of Lemma \[Drinfeld-formula\].
We say that $f$ is *Whittaker normalized* if $\eta$ has finite order and $C_f=1$. We say that $f$ is $L^2$-normalized if $\eta$ is unitary and for some (equivalently any) $d\in \mathbb Z$
$$\int_{PGL_2(F) \backslash PGL_2(\mathbb A_F) } |f|^2 =1.$$
Here we note that $|f|$ is a well-defined function on $PGL_2(F) \backslash PGL_2(\mathbb A_F)$ because $f$ has unitary central character, and we take the integral is against the uniform measure on $PGL_2(\mathbb A_F)$ which assigns measure $1$ to $P \Gamma_1(N)$.
Acknowledgments
---------------
I would like to thank Simon Marshall, Farell Brumley, Paul Nelson, and Ahbishek Saha for helpful discussions on the sup-norm problem, and Takeshi Saito for helpful discussions on the characteristic cycle.
This research was conducted during the period the author was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation, and, later, during the period the author served as a Clay Research Fellow.
Characteristic cycles of natural sheaves on the symmetric power of a curve {#symmetric-powers}
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Let us review the definitions of the characteristic cycle and singular support, and some related definitions, from [@Beilinson] and [@saito1].
\[C-transversal-1\] [@saito1 Definition 3.5(1)] Let $X$ be a smooth scheme over $k$ and let $C \subseteq T^* X$ be a closed conical subset of the cotangent bundle. Let $f: X\to Y$ be a morphism of smooth schemes over $k$.
We say that $f: X \to Y$ is *$C$-transversal* if the inverse image $df^{-1}(C)$ by the canonical morphism $X \times_Y T^* Y \to T^* X$ is a subset of the zero-section $X \subseteq X \times_Y T^* Y$.
[@saito1 Definition 3.1] Let $X$ be a smooth scheme over $k$ and let $C \subseteq T^* X$ be a closed conical subset of the cotangent bundle. Let $h: W \to X$ be a morphism of smooth schemes over $k$.
Let $h^* C$ be the pullback of $C$ from $T^* X$ to $W \times_X T^* X$ and let $K$ be the inverse image of the $0$-section $W \subseteq T^* W$ by the canonical morphism $dh: W \times_X T^* X \to T^* W$.
We say that $h: W\to X$ is *$C$-transversal* if the intersection $h^* C \cap K$ is a subset of the zero-section $W \subseteq W \times_X T^* X$.
If $h: W \to X$ is $C$-transversal, we define a closed conical subset $h^\circ C \subseteq T^* W$ as the image of $h^* C$ under $dh$ (it is closed by [@saito1 Lemma 3.1]).
[@saito1 Definition 3.5(2)] We say that a pair of morphisms $h: W \to X$ and $f:W\to Y$ of smooth schemes over $k$ is *$C$-transversal*, for $C \subseteq T^* X$ a closed conical subset of the cotangent bundle, if $h$ is $C$-transversal and $f$ is $h^\circ C$-transversal.
[@Beilinson 1.3] For $K \in D^b_c(X, \mathbb F_\ell)$, let the *singular support $SS(K)$ of $K$* be the smallest closed conical subset $C \in T^* X$ such that for every $C$-transversal pair $h: W \to X$ and $f: W\to Y$, the morphism $f: W\to Y$ is locally acyclic relative to $h^* K$.
[@saito1 Definition 5.3(1)] Let $X$ be a smooth scheme of dimension $n$ over $k$ and let $C\subseteq T^* X$ be a closed conical subset of the cotangent bundle. Let $Y$ be a smooth curve over $k$ and $f: X\to Y$ a morphism over $k$.
We say a closed point $x \in X$ is at most an *isolated $C$-characteristic point* of $f$ if $f$ is $C$-transversal when restricted to some open neighborhood of $x$ in $X$, minus $x$. We say that $x\in X$ is an *isolated $C$-characteristic point* of $f$ if this holds, but $f$ is not $C$-transversal when restricted to any open neighborhood of $X$.
For $V$ a representation of the Galois group of a local field over $\mathbb F_\ell$ (or a continuous $\ell$-adic representation), we define $\operatorname{dimtot} V$ to be the dimension of $V$ plus the Swan conductor of $V$. For a complex $W$ of such representations, we define $\operatorname{dimtot}W $ to be the alternating sum $\sum_i (-1)^i \operatorname{dimtot} \mathcal H^i(W)$ of the total dimensions of its cohomology objects.
[@saito1 Definition 5.10] Let $X$ be a smooth scheme of dimension $n$ over $k$ and $K$ an object of $D^b_c(X, \mathbb F_\ell)$. Let the *characteristic cycle of $K$*, $CC(K)$ , be the unique $\mathbb Z$-linear combination of irreducible components of $SS(K)$ such that for every étale morphism $j: W \to X$, every morphism $f: W\to Y$ to a smooth curve and every at most isolated $h^\circ SS(\mathcal F)$-characteristic point $u \in W$ of $f$, we have
$$- \operatorname{dimtot} \left( R \Phi_f(j^* K) \right)_u = (j^* CC(K), (df)^*\omega )_{T^*W,u}$$ where $\omega$ is a meromorphic one-form on $Y$ with no zero or pole at $f(u)$.
Here the notation $(,)_{T*W ,u}$ denotes the intersection number in $T^* W$ at the point $u$.
The existence and uniqueness is [@saito1 Theorem 5.9], except for the fact that the coefficients lie in $\mathbb Z$ and not $\mathbb Z[1/p]$, which is [@saito1 Theorem 5.18] and is due to Beilinson, based on a suggestion by Deligne.
Let $C$ be a smooth curve and $n$ a natural number. Let $C^{(n)}$ be the $n$th symmetric power of $C$. We begin by proving a general result computing the characteristic cycles of some natural perverse sheaves on $C^{(n)}$, which we will later use a special case of.
At a point in $C^{(n)}$ corresponding to an ideal sheaf $\mathcal I$ of codimension $n$ in $\mathcal O_C$, the tangent space to $C^{(n)}$ at $I$ can be viewed as $H^0(C, \mathcal I^\vee / \mathcal O_C)$. Here at a point $(x_1,\dots,x_n)$ of $C^n$, the derivative of the natural map $sym: C^n \to C^{(n)}$ is given by $\sum_{i=1}^{n} \frac{d x_i}{x_i}$ where $d x_i$ is calculated by some local coordinate at $x_i$.
Let $K$ be a perverse sheaf on $C$. Let $\Sing$ be the singular locus of $K$. Let $\operatorname{rank}$ be the generic rank of $K$ and for $x \in T$, let $c_{x}$ be the multiplicity of the contangent space at $x$ in the characteristic cycle of $K$ (i.e. the generalized logarithmic Artin conductor of $K$ at $x$).
Let $\rho$ be a representation of $S_{n} $.
Let $(e_x)_{x \in \Sing} $ be a tuple of natural numbers indexed by $\Sing$ and let $( w_k)_{k \in \mathbb N^+} $ be a tuple of natural numbers indexed by positive natural numbers such that $\sum_{x \in X} e_x + \sum_k k w_k =n$. We will refer to the tuples as $(e_x)$ and $(w_k)$, for short, and their individual elements as $e_x$ and $w_k$.
Define a closed subset $A_{(e_x), (w_k)} $ of $C^{(n)}$ as the image of the map from $\prod_{k=1}^{\infty} C^{(w_k)}$ to $C^{(n)}$ that sends a tuple $(D_k)_{k \in \mathbb N^+}$ of effective divisors, with $\deg D_k =w_k$, to $ \sum_{x \in \Sing} e_x [x] + \sum_{k=1}^{\infty} k [D_k]$.
Define a closed subset $B_{(e_x), (w_k)}$ of $T^* C^{(n)}$ as the image under the same map of the vector bundle over $\prod_{k=1}^{\infty} C^{(w_k)} $ consisting of linear forms on $H^0(C, \mathcal I^\vee / \mathcal O_C)$ that vanish on all elements whose divisor of poles is at most $\sum_{k=1}^{\infty} [D_k]$, where $\mathcal I$ is the ideal corresponding to the divisor $\sum_{k=1}^{\infty} k D_k$.
It is not hard to see that $B_{(e_x),(w_k)}$ is a closed conical cycle on $T^* C^{(n)}$ of dimension $n$.
Note that the map $\prod_{k=1}^{\infty} C^{ (w_k)} \to C^{(n)}$ used in this definition is generically injective, so that the cycle $B_{(e_x),(w_k)}$ is an irreducible closed conical subset of $T^* C^{(n)}$ with multiplicity one.
The characteristic cycle in characteristic $p$ is not necessarily Lagrangian. For instance, once can see that $B_{(e_x),(w_k)}$ if Lagrangian if and only if $w_k=0$ for all $k$ a multiple of $p$, and we will see in Theorem \[characteristic-cycle-sym\] that $B_{(e_x), (w_k)}$ will appear as the irreducible components of some natural sheaves.
Let $$M_{ \rho, K} ( (e_x), (w_k)) =
\dim \left ( \bigotimes_{x \in \Sing } ( \mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k } } \otimes \rho \right)^{ \prod_{x \in \Sing} S_{e_{x} } \times \prod_{k =1}^{\infty} S_{k }^{w_{k } } } .$$
Here $\prod_{x \in \Sing} S_{e_{x} } \times \prod_{k =1}^{\infty} S_{k }^{w_{k } } $ embeds into $S_{n} $ by acting as the group preserving the partition of $\{1,\dots,n\}$ into one part of size $e_{x} $ for each $x\in \Sing$ and $w_k$ parts of size $k $ for each $k$ in $\mathbb N^+$, and acts on $ \bigotimes_{x \in \Sing } ( \mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k }}$ where $S_{e_x}$ permutes the factors of the form $\mathbb C^{c_x}$ and each copy of $S_k$ permutes $k$ factors of the form $\mathbb C^{\operatorname{rank}}$.
\[characteristic-cycle-sym\] We have $$CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right) = \sum_{\substack{ (e_{x}) : \Sing \to \mathbb N \\ (w_{k} ): \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}M_{ \rho, K} ( (e_x), (w_k)) [ B_{(e_x),(w_k)} ] .$$
To prove Theorem \[characteristic-cycle-sym\] will take several steps. We will first verify that the singular support of $sym_* K^{\boxtimes n}$ is contained in $\bigcup B_{ (e_x), (w_k)}$, so it suffices to check that the multiplicity of each irreducible component of the singular support in the characteristic cycle is as stated. We will next set up an inductive system where knowing this multiplicity identity for lesser $n$ lets us deduce it for most irreducible components for the original $n$. Then we will use the index formula to verify that if the identity holds for all but one irreducible component, then it holds for all irreducible components. Finally we will use a series of examples, as well as the étale-local nature of the characteristic cycle, to deduce the identity for all possible irreducible components.
\[singular-support\] We have
$$SS \left( ( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right)\subseteq \bigcup_ {\substack{ e_{x} : \Sing \to \mathbb N \\ w_{k} : \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n } } B_{ (e_x), (w_k)} .$$
Because tensoring with $\rho$ and taking $S_n$-invariants both preserve local acyclicity along any map, they can only shrink the singular support, and so it suffices to prove
$$SS \left( sym_* K^{\boxtimes n} \right)\subseteq \bigcup_ {\substack{ (e_{x}) : \Sing \to \mathbb N \\ (w_{k}) : \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n } }B_{(e_x),(w_k)} .$$
We apply [@Beilinson Lemma 2.2(ii)] to the map $sym$. This says that $$SS \left( sym_* K^{\boxtimes n} \right)\subseteq sym_\circ SS(K^{\boxtimes n})$$ where $sym_\circ$ of a cycle means the image under $(T^* C^{(n)} \times_{C^{(n)}} C^n) \to T^* C^{(n)}$ of the inverse image under $d(sym): (T^* C^{(n)} \times_{C^{(n)}} C^n) \to T^* C^n$ of the cycle. By [@saito2 Theorem 2.2(3)], the singular support of $K^{\boxtimes n}$ inside $T^* (C^n) = (T^* C)^n$ is the $n$-fold product of the union of the zero section with the inverse image of $Sing$ in $T^* C$. A linear form on the tangent space at an $n$-tuple of points in $C$ lies in $SS(K^{\boxtimes n})$ if and only if, for all points on the tuple that don’t lie in $\Sing$, the linear form vanishes on the tangent line at that point. Hence a linear form on $H^0(\mathcal I^\vee / \mathcal O)$ lies in $ sym_\circ SS(K^{\boxtimes n})$ if and only if, for all points where $\mathcal I$ vanishes not contained in $\Sing$, the linear form vanishes on functions with a pole of order $1$ at that point and no poles elsewhere.
Given a point of $C^{(n)}$ corresponding to an unordered $n$-tuple of points in $C$, for $x \in \Sing$ let $e_x$ be the multiplicity of $x$, and for any positive integer $k$ let $w_k$ be the number of points outside $\Sing$ that occur with multiplicity $k$ in the tuple. Then we can write our point of $C^{(n)}$, viewed as a divisor, in the form $ \sum_{x \in \Sing} e_x [x] + \sum_{k=1}^{\infty} k [D_k]$ where $D_k$ is the divisor of degree $w_k$ consisting of all points outside $\Sing$ of multiplicity $k$, so this point lies is $A_{(e_x),(w_k)}$. A linear form on $H^0(\mathcal I^\vee / \mathcal O)$ lies in $ sym_\circ SS(K^{\boxtimes n})$ at this point if and only if it vanishes on the space of functions with poles of order at most $1$ at the support of $D_k$ and no poles elsewhere, which is the space of functions whose divisor of poles is at most $\sum_k [D_k]$ because all the points in the support of all the $D_k$ are distinct. Hence a linear form over this point lying in $sym_\circ SS(K^{\boxtimes n})$ must also lie in $B_{(e_x),(w_k)}$.
Thus $$sym_\circ SS(K^{\boxtimes n})\subseteq \bigcup_ {\substack{( e_{x}) : \Sing \to \mathbb N \\ (w_{k}) : \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n } }B_{(e_x),(w_k)}$$ and we are done.
\[splitting-1\] Let $(e_{x,1})$ and $(e_{x,2})$ be two tuples of natural numbers indexed by $\Sing$, such that $e_{x,1}e_{x,2}=0$ for all $x$. Let $(w_{k,1})$ and $(w_{k,2})$ be two tuples of natural numbers indexed by $\mathbb N^+$. Let $n_i = \sum_{x \in \Sing} e_{x,i} + \sum_{k=1}^\infty k w_{k,i}$, and let $n=n_1+n_2$.
Then:
1. Near a general point of $A_{(e_{x,1}), (w_{k,1})} \times A_{(e_{x,2}), (w_{k,2})}$, the natural map $s: C^{(n_1)} \times C^{(n_2)} \to C^{(n)}$ is étale.
2. Over the locus where $s$ is étale, the multiplicity of $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}), (w_{k,2})}$ in $ s^* B_{(e_x),(w_k)}$ is $1$ if $ e_x= e_{x,1} +e_{x,2} $ and $w_k = w_{k,1} +w_{k,2} $ and zero otherwise.
To prove part (1), we observe that $s$ is étale at a point $(y_1,y_2) \in C^{(n_1)} \times C^{(n_2)}$ when the $n_1$ points of $C$ that make up $y_1$ are distinct from the the $n_2$ points of $C$ that make up $y_2$. At a generic point of $A_{(e_{x,1}), (w_{k,1})} \times A_{(e_{x,2}), (w_{k,2})}$, all the points in the $D_k$s are distinct from each other and from $\Sing$, and by the assumption $e_{x,1}e_{x,2}=0$, all the points in $\Sing$ that make up $y_1$ are distinct from all the points of $\Sing$ that make up $y_2$. So $s$ is indeed étale at this point.
To prove part (2), first observe that the image of a general point of $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}),(w_{k,2})} $ under $s$ is a general point of $B_{( e_{x,1}+e_{x,2}), (w_{k,1} + w_{k,2})}$. Because $B_{(e_x),(w_k)}$ are distinct for distinct $(e_x), (w_k)$, and equidimensional, the pullback of any other $B_{(e_x),(w_k)}$ includes $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}),(w_{k,2})}$ with multiplicity zero. Furthermore, because $s$ is étale here, it preserves multiplicity of cycles, so the pullback of $B_{ (e_{x,1}+e_{x,2}), (w_{k,1} + w_{k,2})}$ must include $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}),(w_{k,2})} $ with multiplicity one.
\[splitting-induction\] Assume Theorem \[characteristic-cycle-sym\] holds for $n'< n$. Let $(e_x), (w_k)$ satisfy $$\sum_{x \in \Sing} e_x + \sum_{k=1}^{\infty} k w_k = n.$$
Then the multiplicity of $B_{(e_x),(w_k)}$ inside $$CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right)$$ is equal to $$M_{\rho,k}((e_x), (w_k))$$ unless $e_x=n$ for some $x$ or $w_n=1$.
We have $$\sum_{x \in \Sing} e_x + \sum_{k=1}^{\infty} k w_k = n.$$ Unless $e_x=n$ for some $n$ or $w_n=1$, we can find $(e_{x,1}), (e_{x,2}), (w_{k,1}),(w_{k,2})$ with $$e_{x,1}+e_{x,2}=e_x$$ $$w_{k,1} + w_{k,2} = w_{k,x}$$ $$e_{x,1}e_{x,2}=0$$ $$0< \sum_{x \in \Sing} e_{x,i} + \sum_{k=1}^{\infty} k w_{k,i}< n.$$
Let $n_i= \sum_{x \in \Sing} e_{x,i} + \sum_{k=1}^{\infty} k w_{k,i}.$
We calculate the multiplicity of $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}), (w_{k,2})}$ in $CC \left( s^* ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right) $ in two ways.
First note that $s$ is étale at the generic point of $A_{(e_{x,1}), (w_{k,1})} \times A_{(e_{x,2}), (w_{k,2})}$ by Lemma \[splitting-1\](1). So by [@saito1 Lemma 5.11(2)], we have $$CC \left( s^* ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right) = s^* CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right).$$ By Lemma \[splitting-1\](2), the multiplicity of $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}), (w_{k,2})}$ in this pullback is equal to the multiplicity of $B_{(e_x),(w_k)}$ inside $ CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right)$.
Second note that $$s^* ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} = ( sym_{n_1*} K^{\boxtimes n_1} \boxtimes sym_{n_2 *} K^{\boxtimes n_2} \otimes \rho)^{S_{n_1} \times S_{n_2}}.$$ If we write $$\rho=\bigoplus_{a} \rho_{a,1} \otimes \rho_{a,2}$$ where $\rho_{a,i}$ are irreducible representations of $S_{n_i}$, then $$= ( sym_{n_1*} K^{\boxtimes n_1} \boxtimes sym_{n_2 *} K^{\boxtimes n_2} \otimes \rho)^{S_{n_1} \times S_{n_2}}= \bigoplus_{a} ( sym_{n_1*} K^{\boxtimes n_1}\otimes \rho_{a,1} ) \boxtimes ( sym_{n_1*} K^{\boxtimes n_1}\otimes \rho_{a,2} ) .$$
The characteristic cycle of this complex is [@saito2 Theorem 2.2.2] $$\sum_a CC( sym_{n_1*} K^{\boxtimes n_1}\otimes \rho_{a,1} ) \boxtimes CC( sym_{n_1*} K^{\boxtimes n_1}\otimes \rho_{a,2} ).$$ By our assumption about Theorem \[characteristic-cycle-sym\], the multiplicity of $B_{(e_{x,1}), (w_{k,1})} \times B_{(e_{x,2}), (w_{k,2})}$ inside this characteristic cycle is $$\sum_a M_{K, \rho_{a,1}} ((e_{x,1}),(w_{k,1})) M_{K, \rho_{a,2}} ((e_{x,2}),(w_{k,2})) .$$
So it remains to check that $$M_{K, \rho} ((e_{x}),(w_k)) = \sum_a M_{K, \rho_{a,1}} ((e_{x,1}),(w_{k,1})) M_{K, \rho_{a,2}} ((e_{x,2}),(w_{k,2})) .$$ To do this, we recall the definition $$M_{ \rho, K} ( (e_x), (w_k)) =
\dim \left ( \bigotimes_{x \in \Sing } ( \mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k } } \otimes \rho \right)^{ \prod_{x \in \Sing} S_{e_{x} } \times \prod_{k =1}^{\infty} S_{k }^{w_{k } } } .$$
Next note that everything in this formula splits, i.e. $$\prod_{x \in \Sing} S_{e_x} \times \prod_{k=1}^{\infty} S_k^{w_k} = \left( \prod_{x \in \Sing} S_{e_{x,1} } \times \prod_{k=1}^{\infty} S_k^{w_{k,1}}\right) \times \left( \prod_{x \in \Sing} S_{e_{x,2} } \times \prod_{k=1}^{\infty} S_k^{w_{k,2}}\right)$$ and $$(\mathbb C^{c_{x}})^{\otimes e_{x} } \otimes ((\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k } } = \left( (\mathbb C^{c_{x}})^{\otimes e_{x,1 } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,1 } } \right) \otimes \left( \mathbb C^{c_{x}})^{\otimes e_{x,2 } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,2 } } \right).$$ compatibly with the actions of these groups on these vector spaces. Furthermore, note that the splitting of groups is compatible with the embedding of $\prod_{x \in \Sing} S_{e_x} \times \prod_{k=1}^{\infty} S_k^{w_k} $ in $S_n$ and $\prod_{x \in \Sing} S_{e_{x,i}} \times \prod_{k=1}^{\infty} S_k^{w_{k,i}} $ in $S_{n_i}$ in the sense that it commutes with the natural embedding $S_{n_1} \times S_{n_2} \subset S_n$.
Because these splittings are compatible with this embedding,
$$M_{ \rho, K} ( (e_x), (w_k)) =
\dim \left ( \bigotimes_{x \in \Sing } ( \mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k } } \otimes \rho \right)^{ \prod_{x \in \Sing} S_{e_{x} } \times \prod_{k =1}^{\infty} S_{k }^{w_{k } } }$$
$$= \dim \left ( \bigotimes_{i \in \{1,2\}} \left( (\mathbb C^{c_{x}})^{\otimes e_{x,i } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,i } } \right) \otimes \rho \right)^{ \prod_{i \in \{1,2\} } \left( \prod_{x \in \Sing} S_{e_{x,i} } \times \prod_{k=1}^{\infty} S_k^{w_{k,i}}\right) }$$
$$= \dim \bigoplus_a \left ( \bigotimes_{i \in \{1,2\}} \left( (\mathbb C^{c_{x}})^{\otimes e_{x,i } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,i } } \right) \otimes \rho_{a,1} \otimes \rho_{a,2} \right)^{ \prod_{i \in \{1,2\} } \left( \prod_{x \in \Sing} S_{e_{x,i} } \times \prod_{k=1}^{\infty} S_k^{w_{k,i}}\right) }$$
$$= \dim \bigoplus_a \left ( \bigotimes_{i \in \{1,2\}} \left( (\mathbb C^{c_{x}})^{\otimes e_{x,i } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,i } } \otimes \rho_{a,i} \right) \right)^{ \prod_{i \in \{1,2\} } \left( \prod_{x \in \Sing} S_{e_{x,i} } \times \prod_{k=1}^{\infty} S_k^{w_{k,i}}\right) }$$
$$= \dim \bigoplus_a \bigotimes_{i \in \{1,2\}} \left( (\mathbb C^{c_{x}})^{\otimes e_{x,i } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,i } } \otimes \rho_{a,i} \right) ^{ \prod_{x \in \Sing} S_{e_{x,i} } \times \prod_{k=1}^{\infty} S_k^{w_{k,i}} }$$
$$=\sum_a \prod_{i\in \{1,2\}} \dim \left( (\mathbb C^{c_{x}})^{\otimes e_{x,i } } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k,i } } \otimes \rho_{a,i} \right) ^{ \prod_{x \in \Sing} S_{e_{x,i} } \times \prod_{k=1}^{\infty} S_k^{w_{k,i}} }$$
$$= \sum_a M_{K, \rho_{a,1}} ((e_{x,1}),(w_{k,1})) M_{K, \rho_{a,2}} ((e_{x,2}),(w_{k,2}))$$
as desired.
\[basic-index-count\] We have $$( B_{(e_x),(w_k)}, C^{(n)})_{ T^* C^{(n)}} = \frac{ (2g-2)! } { (2g-2 - \sum_{k=1}^{\infty} w_k)! \prod_{k=1}^{\infty} w_k! } = \frac{\prod_{i=0}^{-1+ \sum_{k=1}^{\infty} w_k} (2g-2-i) }{ \prod_{k=1}^{\infty}w_k! } .$$
Here the second formula is to be used to ensure that the expression is well-defined in the case that $2g-2 - \sum_{k=1}^{\infty} w_k$ is negative.
The inverse image of $T^*C^{(n)} $ on $\prod_{k=1}^{\infty}C^{(w_k)} $ under the map that sends a tuple of divisors $D_k$ of degree $k$ to the divisor $\sum_x e_x [x] + \sum_k k D_k$ is the vector bundle $$H^0\left( \mathcal O_C \left( \sum_x e_x [x] + \sum_k k D_k\right) / \mathcal O_C\right) ^\vee .$$ We have a surjective map $$H^0\left( \mathcal O_C \left( \sum_x e_x [x] + \sum_k k D_k\right) / \mathcal O_C\right) ^\vee \to H^0\left( \mathcal O_C \left( \sum_k D_k\right) / \mathcal O_C\right)^\vee .$$ Let $W$ be the kernel, forming a short exact sequence. By definition, $B_{(e_x),(w_k)}$ is the pushforward of the class of $W$ to $ T^* C^{(n)}$. Hence by the push-pull formula $$( B_{(e_x),(w_k)}, C^{(n)})_{ T^* C^{(n)}} = (W, \prod_{k=1}^{\infty} C^{(w_k)})_{ H^0( \mathcal O_C ( \sum_x e_x [x] + \sum_k k D_k) / \mathcal O_C) ^\vee}.$$
By [@Fulton Example 6.3.5], the class of $W$ is simply the top Chern class of $H^0( \mathcal O_C ( \sum_k D_k) / \mathcal O_C)^\vee$. So it suffices to show that degree of the top Chern class of the vector bundle $H^0( \mathcal O_C ( \sum_k D_k) / \mathcal O_C)^\vee$ on $\prod_{k=1}^\infty C^{(w_k)}$ is $ \frac{ (2g-2)! } { (2g-2 - \sum_{k=1}^{\infty} w_k)! \prod_{k=1}^{\infty} w_k! } .$
To do this, note that this vector bundle is the pull back of $H^0 (\mathcal O_C(D) /\mathcal O_C)^\vee$ from $C^{ (\sum_{k=1}^{\infty} w_k )} $. Because this is the pullback of a map of degree $\frac{ (\sum_{k=1}^{\infty} w_k)!}{ \prod_{k=1}^{\infty} w_k!}$, it suffices to show that the top Chern class of this vector bundle on $C^{ (\sum_{k=1}^{\infty} w_k )} $ is $${ 2g-2 \choose \sum_{k=1}^{\infty} w_k} .$$
This vector bundle is simply the cotangent bundle of $C^{ (\sum_{k=1}^{\infty} w_k )} $, so its top Chern class is the topological Euler characteristic, which is indeed ${ 2g-2 \choose \sum_{k=1}^{\infty} w_k}$. We can view this binomial coefficient as a polynomial in $g$, in which case it will still correctly calculate the Euler characteristic outside the range where the binomial coefficient is usually defined. Thus our formula will hold in general as well.
\[advanced-index\] We have
$$\left( CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right), C^{(n)}\right) =\left( \sum_{\substack{( e_{x}) : \Sing \to \mathbb N \\ (w_{k} ): \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}M_{ \rho, K} ( (e_x), (w_k)) [ B_{(e_x),(w_k)} ] , C^{(n)}\right) .$$
For the left side, by [@saito1 Theorem 6.13] we have
$$\left(CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right), C^{(n)}\right) = \chi ( C^{(n)}, (sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} ) = \chi \left( \left( H^* ( C, K)^{\otimes n} \otimes \rho \right)^{S_n} \right).$$
Now $\chi(C,K) = (2g-2) \rank + \sum_{x\in \Sing} c_x$ by the Grothendieck-Ogg-Shafarevich formula.
We have $$\chi \left( \left( H^* ( C, K)^{\otimes n} \otimes \rho \right)^{S_n} \right)= \frac{1}{n!} \sum_{\sigma \in S_n} \tr(\sigma, \rho) \tr \left( \sigma, H^* ( C, K)^{\otimes n},\right) = \frac{1}{n!} \sum_{\sigma \in S_n} \ \tr(\sigma, \rho) \chi(C,K)^{\#\textrm{ of orbits of }\sigma}$$ is a polynomial in $\chi(C,K)$, thus a polynomial in $g$. By Lemma \[basic-index-count\], $ (B_{(e_x),(w_k)}, C^{(n)})$ is a polynomial in $g$. Thus the identity to prove is an identity between two polynomials in $g$. Hence we may assume that $g>0$. Because $g>0$, and because we may freely twist by a rank one lisse sheaf, we may assume that $H^*(C,K)$ is supported in degree zero, so that $$\chi \left( \left( H^* ( C, K)^{\otimes n} \otimes \rho \right)^{S_n} \right) = \dim \left( H^0(C,K)^{\otimes n} \otimes \rho\right)^{S_n}.$$
Then $H^0(C,K)$ is a vector space of dimension $(2g-2) \rank + \sum_{x \in \Sing} c_x$. We can partition a basis for $H^0(C,K)$ into $2g-2$ parts of cardinality $\rank$ and one part of cardinality $c_x$ for each $x \in \Sing$. Having done this, we obtain a basis for $H^0( C, K)^{\otimes n}$ by tensor products $v$ of ordered tuples of $n$ basis vectors. For each tensor product of $n$ basis vectors $v$, let $k_i(v)$ be the number of basis vectors in the $i$th part of size $\rank$ and let $e_x(v)$ be the number of basis vectors in the part of size $c_x$.
For each pair of tuples $(k_i), (e_x)$, the tensor products of $n$ vectors with that $(k_i(v))=(k_i)$ and $(e_x(v))=(e_x)$ generate a $S_n$-stable subspace of $H^0(C,K)^{\otimes n}$. Because each basis vector lies in exactly one of these generating sets, the sum of all these spaces is exactly $H^0(C,K)^{\otimes n}$. Thus we can write $\dim \left( H^0 ( C, K)^{\otimes n} \otimes \rho \right)^{S_n}$ as a sum over tuples $((k_i),(e_x))$ of the contribution to the dimension from the corresponding subspace $H^0 ( C, K)^{\otimes n}$.
As a representation of $S_n$, the subspace of $H^0 ( C, K)^{\otimes n} $ corresponding to $((k_i), (e_x))$ is $$\Ind_{ \prod_{x\in \Sing} S_{e_x} \times \prod_{i=1}^{2g-2} S_{k_i} }^{S_n} \bigotimes_{x\in \Sing} (\mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{i=1 }^{2g-2} k_i }.$$
Hence tensoring with $\rho$ and taking $S_n$-invariants, the contribution of this subspace to $\dim \left( H^0 ( C, K)^{\otimes n} \otimes \rho \right)^{S_n}$.
$$\left( \bigotimes_{x\in \Sing}( \mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{i=1 }^{2g-2} k_i } \otimes \rho \right)^{ \prod_{x\in \Sing} S_{e_x} \times \prod_{i=1}^{2g-2} S_{k_i} }$$ $$= M_{K,\rho}((e_x), (w_k))$$ where $w_k$ is the number of $i$ with $k_i=k$.
The number of tuples $(k_i)$ that produce a fixed sequence $(w_k)$ is $${\prod_{i=0}^{-1+ \sum_{k=1}^{\infty} w_k} (2g-2-i) }{ \prod_{k=1}^{\infty}w_k! }$$ so by Lemma \[basic-index-count\] the contribution of all these tuples to $\dim \left( H^0 ( C, K)^{\otimes n} \otimes \rho \right)^{S_n}$ is exactly equal to $$M_{ \rho, K} ( e_x, w_k) ( B_{(e_x),(w_k)} , C^{(n)})_{T^* C^{(n)}}.$$ Summing, we get
$$\left(CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right), C^{(n)}\right) =\dim \left( H^0(C,K)^{\otimes n} \otimes \rho\right)^{S_n}$$ $$= \sum_{\substack{( e_{x}) : \Sing \to \mathbb N \\ (w_{k} ): \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}\left( M_{ \rho, K} ( (e_x), (w_k)) [ B_{(e_x),(w_k)} ] , C^{(n)}\right)$$ $$=\left( \sum_{\substack{( e_{x}) : \Sing \to \mathbb N \\ (w_{k} ): \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}M_{ \rho, K} ( (e_x), (w_k)) [ B_{(e_x),(w_k)} ] , C^{(n)}\right) .$$
\[sym-induction-step\] Assume Theorem \[characteristic-cycle-sym\] holds for $n'< n$. Then for any $e_x, w_k$ with $\sum_{x\in \Sing} e_x + \sum_{k=1}^{\infty} w_k=n$, the multiplicity of $B_{(e_x),(w_k)}$ inside $$CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right)$$ is equal to $$M_{K,\rho}((e_x), (w_k)) .$$
Let $\operatorname{mult}_{K,\rho}((e_x),(w_k))$ be the multiplicity of $B_{(e_x),(w_k)}$ inside $CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right) $. We aim to show that $$\operatorname{mult}_{K,\rho}((e_x),(w_k))= M_{K,\rho}((e_x), (w_k))$$ for all tuples $(e_x), (w_k)$. Our main tools will be Lemma \[splitting-induction\], which shows that this holds unless $w_n=1$ or $e_x=1$ for some $n$, and Lemma \[advanced-index\], which implies that $$\sum_{\substack{( e_{x}) : \Sing \to \mathbb N \\ (w_{k} ): \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}\operatorname{mult}_{ \rho, K} ( (e_x), (w_k)) \left( [ B_{(e_x),(w_k)} ] , C^{(n)} \right) = \sum_{\substack{( e_{x}) : \Sing \to \mathbb N \\ (w_{k} ): \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}M_{ \rho, K} ( (e_x), (w_k)) \left( [ B_{(e_x),(w_k)} ] , C^{(n)} \right) .$$ This uses the fact, from Lemma \[singular-support\], that $ [ B_{(e_x),(w_k)} ] $ are the only irreducible components that appear in $CC \left( ( sym_* K^{\boxtimes n} \otimes \rho)^{S_{n }} \right) $.
We can cancel from the left side and right side terms where we already know that $\operatorname{mult}_{K,\rho}((e_x),(w_k))= M_{K,\rho}(e_x, w_k)$. Our goal will be to get to a situation where only one term remains, and the intersection number $\left( [ B_{(e_x),(w_k)} ] , C^{(n)} \right)$ is nonzero, so we can divide by $\left( [ B_{(e_x),(w_k)} ] , C^{(n)} \right)$ and get our desired identity for that term.
We now turn to the proof.
Let us first handle the case that $w_n=1$. Because $\sum_{x\in \Sing} e_x + \sum_{k=1}^{\infty} w_k=n$, we must have $e_x=0$ for all $x$ and $w_k=0$ for all $k \neq n$. Thus $B_{(e_x),(w_k)}$ is supported over the diagonal curve $C \subseteq C^{(n)}$. Because the characteristic cycle is preserved by étale pullbacks, as is $M_{\rho,k}$, we may work in an étale-neighborhood of a general point of that curve, which we take to be $\Sym^n$ of an étale neighborhood of a general point of $C$. By working étale-locally at a generic point, we may assume that $K$ is lisse, so that $\Sing$ is empty, and further assume that $g \neq 1$. It follows from Lemma \[splitting-induction\] that the multiplicity of $B_{\empty,(w_k')}$ in the characteristic cycle is $M_{K,\rho} (\empty,(w_k'))$ for all $w_k'\neq w_k$. Hence we can cancel all terms from Lemma \[advanced-index\] except the contribution of $(w_k)$, obtaining $$\operatorname{mult}_{ \rho, K} ( \empty, (w_k)) \left( [ B_{\empty,(w_k)} ] , C^{(n)} \right) = M_{ \rho, K} ( \empty, (w_k)) \left( [ B_{\empty, (w_k)} ] , C^{(n)} \right) .$$
By Lemma \[basic-index-count\] $\left( [ B_{(e_x),(w_k)} ] , C^{(n)} \right) = 2g-2$, and we assumed $g\neq 1$, so $\left( [ B_{(e_x),(w_k)} ] , C^{(n)} \right)\neq 0$. Hence we may divide by it, obtaining $$\operatorname{mult}_{K,\rho}(\empty,(w_k))= M_{K,\rho}(\empty, (w_k))$$ and thus finishing the case $w_n=1$.
Next let us handle the case when $e_x=n$, $K$ has tame ramification around $x$, and $K$ is locally around $x$ a middle extension from the open set where it is lisse. Because $\sum_{x\in \Sing} e_x + \sum_{k=1}^{\infty} w_k=n$, we must have $e_{x'}=0$ for all $x' \neq x$ and $w_k=0$ for all $k$. Thus $B_{(e_x),(w_k)}$ is supported at the point $n[x]$ in $C^{(n)}$. Because this identity is an étale-local question on $C$ in a neighborhood of $x$, we may assume that $C = \mathbb P^1$, $x=0$, and $K$ is a lisse sheaf on $\mathbb G_m$ with tame ramification at $0$ and $\infty$, placed in degree $-1$ and middle extended from $\mathbb G_m$ to $\mathbb P^1$. (This is because of the equivalence of categories between tame lisse sheaves on $\mathbb G_m$ and tame lisse sheaves on the punctured local ring of $\mathbb P^1$ at $0$.) By Lemma \[splitting-induction\] and the previous case, we have $$\operatorname{mult}_{K,\rho}((e_x'),(w_k'))= M_{K,\rho}(e_x'), (w_k'))$$ for all tuples $(e_x'),(w_k')$ except for those with $w_k'=0$ and $e_0'=n, e_\infty'=0$ or $e_0'=1, e_\infty'=n$. There are two tuples with this property. Call them $((e_{x,1}),(w_{k,1}))$ and $((e_{x,2}),(w_{k,2}))$. Lemma \[advanced-index\] now gives $$\operatorname{mult}_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) \left( [ B_{(e_{x,1}) , (w_{k,1})} ] , C^{(n)} \right) + \operatorname{mult}_{ \rho, K} ( (e_{x,2}) , (w_{k,2})) \left( [ B_{(e_{x,2}) , (w_{k,2})} ] , C^{(n)} \right) =M_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) \left( [ B_{(e_{x,1}) , (w_{k,1})} ] , C^{(n)} \right) + M_{ \rho, K} ( (e_{x,2}) , (w_{k,2})) \left( [ B_{(e_{x,2}) , (w_{k,2})} ] , C^{(n)} \right) .$$
Because $(w_{k,i})=0$ for $i=1,2$, $B_{(e_{x,i}),( w_{k,i})}$ is simply the fiber of the cotangent bundle over a point, and this gives $$\operatorname{mult}_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) + \operatorname{mult}_{ \rho, K} ( (e_{x,2}) , (w_{k,2}))$$ $$=M_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) + M_{ \rho, K} ( (e_{x,2}) , (w_{k,2})) .$$
Next we can check that $$\operatorname{mult}_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) = \operatorname{mult}_{ \rho, K} ( (e_{x,2}) , (w_{k,2}).$$ By the classification of lisse sheaves on $\mathbb G_m$, $K$ in a neighborhood of $0$ is geometrically isomorphic to the dual of $K$ in a neighborhood of $\infty$, so $ (K^{\boxtimes n} \otimes \rho)^{S_n}$ in a neighborhood of $0$ is isomorphic to the dual of $ (K^{\boxtimes n} \otimes \rho)^{S_n}$ in a neighborhood of $\infty$. Because they are dual, they have the same characteristic cycle [@saito1 Lemma 4.13.4].
Furthermore, for both these $e_x',w_k'$, $$M_{K,\rho, e_{x'}, w_k'}) = \left( \left( \mathbb C^{c_0} \right)^{\otimes n} \otimes \rho \right)^{S_n} = \left( \left( \mathbb C^{c_0} \right)^{\otimes n} \otimes \rho \right)^{S_\infty}.$$ so we have $$M_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) = M_{ \rho, K} ( (e_{x,2}) , (w_{k,2})) .$$
It now follows by division by $2$ that $$\operatorname{mult}_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) =M_{ \rho, K} ( (e_{x,1}) , (w_{k,1})) .$$ This completes the case that $e_x=n$ for some $x$ where $K$ has tame ramification and is a middle extension sheaf.
Let us finally handle the general case when $e_x=n$. Again we may pass to an étale neighborhood of $x$. Katz and Gabber showed that any lisse sheaf on the punctured spectrum of the étale local ring at $0$ of $\mathbb P^1$ can be extended to a lisse sheaf on $\mathbb G_m$ with tame ramification at $\infty$ [@KatzCanonical Theorem 1.4.1]. It follows that any perverse sheaf on the spectrum of the étale local ring at $0$ can be extended to a sheaf on $\mathbb A^1$, lisse on $\mathbb G_m$, and with tame ramification at $\infty$. We can furthermore middle extend from $\mathbb A^1$ to $\mathbb P^1$. Working locally, we may assume $C$ is $\mathbb P^1$ and $K$ has this form.
We now argue as before. By Lemma \[splitting-induction\] and the previous cases, the multiplicity of $B_{ e_{x}', w_k'}$ in the characteristic cycle is equal to $M_{K,\rho} (e_{x}',w_k')$ for all $e_{x}',w_k'$ except the one whose multiplicity we would like to compute. Because the intersection number of $B_{ e_{x}', w_k'}$ with the zero-section is $1$, we can cancel all the other terms in Lemma \[advanced-index\] and extract our desired identity.
Theorem \[characteristic-cycle-sym\] now follows by induction on $n$, with Lemma \[sym-induction-step\] as the induction step and either $n=0$ or $n=1$, depending on preference, as the base case.
Geometric setup
===============
We now specialize to the genus zero case. Let $C = \mathbb P^1$. Then $C^{(n)} = \mathbb P^n$. Specifically, this isomorphism comes from viewing $\mathbb P^n$ as the projectivization of the vector space $H^0( \mathbb P^1, \mathcal O(n))$. Let $\mathbb P^\vee$ be the projectivization of the dual vector space. Let $Y \subseteq \mathbb P^\vee \times C^{(n)}$ be the graph of the universal family of hyperplanes, with projection $p_1$ to $\mathbb P^\vee$ and $p_2$ to $C^{(n)}$.
Let $\mathcal F$ be a rank two middle extension sheaf on $C$, pure of weight $0$. For each geometric point $x$ in the singular locus $\Sing$ of $\mathcal F$, let $c_x$ be the Artin conductor of $\mathcal F$ at $x$, and let $N = \sum_x m_x[x]$ be the associated divisor. Let $$K_n= R p_{1*} p_2^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} [2n-1] .$$
\[Radon-trace-function\] The trace of $\Frob_q$ on the stalk of $K_n$ at a $\mathbb F_q$-point of $\mathbb P^\vee$ corresponding to a linear form $\alpha$ on $H^0(\mathbb P^1, \mathcal O(n))$ is equal to $$- \sum_{\substack{ D \textrm{ on } \mathbb P^1 \\D\textrm{ effective} \\ \deg(D) =n \\ \alpha(f)=0\textrm{ for all }f\in H^0(\mathbb P^1, \mathcal O(n))\textrm{ with } \div(f) = D}} r_{\mathcal F(D)} .$$
By the Lefschetz fixed point formula, the trace of $\Frob_q$ on the stalk of $K_n$ at $\alpha$ is minus the sum of the trace of $\Frob_q$ on the stalk of $p_2^* ( sym_* \mathcal F^{\boxtimes n})^{S_n}$ over $p_1^{-1}(\alpha) (\mathbb F_q)$. Because $p_1^{-1}(\alpha)$ is the universal hyperplane associated to $\alpha$, which is the set of divisors of degree $n$ on $\mathbb P^1_{{\mathbb F}_q} $ such that $\alpha$ vanishes on all functions in $H^0(\mathbb P^1, \mathcal O(n))$ with divisor $D$, it suffices to prove that the trace of Frobenius on the stalk of $( sym_* \mathcal F^{\boxtimes n})^{S_n}$ at a divisor $D$ is $r_{\mathcal F}(D)$.
If we write $D = \sum_i m_i x_i$ for distinct geometric points $x_i$ with multiplicities $m_i$, $$\left( sym_* \mathcal F^{\boxtimes n}\right)_D = \bigoplus_{ \substack{ (y_1,\dots, y_n) \in \mathbb P^1(\overline{\mathbb F}_q) \\ | \{ j | y_j = x_i \} |= m_i \textrm{ for all }i}} \bigotimes_{j=1}^n \mathcal F_{y_j} .$$
The group $S_n$ acts transitively on the set of such tuples $y_j$, with stabilizer $\prod_i S_{m_i}$, so we can write this sum as an induced representation $$\left( sym_* \mathcal F^{\boxtimes n}\right)_D = \operatorname{Ind}_{\prod_i S_{m_i}}^{S_n} \bigotimes_{i} \mathcal F_{x_i}^{\otimes m_i}$$ and thus $$\left( sym_* \mathcal F^{\boxtimes n}\right)_D^{S_n} = \left( \bigotimes_{i} \mathcal F_{x_i}^{\otimes m_i} \right)^{ \prod_{i} S_{m_i}} = \bigotimes_i \Sym^{m_i} \mathcal F_{x_i}$$
Now $\Frob_q$ acts on $\{x_i\}$ with one orbit of size $d$ for each closed point in $D$ of degree $d$. Thus the trace of $\Frob_q$ on $\bigotimes_i \Sym^{m_i} \mathcal F_{x_i}$ is the product over the set of orbits of the trace of $\Frob_q$ on the tensor product of $\Sym^{m_i} \mathcal F_{x_i}$ for $x_i$ in that orbit. For an orbit of degree $d$, the trace of $\Frob_q$ on the tensor product of the $d$ terms in the orbit is simply the trace of $\Frob_q^d$ on a single term $\Sym^{m_i} \mathcal F_{x_i}$. This orbit corresponds to a closed point $v$ of degree $d$ with multiplicity $m_i$ in $D$, and the local contribution is $$\tr (\Frob_{|\kappa_v|}, \Sym^{m_i} \mathcal F_v) = r_{\mathcal F} (m_v [v] )$$ by the generating function identity $$\frac{1}{ \det( 1- u \Frob_{|\kappa_v | }, \mathcal F_v)}= \sum_{m=0}^{\infty} \tr (\Frob_{|\kappa_v|}, \Sym^{m_i} \mathcal F_v) u^m.$$ Thus the total trace at $D$ is $$\prod_v r_{\mathcal F} (m_v [v] ) = r_{\mathcal F} ( \sum_v m_v[v]) = r_{\mathcal F}(D),$$ as desired.
\[Radon-perverse-pure\] The complex $K_n$ is perverse and pure of weight $2n-1$.
We first verify that $K_n$ is pure of weight $2n-1$. Because $\mathcal F$ is pure of weight $0$, $\mathcal F^{\boxtimes n}$ is pure of weight $0$. Now note that smooth pullbacks, proper pushforwards, and taking summands all preserve perversity. Because $sym$ and $p_1$ are proper and $p_2$ is smooth,
Because $sym$ is proper, $sym_* \mathcal F^{\boxtimes n}$ is pure of weight $0$, and thus $p_{1*} p_2^*\left( sym_* \mathcal F^{\boxtimes n}\right)^{S_n}$ is pure of weight $0$, so $K_n=p_{1*} p_2^*\left( sym_* \mathcal F^{\boxtimes n}\right)^{S_n}[2n-1]$ is pure of weight $2n-1$.
We will prove perversity by comparing $K_n$ to a Fourier transform, as follows:
Let $Y$ be the blowup of the affine space $H^0( \mathbb P^1, \mathcal O(n))$ at the point $0$. Let $\pi: Y \to H^0( \mathbb P^1, \mathcal O(n))$ be the blowup morphism and let $d: Y \to C^{(n)}$ be the projection onto the special fiber.
Consider the dual affine space $ \mathbb A^{n+1} = H^0( \mathbb P^1, \mathcal O(n))^\vee$. Let $u: \mathbb A^{n+1} - \{0\} $ be the open immersion. Let $s: \mathbb A^{n+1} - \{0\} \to \mathbb P^\vee$ be the quotient by the dilation action of $\mathbb G_m$.
We will prove this perversity and purity simultaneously with the identity $$s^* K_n = u^* FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} (1)[n]$$ where $FT_{\psi_0}$ is the $\ell$-adic Fourier transform with respect to the character $\psi_0$. Here we have normalized the Fourier transform on $\mathbb A^{n+1}$ to include a shift, so it preserves perversity, but not a Tate twist, so it does not preserve weight.
To do this:
- Because $\mathcal F$ is a middle extension sheaf pure of weight $0$, $\mathcal F[1]$ is perverse and pure of weight $1$.
- Thus $\mathcal F^{\boxtimes n}[n]$ is perverse and pure of weight $n$.
- Thus $ sym_* \mathcal F^{\boxtimes n}[n]$ is perverse and pure of weight $n$ because $sym$ is finite,
- Thus $ ( sym_* \mathcal F^{\boxtimes n})^{S_n} [n]$ is perverse and pure of weight $n$.
- Thus $d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n}[n+1]$ is perverse and pure of weight $n+1$ because $d$ is smooth of relative dimension $1$.
- Thus $R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} [n+1]$ is the sum of a perverse sheaf pure of weight $n+1$ with a complex supported at the origin (using the decomposition theorem and the fact that $\pi$ is an isomorphism away from zero). Furthermore, the summand supported at zero, if nontrivial, must have its highest component in perverse degree nonnegative. But this is impossible because the stalk cohomology at zero is the cohomology of the fiber at zero, which is $$H^* ( C^{(n)} , ( sym_* \mathcal F^{\boxtimes n})^{S_n} [n+1]) \subseteq H^* ( C^n, \mathcal F^{\boxtimes n} [n+1])$$ which vanishes in degrees $\geq 0$ because $H^* (\mathcal F)$ vanishes in degrees $>1$. So in fact $R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} [n+1]$ is perverse and pure of weight $n+1$.
- Thus $FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} [n+1]$ is perverse and pure of weight $2n+2$ because it is a Fourier transform over an affine space of dimension $n+1$.
- Tate twisting, we see that $FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} (1)[n+1]$ is perverse and pure of weight $2n$.
- Because $u$ is an open immersion, $u^* FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} (1)[n+1]$ is perverse and pure of weight $2n$.
Now $s^* K_n $ and $ u^* FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} (1)[n] $ are each pure complexes of weight $2n-1$. To check that they are equal, up to semisimplification, it suffices to check that they have the same trace functions. (We could also check that they are equal by using properties of the six functors formalism, but this would be more laborious )
To check that they have the same trace function, observe that the trace function of $s^* K_n$ at a point $\alpha \in H^0( \mathbb P^1, \mathcal O(n))^\vee$ as $$- \sum_{\substack{ D \textrm { on }\mathbb P^1 \\ D \textrm { effective} \\ \deg(D) = n\\ \alpha(f)=0 \textrm { for all } f \in H^0(\mathbb P^1, \mathcal O(n))\textrm{ with } \div(f) = D} }\tr \left( \Frob_q, \left(( sym_* \mathcal F^{\boxtimes n})^{S_n} \right)_D \right) .$$
On the other hand, we can view $Y(\mathbb F_q)$ as the set of pairs of a divisor $D$ of a degree $n$ and $f \in H^0(\mathbb P^1, \mathcal O(n))$ with $\div(f) \geq D$. So the trace function of $R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n}$ at a point $f \in H^0(\mathbb P^1, \mathcal O(n)) $ is $$\sum_{\substack{ D \textrm { on }\mathbb P^1 \\ D \textrm { effective} \\ \deg(D) = n \\ D \leq \div f}} \tr \left( \Frob_q, \left(( sym_* \mathcal F^{\boxtimes n})^{S_n} \right)_D \right).$$ Hence the trace function of $u^* FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n}$ at $\alpha$ is $$(-1)^n \sum_{ f \in H^0(\mathbb P^1, \mathcal O(n))} \psi_0 ( \alpha(f)) \sum_{\substack{ D \textrm { on }\mathbb P^1 \\ D \textrm { effective} \\ \deg(D) = n \\ D \leq \div f}} \tr \left( \Frob_q, \left(( sym_* \mathcal F^{\boxtimes n})^{S_n} \right)_D \right)$$ $$= \sum_{\substack{ D \textrm { on }\mathbb P^1 \\ D \textrm { effective} \\ \deg(D) = n }} \tr \left( \Frob_q, \left(( sym_* \mathcal F^{\boxtimes n})^{S_n} \right)_D \right) \sum_{ \substack{ f \in H^0(\mathbb P^1, \mathcal O(n)) \\ D \leq \div f}} \psi_0 ( \alpha(f)) .$$ For $D$ an effective divisor of degree $n$, the set of $f \in H^0(\mathbb P^1, \mathcal O(n))$ is a line, and the sum of $\psi_0(\alpha(f))$ over that line is $q$ if $\alpha(f)=0$ for all $f \in H^0(\mathbb P^1, \mathcal O(n))$ with $\div(f) =D$ and $0$ otherwise, so this is $$(-1)^n q \sum_{\substack{ D \textrm { on }\mathbb P^1 \\ D \textrm { effective} \\ \deg(D) = n\\ \alpha(f)=0 \textrm { for all } f \in H^0(\mathbb P^1, \mathcal O(n))\textrm{ with } \div(f) = D} }\tr \left( \Frob_q, \left(( sym_* \mathcal F^{\boxtimes n})^{S_n} \right)_D \right) .$$ Then the shift and the Tate twist multiplies this trace function by $(-1)^{n+1} q^{-1}$, making it equal the trace function of $s^* K_n$. Thus $$s^* K_n = u^* FT_{\psi_0} R \pi_* d^* ( sym_* \mathcal F^{\boxtimes n})^{S_n} (1)[n] ,$$ up to semisimplification. Hence $$s^* K_n [1]$$ is perverse. Because $s$ is smooth of dimension $1$ and surjective, $K_n$ is perverse as well.
Calculating the characteristic cycle
====================================
We continue to use closed subsets $B_{(e_x),(w_k)}$ defined in Section \[symmetric-powers\].
Viewing points of $C^{(n)}$ as nonzero sections of $H^0( C, \mathcal O(n))$ up to scaling, we can view $A_{e_x,w_k}$ as the set of sections of the form $\left( \prod_{x \in \Sing} l_x^{e_x} \right) \prod_{k=1}^{\infty} f_k^k$ where $l_x$ is a fixed section of $\mathcal O(1)$ vanishing at $x$ and $f_k$ is an arbitrary section of $\mathcal O(w_k)$.
We can view the tangent space at a point of $C^{(n)}$ corresponding to a section $g$ as the space of sections of $\mathcal O(n)$ modulo $g$. The isomorphism to our earlier description of the tangent space as $H^0 ( \mathcal I^\vee/\mathcal O)$ is given by dividing a degree $n$ polynomial by $g$, producing a section of the dual of the ideal sheaf generated by $g$, modulo $\mathcal O$.
Over the point $\left( \prod_{x \in \Sing} l_x^{e_x} \right) \prod_{k=1}^{\infty} f_k^k$, $B_{(e_x),(w_k)}$ is the set of linear forms on the tangent space that vanish on all elements whose divisor of poles is at most the divisor of $\prod_k f_k$, which is the set of linear forms vanishing on all multiples of $\left( \prod_{x \in \Sing} l_x^{e_x} \right) \prod_{k=1}^{\infty} f_k^{k-1}$ by a section of $\mathcal O(\sum_k w_k)$.
\[constant-sheaf-generic\] Let $\mathcal F' =\mathbb Q_\ell^2 + \sum_{x \in \Sing} \delta_x^{c_x}[-1]$. The Euler characteristic of the stalk at the generic point of $$R p_{1*} p_2^* ( sym_*\mathcal F^{'\boxtimes n})^{S_n} [2n-1]$$ is the coefficient of $u^n$ in the generating series $-\frac{2u (1-u)^{\sum_{x\in \Sing} c_x} } {(1-u)^4 (1+u)}$.
For $a,b$ natural numbers with $a+b + \sum_{x \in |N| } e_x = n$, let $ f_{a,b,(e_x)}$ be the map $C^{(a)} \times C^{(b)} \to C^{(n)}$ given by adding the two divisors together and then adding $\sum_x e_x [x]$.
First we check that $$\label{pushforward-constant-sheaf}( sym_* \mathcal F^{'\boxtimes n})^{S_n}[n] = \bigoplus_ {\substack{( e_x) : \Sing \to \mathbb N \\ e_x \leq c_x \\ a, b \in \mathbb N \\ \sum_{a+b+ x \in |N|} e_x =n } } \left( f_{a,b,(e_x) *} \mathbb Q_\ell [a+b] \right)^{\prod_{x \in \Sing } {c_x \choose e_x} } .$$ To do this, write $\mathcal F' = \bigoplus_{i=1}^{ 2+ \sum_{x \in |N| } c_x} \mathcal F_i$ where $\mathcal F_1 = \mathcal F_2= \mathbb Q_\ell$ and the remaining summands are skyscraper sheaves supported at the points of $x$. Then $\mathcal F^{' \boxtimes n}$ is a sum over $n$-tuples $t_i$ of numbers from $1$ to $ 2+ \sum_{x \in |N| } c_x$ of $\boxtimes_{i=1}^n \mathcal F_{t_i}$. Thus $sym_* \mathcal F^{'\boxtimes n}$ is the sum over these $n$-tuples $t_i$ of $( sym_* \boxtimes_{i=1}^n \mathcal F_{t_i})$. Then $S_n$ acts by permuting the $n$-tuples, so the $S_n$-invariants of $sym_* \mathcal F^{'\boxtimes n}$ can be viewed as a sum over unordered $n$-tuples of the $S_n$-invariants of the sum of $sym_* \boxtimes_{i=1}^n \mathcal F_{t_i}$ for all orderings $t_i$ of that unordered tuple. Viewing this as an induced representation, the $S_n$-invariants of this sum will equal the invariants of $sym_* \boxtimes_{i=1}^n \mathcal F_{t_i}$ under the stabilizer in $S_n$ of this tuple $t_i$.
If any number greater than two occurs at least twice among the $t_i$, a transposition swapping two occurances will act as $-1$ on $sym_* \boxtimes_{i=1}^n \mathcal F_{t_i}$, because the tensor product of two skyscraper sheaves is a single skyscraper sheaf so the action is by the Koszul sign for a tensor product of complexes, which is $-1$ because the skyscraper sheaves are in degree $1$. Hence there are no invariants under the stabilizer unless each $t_i$ greater than $2$ occurs at most once. For each unordered tuple, let $a$ be the number of $i$ with $t_i=1$, $b$ be the number of $i$ with $t_i=2$, and $e_x$ be the number of $i$ with $\mathcal F_{t_i} = \delta_x[-1]$. Then the number of unordered tuples attaining $(a,b,(e_x))$ is $\prod_{x \in \Sing } {c_x \choose e_x}$, and each tuple produces the $S^a \times S^b$ invariants of the pushforward from $C^a \times C^b$ of $\mathbb Q_\ell$, which is the pushforward from $C^{(a)} \times C^{(b)}$ along $ f_{a,b,(e_x)}$ of $\mathbb Q_\ell$.
Having verified Equation , we next observe that the Euler characteristic of the Radon transform of $f_{a,b,(e_x) *} \mathbb Q_\ell [a+b] $ is $(-1)^{-1+a+b}$ times the Euler characteristic of the inverse image under $f_{a,b,(e_x)}$ of a general hyperplane. The inverse image of a general hyperplane is a $(1,1)$-hypersurface in $C^{(a)} \times C^{(b)} = \mathbb P^a \times \mathbb P^b$. Viewing this as a $\mathbb P^{\max(a,b)}$-bundle on $\mathbb P^{\min(a,b)}$, we see that if the hypersurface does not contain any fiber, then it is a $\mathbb P^{\max(a,b)-1}$ bundle on $\mathbb P^{\min(a,b)}$ and hence has Euler characteristic $(\min(a,b)+1) (\max(a,b)) =ab+ \max(a,b)$.
To check that the inverse image of a general hyperplane does not contain any fiber, we must check that for a generic linear form on polynomials of degree $n$, there is no polynomial $f$ of degree $\min(a,b)$ such that the linear form vanishes on all multiples of $f \prod_{x} l_x^{e_x}$ by polynomials of degree $\max(a,b)$. The space of such linear forms has dimension $\min(a,b) + \sum_x e_x$ and the choices of polynomials, up to scaling, are $\min(a,b)$-dimensional, so dimension of the space of linear forms is $2 \min(a,b) + \sum_x e_x \leq a+b + \sum_x e_x =n$, which is less than the $n+1$-dimensional space of all linear forms, so indeed a generic linear form does not vanish in this way, and the general Euler characteristic is $ab+ \max(a,b)$, so the contribution to the Euler characteristic is $(-1)^{n-1+a+b} (ab+ \max(a,b))$. Hence the Euler characteristic is $$\sum_{\substack{ (e_x) : \Sing \to \mathbb N \\ e_x \leq c_x \\ a, b \in \mathbb N \\ a=b \sum_{x \in |N|} e_x =n } } (-1)^{n-1+a+b} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) (ab+ \max(a,b))$$
Using the generating series $$F(u,v)= \sum_{a,b\in \mathbb N} (ab+ \max(a,b)) u^{a} v^{b},$$ this is the coefficient of $u^n$ in $ - (1-u )^{\sum_x c_x} F(u,u)$. Because $ab+\max(a,b)=0$ if $\min(a,b)=-1$, we have $$(1-uv) F(u,v) = \sum_{a,b\in \mathbb N} (ab+ \max(a,b) - (a-1) (b-1) - \max(a-1,b-1) ) u^{a} v^{b}= \sum_{a,b \in \mathbb N} (a+b-1+1) u^a v^b$$ $$=\sum_{a,b\in \mathbb N } a u^a v^b + \sum_{a,b,\in \mathbb N} b u^a v^b = \frac{u}{ (1-u)^2 (1-v)} +\frac{v}{(1-v)^2(1-u)}$$ so $$F(u,u) = \frac{ 2u}{ (1-u)^3 (1-u^2)} = \frac{ 2u}{ (1-u)^4 (1+u)} .$$ Plugging in, we see that the Euler characteristic is the coefficient of $u^n$ in $-\frac{2u (1-u)^{\sum_{x\in \Sing} c_x} } {(1-u)^4 (1+u)} $
\[cycle-rank-two\] The characteristic cycle of $ ( sym_* \mathcal F^{\boxtimes n})^{S_n}[n]$ on $C^{(n)}$ is $$\sum_{\substack{ e_x : \Sing \to \mathbb N \\ e_x \leq c_x \\ w_k : \{1,2 \} \to \mathbb N \\ \sum_{x \in |N|} e_x + w_1 + 2w_2 =n } }2^{w_1} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) [ B_{(e_x),(w_k)} ] .$$
To prove this, we apply Theorem \[characteristic-cycle-sym\].
We have $$( sym_* \mathcal F^{\boxtimes n})^{S_n}[n] = ( sym_* (\mathcal F[1])^{\boxtimes n} \otimes \sgn )^{S_n}$$ so we may take $K= \mathcal F[1]$ and $\rho=\sgn$. Then Theorem \[characteristic-cycle-sym\] guarantees that the characteristic cycle of this complex is $$\sum_{\substack{ e_{x} : \Sing \to \mathbb N \\ w_{k} : \mathbb N^+ \to \mathbb N \\ \sum_{ x\in \Sing} e_{x} + \sum_{k =1}^{\infty} k w_{k} = n }}M_{ \rho, K} ( e_x, w_k) [ B_{(e_x),(w_k)} ]$$ so we must check that $M_{\rho,K}$ vanishes if $e_x \geq c_x$ or $w_k >0$ for $k>2$ and it equal to $2^{w_1} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) $ otherwise. By definition,
$$M_{ \rho, K} ( (e_x), (w_k)) =
\dim \left ( \bigotimes_{x \in \Sing } ( \mathbb C^{c_{x}})^{\otimes e_{x} } \otimes (\mathbb C^{\operatorname{rank}} ) ^{ \otimes \sum_{k = 1 }^{\infty} k w_{k } } \otimes \rho \right)^{ \prod_{x \in \Sing} S_{e_{x} } \times \prod_{k =1}^{\infty} S_{k }^{w_{k } } } .$$
Tensoring with $\sgn$ and taking symmetric group invariants is the same as taking a wedge power, and $\rank=2$, so this is $$\dim \left ( \bigotimes_{x \in \Sing} \wedge^{e_x} ( \mathbb C^{c_{x}}) \otimes \bigotimes_{k=1}^{\infty} \left( \wedge^k ( \mathbb C^{2} )\right)^{\otimes w_k} \right)$$ $$=\left( \prod_{x \in \Sing} {c_x \choose e_x} \right) 2^{w_1} 1^{w_2} \prod_{k=2}^{\infty} 0^{w_k}$$ as desired.
We can view the tangent space of $\mathbb P^\vee$ at a point corresponding to a linear form $l$ on polynomials as the space of linear forms on polynomials modulo $l$. Hence we can view the cotangent space at this point as the space of polynomials which $l$ vanishes on.
Let $B_{(e_x),(w_k)}^\vee$ be the closed subset of $T^* \mathbb P^\vee$ defined as the set of pairs of a linear form vanishing on all polynomial multiples of $\left( \prod_{x \in \Sing} l_x^{e_x} \right) \prod_{k=1}^{\infty} f_k^{k-1}$ with a cotangent vector that is a scalar multiple of $\left( \prod_{x \in \Sing} l_x^{e_x} \right) \prod_{k=1}^{\infty} f_k^{k}$.
\[characteristic-cycle-Whittaker\] The characteristic cycle of $ K_n$ is $$\left( 2\sum_{k=0}^{n-1} { \deg N -4 \choose k} \right) [\mathbb P^\vee ] + \sum_{\substack{ e_x : \Sing \to \mathbb N \\ e_x \leq c_x \\ w_k : \{1,2 \} \to \mathbb N \\ \sum_{x \in |N|} e_x + w_1 + 2w_2 =n } }2^{w_1} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) [ B_{(e_x),(w_k)}^\vee ]$$
This follows from [@saito1 Corollary 6.12], which says that the characteristic cycle of the Radon transform of a complex is the Legendre transform of the characteristic cycle of that complex, where the Radon transform of a complex $\mathcal F$ is $p_{1*} p_2^{*} \mathcal F[n-1]$ and the Legendre transform of a cycle defined in [@saito1 (6.16)]. By definition, $K_n$ is the Radon transform of $( sym_* \mathcal F^{\boxtimes n})^{S_n}[n]$, so by Lemma \[cycle-rank-two\] it suffices to compute the Legendre transform of $$\sum_{\substack{ e_x : \Sing \to \mathbb N \\ e_x \leq c_x \\ w_k : \{1,2 \} \to \mathbb N \\ \sum_{x \in |N|} e_x + w_1 + 2w_2 =n } }2^{w_1} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) [ B_{(e_x),(w_k)} ] .$$
We will do this in two parts. We will first check that the Legendre transform of $[ B_{(e_x),(w_k)} ]$ is $[ B_{(e_x),(w_k)}^\vee ] $ plus some multiple of the zero-section, and we will then compute the multiple of the zero-section. The first part is [@saito-notes Corollary 1.2.4].
For the second part, we observe that the multiplicity of the zero section is given by some intersection-theoretic formula involving the characteristic cycle. We can therefore replace $( sym_* \mathcal F^{\boxtimes n})^{S_n}[n]$ by any complex which has the same characteristic cycle. Letting $\mathcal F' =\mathbb Q_\ell^2 + \sum_{x \in \Sing} \delta_x^{c_x}[-1]$, we observe that $\mathcal F'[1]$ is a perverse sheaf and has the same rank and conductors as $K[1]$, so by Theorem \[characteristic-cycle-sym\] $$CC ( ( sym_* \mathcal F^{\boxtimes n})^{S_n}[n] ) = CC ( ( sym_* \mathcal F^{'\boxtimes n})^{S_n}[n] ).$$
Now the multiplicity of the zero-section in the characteristic cycle of the Radon transform of $(sym_* \mathcal F^{'\boxtimes n})^{S_n}[n]$ is the $(-1)^n$ times the generic Euler characteristic of that Radon transform. Hence by Lemma \[constant-sheaf-generic\] the multiplicity of the zero-section is $(-1)^n$ times the coefficient of $u^n$ in $$-2u (1-u)^{ \deg N -4} / (1+u)$$ which is the coefficient of $u^n$ in $2u (1-u)^{ \deg N- 4} / (1-u)$ which by the power series of $1/(1-u)$ and the binomial theorem is $$2\sum_{k=0}^{n-1} { \deg N -4 \choose k} .$$
Calculating the polar multiplicities
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We recall the definition of the polar multiplicities from [@mypaper §3].
Let $Y$ be a smooth variety with a map $f$ to a variety $X$ (which may be the identity), and let $x$ be a point on $X$. Let $C_1, C_2$ be algebraic cycles on $Y$ of total dimension $\dim Y$ such that $C_1 \cap C_2 \cap f^{-1}(x)$ is proper. Assume that all connected components of $C_1 \cap C_2$ are either contained in $f^{-1}(x)$ and proper or disjoint from $X$. We define their intersection number locally at $x$ $$(C_1,C_2)_{Y, x}$$ to be the sum of the degrees of the refined intersection $C_1 \cdot C_2$ [@Fulton p. 131] on all connected components of $C_1 \cap C_2$ contained in $f^{-1}(x)$.
\[polar-multiplicity-2\] Let $X$ be a smooth variety. Let $C$ be a $\mathbb G_m$-invariant cycle on the cotangent bundle $T^* X$ of $X$ of dimension $\dim X$ and let $x$ be a point on $X$.
For $0 \leq i< \dim X$, let $Y$ be a sufficiently general smooth subvariety of $X$ of codimension $i$ passing through $x$ and let $V$ be a sufficiently general sub-bundle of $T^* X$ over $Y$ with rank $i+1$. Define the $i$th polar multiplicity of $C$ at $x$ to be the intersection number $$( \mathbb P(C) , \mathbb P(V))_{ \mathbb P(T^* X), x}$$ where $\mathbb P(T^* X)$ is the projectivization of the vector bundle $T^* X$.
Here “sufficiently general" means that the strict transform of $\mathbb P(V)$ in the blowup of $\mathbb P(T^* X)$ at the fiber over $x$ does not intersect the strict transform of $\mathbb P(C)$ in that same blowup within the fiber over $x$.
For $i=\dim X$, define the $i$th polar multiplicity of $C$ at $x$ to be the multiplicity of the zero section in $C$.
View $\mathbb A^{d+1}$ as the space of linear forms on $H^0(\mathbb P^1, \mathcal O(d))$, and view the cotangent space at $\mathbb A^{d+1}$ as $H^0(\mathbb P^1, \mathcal O(d))$.
For $0 \leq r \leq d/2$, let $B_{d,r}'$ be the cycle in the cotangent bundle of $\mathbb A^{d+1}$ consisting of all pairs of linear form vanishing on all multiples of $f_2$ and the section $f_1f_2^2$, where $ f_2 \in H^0( \mathbb P^1, \mathcal O(r)) $ and $ f_1 \in H^0(\mathbb P^1, \mathcal O(d-2r))$.
Consider the map $loc_d$ from $\mathbb A^{d+1}$ to the moduli stack $\Bun_2$ of vector bundles on $\mathbb P^1$ that sends a linear form on $H^0 ( \mathbb P^1, \mathcal O(d))$ to the extension $0 \to \mathcal O \to V \to \mathcal O( d+2) \to 0$ arising from the corresponding class in $$H^0 ( \mathbb P^1, \mathcal O(d))^\vee = H^1(\mathbb P^1, \mathcal O(-d-2)) = \operatorname{Ext}^1 ( \mathcal O(d+2), \mathcal O).$$
\[local-model-schematic-smooth\] The map $loc_d$ is schematic, locally of finite type, and smooth.
The map $loc_d$ is schematic and locally of finite type because it is a map from a scheme of finite type to an Artin stack.
Let $L_1 = \mathcal O$ and let $L_2 = \mathcal O(d+1)$. For a vector bundle $V$, the tangent space to $V$ in $\Bun_2$ is given by $H^1( \mathbb P^1, V \otimes V^\vee )$. If we write $V$ as an extension $0 \to L_1 \to V \to L_2 \to 0$, the tangent space to the space of extensions of $L_2$ by $L_1$ is $\operatorname{Ext}^1 ( L_2, L_1) = H^1 ( \mathbb P^1, L_1 \otimes L_2^{-1} )$. Furthermore, the derivative at $V$ of the map from the space of the space of extensions to $\Bun_2$ is the map $ H^1 ( \mathbb P^1, L_1 \otimes L_2^{-1} ) \to H^1( \mathbb P^1, V \otimes V^\vee )$ induced by the map $L_1 \otimes L_2^{-1} \to V \otimes V^{\vee}$ given by embedding $L_1$ into $V$ and $L_2^{-1}$ into $V^\vee$. (This can be checked by working with vector bundles over $k[\epsilon]/\epsilon^2$, say.)
Hence the derivative map is surjective as long as $$(H^1 (\mathbb P^1, (V \otimes V^{\vee} ) / L_1 \otimes L_2^{-1} ))=0.$$ The quotient $$(V \otimes V^{\vee} ) /( L_1 \otimes L_2^{-1} )$$ is the extension of $L_2 \otimes L_1^{-1}$ by $L_1 \otimes L_1^{-1} + L_2 \otimes L_2^{-1}$, and so this cohomology group vanishes as soon as $\deg L_2 - \deg L_1 > -2$, which is automatic in our case as $\deg L_2 = d+2$ and $\deg L_1 =0$.
Let $W_r$ be the space of linear forms on $H^0(\mathbb P^1, \mathcal O(n))$ such that there exists nonzero $f_2$ in $H^0 (\mathbb P^1, \mathcal O(r))$ where the linear form vanishes on all multiples of $f_2$. Because $W_r$ is the projection from $ \mathbb P ( H^0 ( \mathbb P^1, \mathcal O(r)) )\times \mathbb A^{d+1}$ to $\mathbb A^{d+1}$ of a closed set, $W_r$ is closed.
\[local-model-conormal\] The closed set $B_{d,r}'$ is the conormal bundle to $W_r$.
The set $W_r$ is the projection from $ \mathbb P ( H^0 ( \mathbb P^1, \mathcal O(r)) \times \mathbb A^{d+1}$ to $\mathbb A^{d+1}$ of the set $Z_r$ of pairs of a polynomial $f_2$ and a linear form vanishing on multiples of $f_2$. For a point $(f_2, \alpha)$ in this closed set, the projection of the tangent space of $Z_r$ to $\mathbb A^{d+1}$ is the set of coefficients of $\epsilon$ in linear forms $H^0 (\mathbb P^1, \mathcal O(n)) \otimes k[\epsilon]/(\epsilon^2) \to k[\epsilon]/(\epsilon^2)$ vanishing on multiples of $f_2+ \epsilon f_3$ for some $f_3$ that mod $\epsilon$ are $\alpha$. In particular, because these linear forms vanishes on all multiples of $f_2 + \epsilon f_3$, they vanish on all multiples of $(f_2+ \epsilon f_3) (f_2 - \epsilon f_3) = f_2^2$ and thus are contained in the space of linear forms vanishing on all multiples of $f_2^2$.
For $\alpha$ a generic point of $W_r$, the map $Z_r \to W_r$ is étale over $\alpha$, so the tangent space of $W_r$ is contained in the space of linear forms vanishing on multiples of $f_2^2$. Since $W_r$ is $2r$-dimensional, so the tangent space of $W_r$ is $2r$-dimensional, and the dimension of the space of linear forms vanishing on all multiples of $f_2^2$ is $2r$, the tangent space of $W_r$ at generic point is equal to the space of linear forms vanishing on all multiples of $f_2^2$. Hence the tangent space of $W_r$ at a generic point is the perpendicular space to the fiber of $B_{d,r}'$ over that point.
The conormal bundle of a singular variety is defined as the closure of the conormal bundle of its smooth locus. Because $B_{d,r}'$ is an irreducible closed variety, it is the closure of any open subset of itself. So because it is equal to the conormal over an open set, it is equal to the conormal bundle everywhere.
\[pullback-bun-stratification\] For $0 \leq r <d/2$, $W_r$ is the inverse image under $loc_d$ of the locus in $\Bun_2$ consisting of line bundles $\mathcal O(a) + \mathcal O(b)$ with $$a \leq r \leq 2+d-r \leq b$$
Let $V = loc_d(\alpha)$. Then $V$ lies in an exact sequence $0 \to \mathcal O \to V \to \mathcal O(d+2) \to 0 $, so $V$ has degree $d+2$. We can write $V$ as $\mathcal O(a) + \mathcal O(b)$ with $ a \leq r \leq 2+d-r \leq b $ if and only if $V$ admits a nonzero map from $\mathcal O(d-r+2)$. (The if direction because if there is a map from $\mathcal O(d-r+2)$ to $V$, the saturation of its image is a subbundle with degree $\geq d-r+2$, and then the quotient has degree $\leq r < d-r+2$ so the extension splits. The only if direction is because we can map $\mathcal O(d-r+2)$ to $\mathcal O(b)$.)
Composing a map $\mathcal O(d-r+2)\to V$ with the map $V \to \mathcal O(d+2) $ from the short exact sequence, we get a map $\mathcal O(d-r+2) \to \mathcal O(d+2)$. If this map is zero, we get a nonzero map $\mathcal O(d-r+2) \to \mathcal O$, which is impossible as $d-r+2>0$.
We can view the set of nonzero maps $\mathcal O(d-r+2) \to \mathcal O(d+2)$ as the set of nonzero sections $f_2$ of $\mathcal O(r)$. Given any such map, we can pull back the class $\alpha$ in $\Ext^1 (\mathcal O(d+2), \mathcal O)$ to obtain a class in $\Ext^1( \mathcal O(d-r+2), \mathcal O)$, which concretely corresponds to the fiber product of $V$ and $\mathcal O(d-r+2)$ over $\mathcal O(d+2)$. This new extension splits if and only if $f_2$ lifts to a map $\mathcal O(d-r+2)\to V$ (Because a lift of a map $A\to C$ along a map $B\to C$ is equivalent to a section of the natural map $A \times_C B \to A$ by the universal property.)
By Serre duality, the pulled-back class in $\Ext^1 (\mathcal O(d+2), \mathcal O)$ vanishes if and only if the linear form on $\mathcal O(d-r)$ induced by composing $\alpha$ with multiplication by $f_2$ vanishes, which happens only if $\alpha$ vanishes on all multiples of $f_2$.
Let $\alpha$ be a linear form on polynomials of degree $n$. Fix $(e_x): \Sing \to \mathbb N$. Let $m_{\alpha, ( e_x)} $ be the minimum $m$ such that $\alpha$ vanishes on all multiples of $f \prod_{x\in \Sing } l_x^{e_x} $ by an element of $H^0 (\mathbb P^1, \mathcal O (n- m - \sum_x e_x))$, for some $f$ of degree $m$. Let $d_{\alpha, (e_x)} = n -2m_{ \alpha, (e_x)} - \sum_x e_x$.
\[local-model-level\] The $i$th polar multiplicity of $B_{(e_x),(w_k)}^\vee$ at a linear form $\alpha\in \mathbb P^\vee$ is equal to the $i+1-2m_{\alpha, (e_x)} - \sum_x e_x$th polar multiplicity of $B_{d,r}'$ at $0 \in \mathbb A^{d+1}$, where $d = d_{\alpha, (e_x)} $ and $r = w_2 - m$. In particular it vanishes if $r<0$.
In this proof, we simplify notation by writing $m$ for $m_{\alpha, (e_x)}$.
The vanishing if $r<0$ is clear because if $m>w_2$ then $B_{(e_x), w_k}^{\vee}$ does not intersect the fiber over $\alpha$, because $\alpha$ does not vanish on all multiples of any polynomial of degree $w_2$. Hence we may assume $m \leq w_2 \leq (n- \sum_x e_x) /2$.
We have a map $$\mathbb P^\vee = ( H^0 ( \mathbb P^1, \mathcal O(n) )^\vee - \{0\} )/ \mathbb G_m \to (H^0 ( \mathbb P^1, \mathcal O(n- \sum_x e_x) ))^\vee / \mathbb G_m$$ where we map linear forms on $\mathcal O(n)$ to linear forms on $\mathcal O(n- \sum_x e_x) $ by composing with multiplication by $\prod_x l_x^{e_x}$.
The map $loc_{n- \sum_x e_x} : H^0 (\mathbb P^1, \mathcal O(n- \sum_x e_x ))^\vee \to \Bun_2$ is invariant under scaling $H^0 (\mathbb P^1, \mathcal O(n))^\vee$, because scaling an Ext class gives an isomorphic extension. Hence we can compose the map$$\mathbb P^\vee \to H^0 ( \mathbb P^1, \mathcal O(n- \sum_x e_x) )^\vee / \mathbb G_m$$ with the descended form of $loc_{n- \sum_x e_x} $ to obtain a map $loc_{n ,(e_x)}: \mathbb P^\vee \to \Bun_2$.
The linear form $\alpha$ is sent to a vector bundle by $loc_{n, (e_x)}$ . We can write this vector bundle as a sum of line bundles $\mathcal O(a) + \mathcal O(b)$ with $a\leq b$. By the definition of $m$ and Lemma \[pullback-bun-stratification\], we have $a \leq m$ but $a > m-1$, so we must have $a=m$. Then we have $b= n+2-m$.
Let $loc_{n,(e_x), m}$ be $loc_{n, (e_x)}$ but with the resulting vector bundle twisted by $\mathcal O(-m)$, so $loc_{n,(e_x),m}(\alpha) = \mathcal O + \mathcal O(n+2-2m-\sum_x e_x )$. Letting $d=n-2m-\sum_x e_x$, we see that $d\geq 0$ by our earlier assumption on $m$, and that $loc_d(0) = \mathcal O + \mathcal O(n+2-2m -\sum_x e_x)$. Let $$Y_{n,(e_x), m} = \left( H^0 (\mathbb P^1, \mathcal O(d))^{\vee} \right) \times_{\Bun_2} \mathbb P^\vee ,$$ using $loc_d$ and $loc_{n,(e_x),m}$ to define the fiber product. Let $\mu_1$ and $\mu_2$ be the induced maps $Y_{n,(e_x), m} \to H^0 (\mathbb P^1, \mathcal O(d))^{\vee} $ and $\mathbb P^\vee$, respectively. Let $y \in Y_{n,m}$ be a point sent to $0$ by $\mu_1$ and to $\alpha$ by $\mu_2$.
By Lemma \[local-model-schematic-smooth\],$Y_{n,(e_x), m} $ is a scheme and $\mu_1$ and $\mu_2$ are smooth.
Let us check that $$\mu_1^! B_{d,r}' = \mu_2^! B_{(e_x), (w_k)}^\vee.$$
By Lemma \[local-model-conormal\], both $\mu_1^! B_{d,r}' $ and $mu_2^! B_{(e_x), (w_k)}^\vee$ are conormal bundles to their supports, so their pullbacks under a smooth map are the conormal bundles of the pullbacks of their support. By Lemma \[pullback-bun-stratification\] the support of $\mu_1^! B_{d,r}'$ is the pullback of $W_r$ under $loc_d\circ \mu_1 $ and the support of $\mu_2^! B_{(e_x), (w_k)}$ is the pullback of $W_r$ under $loc_{n, (e_x),m } \circ \mu_2$. By the commutative diagram $$\begin{tikzcd} H^0 (\mathbb P^1, \mathcal O(d))^{\vee} \arrow[ r, "loc_d" ] & \Bun_2 \\ Y_{n,m} \arrow[u, "\mu_1"] \arrow[r, "\mu_2"] & \mathbb P^\vee \arrow[u, swap,"loc_{n, (e_x), m}"] \end{tikzcd}$$ these maps are equal, so the supports are equal, and thus the cycles are equal. Because $\mu_2$ is smooth, the $i$th polar multiplicity $B_{(e_x), (w_k)}^{\vee}$ at $\alpha$ is the $i -n + \dim Y_{n,m}$th polar multiplicity of $\mu_2^! B_{(e_x),(w_k)}^\vee $ at $y$. By the identity of cycles we just proved, this is also equal to the $i +n - \dim Y_{n,m}$th polar multiplicity of of $\mu_1^! B_{d,r}'$ at $y$. Now by the smoothness of $\mu_1$, this is the $i- n + \dim Y_{n,m} - \dim Y_{n,m}+(n-2m+1- \sum_x e_x)$th polar multiplicity of $B_{d,r}'$ at zero, as desired because $$i- n + \dim Y_{n,m} - \dim Y_{n,m}+(n-2m+1- \sum_x e_x )=i+1-2m - \sum_x e_x$$.
\[local-polar-multiplicity\] The $i$th polar multiplicity of $B_{d,r}'$ at $0$ is $$2^{2r-i}{ d-i\choose d-2r} {d+1-r \choose d+1-i}.$$
Here we take binomial coefficients to vanish if evaluated outside the range where they are normally defined.
We can express $B_{d,r}'$ as the product of two vector bundles on $\mathbb P^r$, where $\mathbb P^r$ paramaterizes $f_2 \in H^0(\mathcal O(r))$ (up to scaling). The first vector bundle $V_1$ has rank $r$ and consists of linear forms on $H^0(\mathcal O(d))$ that vanish on multiples of $f_2$, while the second $V_2$ has rank $d+1-2r$ and consists of multiples of $f_2^2$ by a polynomial of degree $d$.
By definition, the polar multiplicity is the local intersection number of $V_1 \times_{\mathbb P^r} \mathbb P(V_2)$ with a general codimension $i$ subspace of $H^0(\mathcal O(d))^\vee$ and a general codimension $d-i$ subspace of $H^0( \mathcal O(d))$. It is equivalent to intersect $\mathbb P(V_1) \times_{\mathbb P^r} \mathbb P(V_2)$ with a general codimension $i-1$ subspace of $H^0(\mathcal O(d))^\vee$ and a general codimension $d-i$ subspace of $H^0( \mathcal O(d))$. In other words, this is the degree on $\mathbb P(V_1) \times_{\mathbb P^r} \mathbb P(V_2)$ of the $i-1$st power of the hyperplane class of $\mathbb P(V_1)$ times the $d-i$th power of the hyperplane class of $\mathbb P(V_2)$.
For a vector bundle $V$ of rank $w$ on $\mathbb P^r$ with total Chern class $c(V) = 1 + c_1(V) + \dots + c_w(V)$, the Segre class in $A^* (\mathbb P^r)$ is equal to $ c(V)^{-1}$ [@Fulton Proposition 4.1(a)]. Furthermore, the Segre class is equal to the sum over $j$ of the pushforward of the $j$th power of $c_1 (\mathcal O(1))$ from the projectivization $\mathbb P(V)$ to $\mathbb P^r$ [@Fulton Example 4.1.2]. In particular, the pushforward of the $j$th power of the hyperplane class is the codimension $j+1-w$ part of the Segre class, and therefore is the codimension $j+1-w$ part of $ c(V)^{-1}$.
Observe that $V_2$ is the sum of $d+1-2r$ copies of the line bundle of scalar multiples of $f_2$, which is $\mathcal O(-2)$, so $c(v_2) = (1 - 2H)^{d+1-2r}$, with $H$ the hyperplane class of $\mathbb P^r$, so the pushforward of the $d-i$th power of the hyperplane class is the degree $$(d-i) +1 - (d+1-2r) = (2r-i)$$ part of $1/(1- 2H)^{d+1-2r}$ and thus is $ (2H)^{2r-i} { d-i\choose d-2r}$.
On the other hand, $V_1$ is dual to the complement of the sum of $d+1-r$ copies of the line bundle of scalar multiples of $f_2$, which is $\mathcal O(-1)$, so $$c(V_1) = c ( \mathcal O(1)^{ d+1-r} )^{-1} = (1+H)^{- (d+1-r)},$$ so the pushforward of the $i-1$st power of the hyperplane class from $\mathbb P (V_1) $ is the degree $$i-1+ 1 - r = i-r$$ part of $(1+H)^{d+1-r}$ , which is ${d+1-r \choose d+1-i} H^{i-r}$
Hence their product is $ 2^{2r-i}{ d-i\choose d-2r} {d+1-r \choose d+1-i} H^{r}$, whose degree is $2^{2r-i}{ d-i\choose d-2r} {d+1-r \choose d+1-i}$.
This handles the case $i \neq d+1$. If $i=d+1$, the polar multiplicity is defined as the multiplicity of the zero section, which vanishes because $B_{d,r}'$ is not the zero section, and the stated formula for the polar multiplicity vanishes also, so the stated formula remains valid in this case.
\[local-polar-generating-formula\] We have $$\sum_{d=0}^{\infty} \sum_{i=0}^{d+1} \sum_{r=0}^{\lfloor d/2 \rfloor} 2^{2r-i}{ d-i\choose d-2r} {d+1-r \choose d+1-i} u^d v^r w^i = \frac{1}{(1-u^2 v w^2) (1 - u - 2u^2 vw - u^2v w^2 )}.$$
The summand vanishes unless $r \leq i \leq 2r \leq d$. We can reparameterize so that $a=i-r, b=2r-i, c= d-2r$ so $i = 2a+b$, $r=a+b$, $d=c+2a+2b$. Then the sum is
$$\sum_{a,b,c=0}^{\infty} 2^{b}{ b+c \choose c} {a+b+c+1 \choose b+c+1 } u^{c+2a+2b} v^{a+b} w^{2a+b} .$$
We have $$\sum_{a=0}^{\infty}{a+b+c+1 \choose b+c+1 } (u^2 v w^2)^a = 1/ (1- u^2 v w^2)^{b+c+2}$$
so this sum is (taking $n=b+c$) $$\sum_{b,c=0}^{\infty} {b+c \choose c} 2^b u^{c+2b} v^{b} w^{b} / (1- u^2 v w^2)^{b+c+2} = \sum_{ n=0}^{\infty} \frac{ ( u+ 2 u^2 vw)^n }{(1 - u^2 vw^2 )^{n+2}}$$
$$= \frac{1}{1- (u + 2u^2 vw) /(1- u^2 v w^2)} \frac{1}{ (1- u^2 v w^2)^2} = \frac{1}{(1-u^2 v w^2) (1 - u - 2u^2 vw - u^2v w^2 )}.$$
\[level-polar-multiplicity\] The $i$th polar multiplicity of $$\sum_{ \substack{ w_1, w_2 \in \mathbb N \\ \sum_{x } e_x + w_1+2w_2 = n }} 2^{w_1} [ B_{(e_x),(w_k)}^\vee ]$$ at $\alpha$ is equal to the coefficient of $$u^{d_{\alpha, (e_x)} } w^{i +1+d_{\alpha, (e_x)} -n }$$ in $$\frac{1 }{(1-u^2 w^2) (1 -2 u - 2u^2 w - u^2 w^2 )}$$
By Lemma \[local-model-level\], this is the same as the $i+1-2m-\sum_x e_x$th polar multiplicity of $$\sum_{ \substack{ w_1, w_2 \in \mathbb N \\ \sum_{x } e_x + w_1+2w_2 = n }} 2^{w_1} [ B_{ d_{\alpha, (e_x)} , w_2-m}' ] = \sum_{w_2 =m_{\alpha, (e_x)} }^{ \lfloor \frac{n - \sum_x e_x}{2} \rfloor } 2^{n - 2w_2 - \sum_x e_x } [ B_{ d_{\alpha, (e_x)} , w_2-m_{\alpha,e_x}}' ] = \sum_{r=0 }^{ \lfloor \frac{d_{\alpha, (e_x)} }{2} \rfloor } 2^{d_{\alpha, (e_x)} -2r } [ B_{ d_{\alpha, (e_x)} , r}' ] .$$
By Lemma \[local-polar-multiplicity\], the $j$th polar multiplicity of this cycle at $0$ is the same as $$\sum_{r =0 }^{\lfloor \frac{d_{\alpha, (e_x)} }{2} \rfloor} 2^{d_{\alpha, (e_x)} -2r } 2^{2r-j}{ d_{\alpha, (e_x)} -j\choose d_{\alpha, (e_x)} -2r} {d_{\alpha, (e_x)} +1-r \choose d_{\alpha, (e_x)} +1-j}.$$
Taking the sum over $r$ in Lemma \[local-polar-generating-formula\] and plugging in $1/4$ for $v$, we see that the $j$th polar multiplicity is $2^{d_{\alpha, (e_x)} } $ times the coefficient of $u^{d_{\alpha, (e_x)} }w^j$ in $$\frac{1 }{(1-u^2 w^2/4) (1 - u - u^2 w/2 - u^2 w^2/4 )},$$ which is the coefficient of $u^{d_{\alpha, (e_x)} } w^j$ in $$\frac{1 }{(1-u^2 w^2) (1 -2 u - 2u^2 w - u^2 w^2 )}.$$ Plugging in $$j= i-2m-\sum_x e_x+1 = i+d_{\alpha, (e_x)} +1-n,$$ we get the stated formula.
Let $\mathcal B(d)$ be the coefficient of $u^d$ in $$\frac{1}{(1-u^2) (1 - 2 \sqrt{q} u - (2\sqrt{q}+1) u^2)} = \frac{1}{ (1-u) (1+u)^2 (1 - (2\sqrt{q}+1)u ) } .$$
\[final-Radon-bound\] The trace of $\Frob_q$ on the stalk of $K_n$ at a point $\alpha \in \mathbb P^\vee$ is at most
$$\left( 2 q^{\frac{n-1}{2} } \sum_{k=0}^{n-1} { \deg N -4 \choose k} \right) + \sum_{\substack{ (e_x) : \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) q^{\frac{n}{2}}\mathcal B( d_{\alpha,(e_x)} ) .$$
By definition, the trace of $\Frob_q$ on $\mathcal H^* ( K_n)_\alpha$ is $\sum_i (-1)^i \tr (\Frob_q, \mathcal H^{-i} (K_n)_\alpha)$. Because $K_n$ is pure of weight $2n-1$ by Lemma \[Radon-perverse-pure\], the eigenvalues of Frobenius on $\mathcal H^{-i}$ have size at most $q^{ \frac{2n-1-i}{2}}$, so this sum is at most $$\sum_i q^{ \frac{2n-1-i}{2}} \dim \mathcal H^{-i} (K_n)_\alpha.$$
Now by [@mypaper Theorem 1.4], $\dim \mathcal H^{-i} (K_n)_\alpha$ is at most the $i$th polar multiplicity of $CC(K_n)$ at $\alpha$. By Lemma \[characteristic-cycle-Whittaker\], the characteristic cycle of $K_n$ is $$\left( 2\sum_{k=0}^{n-1} { \deg N -4 \choose k} \right) [\mathbb P^\vee ] + \sum_{\substack{ e_x : \Sing \to \mathbb N \\ e_x \leq c_x \\ w_k : \{1,2 \} \to \mathbb N \\ \sum_{x \in |N|} e_x + w_1 + 2w_2 =n } }2^{w_1} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) [ B_{(e_x),(w_k)}^\vee ]$$
For $i=n$, the polar multiplicity is simply the multiplicity of the zero-section, which gives the first term. For all other $i$, we observe that the polar multiplicity is the sum over $e_x$ of $\left (\prod_{x \in \Sing } {c_x \choose e_x} \right) $ times the $i$th polar multiplicity of $$\sum_{ \substack{ w_1, w_2 \in \mathbb N \\ \sum_{x } e_x + w_1+2w_2 = n }} 2^{w_1} [ B_{(e_x),(w_k)}^\vee ].$$ By Lemma \[level-polar-multiplicity\], the term corresponding to $(e_x)$ in the $i$th polar multiplicity is the coefficient of $$u^{d_{\alpha, (e_x)} } w^{i +d_{\alpha, (e_x)} +1 -n }$$ in $$\frac{1 }{(1-u^2 w^2) (1 -2 u - 2u^2 w - u^2 w^2 )}.$$
Hence the term corresponding to $(e_x),i$ in our bound for the trace is this coefficient times $q^{ \frac{2n-1-i}{2}} = q^{ - \frac{i + d_{\alpha, (e_x)} +1-n}{2} } q^{ \frac{ d_{\alpha, (e_x)} }{2}} q^{n/2}$. Thus the term corresponding to $(e_x),i$ is $q^{n/2 }$ times the coefficient of $$u^{d_{\alpha, (e_x)} } w^{i +d_{\alpha, (e_x)} +1-n }$$ in $$\frac{1 }{(1-u^2 w^2) (1 -2 \sqrt{q} u - 2\sqrt{q} u^2 w - u^2 w^2 )},$$ having multiplied $u$ by $\sqrt{q}$ and divided $w$ by $\sqrt{q}$. Summing over all values of $i$ is equivalent to summing over all powers of $w$, which is the same as substituting $w=1$. So the term corresponding to $(e_x)$ in our bound for the trace is $q^{n/2}$ times the coefficient of $u^{d_{\alpha,j}}$ in $\frac{1}{(1-u^2) (1 - 2 \sqrt{q} u - (2\sqrt{q}+1) u^2)} $. This is exactly $q^{n/2}$ times $\mathcal B(d_{\alpha,(e_x)})$. Summing over all values of $e_x$, we get the stated formula.
\[first-newform-bound\] Let $f$ be a cuspidal newform of level $N$ whose central character has finite order.
For $a,z,b \in \mathbb A_F$, let $n$ be $\deg (a/b) - 2$. Fix an isomorphism between $\mathcal O(n)$ and the line bundle whose divisor is $\operatorname{div}( \omega_0a/b)$. Let $\alpha_{a,b,z}$ be the linear form on $H^0( \mathcal O(n))$ whose value on a section $s$ is $\langle z, \omega_0 s \rangle$.
$$\left| f \left( \begin{pmatrix} a & az \\ 0 & b \end{pmatrix} \right) \right| \leq |C_f| \left(q^{1/2} 2 ^{ \deg N-3} + q \sum_{\substack{ (e_x ): \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) \mathcal B( d_{\alpha_{a,b,z},(e_x)}) \right) .$$
By Lemma \[Drinfeld-formula\].
$$f \left( \begin{pmatrix} a & bz \\ 0 & b \end{pmatrix} \right) =C_f q^{ - \deg (\omega_0a/b)/2} \eta(b)^{-1} \sum_{\substack{ w \in F^\times \\ \operatorname{div} (w\omega_0a/b) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div} (wa\omega_0/b) ) .$$
Now $\{ w \in F^\times | \operatorname{div} (w\omega_0a/b) \geq 0\}$ is simply the set of nonzero elements in $H^0 ( \mathcal O(n))$, and for such a $w$, $ \operatorname{div} (w\omega_0a/b) $ is its divisor as a section of $\mathcal O(n)$. In particular, $r_{\mathcal F} (\operatorname{div} (w\omega_0a/b) )$ is invariant under scalar multiplication. Hence we can view this sum as a sum over divisors $D$ of degree $n$ of $r_{\mathcal F}(D)$ times the sum over $w$ a nonzero section of $H^0( \mathcal O(n))$ corresponding to that divisor of $\psi(wz ) = \psi_0 ( \langle \alpha_{a,b,z} , w\rangle)$. The sections corresponding to a given divisor form a line in $H^0(\mathcal O(n))$, less the origin, and hence the sum of $\psi(wz)$ over that line is $q-1$ if $\alpha_{a,b,z} \cdot w = 0$ and is $-1$ otherwise.
Hence if $\alpha_{a,b,z} \neq 0$ then $$\sum_{\substack{ w \in F^\times \\ \operatorname{div} (wa \omega_0/b) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div} (w a \omega_0 /b )$$ is $$q \sum_{\substack{ D \textrm{ on } \mathbb P^1 \\D\textrm{ effective} \\ \deg(D) =n \\ \alpha(f)=0\textrm{ for all }f\in H^0(\mathbb P^1, \mathcal O(n))\textrm{ with } \div(f) = D}} r_{\mathcal F(D)} - \sum_{\substack{ D \textrm{ on } \mathbb P^1 \\D\textrm{ effective} \\ \deg(D) =n } } r_{\mathcal F(D)}$$ $$= -q \tr(\Frob_q, (K_n)_{\alpha_{a,b,z} })- \sum_{ D \textrm{ effective, degree }n} r_{\mathcal F} (D)$$ by Lemma \[Radon-trace-function\].
We have $$\sum_{ D \textrm{ effective, degree }n} r_{\mathcal F} (D) \leq { \deg N -4 \choose n} q^{n/2}$$ by the Riemann hypothesis for the $L$-function of $\mathcal F$. By Lemma \[final-Radon-bound\], we have
$$q\tr(\Frob_q, (K_n)_{\alpha_{a,b,z}} )) \leq \left( 2 q^{\frac{n+1}{2} } \sum_{k=0}^{n-1} { \deg N -4 \choose k} \right) + \sum_{\substack{( e_x) : \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) q^{\frac{n+1}{2}} B( d_{\alpha_{a,b,z},(e_x)} ) .$$
Hence
$$\sum_{\substack{ w \in F^\times \\ \operatorname{div} (w a \omega_0/b) \geq 0}} \psi(wz ) r_{\mathcal F} (w a \omega_0/b)$$ $$\leq \left( 2 q^{\frac{n+1}{2} } \sum_{k=0}^{n} { \deg N -4 \choose k} \right) + \sum_{\substack{ (e_x ): \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) q^{\frac{n+2}{2}} \mathcal B( d_{\alpha_{a,b,z},(e_x)} ) .$$
Now because $n = \deg \div (a \omega_0/b)$ and $\eta$ has finite order, we have $ q^{ - \deg (\omega_0a/b)/2}=q^{-n/2}$ and $| \eta(b)^{-1} |=1$ so
$$\left| f \left( \begin{pmatrix} a & bz \\ 0 & b \end{pmatrix} \right) \right| \leq|C_f|\left( 2q^{1/2} \sum_{k=0}^{n} { \deg N -4 \choose k} + q\sum_{\substack{ (e_x ): \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) B( d_{\alpha_{a,b,z},(e_x)} )\right)$$
$$\leq |C_f| \left(q^{1/2} 2 ^{ \deg N-3} + q \sum_{\substack{ (e_x ): \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) B( d_{\alpha_{a,b,z},(e_x)} )\right).$$
On the other hand, if $\alpha_{a,b,z}$ is zero, $$\sum_{\substack{ w \in F^\times \\ \operatorname{div} (wa \omega_0/b) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div} (w a \omega_0 /b))$$ is simply $$(q-1) \sum_{ D \textrm{ effective, degree }n} r_{\mathcal F} (D) \leq { \deg N -4 \choose n} (q-1) q^{n/2},$$ and multiplying by $|C_f|q^{-n/2}$, we obtain a bound of $|C_f|{ \deg N -4 \choose n} (q-1)$ from the Riemann hypothesis for $\mathcal F$. On the other hand, the contribution to our stated upper bound from the terms where $\sum_x e_x = n $ is $|C_f|q {\deg N \choose n} B(0) = |C_f| q {\deg N \choose n}$, which is at least as large, and so the stated upper bound is sufficient.
Heights of virtual cusps
========================
For $ \begin{pmatrix} a & b\\ c & d\end{pmatrix} \in GL_2(\mathbb A_F)$, $(f_1:f_2) \in \mathbb P^1 ( \overline{ \mathbb F}_q(C))$, and $e_x$ a function from $\Sing$ to $\mathbb N$ with $0\leq e_x \leq m_x$, let
$$h \left( \begin{pmatrix} a & b\\ c & d \end{pmatrix}, (f_1:f_2), (e_x) \right)$$ $$=2\deg ( \min( \operatorname{div} (af_1+cf_2), \operatorname{div}(bf_1+ df_2)+ \sum_x e_x [x] )) - \deg ( \operatorname{div}(ad-bc) + \sum_x e_x[x]) .$$
Here $af_1+cf_2$ and $bf_1+df_2$ are elements of the adeles of $\overline{\mathbb F}_q (C)$, so their divisors are divisors on $C$ defined over $\overline{\mathbb F}_q$, and the min of two divisors involves taking the min of their multiplicities at a given point.
Let us describe the analogue of this in the classical theory of modular forms. Associated to each point of $\mathbb P^1(\mathbb Q)$ is a cusp on the upper half plane. Associated to the cusp at $\infty$ is the function $\log y$ on the upper half plane which measures the height (in some sense, measures how close a point is to that cusp). We can associate to any other cusp a corresponding function, by choosing any element of $SL_2(\mathbb Z)$ that sends that cusp to $\infty$ and then composing $\log y$ with that element of $SL_2(\mathbb Z)$. One can define this function adelically, and the definition will look similar to our definition of $h$. However, our definition has the additional complexity that $(f_1:f_2)$ need not be defined over $\mathbb F_q(C)$, but in fact over $\overline{\mathbb F}_q(C)$. Thus we are studying “virtual" cusps that appear as actual cusps only over an extension of the constant field.
The height function has an alternate definition in the geometry of vector bundles on $\mathbb P^1$, which we explain next. After proving the equivalence of the two definitions, we will use whichever is more convenient for proving a particular lemma. However, everything can be proved using only the adelic definition or only the geometric one, and readers might want to try to work out the analogue in the other setting of one of the arguments we use.
Let $V_{ (e_x)} \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) $ be the vector bundle on $\mathbb P^1$ whose sections over an open set $U$ consist of all those $f_1, f_2 \in F^2$ such that the restriction of the divisor $\min ( \div ( af_1+cf_2) , \div(bf_1 +df_2) + \sum_x e_x [x] ) $ to $U$ is effective.
Let $L_{ (f_1:f_2), (e_x) } \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) \subset V_{ (e_x)} \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) $ be the sub-line-bundle of $\subset V_{ (e_x)} \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) $ generated by the meromorphic section $(f_1,f_2)$.
\[height-geometric\] We have $$h \left( \begin{pmatrix} a & b\\ c & d \end{pmatrix}, (f_1:f_2), (e_x) \right) = 2 \deg L_{ (f_1:f_2), (e_x) } \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) - \deg \det V_{ (e_x)} \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) .$$
We first can see that $$\deg \min ( \div ( af_1+cf_2) , \div(bf_1 +df_2) + \sum_x e_x [x] ) = \deg L_{ (f_1:f_2), (e_x) } \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right)$$ because this divisor, by construction, is the divisor of the line bundle $L_{ (f_1:f_2), (e_x) } \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) $.
Furthermore, we can see that $$\deg ( \div (ad-bc) + \sum_x e_x)= \deg \det V_{ (e_x)} \left( \scriptscriptstyle{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} }\right) .$$ In the case where $e_x=0$ for all $x$, this is because the matrix $\begin{pmatrix} a &b\\ c & d \end{pmatrix}$ gives gluing data for $V$, so its determinant gives gluing data for $\det V$. In the case where $(e_x) \neq 0$, this simply corresponds to a modification of this vector bundle where we extend it by a length $e_x$ module over $x$, and so the divisor class of its determinant increases by $\sum_x e_x[x]$.
The equality now follows from the definition of $h$).
\[height-max-nonnegative\] For any $\begin{pmatrix} a &b\\ c & d \end{pmatrix} \in GL_2(\mathbb A_F)$, we have $$\max_{(f_1,f_2) \in \mathbb \overline{F}_q (C)} h \left( \begin{pmatrix} a &b\\ c & d \end{pmatrix}, (f_1:f_2), (e_x) \right)\geq 0.$$
In view of Lemma \[height-geometric\], it suffices to prove that every rank two vector bundle $V$ on $\mathbb P^1$ contains a line-sub-bundle $L$ such that $2 \deg L - \deg V \geq 0 $
By the classification of vector bundles on $\mathbb P^1$, $V $ must have the form $ \mathcal O(d_1) + \mathcal O(d_2)$ for integers $d_1$ and $d_2$. Taking $L = \mathcal O(\max(d_1,d_2))$ gives $$2 \deg L - \deg V = | d_1 -d_2 | \geq 0 .$$ (It is easy to check that this choice of $L$ in fact maximizes $\det L$.)
\[height-F-invariance\] $h \left( \begin{pmatrix} a & b\\ c & d \end{pmatrix}, (f_1:f_2), (e_x) \right)$ is invariant under the right action of $\Gamma_1(N)$ on $ \begin{pmatrix} a & b\\ c &d \end{pmatrix}$.
In view of Lemma \[height-geometric\] it suffices to show that the vector bundle $V_{ \begin{pmatrix} a & b\\ c & d \end{pmatrix}, (e_x)}$ is invariant under the right action of $\Gamma_1(N)$ on $ \begin{pmatrix} a & b\\ c &d \end{pmatrix}$. In other words, we must show, for $\begin{pmatrix} a' & b'\\ c' & d' \end{pmatrix} \in \Gamma_1(N)$, that if $\div ( af_1+cf_2) $ and $\div(bf_1 +df_2) + \sum_x e_x [x]$ are effective over an open set $U$, then $\operatorname{div}( a'(af_1+cf_2)+ c' (bf_1+ df_2) )$ and $\operatorname{div}( b'(af_1+cf_2)+ d' (bf_1+ df_2) )+ \sum_x e_x [x] $ are as well. (We only need show the “if" direction) because the “only if" direction follows upon taking the inverse matrix.
At each point $x \in U$, because $ (af_1+cf_2)$ is integral, and $a'$ is integral, $a'(af_1+cf_2)$, and because the order of pole of $(bf_1+df_2)$ is at most $e_x$, and the order of zero of $c'$ is at least $m_x \geq e_x$, $ c' (bf_1+ df_2)$ is integral, so their sum is integral as well.
Similarly, because $b'$ is integral, and $(af_1+cf_2)$ is integral $b'(af_1+cf_2)$ is integral, and because $d'$ is integral, and the order of pole of $(bf_1+ df_2)$ is at most $e_x$, the order of pole of $d' (bf_1+ df_2) $ is at most $e_x$, so the order of pole of $ b'(af_1+cf_2)+ d' (bf_1+ df_2) )$ is at most $e_x$.
\[height-compact-invariance\] $h \left( \begin{pmatrix} a & b\\ c & d\end{pmatrix}, (f_1:f_2), (e_x) \right)$ is invariant under the action of $GL_2(F)$ by left multiplication on $\begin{pmatrix} a & b\\ c& d \end{pmatrix}$ and by multiplication of the inverse transpose on $\begin{pmatrix} f_1 \\ f_2 \end{pmatrix}$.
Note that $$\begin{pmatrix} f_1 \\ f_2 \end{pmatrix}^T \begin{pmatrix} a & b\\ c& d \end{pmatrix} = \begin{pmatrix} af_1 + cf_2 \\ bf_1+df_2 \end{pmatrix}^T$$ so multiplying $\begin{pmatrix} a & b\\ c& d \end{pmatrix}$ on the left by a matrix and $\begin{pmatrix} f_1 \\ f_2 \end{pmatrix}^T$ on the right by the inverse matrix does not affect those terms, and it also does not affect the degree of the determinant $ad-bc$ as the determinant of the matrix being multiplied by lies $\overline{\mathbb F}_q(C)$ and thus has zero degree.
\[height-unique-cusp\] For each $ \begin{pmatrix} a & b\\ c & d \end{pmatrix},( e_x)$, if $(f_1:f_2) \neq (f_3:f_4)$ as points of the projective line, then
$$h \left( \begin{pmatrix} a, b\\ c,d \end{pmatrix}, (f_1:f_2), (e_x) \right) + h \left( \begin{pmatrix} a, b\\ c,d \end{pmatrix}, (f_3:f_4), (e_x) \right)\leq 0$$
In particular, there is at most one $(f_1:f_2)$ such that $h \left( \begin{pmatrix} a, b\\ c,d \end{pmatrix}, (f_1:f_2), (e_x) \right)$ is positive.
In view of Lemma \[height-geometric\], it suffices to show that for $L_{12}, L_{34}$ distinct line sub-bundles of a vector bundle $V$, we have $$(2\deg L_{12} - \deg \det V) + (2 \deg L_{34} - \deg \det V) \geq 0.$$ This follows from the fact that the natural map $L_{12} + L_{34} \to V$ is injective, so $V$ is the extension of $L_{12} +L_{34}$ by a finite length module and thus $\deg \det V \leq \deg L_{12} + \deg L_{34}$.
\[height-mountain-shape\] For each $\begin{pmatrix} a & b\\ c & d \end{pmatrix}, (f_1:f_2)$, there is a unique $(e_x(f_1:f_2))$ and $h^*(f_1:f_2)$ such that $$h \left( \begin{pmatrix} a & b\\ c & d \end{pmatrix}, (f_1:f_2), e_x' \right) = h^*(f_1:f_2) - \sum_{x \in \Sing} | e_x(f_1:f_2) - e_x' |$$ for all $(e_x')$.
First we check that such a representation is unique. This follows because $h^*(f_1:f_2)$ is identifiable as the maximum value of $h$ with varying $(e_x)$, and $(e_x(f_1:f_2))$ is identifiable as the location of the maximum value.
Then we check existence. Note that $h$ may be written as a sum over the closed points of $C_{\overline{\mathbb F}_q}$ of the contribution of that point to the degrees of the relevant divisors. Only the contribution from the point $x$ depends on $e_x$, so $h$, viewed as a function of the tuple $(e_x)$, is a sum of functions depending on the individual $e_x$, plus a constant. It suffices to check that these individual functions are piecewise linear with slopes $1$ and $-1$ (in order). The multiplicity of $x$ in the divisor $ \min( \operatorname{div} (af_1+cf_2), \operatorname{div}(bf_1+ df_2)+ \sum_x e_x [x] )$ is a piecewise linear function with slopes $1$ and $0$. Doubling it gives slopes $2$ and $0$ and then subtracting the $e_x$ appearing in the $$\deg ( \operatorname{div}(ad-bc) + \sum_x e_x[x])= \deg (ad-bc) + \sum_x e_x$$ term gives slopes $1$ and $-1$.
\[height-orbit-size\] If $h^*(f_1:f_2)>0$ then the size of the orbit of $\Gal(\mathbb F_q)$ on $(f_1:f_2)$ is equal to the size of the orbit of $\Gal(\mathbb F_q)$ on $e_x(f_1:f_2)$.
This is immediate because, by Lemmas \[height-unique-cusp\] and \[height-mountain-shape\], $e_x(f_1:f_2)$ is uniquely determined by $(f_1:f_2)$ and $(f_1:f_2)$ is uniquely determined by $e_x(f_1:f_2)$.
\[height-triangularizable-characterization\] There exists $\bfg \in \Gamma_1(N)$, $\gamma \in GL_2(F)$ such that $\gamma \begin{pmatrix} a, b\\ c,d \end{pmatrix} \bfg$ is upper triangular and $\gamma^{-T} \begin{pmatrix} f_1 \\ f_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} $ if and only if $e_x(f_1:f_2) =m_x$.
We check the case “only if" first. Using Lemmas \[height-F-invariance\] and \[height-compact-invariance\], we can multiply on the left by $\gamma$ and on the right by $\bfg$, preserving the height function, in doing so, and thus preserving $e_x(f_1:f_2)$. Having done this, by our assumptions we have $f_1=0, f_2=1$ and $c=0$.
In this case $$h \left( \begin{pmatrix} a & b\\ c & d \end{pmatrix}, (f_1:f_2), (e_x) \right)= h \left( \begin{pmatrix} a & b\\ 0 & d \end{pmatrix}, (0:1), (e_x) \right)$$ $$=2\deg ( \min( \operatorname{div} (0), \operatorname{div}(d)+ \sum_x e_x [x] )) - \deg ( \operatorname{div}(ad) + \sum_x e_x[x])$$ $$=2 \deg ( \operatorname{div}(d)+ \sum_x e_x [x] ) - \deg ( \operatorname{div}(ad) + \sum_x e_x[x]) = \deg( \div{d/a}) + \sum_x e_x[x] ,$$ because $\div(0)=\infty$. The sum $\deg( \div{d/a}) + \sum_x e_x[x]$ is maximized when $e_x$ is maximized for all $x$. So indeed $e_x(f_1:f_2) = m_x$.
For the “if" direction, first observe that by Lemma \[height-orbit-size\], $(f_1:f_2)$ is fixed by $\Gal(\mathbb F_q)$. Hence by the action of a suitable element of $GL_2(F)$, we may assume $f_1=0$. Now by assumption $$2\deg ( \min( \operatorname{div} (cf_2), \operatorname{div}(df_2)+ \sum_x e_x [x] )) - \deg ( \operatorname{div}(ad) + \sum_x e_x[x])$$ is an increasing function of $e_x$ as $e_x \leq m_x$, which implies that for all $x \in \Sing$, the order of vanishing of $c$ at $x$ is at least the order of vanishing of $d$ plus $e_x$. This means we can multiply on the right by $\begin{pmatrix} 1 & 0 \\ - c_x/d_x & 1 \end{pmatrix}$ to make $c$ vanish.
At all points not in $\Sing$, we use the fact that $GL_2( \mathcal O_{F_v})$ acts transitively on $ \mathbb P^1( \mathcal O_{F_v}) = \mathbb P^1(F_v)$ to multiply on the right by something that makes $c$ vanish.
\[height-volume-comparison\] Let $a,b,z$ be adeles and let $n= \deg a -\deg b-2$.
1. We have $\sum_x e_x>n$ if and only if $$h \left( \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix}, (0:1), (e_x) \right) >-2.$$
2. For all other $(e_x)$, we have
$$d_{\alpha_{a,b,z}, (e_x)} = \max_{(f_1,f_2) \in \overline{\mathbb F}_q (C)} h \left( \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix}, (f_1:f_2), (e_x) \right)-2.$$
For part (1), note that $$h \left( \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix}, (0:1), (e_x) \right) = 2\deg ( \min( \operatorname{div} (0), \operatorname{div}(b)+ \sum_x e_x [x] )) - \deg ( \operatorname{div}(ab) + \sum_x e_x[x])$$ $$=2 \deg ( \operatorname{div}(b)+ \sum_x e_x [x] ) - \deg ( \operatorname{div}(ab) + \sum_x e_x[x]) = \deg(b/a)+ \sum_x e_x[x])$$ $$= \deg(b/a) + \sum_x e_x= \sum_x e_x - n-2$$ so it takes a value $\geq -2$ if and only if $\sum_x e_x>n$.
For part (2), recall that by definition, $d_{\alpha_{a,b,z},(e_x)}$ is $n-2m- \sum_x e_x$ where $m$ is the minimum $m$ such that $\alpha_{a,b,z}$ vanishes on all polynomial multiples of $f \prod_{x\in \Sing } l_x^{e_x} $ for some $f$ of degree $m$.
We can equivalently say that $m$ is the minimum $r$ such that the linear form $g \mapsto \alpha_{a,b,z} ( g \prod_{x\in \Sing } l_x^{e_x} ) $ on $H^0( \mathcal O(n- \sum_x e_x))$ lies in the subset $W_r$. By Lemma \[pullback-bun-stratification\], $W_r$ is the set where the extension $0 \to \mathcal O \to V \to \mathcal O(n+2- \sum_x e_x ) \to 0 $ defined, by Serre duality, from this linear form, has a line bundle summand of degree at most $r$. Because the determinant of this vector bundle $V$ has degree $n+2-\sum_x e_x$, we can see that the maximum of $2 \deg L - \deg \det V$ over line sub-bundles $L$ of $V$ is $ 2 (n+2 - m- \sum_x e_x ) - (n+2-\sum_x e_x) = n+ 2 - m - \sum_x e_x = d_{\alpha_{a,b,z},(e_x)}+2$. So to show that $$d_{\alpha_{a,b,z}, (e_x)} = \max_{(f_1,f_2) \in \overline{\mathbb F}_q (C)} h \left( \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix}, (f_1:f_2), (e_x) \right)-2$$ it suffices to show that $$V \cong V_{(e_x)} \left( {\scriptscriptstyle \begin{pmatrix} a & bz \\ 0 & b \end{pmatrix} } \right)$$ up to twisting by line bundles.
To do this, we show that they are each constructed from the same $\Ext$ class, using the adelic construction of Serre duality. For a line bundle $L$, we can express $H^1(L)$ as $(\mathbb A_F \otimes L) / (F \otimes L + ( \prod_d \mathcal O_{F_v}) \otimes L) $ by taking a torsor, trivializing it over the generic point and over a formal neighborhood of each point, and viewing the discrepancy between those two trivializations in the punctured formal neighborhood of each point as an element of $\mathbb A_F \otimes L $. Then the Serre duality pairing between $H^1(L)$ and $H^0( K L^{-1})$ is the residue pairing between these adeles and global sections of $KL^{-1}$. In our case, the relevant $H^1$-torsor is the extension class of $V_{ (e_x)} \left( \scriptscriptstyle{ \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix} }\right) / L_{ (0:1), (e_x) } \left( \scriptscriptstyle{ \begin{pmatrix} a & bz\\0 & b \end{pmatrix} }\right) $ by $L_{ (0:1), (e_x) } \left( \scriptscriptstyle{ \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix} }\right) $. We can trivialize this torsor by splitting the extension. Over the generic point, we choose the splitting $F^2 = F + F$. Over a formal neighborhood of each point, we choose the inverse image under $ \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix}^T $ of the splitting of $\mathcal O_{F_x} + \pi_x^{-e_x} \mathcal O_{F_x}$ into $\mathcal O_{F_x} $ and $ \pi_x^{-e_x} \mathcal O_{F_x}$. Because this is an extension of $\mathcal O ( \div a) $ by $\mathcal O( \div b+ \sum_x e_x [x])$, the relevant line bundle is $\mathcal O ( \div{b/a} + \sum_x e_x[x])$. The torsor is represented by the adele $z$, because the image of the vector $(1,0)$ generates the splitting over $F$ and $(1, -z) = \begin{pmatrix} a & bz\\ 0 & b \end{pmatrix}^{-T} (a,0)$ generates the local splitting, so the discrepancy is $z$. When we view this torsor as a linear form on $\mathcal O ( \div{a \omega_0/b} - \sum_x e_x[x]) = \mathcal O(n- \sum_x e_x[x])$ using Serre duality / the residue pairing, we will obtain exactly the definition of $\alpha_{a,b,z}$.
\[cusp-newform-bound\] Let $f$ be a cuspidal newform of level $N$ whose central character has finite order. We have
$$\frac{ \left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right|}{ |C_f| } \leq q^{1/2} 2 ^{ \deg N-3} +$$ $$q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2 \\ (e_x(f_1:f_2)) \neq (c_x) }} \sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n\\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B\left( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)| \right) .$$
We will show how this follows from Lemma \[first-newform-bound\]. First note that the function $f$ and our putative bound for it are left-invariant under $GL_2(\mathbb F_q(C))$ and right invariant under $\Gamma_1(N)$.
Let us first check that we may assume that $c=0$ and that, for any $(f_1:f_2) \in \mathbb P^1 ( \overline{\mathbb F}_q(C))$ with $h^*(f_1:f_2) \geq 2$ and $e_x(f_1:f_2)=c_x$, we have $f_1=0$. First suppose that there is such a $(f_1:f_2)$. Then by Lemma \[height-triangularizable-characterization\], we may act on the left by an element of $GL_2(\mathbb F_q(C))$ that sends $c$ and $f_1$ to $0$. Then by Lemma \[height-unique-cusp\], there is no other $(f_1:f_2)$ with $h^*(f_1:f_2) \geq 2$ and $e_x(f_1:f_2)=c_x$. Next suppose that there is no $(f_1:f_2)$ satisfying the hypotheses. Then we can still take $(f_1:f_2)$ with $e_x(f_1:f_2)=c_x$ and use Lemma \[height-triangularizable-characterization\] to force $c=0$, and our other claim is vacuously true.
In view of the upper-bound of Lemma \[first-newform-bound\], it suffices to show that
$$\left(q^{1/2} 2 ^{ \deg N-3} + q \sum_{\substack{ (e_x ): \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) \mathcal B( d_{\alpha_{a,b,z},(e_x)}) \right) \leq q^{1/2} 2 ^{ \deg N-3} +$$ $$q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2 \\ (e_x(f_1:f_2)) \neq (c_x) }} \sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n\\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B\left( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)| \right) .$$
In other words, it suffices to show that
$$\sum_{\substack{ (e_x ): \Sing \to \mathbb N \\ e_x \leq c_x \\ \sum_x e_x \leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x} \right) \mathcal B( d_{\alpha_{a,b,z},(e_x)})$$ $$\leq \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2 \\ (e_x(f_1:f_2)) \neq (c_x) }} \sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n\\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B\left( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)| \right) .$$
We can rearrange the sums on both sides to look more similar. It suffices to show:
$$\sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x '\leq n }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B( d_{\alpha_{a,b,z},(e_x')})$$ $$\leq\sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n}} \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2 \\ (e_x(f_1:f_2)) \neq (c_x) \\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B\left( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)| \right) .$$
For this, it is clearly sufficient to show that for every $(e_x'): \Sing \to \mathbb N$ with $e_x' \leq c_x$, $\sum_x e_x' \leq n$, and $ d_{\alpha_{a,b,z},(e_x')} \geq 0$ , there is $(f_1,f_2) \in \mathbb P^1(\overline{\mathbb F}_q(X))$ with $$h^*(f_1:f_2) \geq 2,$$ $$(e_x(f_1:f_2)) \neq (c_x) ,$$ $$\sum_x |e_x' - e_x(f_1:f_2) | \leq h^* (f_1:f_2) - 2 ,$$$$h^*(f_1:f_2) - 2- \sum_x |e_x' - e_x (f_1:f_2) | = d_{\alpha_{a,b,z},(e_x')},$$ as all the other terms that may appear on the right side are nonnegative.
So fix such an $(e_x')$. By Lemma \[height-volume-comparison\](2) and Lemma \[height-mountain-shape\], we have $$d_{\alpha_{a,d,b/d},(e_x')}= \max_{(f_1:f_2) \in \overline{\mathbb F}_q (C)} h \left( \begin{pmatrix} a & bz\\ 0 & z \end{pmatrix}, (f_1:f_2'), (e_x) \right)-2$$ $$= \max_{(f_1:f_2) \in \overline{\mathbb F}_q (C)} ( h^* ( f_1:f_2) -2 \sum_{x\in\Sing} |e_x (f_1:f_2) -e_x'| )$$
Take $(f_1:f_2)$ maximizing this function. Because we have assumed that $d_{\alpha_{a,b,z},(e_x') }\geq 0$, we have $$h^*(f_1:f_2) \geq h \left( \begin{pmatrix} a & bz\\ 0 & z \end{pmatrix}, (f_1:f_2'), (e_x) \right) \geq 2.$$ By Lemma \[height-volume-comparison\](1), because $\sum_x e_x \leq n$ and $h \left( \begin{pmatrix} a & bz\\ 0 & z \end{pmatrix}, (f_1:f_2'), (e_x) \right)\geq 2$, we have $(f_1:f_2) \neq (0:1)$. Because of this, and our earlier assumption, we conclude that $(e_x(f_1:f_2) ) \neq c_x$. This gives all the desired properties of $(e_x'), (f_1:f_2)$.
Atkin-Lehner operators and conclusion
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\[character-existence-lemma\] Let $\eta$ be a continuous character of $F^\times \backslash \mathbb A_F^\times $. Assume that the restriction of $\eta$ to $F_v^\times$ is is trivial on $\mathbb F_q^\times \subseteq F_v^\times$. Then there exists a finite order character $\theta$ of $F^\times \backslash \mathbb A_F^\times$, unramified away from $v$, that agrees with $\eta$ on $\mathcal O_{F_v}^\times$.
Consider the map $$\mathcal O_{F_v}^\times \to F^{\times} \backslash \mathbb A_F^\times / \prod_{ \substack{ w \in |C| \\ w\neq v}} \mathcal O_{F_w}^\times.$$ Its kernel is $F^\times \cap \prod_{ \substack{ w \in |C| }} \mathcal O_{F_w}^\times = \mathbb F_q^\times$. Its cokernel is $ F^{\times} \backslash \mathbb A_F^\times / \prod_{ w \in |C| } \mathcal O_{F_w}^\times =\mathbb Z$. So any continuous character of $\mathcal O_{F_v}^\times$, trivial on the kernel, can be extended to a continuous character of $\mathbb A_F^\times / \prod_{ \substack{ w \in |C| \\ w\neq v}} \mathcal O_{F_w}^\times$ that vanishes on some fixed inverse image of the generator of $\mathbb Z$. Because the restriction of $\eta$ to $\mathcal O_{F_v}^\times $ is a continuous character of a compact group, it has finite order. Because $\theta$ is trivial on some element whose image generates the quotient group, the order of $\theta$ equals the order of $\eta$ and is finite.
, it had finite order, and because the new character is trivial on a generator of the quotient group, it also has finite order
\[e-switching-lemma\] Fix a closed point $v \in |N|$ and $(e_x): \Sing \to \mathbb N$ with $e_x\leq c_x$.
Let $\tilde{e}_x = e_x$ for $x$ not lying over $v$ and equals $c_x -e_x$ for $x$ lying over $v$. Then
$$h \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 & \pi_v^{-c_v} \\ 1& 0 \end{pmatrix} , (f_1:f_2), (e_x)\right) = h \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} , (f_1:f_2), (\tilde{e}_x)\right)$$ where
We must show that $$2\deg ( \min( \operatorname{div} (af_1+cf_2), \operatorname{div}(bf_1+ df_2)+ \sum_x e_x [x] )) - \deg ( \operatorname{div}(ad-bc) + \sum_x e_x[x])$$
$$2\deg ( \min(\operatorname{div}(bf_1+ df_2), \operatorname{div} (\pi_v^{-c_v} af_1+\pi_v^{-c_v} cf_2)+ \sum_x \tilde{e}_x [x] )) - \deg ( \operatorname{div}( \pi_v^{-c_v} (ad-bc) + \sum_x \tilde{e}_x[x]).$$
It suffices to show that the contributions to the degrees from the valuations at each place are equal. At places not over $v$, this is immediate, so fix $x$ over $v$. We must show $$2 \min ( v_x ( af_1+cf_2), v_x(bf_1+df_2) +e_x) - (v_x(ad-bc) + e_x)$$$$= 2 \min( v_x(bf_1+df_2), v_x(a f_1 + c f_2) - c_v + c_v-e_x) - ( v_x(ad-bc) - c_v +c_v-e_x) .$$
This is straightforward because
$$2 \min( v_x(bf_1+df_2), v_x(a f_1 + c f_2) - c_v + c_v-e_x) - ( v_x(ad-bc) - c_v +c_v-e_x)$$ $$=2 \min( v_x(bf_1+df_2), v_x(a f_1 + c f_2) -e_x) - ( v_x(ad-bc) -e_x)$$ $$=2 \min( v_x(bf_1+df_2) +e_x, v_x(a f_1 + c f_2) ) - ( v_x(ad-bc) -e_x)-2e_x$$ $$= 2 \min ( v_x ( af_1+cf_2), v_x(bf_1+df_2) +e_x) -( v_x(ad-bc) + e_x) .$$
\[Atkin-Lehner-bound\] Let $f$ be.a cuspidal newform of level $N$ whose central character has finite order.
Assume that, for all closed points $v$ in the support of $N$, the restriction of $\eta$ to the global units $\mathbb F_q^\times \subset F_v^\times$ is trivial. Then
$$\frac{ \left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right| }{ |C_f| } \leq 2 ^{ \deg N-3} q^{1/2} +$$ $$\hspace*{-1cm} q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2 }} \sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n\\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B\left( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)|\right)$$ $$-q \max_{\substack{ (f_1:f_2) \in \mathbb P^1(\mathbb F_q(C)) \\ h^* (f_1:f_2) \geq 2 \\ e_x(f_1:f_2) \in \{0,c_x\} \textrm{ for all }x}} \sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n\\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B\left( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)| \right) .$$
Fix $(f_1:f_2)$ maximizing $$\sum_{\substack{ (e_x '): \Sing \to \mathbb N \\ e_x' \leq c_x \\ \sum_x e_x' \leq n\\ \sum_x |e_x' -e_x(f_1:f_2)| \leq h^*(f_1:f_2) -2 }} \left (\prod_{x \in \Sing } {c_x \choose e_x'} \right) \mathcal B( h^*(f_1:f_2) -2 - \sum_x |e_x' -e_x(f_1:f_2)|) .$$ We prove this by induction on the number of geometric points $x$ with $e_x(f_1:f_2)=0$.
First assume that this number is zero. Because there is a unique $(f_1:f_2)$ with $h^*(f_1:f_2) \geq 2$ and $e_x(f_1:f_2)=c_x$ by Lemma \[height-unique-cusp\], the stated bound is exactly the bound of Lemma \[cusp-newform-bound\], where the $\max$ term cancels the unique term in the sum with $e_x(f_1:f_2)= c_x$.
Next assume that it is positive. By Lemma \[height-orbit-size\], because $(f_1:f_2)$ is defined over $\mathbb F_q$, $e_x$ is stable under the Galois action, and so there is some place $v$ with $c_x=0$ for $x \in v$. Because $\eta| F_v^\times$ is trivial on $\mathbb F_q^\times$, by Lemma \[character-existence-lemma\] we can find a finite order character $\theta$ of $\mathbb A_F^\times / F^\times$, unramified away from $v$, that agrees with $\eta$ on $\mathcal O_{F_v}^\times$. Then $$f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \theta( ad-bc)$$ is left-invariant under $GL_2(F)$, right invariant under $\Gamma_1(N)$ away from $z$ and invariant under the subgroup of $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2( \mathcal O_{ F_v})$ where $c \equiv 0 \mod \pi_v^{c_v}$, $a\equiv 1 \mod \pi_v^{c_v}$.
Let $$f' \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 & 1 \\ \pi_v^{c_v} & 0 \end{pmatrix} \right) \theta( ad-bc) .$$
Then $f'$ is left invariant under $GL_2(F)$, right invariant under $\Gamma_1(N)$, cuspidal, and a Hecke eigenform. Its Langlands parameter is simply the parameter $V$ of $f$ twisted by the character $\theta^{-1}$, which as an inertia representation at $v$ is the dual representation. Thus, because the restriction to inertia of the Langlands parameter of $f$ agrees with either the Langlands parameter of $f$ or its dual at every place, the conductor of the Langlands parameter of $f'$ is $N$, and so the level of $f'$ is equal to the conductor of the Langlands parameter of $f'$, and thus $f'$ is a newform.
The $L^2$-norm of $f$ equals the $L^2$ norm of $f'$, and the adjoint $L$-function of $f$ equals the adjoint $L$-function of $f'$. By Lemma \[rankin-selberg-normalized\], it follows that $|C_f|=|C_{f'}|$.
Finally we have
$$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right| = \left| f' \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 & \pi_v^{-c_v} \\ 1& 0 \end{pmatrix} \right) \right|.$$
We will prove our bound for $f$ by inductively using the corresponding bound for $f'$. Because of Lemma \[e-switching-lemma\], $h^*(f_1:f_2)$ is preserved by this operation and $e_x(f_1:f_2)$ is sent to $\tilde{e}_x(f_1:f_2)$. So our primitive bound is preserved, but the number of $x$ with $e_x(f_1:f_2)=0$ for the maximizing $(f_1:f_2)$ is reduced by the number of $x$ lying over $v$, which is at least one, so we can apply the induction hypothesis and conclude the induction step.
Let $\mathcal S(a,b)$ be the coefficient of $u^a$ in $\frac{(1+u)^b}{ (1-u)(1+u)^2 (1- (2\sqrt{q}+1) u)}$.
\[local-squarefree-bound\] Let $f$ be.a cuspidal newform of level $N$ whose central character has finite order.
Assume that $N$ is squarefree (i.e. $c_x=1$ for all $x \in \Sing$) and for all closed points $v$ in the support of $N$, the restriction of $\eta$ to the global units $\mathbb F_q^\times \subset F_v^\times$ is trivial.
Then $$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right| \leq |C_f| \left( 2 ^{ \deg N-3} q^{1/2} + q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ \deg N /2 \geq h^* (f_1:f_2) \geq 2 }} \mathcal S( h^*(f_1:f_2) -2 , \deg N ) \right)$$
Because $c_x=1$ for all $x$, $e_x=0$ or $1$ for all $x$, thus $e_x=0$ or $c_x$ for all $x$. Because of this, Lemma \[Atkin-Lehner-bound\] reduces to
$$\label{squarefree-simplified-bound} \begin{aligned} \left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right|/ |C_f| \hspace{300pt}\\
\leq 2 ^{ \deg N-3} q^{1/2} + q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2 }} \mathcal S( h^*(f_1:f_2) -2 , \deg N ) - q \max_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 2} } \mathcal S( h^*(f_1:f_2) -2 , \deg N ) .\end{aligned}$$
This uses the fact that $\mathcal S(a,b) = \sum_{k=0}^b {b\choose k} \mathcal B(a-k)$ and so $\mathcal S(h^*(f_1:f_2)-2, \deg N)$ matches exactly the sum over $(e_x')$, because there are ${\deg N \choose k}$ tuples $(e_x')$ with $\sum_x |e_x(f_1:f_2) -e_x'| = k$.
Next observe that there is at most one $(f_1:f_2)$ with $\deg N/2 < h^*(f_1:f_2)$: Given two such $(f_1:f_2)$ and $(f_3:f_4)$, we have $$\sum_{\Sing} | e_x(f_1:f_2) - e_x(f_3:f_4) | \leq \sum_{x\in \Sing} 1 = \deg N$$ so by Lemma \[height-unique-cusp\]. and Lemma \[height-mountain-shape\] $$0 \geq h \left( \begin{pmatrix} a, b\\ c,d \end{pmatrix}, (f_1:f_2), (e_x(f_1:f_2)) \right) + h \left( \begin{pmatrix} a, b\\ c,d \end{pmatrix}, (f_3:f_4), (e_x(f_1:f_2)) \right)$$ $$= h^*(f_1:f_2) + h^*(f_3:f_4) - \sum_{x \in \Sing} | e_x(f_1:f_2) - e_x(f_3:f_4) | > \frac{\deg N}{2} + \frac{\deg N}{2} - \deg N ,$$ a contradiction.
Because there is at most one such $(f_1:f_2)$, if there is any, it is Galois-invariant by Lemma \[height-orbit-size\]. Thus it lies in $\mathbb P^1(\mathbb F_q(t))$ and it is canceled (at least) by the $\max$ term in Equation . So we may remove it from the summation, obtaining $$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right| \leq |C_f| \left( 2 ^{ \deg N-3} q^{1/2} + q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ \deg N /2 \geq h^* (f_1:f_2) \geq 2 }} \mathcal S( h^*(f_1:f_2) -2 , \deg N ) \right).$$
If we do not make assumptions on the level or central character but assume that $ h^* (f_1:f_2) \leq \deg N/2$ for all $(f_1:f_2) \in \mathbb P^1 (\mathbb F_q(t))$, a similar bound as Lemma \[local-squarefree-bound\] holds, for the same logic that there is at most $(f_1:f_2)$ with $\deg N/2 < h^*(f_1:f_2)$ and therefore it must be defined over $\mathbb F_q(t)$. This is equivalent to assuming that the point $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is not too close to any cusp. However, the definition of $\mathcal S(a,b)$ must be modified to depend on $(e_x ( f_1:f_2) )$.
\[height-packing-bound\] Assume that $N$ is squarefree. For any $\begin{pmatrix} a& b \\ c& d \end{pmatrix}$, we have $$\sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ h^* (f_1:f_2) \geq 1 }} \sum_{k=0}^{ h^* (f_1:f_2)-1} {\deg N \choose k} \leq 2^{\deg N}.$$
By Lemma \[height-mountain-shape\], $\sum_{k=0}^{ h^* (f_1:f_2)-1} {\deg N \choose k} $ is the size of the set of $e_x$ such that $$h \left( \begin{pmatrix} a& b \\ c& d \end{pmatrix}, (f_1:f_2), (e_x)\right)>0.$$ By Lemma \[height-unique-cusp\], these sets of $e_x$ do not overlap, so their total size is at most the total number of $(e_x)$, which is $2^{\deg N}$.
\[B-increasing\] $$\frac{ \mathcal S(a, \deg N) }{ \sum_{k=0}^{a+1} {\deg N \choose k} }$$ is increasing as a function of $a$ when $a$ ranges over integers at least $-1$.
We prove this by induction on $\deg N$, starting at $\deg N=2$.
By definition we have $\sum_a \mathcal S(a,2) u^a = \frac{1}{ (1-u) (1-(2\sqrt{q}+1)u)}$ so we have $\sum_a (\mathcal S(a,2) - \mathcal S(a-1,2)) u^a = \frac{1}{ 1-(2\sqrt{q} +1)u}$ and thus $\mathcal S(a,2) = \mathcal S(a-1,2)+ (2\sqrt{q} +1)^a > \mathcal S(a-1,2)$. Now for $a$ at least $1$, the denominator $\sum_{k=0}^{a+1} {2 \choose k} $ is constant, so the ratio $\frac{ \mathcal S(a, \deg N) }{ \sum_{k=0}^{a+1} {\deg N \choose k} }$ is increasing for $a$ at least $1$. For $a=-1, 0,1$ this ratio is $0, 1/3, (2\sqrt{q}+2)/4$ respectively so in fact the sequence is increasing for all $a$.
For the induction step, we use the identities $$\mathcal S(a, \deg N ) = \mathcal S(a-1, \deg N-1)+ \mathcal S(a, \deg N-1)$$ and $\sum_{k=0}^{a+1} {\deg N \choose k} = \sum_{k=0}^{a} {\deg N-1 \choose k} + \sum_{k=0}^{a +1} { \deg N-1 \choose k} $. These identities make $\mathcal S(a,\deg N)$ a convex combination of $\mathcal S(a-1,\deg N-1)$ and $\mathcal S(a, \deg N)$, so assuming $\mathcal S(a,\deg N-1)$ is increasing in $a$, we have $\mathcal S( a-1, \deg N-1) < \mathcal S(a, \deg N) < \mathcal S(a, \deg N-1)$. This shows that $\mathcal S(a,\deg N)$ is increasing for $a$ at least zero, and because $\mathcal S(-1,\deg N)=0$ while $\mathcal S(0,\deg N)=1$, it is increasing for $a$ at least $-1$, giving the induction step.
\[B-binomial-estimate\] We have $$\mathcal S(a,b) \leq \frac{1}{ 2\sqrt{q}} \left( \frac{ (2\sqrt{q}+2)^{b-2} }{ (2\sqrt{q}+1)^{b-1-a}} +2^{b-2} \right).$$
Now $\mathcal S(a, b)$ is the coefficient of $u^a$ in $$\frac{ (1+u)^{b-2} } { (1-u) (1 - (2\sqrt{q}+1) u)} .$$ We have $$\frac{1}{ (1-u) (1 - (2\sqrt{q}+1) u} = \frac{1}{2 \sqrt{q} u } \left( \frac{1}{ 1- (2\sqrt{q}+1) u} - \frac{1}{ 1-u} \right),$$ so this is $1/(2\sqrt{q})$ times the coefficient of $u^{a+1} $ in $\frac{ (1+u)^{b-2} }{ 1- (2\sqrt{q}+1) u}$ minus the coefficient of $u^{a+1}$ in $\frac{ (1+u)^{b-2} }{ 1-u}$ . The coefficient of $u^{a+1}$ in $\frac{ (1+u)^{b-2} }{ 1-u}$ so subtracting it can only lower our bound and thus we can ignore it.
The coefficient of $u^{a+1}$ in $\frac{ (1+u)^{b-2} }{ 1- du} $ is $ \sum_{k=0}^{\min( b-2, a+1)} {b-2 \choose k} d^{a+1-k} \leq \sum_{k=0}^{b-2} {b-2 \choose k} d^{a-1-k}= \frac{(d+1)^{b-2}}{d^{b-3-a}}$ so this is at most
$$\frac{1}{ 2\sqrt{q}} \frac{ (2\sqrt{q}+2)^{b-2} }{ (2\sqrt{q}+1)^{b-3-a}} .$$
\[final-Whittaker-bound\] Let $f$ be.a cuspidal newform of level $N$ whose central character has finite order.
Assume that $N$ is squarefree (i.e. $c_x=1$ for all $x \in \Sing$) and for all closed points $v$ in the support of $N$, the restriction of $\eta$ to the global units $\mathbb F_q^\times \subset F_v^\times$ is trivial.
Then $$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right|= O \left( |C_f | \left( \frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}}\right)^{\deg N} \right).$$where the constant in the big $O$ is completely uniform.
By Lemma \[local-squarefree-bound\], Lemma \[B-increasing\], Lemma \[height-packing-bound\] and Lemma \[B-binomial-estimate\] we have
$$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right| \leq |C_f| \left( 2 ^{ \deg N-3} q^{1/2} + q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ \deg N /2 \geq h^* (f_1:f_2) \geq 2 }} \mathcal S( h^*(f_1:f_2) -2 , \deg N ) \right)$$
$$\leq |C_f| \left( 2 ^{ \deg N-3}q^{1/2} +q \sum_{\substack{ (f_1:f_2) \in \mathbb P^1(\overline{\mathbb F}_q(C)) \\ \deg N /2 \geq h^* (f_1:f_2) \geq 2 }} \frac{ \mathcal S(\lfloor \deg N/2\rfloor-2, \deg N) } {\sum_{k=0}^{\lfloor \deg N/2\rfloor-1} {\deg N \choose k} } \sum_{k=0}^{ h^*(f_1,f_2)-1} {\deg N \choose k} \right).$$
$$\leq |C_f|\left(q^{1/2} 2 ^{ \deg N-3} +q 2^{\deg N} \frac{ \mathcal S(\lfloor \deg N/2\rfloor-2, \deg N) } {\sum_{k=0}^{\lfloor \deg N/2\rfloor-1} {\deg N \choose k} } \right) .$$
$$\leq |C_f|. \left( 2 ^{ \deg N-3} q^{1/2}+\frac{q 2^{\deg N}}{ \sum_{k=0}^{\lfloor \deg N/2\rfloor-1} {\deg N \choose k} }\frac{1}{ 2 \sqrt{q}} \frac{ (2\sqrt{q}+2)^{\deg N-2} }{ (2\sqrt{q}+1)^{\lceil \deg N/2 \rceil-1 }} \right)$$
$$\leq |C_f|. q^{1/2} \left( 2 ^{ \deg N-3} +\frac{ 2^{\deg N-1}}{ \sum_{k=0}^{\lfloor \deg N/2\rfloor-1} {\deg N \choose k} } \frac{ (2\sqrt{q}+2)^{\deg N-2} }{ (2\sqrt{q}+1)^{\lceil \deg N/2 \rceil-1 }} \right)$$
Now because $\deg N \geq 4$, $$\frac{ 2^{\deg N-1}}{ \sum_{k=0}^{\lfloor \deg N/2\rfloor-1} {\deg N \choose k} } = O(1).$$
We have $$\frac{ (2\sqrt{q}+2)^{\deg N-2} }{ (2\sqrt{q}+1)^{\lceil \deg N/2 \rceil -1 }} \leq \left( \frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}}\right)^{\deg N-2} .$$
This gives
$$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right| \leq |C_f| q^{1/2}\left( 2 ^{ \deg N-3} +O \left( \frac{ (2\sqrt{q}+2)^{\deg N-2} }{ (2\sqrt{q}+1)^{\lceil \deg N/2 \rceil-1 }} \right) \right).$$
Note that $$\frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}} \geq 2$$ because $4 q + 8 \sqrt{q} +4 \geq 8 \sqrt{q} + 4$ and thus $$2 ^{ \deg N-3} \leq 2^{\deg N -2} \leq \left( \frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}}\right)^{\deg N-2} .$$ so $$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right|= O \left( |C_f| q^{1/2} \left( \frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}}\right)^{\deg N-2} \right).$$
Note in addition that
$$\left( \frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}} \right)^2 \geq q^{1/2}$$ because $4q + 8 \sqrt{q} +4 \geq 2 q + \sqrt{q}$. Thus $$\left| f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) \right|= O \left( |C_f | \left( \frac{2 \sqrt{q}+2}{\sqrt{ 2 \sqrt{q} +1}}\right)^{\deg N} \right).$$ as desired.
We now recall the statement of Theorem \[sup-norm-intro\], and prove it.
\[sup-norm-final\]\[Theorem \[sup-norm-intro\]\] Let $ F = \mathbb F_q(t)$, let $N$ be a squarefree effective divisor on $\mathbb P^1$, and let $f: GL_2(\mathbb A_F) \to \mathbb C$ be a cuspidal newform of level $N$ with unitary central character. Assume that for each place $v$ in the support of $N$, the restriction of the central character of $f$ to $\mathbb F_q^\times \subset F_v^\times$ is trivial. Then $$||f||_{\infty} = O \left( \left(\frac{ 2 \sqrt{q} +2}{ \sqrt{ 2 \sqrt{q}+ 1}}\right)^{ \deg N} \right)$$ if $f$ is Whittaker normalized and $$||f||_{\infty} = O \left( \left(\frac{ 2 (1+ q^{-1/2} ) }{\sqrt{ 2 \sqrt{q}+ 1}}\right)^{ \deg N} \log(\deg N)^{3/2} \right)$$ if $f$ is $L^2$-normalized.
We see that the assumptions of Theorem \[sup-norm-intro\] match exactly the assumptions of Lemma \[final-Whittaker-bound\], except that we have a bound in the case of a central character of finite order and we wish to prove a bound in the case of a unitary central character. To reduce to the finite order case we observe that if $f$ has unitary central character then $$f\left( \begin{pmatrix} a & b\\ c & d \end{pmatrix} \right) \alpha^{ \deg (ad-bc)} ,$$ where $\alpha$ is the square root of the value of $\eta$ on some fixed adele of degree $1$, has finite order central character and the same maximum value as $f$.
If $f$ is $L^2$-normalized then it follows from Lemma \[rankin-selberg-normalized\] that $$1= 2 |C_f|^2 q^{2g-2 + \deg N } L (1 , \operatorname{ad} \mathcal F ) \prod_{v | N} \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) ) / (1 -q^{- \deg v} ) .$$
Furthermore, because $N$ is squarefree, the local monodromy representation of $\mathcal F$ at any point in the support of $N$ is either a rank two unipotent representation or a trivial representation plus a one-dimensional character. In either case, one can check that $$(\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} )$$ is one-dimension with trivial $\Frob_q$ action. This, and the fact that $g=0$, gives
$$1= 2 |C_f|^2 q^{\deg N - 2} L (1 , \operatorname{ad} \mathcal F ) .$$
$$|C_f| =2^{-1/2} q^{1 - \deg N/2 } L (1 , \operatorname{ad} \mathcal F )^{-1/2} .$$
By Lemma \[good-L-value-bound\], $$L (1 , \operatorname{ad} \mathcal F )^{-1/2} =O ( (\log \deg N)^{3/2} ).$$ This gives
$$|C_f| \leq q^{1 - \deg N/2} (\log \deg N )^{O(1)}$$ and plugging this into Lemma \[final-Whittaker-bound\] gives the stated bound in the $L^2$-normalized case.
We also prove Proposition \[automorphic-cohomology-vanishing\]:
Let $\mathcal F$ be a perverse sheaf on $\Bun_G$ whose characteristic cycle is contained in the nilpotent cone of the moduli space of Higgs bundles. Then for a $G$-bundle $\alpha$ on $X$, $\mathcal H^i(\mathcal F)_{\alpha} $ vanishes for $$i > \dim \{ v \in H^0 ( X, \operatorname{ad} (\alpha) \otimes K_X) | v \textrm { nilpotent} \} - (g-1) \dim G.$$
This follows immediately from [@mypaper Corollary 1.5], which says that the stalk cohomology of a perverse sheaf $\mathcal F$ on $X$ at a point $x$ vanishes in degree greater than the dimension of the fiber of the characteristic cycle of $\mathcal F$ over $x$ minus the dimension of $X$. We take $f: X \to \Bun_G$ a smooth map of relative dimension $r$ from a smooth scheme $X$ and $x \in X$ with $f(x) \alpha$, and apply the theorem to $f^* \mathcal F[r]$ , $x=\alpha$. Because $\dim X= \dim \Bun_G+r = (g-1) \dim G+r$ and because the fiber over $\alpha$ in the nilpotent cone of the space of Higgs bundles is $ \{ v \in H^0 ( X, \operatorname{ad} (\alpha) \otimes K_X) | v \textrm { nilpotent} \} $, the stalk cohomology of $f^*\mathcal F[r]$ vanishes in degrees $> \dim \{ v \in H^0 ( X, \operatorname{ad} (\alpha) \otimes K_X) | v \textrm { nilpotent} \} - (g-1) \dim G$, so the stalk cohomology of $\mathcal F$ vanishes in degrees $> \dim \{ v \in H^0 ( X, \operatorname{ad} (\alpha) \otimes K_X) | v \textrm { nilpotent} \} - (g-1) \dim G$.
Standard analytic number theory in the function field setting
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This appendix contains some now-standard results in analytic number theory - the Fourier expansion of modular forms, a Rankin-Selberg formula for the $L^2$-norm of a modular form in terms of an $L$-function special value, and an estimate for that special value using the Riemann hypothesis - done in the level of generality needed for this paper, except that we do not assume that $C = \mathbb P^1$.
\[Drinfeld-formula\] For any newform $f$ of level $N$ whose central character has finite order, there exists $\mathcal F$ an irreducible middle extension sheaf of rank two on $C$, pure of weight $0$, of conductor $N$, and $C_f\in \mathbb C$ such that
$$f \left( \begin{pmatrix} a & bz \\ 0 & b \end{pmatrix} \right) =C_f q^{- \frac{ \deg (\omega_0 a/b) }{2} } \eta(b)^{-1} \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} (w \omega_0 a/b ) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div}( w \omega_0 a/b )) .$$
$\mathcal F$ and $C_f$ are unique with this property.
This formula is a slight variant of one proved by @Drinfeld [(4)], who handled the case where the level $N$ is trivial. Peter Humphries explained to me where to look in the literature for the tools to perform this calculation.
This follows from the Langlands correspondence for $GL_2(F)$ proven by Drinfeld, though we will find it more convenient to use the version stated by Laurent Lafforgue.
Let $\pi = GL_2(\mathbb A_F) f$ be the representation generated under right translation by $f$. In other words, $\pi$ is a space of functions $f'$ on $GL_2(\mathbb A_F)$, with an action of $GL_2(\mathbb A_F)$ where ${\mathbf g} \in GL_2(\mathbb A_F)$ takes $h \mapsto f'({\mathbf h})$ to $h \mapsto f'({\mathbf h} \bfg)$.
By the strong multiplicity one theorem, $\pi$ is an irreducible automorphic representation of $GL_2(\mathbb A_F)$, in particular a tensor product of irreducible local representations $\pi_v$. Let $c$ be an adele such that the order of pole of $c$ at each place matches the order of vanishing of the meromorphic form $\omega_0$ and let $\psi'(z) =\psi(cz)$, so for each $v$, the maximal $\mathcal O_{K_v}$-lattice in $K_v$ on which $\psi'(z)$ vanishes is $\mathcal O_{K_v}$.
Consider the map that takes a function $f' \in \pi$ to a function $$W_{\psi', f'} (\mathbf g) = \int_{z \in \mathbb A_F/F} f'\left( \begin{pmatrix} 1 & z \\ 0 & 1\end{pmatrix} \begin{pmatrix} c & 0 \\ 0 & 1\end{pmatrix} \bfg \right) \psi(-z) .$$
The image of this map is a space of functions isomorphic to $\pi$ and on which left translation by $\begin{pmatrix} 1 & z \\ 0 & 1\end{pmatrix}$, for $z \in \mathbb A_F$, acts as multiplication by $\psi(cz)$. Thus it is the tensor product of the Whittaker models $\mathcal W(\pi_v, \psi')$ of $\pi_v$, which are the unique spaces of functions stable under right translation, isomorphic to $\pi_v$ as representations of $GL_2( F_v)$, and on which left translation acts by this character.
Now $f$ is invariant under $\Gamma_1(N)$. Hence it is a linear combination of products over $v$ of vectors invariant under $\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2 ( \mathcal O_{F_v} ) \mid b \equiv 0\mod N, d \equiv 1 \mod N \right \}$. Moreover, $N$ is minimal such that there exists a vector of this form. It follows from [@JPSS Theorem in (5.1)] that for each place $v$, the space of such vectors is one-dimensional, generated by the local newform, called the “vecteur essential" by [-@JPSS]. So $f$ is a scalar multiple of the product of these vectors at each place.
Thus $$\int_{z \in \mathbb A_F/F} f\left( \begin{pmatrix} 1 & z \\ 0 & 1\end{pmatrix} \begin{pmatrix} c & 0 \\ 0 & 1\end{pmatrix} \bfg \right) \psi(-z)= W_{\psi', f} ({\mathbf g}) = C \prod_{v} W_v(\mathbf g)$$ where $W_v$ is the Whittaker function of the local newform at $v$ for some constant $C$.
Now by [@JPSS Theorem in (4.1)] (whose proof is corrected in [@JacquetCorrection Theorem 1], the function $W_v$ has the following properties:
1. For $a \in \mathcal O_{F_v}^\times $, $$W_v \left( \bfg \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = W_v( \bfg).$$
2. We have $$\int_{ a \in F_v^\times} W_v \left( \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) |a|^{s-1/2} = L(s, \pi_v).$$
Here for convenience the integral is taken over an invariant measure where $\mathcal O_{F_v}^\times$ has measure one. To obtain the second statement, one has to observe that the function $W$ defined in [@JPSS (3.3)] takes $a$ to $X_1^{\deg a}$ and plug in $X_1=1$.
By [@Lafforgue Theorem VI.9], there is associated a two-dimensional representation $V$ of the Galois group of $F$, unramified outside the support of $V$, whose $L$-factors and $\epsilon$-factors agree with those of $\pi$. Because $V$ is a representation of $\pi_1 ( X - N)$, it defines a lisse sheaf on $X-N$. Let $\mathcal F$ be the middle extension of this lisse sheaf to $X$. By definition, the local $L$-factor of $V$ at the place $v\in |X|$ is $$\frac{1}{ \det ( 1 - |\kappa_v|^{-s} \operatorname{Frob}_{ |\kappa_v|}, \mathcal F_v)} = \sum_{n=0}^{\infty} r_{\mathcal F} (n [v]) |\kappa_v|^{-ns} .$$
Because $|a| = \kappa_v^{-\deg a}$, we can observe that the coefficient of $\kappa_v^{-ns}$ in the integral of (2) is simply the restriction of the integral to $a$ of degree $n$, where by (1) it takes a constant value, which must therefore be $ r_{\mathcal F} (n [v]) |a|^{1/2}$.
So we have $$W_v \left( \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = r_{\mathcal F} (\deg a [v] ) |a|^{1/2}$$ if $\deg a\geq 0$ and $0$ otherwise.
Multiplying, for $a \in \mathbb A_F^\times,$ $$W_{\psi',f} \left( \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right)= C r_{\mathcal F} (\operatorname{div} a) |a|^{1/2} = C r_{\mathcal F} (\operatorname{div} a) q^{-\deg (a)/2} .$$
Now by definition
$$W_{\psi,f} \left( \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = \int_{z \in \mathbb A_F/F} f\left( \begin{pmatrix} 1 & z \\ 0 & 1\end{pmatrix} \begin{pmatrix} ac & 0 \\ 0 & 1 \end{pmatrix} \right) \psi(-z) .$$
Now observe that by Fourier analysis on $\mathbb A_F/ F$,
$$f\left( \begin{pmatrix} a & z \\ 0 & 1\end{pmatrix} \right) = f\left( \begin{pmatrix} 1 & z \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = \sum_{ w \in F} \psi( w z) \int_{z' \in \mathbb A_F/F} f\left( \begin{pmatrix} 1 & z' \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) \psi(-w z').$$
Now if $w=0$ the integral vanishes by cuspidality. For all other $w$ by left invariance under $GL_2(F)$ we have
$$f\left( \begin{pmatrix} 1 & z' \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = f\left( \begin{pmatrix} w & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & z' \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = f\left( \begin{pmatrix} 1 & wz' \\ 0 & 1\end{pmatrix} \begin{pmatrix} wa & 0 \\ 0 & 1 \end{pmatrix} \right)$$ so we obtain
$$f\left( \begin{pmatrix} 1 & z \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \right) = \sum_{ w \in F^{\times} } \psi( w z) W_{\psi', f} \left( \begin{pmatrix} wa/c & 0 \\ 0 & 1 \end{pmatrix} \right)$$
$$=C \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} w \omega_0 a \geq 0}} \psi(wz) r_{\mathcal F} (\operatorname{div} w \omega_0 a ) q^{ - \deg( w \omega_0 a)/2}.$$
To calculate $f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right)$, we divide all entries by $b$, and use the definition of the central character, getting $$f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right) =C \eta(b)^{-1} \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} q^{ -\frac{ \deg( w \omega_0 a/b) }{2} } (w \omega_0 a/b ) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div} (w \omega_0 a/b) ) .$$ Because $w$ is meromorphic, $\deg w=0$, so $\deg(w\omega_0 a/b) = \deg(\omega_0 a/b)$. Thus we have $$f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right) =C q^{ -\frac{ \deg( \omega_0 a/b) }{2} } \eta(b)^{-1} \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} (w \omega_0 a/b ) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div} (w \omega_0 a/b) ) .$$
We now describe why $\mathcal F$ has the stated properties. It is irreducible because the Galois representation $V$ is irreducible. It is pure of weight zero because its central character matches $\eta$ and hence has finite order. Its conductor is $N$ because the multiplicity of a place in the Artin conductor of.a Galois representation is the exponent of the local $\epsilon$-factor of the Galois representation, which by Lafforgue’s theorem matches the exponent in the local $\epsilon$-factor of the automorphic form, which is the level.
The uniqueness of $\mathcal F$ follows from the fact that its trace function is determined by the Hecke eigenvalues of $\mathcal F$, and the uniqueness of $C$ is clear once $\mathcal F$ is fixed, because $f\neq 0$ so $f$ is not preserved by multiplication by any nontrivial scalar.
\[rankin-selberg-method\] Let $f$ be a cuspidal newform of level $N$ with central character of finite order.
Then for any $d \in \mathbb Z$, $$\int_{\substack{ GL_2(F) \backslash GL_2(\mathbb A_F) / \Gamma_1(N)\\\deg \det=d}} |f|^2$$ $$= |C_f|^2 \frac{ |J_C(\mathbb F_q) |} {1-q^{-1}} q^{2g-3 + 2 \deg N } L (1 , \operatorname{ad} \mathcal F ) \prod_{v | N} \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) ).$$
Here we take the integral against the measure that assigns mass to any coset in $$GL_2(F) \backslash GL_2(\mathbb A_F) / \Gamma_1(N)$$ equal to $1$ over the order of its automorphism group.
First observe that $$\sum_{D} |r_\mathcal F(D)|^2 q^{-s \deg D} = \prod_v \sum_n |r_\mathcal F(n[v])|^2 q^{-s n \deg v}$$ and that if $v$ is unramified $$\sum_n |r_\mathcal F(n[v])|^2 q^{ -s \deg v} = \frac{1- q^{-2s\deg v} }{ \det(1 - q^{-s\deg v } \Frob_{|\kappa_v|} \mathcal F_v \otimes \mathcal F^\vee_v }$$ while if $v$ is ramified, $$\sum_n |r_\mathcal F(n[v])|^2 q^{-s n \deg v} = \frac{1 }{ \det(1 - q^{-s\deg v } \Frob_{|\kappa_v|} \mathcal F_v \otimes \mathcal F^\vee_v }.$$ This means that, for $v$ unramified $$\sum_n |r_\mathcal F(n[v])|^2 q^{-s n \deg v}$$ is the local factor of the $L$-function $$\frac{ \zeta_C(s) L (s , \operatorname{ad} \mathcal F ) }{\zeta_C(2s) } .$$ For $v$ ramified, the local factor of this zeta function is $$\frac{ 1 - q^{ -2 s \deg v}} { \det(1 - q^{-s \deg v} \Frob_{|\kappa_v|} (\mathcal F \otimes \mathcal F^\vee)^{I_v})}$$ so $\sum_n |r_\mathcal F(n[v])|^2 q^{-s n \deg v} $ is equal to the local factor of this $L$-function times $$\frac{ \det( 1- q^{-s \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) } {1 -q^{-2s \deg v} }.$$ Hence, the product of this local factor over all $v$ is $$\sum_{D} |r_\mathcal F(D)|^2 u^{\deg D} = \prod_v \sum_n |r_\mathcal F(n[v])|^2 q^{ -s \deg v}$$ $$=\frac{ \zeta_C(s) L (s , \operatorname{ad} \mathcal F ) }{\zeta_C(2s) }\prod_{v | N} \frac{ \det( 1- q^{-s \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) } {1 -q^{-2s \deg v} }.$$
Taking a residue at $s=1$ (i.e. dropping the $\frac{1}{ 1- q^{1-s}}$ factor in $\zeta_C(s)$ and then substituting $1$ for $s$), we obtain $$\label{l-function-limit} \lim_{n \to \infty} q^{-n} \sum_{\substack {D \\ \deg D=n}} |r_{\mathcal F}(D)|^2$$ $$= \frac{ |J_C(\mathbb F_q)| L (1 , \operatorname{ad} \mathcal F ) }{(1-q^{-1} ) q^g \zeta_C(2) } \prod_{v | N} \frac{ \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) } {1 -q^{- \deg v} }.$$
On the other hand, by Lemma \[Drinfeld-formula\], we have $$f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right) =C_f q^{ -(\deg (\omega_0 a/b)/2} \eta(b)^{-1} \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} (w\omega_0 a/b) \geq 0}} \psi(wz ) r_{\mathcal F} (\operatorname{div} (w\omega_0 a/b)) .$$
For $w$ such that $\operatorname{div} (w a \omega_0/b) \geq 0$, we have $\psi(wz)=0$ as long as $z \in F$ or $z \in (a/b) \prod_v \mathcal O_{F_v}$. Thus the dual vector space to the space of $w$ such that $\operatorname{div} (w a \omega_0/b) \geq 0$ is $ \mathbb A_F / (F + (a/b) \prod_v \mathcal O_{F_v}$. Furthermore note that the value of $f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right)$ depends only on the equivalence class of $z$ modulo $F + (a/b) \prod_v \mathcal O_{F_v})$, since we can add an element of $F$ to $z$ through left multiplication by an upper unipotent in $GL_2(F)$ and add an element of $(a/b) \prod_v \mathcal O_{F_v}$ to $z$ through right multiplication by an upper unipotent in $\Gamma_1(N)$.
By the Plancherel formula, we have $$\sum_{z \in \mathbb A_F/ (F + (a/b) \prod_v \mathcal O_{F_v} )} \left|f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right)\right|^2$$ $$= |C_f|^2 q^{ - \deg (\omega_0 a/b) + \dim H^0(C, \div(\omega_0 a/b))} \sum_{\substack{ w \in F^{\times} \\ \operatorname{div} (w\omega_0 a/b) \geq 0}} | r_{\mathcal F} (\operatorname{div} (w\omega_0 a/b)|^2$$
Fix $n$ congruent to $d$ mod $2$. We will fix $b$ of degree $(d-n)/2+g-1$ and sum over all classes $a \in \mathbb F^\times\backslash \mathbb A_F^\times / \prod_v \mathcal O_{F_v}^\times$ of degree $(n+d)/2+1-g$
This implies that $\deg (a/b) = n +2-2g$ and $\deg(ab) =d$. Taking $n$ sufficiently large, $$\dim H^0(C, \div{a} + \div \omega_0 - \div b) = n +1-g$$ by Riemann-Roch and so we can simplify $$q^{ - \deg (\omega_0 a/b) + \dim H^0(C, \div(\omega_0 a/b))} = q^{ -n + n+1-g} = q^{ 1-g}.$$
When we sum over $a \in \mathbb F^\times\backslash \mathbb A_F^\times / \prod_v \mathcal O_{F_v}^\times$ of degree $(n+d)/2+1-g$, each divisor of degree $n$ will occur $q-1$ times as $\operatorname{div} (w\omega_0 a/b)$, so we obtain
$$\sum_{\substack { a\in \mathbb F^\times\backslash \mathbb A_F^\times / \prod_v \mathcal O_{F_v}^\times\\ \deg a =(n+d)/2+1-g} } \sum_{z \in \mathbb A_F/ (F + (a/b) \prod_v \mathcal O_{F_v} )} \left|f\left( \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \right)\right|^2 = q^{1-g} (q-1) |C_f|^2 \sum_{\substack {D \\ \deg D=n}} |r_{\mathcal F}(D)|^2$$
So this gives
$$\label{rankin-selberg-fourier-step} \sum_{ \substack{ GL_2(F) \bfg \Gamma_1(N) \\ \deg \bfg = d}} |f(\bfg)|^2 \lim_{n \to \infty} q^{-n} \sum_{ \substack{ a\in F^\times\backslash \mathbb A_F^\times / \prod_v \mathcal O_{F_v}^\times\\ z \in \mathbb A_F/ (F + (a/b) \prod_v \mathcal O_{F_v})\\ \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \in GL_2(F) \bfg \Gamma_1(N) }} 1$$
$$= q^{1-g} (q-1) |C_f|^2 \sum_{\substack {D \\ \deg D=n}} \lim_{n \to \infty} |r_{\mathcal F}(D)|^2$$
Now $$\sum_{ \substack{ a\in F^\times\backslash \mathbb A_F^\times / \prod_v \mathcal O_{F_v}^\times\\ z \in \mathbb A_F/ (F + (a/b) \prod_v \mathcal O_{F_v})\\ \begin{pmatrix} a & bz \\ 0 & b\end{pmatrix} \in GL_2(F) \bfg \Gamma_1(N) }} 1 =\sum_{ \substack{\gamma \in \begin{pmatrix} * & * \\ 0 & 1 \end{pmatrix} \backslash GL_2(F)\\ \bfh \in \Gamma_1(N) / \begin{pmatrix} * & * \\ 0 & 1 \end{pmatrix} \\ \gamma \bfg \bfh = \begin{pmatrix}* & * \\ 0 & b \end{pmatrix}}} \frac{ (q-1) }{ | GL_2(F) \cap \bfg \Gamma_1(N) \bfg^{-1} }$$ because $(a,z)$ determine $\gamma, \bfh$ up to the actions of upper-triangular matrices with a one in the bottom right, and vice verse $\gamma, \bfh$ determine $(a,z)$ up to multiplying $z$ by an element of $\mathbb F_q^\times$.
Furthermore we can count such elements $\gamma \in \begin{pmatrix} * & * \\ 0 & 1 \end{pmatrix} \backslash GL_2(F)$ by their bottom rows $(\gamma_1, \gamma_2)$. Such a row uniquely determines $\bfh$ modulo right multiplication, and a suitable $\bfh$ exists if and only if we can solve $$\begin{pmatrix} * & * \\ \gamma_1 & \gamma_2 \end{pmatrix} \bfg= \begin{pmatrix} * & * \\ 0 & b \end{pmatrix} \bfh^{-1} .$$ This happens if, for each $v$ not in the support of $N$, the minimal valuation of the entries of $$\begin{pmatrix} \gamma_1 & \gamma_2 \end{pmatrix} \bfg$$ is $v(b)$, and for each $v$ in the support of $N$, the first entry has valuation at least $v(b)+ m_v$ and the second is congruent to $b$ mod $\pi_v^{ v(b) + m_v}$.
In other words we are counting $\gamma_1, \gamma_2\in F$ such that $$\begin{pmatrix} \prod_{v \in N} \pi_v^{-m_v} &0 \\ 0 & 1 \end{pmatrix} b^{-1} \bfg^T \begin{pmatrix} \gamma_1 \\ \gamma_2 \end{pmatrix}$$ is integral at every place, nondegenerate at unramified places, and satisfies a congruence condition at ramified places. By Riemann-Roch, the dimension of the space of $\gamma_1, \gamma_2$ in $F$ where this is integral at every place, goes, as $n$ and thus $\deg b^{-1}$ go to $\infty$, to
$$\deg \det \left( \begin{pmatrix} \prod_{v \in N} \pi_v^{-m_v} &0 \\ 0 & 1 \end{pmatrix} b^{-1} \bfg^T\right) +2 - 2g = - \deg N - 2\left((d-n)/2 +g-1\right) + d + 2-2g$$ $$= n + 4 -4g - \deg N .$$
By a sieve using Riemann-Roch argument, the total number of $\gamma_1, \gamma_2$ satisfying the conditions is $q^{ n + 4-4g - \deg N}$ times the product of local densities. The product of the local densities at the ramified primes is $ q^{ - \deg N}$, and the product of the local densities at the unramified primes is $\frac{1}{ \zeta_C(2)\prod_{v|N} (1- q^{-2 \deg v})}.$ Plugging this into Equation , and combining with Equation , we obtain $$\frac{ q^{ 4 -4g - 2\deg N } (q-1) }{ \zeta_C(2)\prod_{v|N} (1- q^{-2 \deg v})} \int_{\substack{ GL_2(F) \backslash GL_2(\mathbb A_F) / \Gamma_1(N)\\\deg \det=d}} |f|^2$$ $$= |C_f|^2q^{1-g} (q-1) \frac{ |J_C(\mathbb F_q)| L (1 , \operatorname{ad} \mathcal F ) }{(1-q^{-1} ) q^g \zeta_C(2) } \prod_{v | N} \frac{ \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) } {1 -q^{- \deg v} }.$$
Canceling like terms, we obtain $$\int_{\substack{ GL_2(F) \backslash GL_2(\mathbb A_F) / \Gamma_1(N)\\\deg \det=d}} |f|^2$$ $$= |C_f|^2 \frac{ |J_C(\mathbb F_q) |} {1-q^{-1}} q^{2g-3 + 2 \deg N } L (1 , \operatorname{ad} \mathcal F ) \prod_{v | N} \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) ).$$
\[rankin-selberg-normalized\] Let $f$ be a cuspidal newform of level $N$ with central character of finite order. Then
$$\int_{ PGL_2(F) \backslash PGL_2(\mathbb A_F) } |f|^2$$ $$= 2 |C_f|^2 q^{2g-2 + \deg N } L (1 , \operatorname{ad} \mathcal F ) \prod_{v | N} \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) ) / (1 -q^{- \deg v} ) .$$
Here we take the integral against the uniform measure on $PGL_2(\mathbb A_F)$ that assigns measure $1$ to the image of $\Gamma_1(N)$ inside $PGL_2(\mathbb A_F)$.
We will deduce this from Lemma \[rankin-selberg-method\] by comparing the two integrals.
We can divide $PGL_2(\mathbb A_F)$ into two components, one consisting of matrices whose determinant has odd degree and one consisting of matrices whose determinant has even degree. It suffices to show that the integral over each of these is $$|C_f|^2 q^{2g-2 + \deg N } L (1 , \operatorname{ad} \mathcal F ) \prod_{v | N} \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) ) / (1 -q^{- \deg v} ) .$$
First note that $$\int_{\substack{ GL_2(F) \backslash GL_2(\mathbb A_F) / \Gamma_1(N)\\\deg \det=d}} |f|^2 = \int_{ \substack{ GL_2(F) \backslash GL_2(\mathbb A_F) \\\deg \det=d}} |f|^2$$ where the measure is the uniform measure that assigns mass $1$ to a right coset of $\Gamma_1(N)$.
Because $|f|^2$ is a pullback from $PGL_2(F) \backslash PGL_2(\mathbb A_F)$, this integral is equal to the integral of the pushforward measure from the degree $d$ part of $GL_2(F) \backslash GL_2(\mathbb A_F)$ to $PGL_2(F) \backslash PGL_2(\mathbb A_F)$ of $|f|^2$. The puhsforward measure is right invariant by the degree zero elements of $GL_2(\mathbb A_F)$, which have two orbits on $PGL_2(\mathbb A_F)$, the even and odd components. If $d$ is even, we get an integral over an even component with a uniform measure, and if $d$ is odd, we get an integral over the odd component with the uniform measure.
Thus, the integral over the even component of $PGL_2(F) \backslash PGL_2(\mathbb A_F)$ of $|f|^2$ against the uniform measure that assigns $1$ to the image of $\Gamma_1(N)$ is equal to the integral over the degree $0$ parts of $GL_2(F) \backslash GL_2(\mathbb A_F)$ of $|f|^2$ against the uniform measure that assigns $1$ to $\Gamma_1(N)$, divided by the number of $\Gamma_1(N)$-cosets among the inverse image of the image of $\Gamma_1(N)$. This count of cosets is the degree zero part of $| F^\times \backslash \mathbb A_F^\times / \mathbb A_F^\times \cap \Gamma_1(N)| $ and thus is $ \frac{ |J_C(\mathbb F_q)| q^{\deg N} \prod_{v | n} (1- q^{-\deg v} ) }{q-1} $. Dividing the formula of Lemma \[rankin-selberg-method\], we see that the integral over the even component is $$|C_f|^2 q^{2g-2 + \deg N } L (1 , \operatorname{ad} \mathcal F ) \prod_{v | N} \det( 1- q^{- \deg v} \Frob_{|\kappa_v|}, (\mathcal F \otimes \mathcal F^\vee)^{I_v} / (\mathcal F^{I_v} \otimes (\mathcal F^\vee)^{I_v} ) ) / (1 -q^{- \deg v} ) ,$$ and the integral over the odd component is given by the same formula, for the same reason.
\[good-L-value-bound\] Assume $C$ admits a degree $d$ map to $\mathbb P^1$ defined over $\mathbb F_q$.
We have $$( \Theta ( \log (3g-3 + 2\deg N )))^{-3d} \leq \left| \log L(1,\operatorname{ad}(\mathcal F)) \right| \leq ( O ( \log (3g-3 + 2\deg N )))^{3d}$$ where the constant depends only on $q$ and $D$.
We have $$\log L(1,\operatorname{ad}(\mathcal F)) = \sum_{n=0}^{\infty} \tr (\Frob_q^n , H^1 (C_{\overline{\mathbb F}_q}, \operatorname{ad}(\mathcal F) )) q^{-n} /n$$
We have the upper bound $$\tr (\Frob_q^n , H^1 (C_{\overline{\mathbb F}_q}, \operatorname{ad}(\mathcal F) )) \leq q^{n/2} \dim H^1 (C_{\overline{\mathbb F}_q}, \operatorname{ad}(\mathcal F) ) \leq q^{n/2} (3g-3 + 2 \deg N)$$ and the upper bound $$\tr (\Frob_q^n , H^1 (C_{\overline{\mathbb F}_q}, \operatorname{ad}(\mathcal F) )) = \sum_{x \in C(\mathbb F_{q^n} )} \tr (\Frob_{q^n}, \ad (\mathcal F)_x) \leq 3 | C( \mathbb F_{q^n})| \leq 3 d (q^n+1)$$
Let $k = \lfloor 2\log_q \frac{ 3g -3 +2 \deg N}{3d} \rfloor$.
We use the first bound for $n> k$ and the second bound for $n \leq k$. The total contribution from the first bound is $$\leq (3g-3 + 2 \deg N) \sum_{n=k+1}^\infty q^{ -n/2} /n \leq ( 3g-3 + 2 \deg N) q^{ - (k+1)/2 } / (k+1) (1-1/\sqrt{q} ) \leq 3 d / (k+1) (1-1/\sqrt{q}).$$ The total contribution from the second bound is $$\sum_{n=1}^k 3 d (1 + q^{-n}) /n \leq 3d \log (1+ q^{-1}) + \sum_{n=1}^k 3d /n$$ so the total contribution from both is $$3d \log (1+ q^{-1}) + \sum_{n=1}^{k+1} 3d /n \leq 3d \log (1+ q^{-1}) + 3d \log (k+1) + 3d .$$ Exponentiating, this is at most $O ( (k+1)^{3d} )= O ( ( \log_q ( 3g-3+ 2\deg N) )^{3d})$, where the constant bounds $e^{3d+ 3d \log (1+q^{-1})}$, and at least the inverse of that same term.
[^1]: I learned the information in this paragraph and the previous one from user:t3suji on mathoverflow
|
---
address: |
Center for Nonlinear Science, School of Physics,\
Georgia Institute of Technology, Atlanta 30332-0430, U.S.A.\
E-mail: [email protected]
author:
- 'Predrag Cvitanovi[ć]{} and Yueheng Lan'
date: 'March 10 2003, printed '
title: Turbulent fields and their recurrences
---
Introduction
============
Chaos is the norm for generic Hamiltonian flows, and for path integrals that implies that instead of a few, or countably many extremal configurations, classical solutions populate fractal sets of saddles. For the path-integral formulation of quantum mechanics such solutions were discovered by Gutzwiller who derived a trace formula that relates a semi-classical approximation of the energy eigenspectrum to the classical periodic solutions. While the theory has worked very well in quantum mechanical applications, these ideas remain largely unexplored in quantum field theory.
The classical solutions for most strongly nonlinear field theories are nothing like the harmonic oscillator degrees of freedom, the electrons and photons of QED; they are unstable and highly nontrivial, accessible only by numerical techniques. The new aspect, prerequisite to a semi-classical quantization of strongly nonlinear field theories, is the need to determine a large number of spatio-temporally periodic solutions for a given classical field theory. Why periodic?
The dynamics of strongly nonlinear classical fields is turbulent, not “laminar”, and how are we to think about turbulent dynamics? Hopf and Spiegel have proposed that the turbulence in spatially extended systems be described in terms of recurrent spatiotemporal patterns. Pictorially, dynamics drives a given spatially extended system through a repertoire of unstable patterns; as we watch a turbulent system evolve, every so often we catch a glimpse of a familiar pattern. For any finite spatial resolution, for a finite time the system follows approximately a pattern belonging to a finite alphabet of admissible patterns, and the long term dynamics can be thought of as a walk through the space of such patterns, just as chaotic dynamics with a low dimensional attractor can be thought of as a succession of nearly periodic (but unstable) motions. So periodic solutions are needed both to quantify “turbulence” in classical field theory, and as a starting point for the semi-classical quantization of a quantum field theory.
There is a great deal of literature on numerical periodic orbit searches. Here we take as the starting point Cvitanović et al. webbook, and in briefly review the Newton-Raphson method for low-dimensional flows described by ordinary differential equations (ODEs), in order to motivate the [**]{} approach that we shall use here, and show that it is equivalent to a minimization method.
The problem one faces with high-dimensional flows is that their topology is hard to visualize, and that even with a decent starting guess for a point on a periodic orbit, methods like the Newton-Raphson method are likely to fail. In we describe a new method for finding spatio-temporally periodic solutions of extended, infinite dimensional systems described by partial differential equations (PDEs), and in we discuss a simplification of the method specific to Hamiltonian flows.
The idea is to make an informed rough guess of what the desired periodic orbit looks like globally, and then use variational methods to drive the initial guess toward the exact solution. Sacrificing computer memory for robustness of the method, we replace a guess that a [*point*]{} is on the periodic orbit by a guess of the [*entire orbit*]{}. And, sacrificing speed for safety, we replace the Newton-Raphson [*iteration*]{} by the , a differential [*flow*]{} that minimizes a computed as deviation of the approximate flow from the true flow along a smooth loop approximation to a periodic orbit.
In the method is tested on several systems, both infinite-dimensional and Hamiltonian, and its virtues, shortcomings and future prospects are discussed in .
Periodic orbit searches {#s:POserchLowDim}
=======================
A periodic orbit is a solution $(\pSpace,\period{})$, $\pSpace \in \reals^{d}$, $\period{} \in \reals$ of the [*periodic orbit condition*]{} f\^() = , > 0 \[e:periodic\] for a given flow or mapping $\pSpace \to f^{t}(\pSpace)$. Our goal here is to determine periodic orbits of flows defined by first order ODEs =() , \^d , (,v) **[T]{} \[p-1\] in many (even infinitely many) dimensions $d$. Here $\pS$ is the phase space (or state space) in which evolution takes place, $\bf{T}\pS$ is the tangent bundle, and the vector field $\pVeloc(\pSpace)$ is assumed smooth (sufficiently differentiable).**
A [*prime*]{} cycle $p$ of period $\period{p}$ is a single traversal of the orbit. A cycle point of a flow which crosses a Poincaré section $\cl{p}$ times is a fixed point of the $f^\cl{p}$ iterate of the Poincaré section return map $f$, hence one often refers to a cycle as a “fixed point”. By [cyclic invariance]{}, stability eigenvalues and the period of the cycle are independent of the choice of an initial point, so it suffices to solve at a single cycle point. Our task is thus to find a cycle point $\pSpace \in p$ and the shortest time $\period{p}$ for which has a solution.
If the cycle is an attracting limit cycle with a sizable basin of attraction, it can be found by integrating the flow for sufficiently long time. If the cycle is unstable, simple integration forward in time will not reveal it, and methods to be described here need to be deployed. In essence, any method for solving numerically the periodic orbit condition $F(\pSpace)=\pSpace-f^{\period{}}(\pSpace)=0$ is based on devising a new dynamical system which possesses the same cycle, but for which this cycle is attractive. Beyond that, there is a great freedom in constructing such systems, and many different methods are used in practice.
in 1 Dimension
---------------
(a)![ (a) Newton method: bad initial guess $\pSpace^{(b)}$ leads to the Newton estimate $\pSpace^{(b+1)}$ far away from the desired zero of $F(\pSpace)$. Sequence $\cdots,\,\pSpace^{(m)},\,\pSpace^{(m+1)},\cdots$, starting with a good guess converges super-exponentially to $\pSpace^*$. (b) : any initial guess $\pSpace(0)$ in the monotone interval $[\pSpace^L,\pSpace^R]$ of $F(\pSpace)$ flows to $\pSpace^*$ exponentially fast. Both methods diverge if they fall into the basin of attraction of a local minimum $\pSpace^{c}$. []{data-label="f:NewtonDescent"}](NewtonMeth.eps "fig:"){width="52.56000%"} (b)![ (a) Newton method: bad initial guess $\pSpace^{(b)}$ leads to the Newton estimate $\pSpace^{(b+1)}$ far away from the desired zero of $F(\pSpace)$. Sequence $\cdots,\,\pSpace^{(m)},\,\pSpace^{(m+1)},\cdots$, starting with a good guess converges super-exponentially to $\pSpace^*$. (b) : any initial guess $\pSpace(0)$ in the monotone interval $[\pSpace^L,\pSpace^R]$ of $F(\pSpace)$ flows to $\pSpace^*$ exponentially fast. Both methods diverge if they fall into the basin of attraction of a local minimum $\pSpace^{c}$. []{data-label="f:NewtonDescent"}](NewtonDescent.eps "fig:"){width="37.44000%"}
Newton’s method for determining a zero $\pSpace^*$ of a function $F(\pSpace)$ of one variable is based on a linearization around a starting guess $\pSpace^{(0)}$: F() F(\^[(0)]{})+F’(\^[(0)]{})(-\^[(0)]{}). \[fctayl\] An improved approximate solution $\pSpace^{(1)}$ of $F(\pSpace)=0$ is then $
\pSpace^{(1)} = \pSpace^{(0)} - F(\pSpace^{(0)})/F'(\pSpace^{(0)})
\,.
%\label{NewtIt}
$ [*Provided*]{} that the $m$th guess is sufficiently close to $\pSpace^*$, the Newton iteration \^[(m+1)]{}=\^[(m)]{}-F(\^[(m)]{}) / F\^(\^[(m)]{}) \[c-1\] converges to $\pSpace^*$ super-exponentially fast (see (a)). In order to avoid jumping too far from the desired $\pSpace^*$, one often initiates the search by the [*damped Newton method*]{}, $$\Delta \pSpace^{(m)}=\pSpace^{(m+1)}-\pSpace^{(m)}=
-{F(\pSpace^{(m)}) \over F^{\prime}(\pSpace^{(m)})}\,\Delta \tau
\,,\qquad
0 < \Delta \tau \leq 1
\,,$$ takes small $\Delta \tau$ steps at the beginning, reinstating to the full $\Delta \tau=1$ jumps only when sufficiently close to the desired $\pSpace^*$.
Let us now take the extremely cautious approach of keeping all steps [*infinitesimally*]{} small, and replacing the discrete sequence $\pSpace^{(m)},
\pSpace^{(m+1)}, \ldots $ by the fictitious time $\tau$ flow $\pSpace=\pSpace(\tau)$: $$d\pSpace=-\frac{F(\pSpace)}{F^{\prime}(\pSpace)} d \tau
\,,\qquad
\tau \in [0,\infty]
\,.
\label{c-2}$$ If a simple zero, $F^{\prime}(\pSpace^*)\neq 0$, exists in any given monotone lap of $F(\pSpace)$, it is the attractive fixed point of the flow (see (b)).
While reminiscent of “gradient descent” methods, this is a flow, rather than an iteration. For lack of established nomenclature we shall refer to this method of searching for zeros of $F(\pSpace)$ as the [**]{}, and now motivate it by re-deriving it from a minimization principle. Rewriting in terms of a “” $F(\pSpace)^2$, $$d\tau=-\frac{F^{\prime}(\pSpace)}{F(\pSpace)} d\pSpace=
- \left(\frac{1}{2}\frac{d}{d\pSpace}\ln F(\pSpace)^2\right)d\pSpace,$$ and integrating, $$\tau=-\frac{1}{2}\int_{\pSpace(0)}^{\pSpace}d\pSpace'
\left(\frac{d}{d\pSpace}\ln F(\pSpace')^2\right)
=-\frac{1}{2}
\ln \frac{F(\pSpace)^2}{F(\pSpace(0))^2},$$ we find that the deviation of $F(\pSpace)$ from $F(\pSpace^*)=0$ decays exponentially with the fictitious time, $$F(\pSpace(\tau))=F(\pSpace(0)) \, e^{-\tau}
\,,\quad
\label{c-3}$$ with the fixed point $
\pSpace^* =\lim _{\tau \rightarrow \infty} \pSpace(\tau)
%, \label{c-4}
$ reached at exponential rate. In other words, the , derived here as an overcautious version of the damped Newton method, is a flow that minimizes the $F(\pSpace)^2$.
Multi-dimensional {#s:MultDimDescent}
------------------
Due to the exponential divergence of nearby trajectories in chaotic dynamical systems, fixed point searches based on direct solution of the fixed-point condition as an initial value problem can be numerically very unstable. Methods that start with initial guesses $\lSpace_i$ for a number of points along the cycle are considerably more robust and safer. Hence we consider next a set of periodic orbit conditions F\_i()=\_i-f\_i()=0 ,\^[nd]{} \[mltDimZero\] where the periodic orbit traverses $n$ Poincaré sections (multipoint shooting method), $f_i(\pSpace)$ is the Poincaré return map from a section to the next one, and the index $i$ runs over $nd$ values, $d$ dimensions for each Poincaré section crossing. In this case the expansion yields the Newton-Raphson iteration \^[(m+1)]{}\_k= \^[(m)]{}\_k - ( [-(\^[(m)]{})]{} )\_[kl]{} F\_l(\^[(m)]{}) , ()\_[kl]{} = [[f\_k()]{} ]{} , \[c-5\] where $
\monodromy(\pSpace)
$\[jacoBcret\] is the \[$d \cl{} \times d \cl{}$\] of the map $f(\pSpace)$, and $\lSpace^{(m)}$ is the $m$th Newton-Raphson cycle point estimate.
The method now takes form d\_j=-F\_i()d . \[c-6\] Contracting both sides with $F_i(\lSpace)$ and integrating, we find that ()=\_[i=1]{}\^[d]{} F\_i()\^2 \[c-7\] can be interpreted as the , also decaying exponentially, $
\costF(\lSpace(\tau))=\costF(\lSpace(0)) \, e^{-2\tau}
$, with the fictitious time gradient flow now taking a multi-dimensional form: = - F() . \[c-8\]
Biham and collaborators (see also ) were the first to introduce a fictitous time flow in searches for periodic orbits of low-dimensional maps, with a diagonal matrix with entries $\pm 1$ in place of the $1/({\partial F / \partial\pSpace})$ matrix in . As we lack good visualizations of high-dimensional flows, for us the ${\partial F / \partial\pSpace}$ matrix is essential in determining the direction in which the cycle points should be adjusted.
Here we have considered the case of the guess $\lSpace$ a vector in a finite-dimensional vector space, with $\costF(\lSpace)$ the penalty for the distance of $F(\lSpace)$ from its zero value at a fixed point $\pSpace^*$. Our next task is to generalize the to a [*al*]{} $\,\,\costF[\lSpace]$ which measures the distance of a loop $\lSpace(s)\in \Loop(\tau)$ from a periodic orbit $\pSpace(t) \in p$.
in Loop Space {#s:POserchLops}
==============
For a flow described by a set of ODEs, multipoint shooting method of can be quite efficient. However, multipoint shooting requires a set of phase space Poincaré sections such that an orbit leaving one section reaches the next one in a qualitatively predictable manner, without traversing other sections along the way. In turbulent, high-dimensional flows such sequences of sections are hard to come by. One cure for this ill might be a large set of Poincaré sections, with the intervening flight segments short and controllable. Here we shall take another path, and discard fixed Poincaré sections altogether.
Emboldened by success of methods such as the multipoint shooting (which eliminates the long-time exponential instability by splitting an orbit into a number of short segments, each with a controllable expansion rate) and the cyclist relaxation methods (which replace map iteration by a contracting flow whose attractor is the desired periodic orbit of the original iterative dynamics), we now propose a method in which the initial guess is not a finite set of points, but an entire smooth, differentiable closed loop.
A general flow has no extremal principle associated with it (we discuss the simplification of our method in the case of Hamiltonian mechanics in ), so there is a great deal of arbitrariness in constructing a flow in a loop space. We shall introduce here the simplest which penalizes mis-orientation of the local loop tangent vector $\lVeloc(\lSpace)$ relative to the dynamical velocity field $\pVeloc(\lSpace)$ of , and construct a flow in the loop space which minimizes this function. This flow is corralled by general topological features of the dynamics, with rather distant initial guesses converging to the desired orbit. Once the loop is sufficiently close to the periodic orbit, faster numerical algorithms can be employed to pin it down.
\(c) ![ (a) A continuous path; (b) a loop $\Loop$ with its tangent velocity vector $\lVeloc$; (c) a periodic orbit $p$ defined by the vector field $\pVeloc(\pSpace)$. []{data-label="f:loops"}](path.eps "fig:"){width="2.5cm"} (b) ![ (a) A continuous path; (b) a loop $\Loop$ with its tangent velocity vector $\lVeloc$; (c) a periodic orbit $p$ defined by the vector field $\pVeloc(\pSpace)$. []{data-label="f:loops"}](loop.eps "fig:"){width="3.5cm"} (c) ![ (a) A continuous path; (b) a loop $\Loop$ with its tangent velocity vector $\lVeloc$; (c) a periodic orbit $p$ defined by the vector field $\pVeloc(\pSpace)$. []{data-label="f:loops"}](porbit.eps "fig:"){width="4.0cm"}
In order to set the notation, we shall distinguish between (see ):
[**closed path:**]{} any closed (not necessarily differentiable) continuous curve $J \subset \pS$.
[**loop:**]{} a smooth, differentiable closed curve $\lSpace(s)\in \Loop \subset
\pS$, parametrized by $s \in [0,2\pi]$ with $\lSpace(s)=\lSpace(s+2\pi)$, with the magnitude of the loop tangent vector fixed by the (so far arbitrary) parametrization of the loop, $$\lVeloc(\lSpace)=\frac{d \lSpace}{ds}\,, \quad \lSpace=\lSpace(s) \in \Loop
\,.$$ [**annulus:**]{} a smooth, differentiable surface $\lSpace(s,\tau)\in \Loop(\tau)$ swept by a family of loops $\Loop(\tau)$, by integration along a fictitious time flow (see (a)) $$\dot{\lSpace}=\frac{\partial \lSpace}{\partial \tau}
\,.$$ [**periodic orbit:**]{} given a smooth vector field $\pVeloc=\pVeloc(\pSpace),\; (\pSpace,\pVeloc) \in {\bf T} \pS$, periodic orbit $\pSpace(t) \in p$ is a solution of $$\frac{dx}{dt}=\pVeloc(\pSpace)
\,,\quad
\mbox{ such that } \pSpace(t)=\pSpace(t+\period{p}),$$ where $\period{p}$ is the shortest period of $p$.
in the Loop Space
------------------
\(a) ![ (a) An annulus $\Loop(\tau)$ with vector field $\dot{\lSpace}$ connecting smoothly the initial loop $\Loop(0)$ to a periodic orbit $p$. (b) In general the orientation of the loop tangent $\lVeloc(\lSpace)$ does not coincide with the orientation of the velocity field $\pVeloc(\lSpace)$; for a periodic orbit $p$ it does so at every $x \in p$. []{data-label="f:velocField"}](tube.eps "fig:"){width="4.0cm"} (b) ![ (a) An annulus $\Loop(\tau)$ with vector field $\dot{\lSpace}$ connecting smoothly the initial loop $\Loop(0)$ to a periodic orbit $p$. (b) In general the orientation of the loop tangent $\lVeloc(\lSpace)$ does not coincide with the orientation of the velocity field $\pVeloc(\lSpace)$; for a periodic orbit $p$ it does so at every $x \in p$. []{data-label="f:velocField"}](velocField.eps "fig:"){width="6.5cm"}
In the spirit of , we now define a al for a loop and the associated fictitious time $\tau$ flow which sends an initial loop $\Loop(0)$ via a loop family $\Loop(\tau)$ into the periodic orbit $p=\Loop(\infty)$, see (a). The only thing that we are given is the velocity field $\pVeloc(\pSpace)$, and we want to “comb” the loop $\Loop(\tau)$ in such a way that its tangent field $\lVeloc$ aligns with $\pVeloc$ everywhere, see (b). The simplest al for the job is = [1 2]{} \_ds(- )\^2 , = ((s,)), = ((s,)) . \[loopCostFct\] As we have fixed the loop period to $s=2\pi$, the parameter $\lambda = \lambda(s,\tau)$ is needed to match the magnitude of the tangent field $\lVeloc$ (measured in the loop parametrization units $s$) to the velocity field $\pVeloc$ (measured in the dynamical time units $t$). $|\pVeloc|$ cannot vanish anywhere along the loop $\Loop(\tau)$, as in that case the loop would be passing through an equilibrium point, and have infinite period. We shall take a very simple choice, and set $\lambda$ to be a global, space-independent loop parameter $\lambda = \lambda(\tau)$. In the limit where the loop is the desired periodic orbit $p$, this $\lambda$ is the ratio of the dynamical period $\period{p}$ to the loop parametrization period $2\pi$, $\lambda={\period{p}}/{2\pi}$. More general choices of the parametrization $s$ will be discussed elsewhere.
Take a derivative of the al $\costF[\lSpace]$ with respect to the (yet undetermined) fictitious time $\tau$, $$%\beq
{d \costF \over d\tau} = {1 \over \pi} \oint_\Loop ds\,
(\lVeloc-\lambda\, \pVeloc)\,
{d~ \over d\tau}
(\lVeloc-\lambda\, \pVeloc)
\,.
%\label{CostFctDer}$$ The simplest, exponentially decreasing al is obtained by taking the $\lSpace(s,\tau)$ dependence on $\tau$ to be point-wise proportional to the deviation of the two vector fields (-)= -(-) , \[fm0\] so the fictitious time flow drives the loop to $\Loop(\infty)=p$, see (a): -= e\^[-]{} (- )|\_[=0]{} . \[fm\]
Making the $\lSpace$ dependence in explicit we obtain our main result, the PDE which evolves the initial loop $\Loop(0)$ into the desired periodic orbit $p$ -- = -, \_[ij]{}(x)= \[bd5\] in the fictitious time $\tau \to \infty$. Here $\Mvar$ is the matrix of variations of the flow (its integral around $p$ yields the linearized stability matrix for the periodic orbit $p$).
Loop Initialization and Numerical Integration {#s:LoopInit}
---------------------------------------------
Replacement of a finite number of points along a trajectory by a closed smooth loop, and of the Newton-Raphson iteration by the flow results in a second order PDE for the loop evolution. The loop parameter $s$ converges (up to a proportionality constant) to the dynamical time $t$ as the loop converges to the desired periodic orbit. The flow parameter $\tau$ plays the role of a fictitious time. Our aim is to apply this method to high-dimensional flows; and thus we have replaced the initial ODE dynamics by a very high-dimensional PDE. And here our troubles start - can this be implemented at all? How do we get started?
A qualitative understanding of the dynamics is the essential prerequisite to successful periodic orbit searches. We start by long-time numerical runs of the dynamics, in order to get a feeling for frequently visited regions of the phase space (“natural measure”), and to search for close recurrences. We construct the initial loop $\Loop(0)$ using the intuition so acquired. Taking a fast Fourier transform of the guess, keeping the lowest frequency components, and transforming back to the initial phase space helps smooth the initial loop $\Loop(0)$. A simple linear stability analysis shows that the smoothness of the loop is maintained by the flow in the fictitious time $\tau$. This, as well as worries about the marginal stability eigenvalues and other details of the numerical integration of the loop flow , are described in . Suffice it to say that after a considerable amount of computation one is rewarded by periodic orbits that could not have been obtained by the methods employed previously.
Extensions of the Method {#s:ExtMeth}
========================
In classical mechanics particle trajectories are solutions of a different variational principle, the Hamilton’s variational principle. For example, one can determine a periodic orbit of a billiard by wrapping around a rubber band of roughly correct topology, and then moving the points along the billiard walls until the length (, the action) of the rubber band is extremal (maximal or minimal under infinitesimal changes of the boundary points). In this case, extremization of action requires only $D$-dimensional ($D = $ degrees of freedom) rather than $2D$-dimensional ($2D = $ dimension of the phase space) variations.
Can we exploit this fact to simplify our calculations in Newtonian mechanics? The answer is yes, and easiest to understand in terms of the Hamilton’s variational principle which states that classical trajectories are extrema of the Hamilton’s principal function (or, for fixed energy, the action) $$R(q_1,t_1;q_0,t_0) = \int_{t_0}^{t_1} \! dt \,{\cal L}(q(t), {\dot q}(t),t)
\,,$$ where $ {\cal L}(q, {\dot q},t)$ is the Lagrangian. Given a loop $\Loop(\tau)$ we can compute not only the tangent “velocity” vector $\lVeloc$, but also the local loop “acceleration” vector $$\tilde{a} =
\frac{d^2 \lSpace}{d s^2}
\,,$$ and indeed, as many $s$ derivatives as needed. Matching the dynamical acceleration $a(\lSpace)$ with the loop “acceleration” $\tilde{a}(\lSpace)$ results in an equation for the evolution of the loop $$\frac{d}{d \tau}(\tilde{a}-\lambda^2 a)=-(\tilde{a}-\lambda^2 a)
\,,$$ where $\lambda^2$ appears instead of $\lambda$ for dimensional reasons. This equation can be re-expressed in terms of loop variables $\lSpace(s)$; the resulting equation is somewhat more complicated than , but the saving is significant - only 1/2 of the phase-space variables appears in the fictitious time flow. More generally, the method works for extremization of functions of form $ {\cal L}(q, {\dot q},{\ddot q},\dots,t)$, with considerable computational savings.
Applications {#s:ApplicLoopDesc}
============
We now offer several examples of the application of the in the loop space, .
Unstable Recurrent Patterns in a Classical Field Theory
-------------------------------------------------------
One of the simplest and extensively studied spatially extended dynamical systems is the Kuramoto-Sivashinsky system u\_t=(u\^2)\_x-u\_[xx]{}-u\_[xxxx]{} \[ks1\] which arises as an amplitude equation for interfacial instabilities in a variety of contexts. The “flame front” $u(x,t)$ has compact support, with $x \in [0,2\pi]$ a periodic space coordinate. The $u^2$ term makes this a nonlinear system, $t$ is the time, and $\nu$ is a fourth-order “viscosity” damping parameter that irons out any sharp features. Numerical simulations demonstrate that as the viscosity decreases (or the size of the system increases), the “flame front” becomes increasingly unstable and turbulent. The task of the theory is to describe this spatio-temporal turbulence and yield quantitative predictions for its measurable consequences.
As was argued in , turbulent dynamics of such systems can be visualized as a walk through the space of unstable spatio-temporally recurrent patterns. In the PDE case we can think of a spatio-temporally discretized guess solution as a surface covered with small but misaligned tiles. Decreasing by the means smoothing these strutting fish scales into a smooth surface, a solution of the PDE in question.
In case at hand it is more convenient to transform the problem to Fourier space. If we impose the periodic boundary condition $u(t,x+2\pi)=u(t,x)$ and choose to study only the odd solutions $u(-x,t)=-u(x,t)$, the spatial Fourier series for the wavefront is $$u(x,t)=i\sum_{k=-\infty}^{\infty} a_k(t) \exp (ikt)
\,,
\label{expan}$$ with real Fourier coefficients $a_{-k}=-a_k$, and takes form $$\dot{a_k}=(k^2-\nu k^4)a_k-k\sum_{m=-\infty}^{\infty}a_m a_{k-m} \,.
\label{ksf}$$ After the initial transients die out, for large $k$ the magnitude of $a_k$ Fourier component decreases exponentially with $k^4$, justifying use of Galerkin truncations in numerical simulations. As in numerical work on any PDE we thus replace by a finite but high-dimensional system of ODEs. The initial searches for the unstable recurrent patterns for this spatially extended system found several hundreds of periodic solutions close to the onset of spatiotemporal chaos, but a systematic exploration of more turbulent regimes was unattainable by the numerical techniques employed.
\(a) ![ (a) An initial guess $\Loop(0)$, and (b) the periodic orbit $p$ of period $\period{p}= 0.5051$ reached by the , the Kuramoto-Sivashinsky system in a spatio-temporally turbulent regime (viscosity parameter $\nu=0.01500$, $d=32$ Fourier modes truncation). In discretization of the initial loop $\Loop(0)$ each point has to be specified in all $d$ dimensions; here the coordinates $\{a_5,a_7,a_8\}$ are picked arbitrarily, other projections from $d=32$ dimensions to a subset of 3 coordinates are equally (un)informative. []{data-label="f:ks1"}](ks2a.eps "fig:"){width="2.0in"} (b) ![ (a) An initial guess $\Loop(0)$, and (b) the periodic orbit $p$ of period $\period{p}= 0.5051$ reached by the , the Kuramoto-Sivashinsky system in a spatio-temporally turbulent regime (viscosity parameter $\nu=0.01500$, $d=32$ Fourier modes truncation). In discretization of the initial loop $\Loop(0)$ each point has to be specified in all $d$ dimensions; here the coordinates $\{a_5,a_7,a_8\}$ are picked arbitrarily, other projections from $d=32$ dimensions to a subset of 3 coordinates are equally (un)informative. []{data-label="f:ks1"}](ks2b.eps "fig:"){width="2.0in"}
With decreasing viscosity $\nu$ the system becomes quite turbulent, with the spatiotemporal portraits of the flame front $u(x,t)$ a complex labyrinth of eddies of different scales and orientations, and its Fourier space dynamics a complicated high-dimensional trajectory.
In we give an example of a calculation for this system, for the viscosity parameter $\nu$ significantly lower than in the earlier investigations. Although the initial guess $\Loop(0)$ is quite far from the final configuration $p=\Loop(\infty)$, the method succeeds in molding the starting loop into a periodic solution of this high dimensional flow. A systematic exploration of the shortest cycles found, and the hierarchy of longer cycles will be reported elsewhere.
Hénon-Heiles and Restricted 3-body Problems
-------------------------------------------
Next we offer two examples of the applicability of the extension of the of to low-dimensional Hamiltonian flows.
Hénon-Heiles Hamiltonian H=(\^2+\^2+x\^2+y\^2)+x\^2 y- \[hheq\] is frequently used in astrophysics. shows an application of the method of to a periodic orbit search restricted to the configuration space.
\(a) ![ (a) An initial loop $\Loop(0)$, and (b) the periodic orbit $p$ reached by the , the Hénon-Heiles system in a chaotic regime, $E=0.1794$. The period was fixed arbitrarily to $\period{}=13.1947$ by taking a fixed value of the scaling $\lambda=2.1$. []{data-label="f:hh"}](heils1.eps "fig:"){width="2.0in"} (b) ![ (a) An initial loop $\Loop(0)$, and (b) the periodic orbit $p$ reached by the , the Hénon-Heiles system in a chaotic regime, $E=0.1794$. The period was fixed arbitrarily to $\period{}=13.1947$ by taking a fixed value of the scaling $\lambda=2.1$. []{data-label="f:hh"}](heils2.eps "fig:"){width="2.0in"}
In the Hénon-Heiles case the acceleration $(a_x,a_y)$ depends only on the configuration coordinates $(x,y)$. More generally, the $a$’s could also depend on $(\dot{x},\dot{y})$, as is the case for the restricted three-body problem equations of motion $$\begin{aligned}
\ddot{x} &=& 2\dot{y}+x-(1-\mu)\frac{x+\mu}{r_1^3}-\mu \frac{x-1+\mu}{r_2^3}
\continue
\ddot{y} &=& -2\dot{x}+y-(1-\mu)\frac{y}{r_1^3}-\mu \frac{y}{r_2^3}
\label{rtbeq}\\
r_1 &=& \sqrt{(x+\mu)^2+y^2}
\,,\qquad
% YL: Jun 19 2003: OLD r_2 \,=\,\sqrt{(x-1-\mu)^2+y^2}
r_2 \,=\,\sqrt{(x-1+\mu)^2+y^2}
\nnu\end{aligned}$$ which describe the motion of a “test particle” in a rotating frame under the influence of the gravitational force of two heavy bodies with masses $1$ and $\mu \ll 1$, fixed at $(-\mu,0)$ and $(1-\mu,0)$ in the $(x,y)$ coordinate frame. The periodic solutions of correspond to periodic or quasi-periodic motion of the test particle in the inertial frame. shows an application of the method to this problem.
\(a) ![ (a) An initial loop $\Loop(0)$, and (b) the periodic orbit $p$ reached by the , the restricted three body problem in the chaotic regime, $\mu=0.04$, $T_p=2.7365$. []{data-label="rtb"}](rtb1.eps "fig:"){width="2.0in"} (b) ![ (a) An initial loop $\Loop(0)$, and (b) the periodic orbit $p$ reached by the , the restricted three body problem in the chaotic regime, $\mu=0.04$, $T_p=2.7365$. []{data-label="rtb"}](rtb2.eps "fig:"){width="2.0in"}
Summary and Future Directions {#s:SummaryLoopDesc}
=============================
The periodic orbit theory approach to classically turbulent field theory is to visualize turbulence as a sequence of near recurrences in a repertoire of unstable spatio-temporal patterns. So far, existence of a hierarchy of spatio-temporally periodic solutions, and applicability of the periodic orbit theory in evaluation of global averages for spatially extended nonlinear system has been demonstrated in one example, the Kuramoto-Sivashinsky system. The parameter ranges previously explored probe the weakest nontrivial “turbulence”, and it is an open question to what extent the approach remains implementable as such classical fields go more turbulent. Kawahara and Kida were able to find two periodic solutions in a turbulent plane Couette flow, in a [*15,422-dimensional*]{} discretization of the full 3-$d$ Navier-Stokes equations, so we remain optimistic.
The bottleneck has been the lack of methods for finding even the simplest periodic orbits in high-dimensional flows, and the lack of intuition as to what such orbits would look like. Here we have formulated the “” method, a very conservative method which emphasizes topological robustness at a considerable cost to numerical speed, and demonstrated that the method enables us to find the shortest spatio-temporally unstable periodic solutions of an (infinite dimensional) classical field theory, as well as several Hamiltonian flows.
In devising the method we have made a series of restrictive choices, many of which could be profitably relaxed.
The choice of a [**Euclidean metric** ]{} $\costF[\lSpace]$ has no compelling merit, other than notational simplicity. For a flow like the Kuramoto-Sivashinsky the $a_1$, $a_2$, $\dots$ directions are clearly more important than $a_k$, $a_{k+1}$, $\dots\;$, $k$ large, and that is not accounted for by the current form of the . A more inspired choice would use intrinsic information about dynamics, replacing $\delta_{ij}F_i F_j$ by a more appropriate metric $g_{ij}F_i F_j$ that penalizes straying away in the unstable directions more than deviations in the strongly contracting ones.
[**Loop parametrization**]{}. Once it is understood that given a vector field $\pVeloc(\pSpace)$, the objective is to determine a loop $\Loop(\infty)=p$ whose tangent vectors point along $\pVeloc(\pSpace)$ everywhere along the loop, there is no reason to fix the loop parameter $s$ by making it proportional to the dynamical system time $t$. Any loop parametrization $s$ will do, and other choice might be more effective for numerical discretizations.
[**Zero modes**]{}. In numerical calculations we eliminate the marginal eigendirection along the loop by “gauge fixing”, fixing one point on the loop by an arbitrary Poincaré section. This seems superfluous and perhaps should be eliminated in favor of some other, more invariant criterion.
The [** method**]{} introduced here replaces the Newton-Raphson iteration by an exponentially contracting flow. Keeping the fictitious time step $d \tau$ infinitesimal is both against the spirit of the Newton method, and not what we do in practice; once the approximate loop is sufficiently close to the desired periodic orbit, we replace $d \tau$ by discrete steps of increasing size $d \tau \rightarrow 1$, in order to regain the super-exponential convergence of the Newton method.
[**Topology**]{}. As for high-dimensional flows we are usually clueless as to what the solutions should look like, currently we have no way of telling to which periodic orbit the loop space flow will take our initial guess, other than to the “nearest” periodic orbit of topology “similar” to the initial loop.
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|
---
abstract: 'We present a field theoretical analysis of the $2+1$ dimensional BF model with boundary in the Abelian and the non-Abelian case based on the Symanzik’s separability condition. Our aim is to characterize the low energy properties of time reversal invariant topological insulators. In both cases, on the edges, we obtain Kač–Moody algebras with opposite chiralities reflecting the time reversal invariance of the theory. While the Abelian case presents an apparent arbitrariness in the value of the central charge, the physics on the boundary of the non-Abelian theory is completely determined by time reversal and gauge symmetry. The discussion of the non-Abelian BF model shows that time reversal symmetry on the boundary implies the existence of counter-propagating chiral currents.'
address:
- '$^1$ Dipartimento di Fisica, Università di Genova,Via Dodecaneso 33, 16146, Genova, Italy'
- '$^2$ INFN, Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy'
- '$^3$ CNR-SPIN, Via Dodecaneso 33, 16146, Genova, Italy'
- '$^4$ NEST, Istituto Nanoscienze - CNR and Scuola Normale Superiore, I-56126 Pisa, Italy'
author:
- '**[A Blasi$^{1,2}$, A Braggio$^{3}$, M Carrega$^{4}$, D Ferraro$^{1,2,3}$, N Maggiore$^{1,2}$ and N Magnoli$^{1,2}$]{}**'
title: 'Non-Abelian BF theory for $2+1$ dimensional topological states of matter'
---
Introduction
============
During the last decades a new paradigm, based only on the topological properties of a system, has been investigated to classify and predict new states of matter. The starting point was, in the early 80’s, the discovery of the quantum Hall effect (QHE), a phase peculiar of $2+1$ dimensions (D) [@Tsui99], where firstly appeared the connection between a macroscopical physical quantity, i.e. the Hall conductivity, and a class of topological invariants [@Thouless82]. This was the first example of new kinds of materials which cannot be described in terms of the principle of broken symmetries. Recently, new class of systems were discovered which, analogously to the QHE, behave like insulators in the bulk, but support robust conducting edge or surface states, hence displaying non trivial topological properties. The low energy sector of this kind of states is well described in terms of topological field theories (TFT) [@Witten88]. The QHE was successfully described by the Abelian Chern Simons (CS) theory both in the integer and in the fractional regime [@Zhang92; @Wen95]. Moreover, the non-Abelian CS theory has been proposed for more exotic fractional states [@Wen95] that are predicted to have excitations with non-Abelian statistics, a key point for the topologically protected quantum computation [@Nayak08]. Restriction to the boundary of these TFT has been discussed in order to describe the dynamics of the edge states of the system [@Wen95]. In contrast with the QHE, where the magnetic field breaks time reversal (T) symmetry, a new class of T invariants systems, i.e. topological insulators (TI), has been predicted [@Kane05; @Bernevig06a] and experimentally observed [@Konig07] in $2+1$ D, leading to the quantum spin Hall effect (QSH). At the boundary of these systems, one has helical states, namely electrons with opposite spin propagating in opposite directions. The presence of these edge currents with opposite chiralities leads to extremely peculiar current-voltage relationship in multi-terminal measurements. The experimental observations carried out by the Molenkamp group [@Konig07; @Roth09] support these theoretical predictions. Also non trivial realization of TI in $3+1$ D have been conjectured and realized [@Qi10; @Hazan10]. States with T and parity (P) symmetry emerge naturally also in lattice models of correlated electrons [@Freedman04; @Levin05] where doubled CS field theories were developed. In the case of $2+1$ D TI an Abelian doubled CS [@Bernevig06b; @Levin09] or, equivalently, a BF effective theory has been introduced [@Cho11; @Santos11]. The latter TFT allows generalizations to higher dimensions and represents a promising candidate for the description of $3+1$ D TI [@Cho11]. Despite the great potential of the BF model in describing T invariants topological states of matter, a careful discussion of these theories in presence of a boundary is still lacking.
In this paper, we present a detailed and systematic derivation of the helical states at the boundary of the $2+1$ D Abelian and non-Abelian BF model, applying the Symanzik’s method [@Symanzik81]. This approach represents the most natural way to introduce a boundary in a quantum field theory and was already considered with success in a different context for the CS case [@Emery91; @Blasi08; @Blasi10]. It is known that $2+1$ D CS theory on a manifold with boundary displays a conformal structure [@Moore86]. This is a general property of topological field theories. The BF theory, once a T invariant boundary is introduced, displays two Kač–Moody algebras with opposite chiralities, according to the physics of the QSH effect [@Bernevig06a; @Konig07], both in the Abelian and in the non-Abelian case [@Maggiore92].
Abelian BF model
================
Here, we recall the main properties of the $2+1$ D Abelian BF model. Its action depends on two Abelian gauge fields $A$ and $B$ and, in the Minkowski spacetime, can be written as S\_[bf]{}= d\^[3]{}x \^ F\_B\_ \[BFaction\] with $F_{\mu \nu}=\partial_{\mu} A_{\nu}- \partial_{\nu} A_{\mu}$ and $k$ integer due to the invariance of the partition function for large gauge transformations, exactly in the same way as it happens in the CS case [@Witten88]. At low energy, the action (\[BFaction\]) is the only renormalizable one, involving two gauge fields, which is invariant under gauge transformations in $2+1$ dimensions. At this point the T invariance appears to be a discrete symmetry of the theory together with the trivial exchange $A_{\mu}\leftrightarrow B_{\mu}$. In order to describe the model in presence of a planar boundary it is useful to rewrite (\[BFaction\]) in the so called light-cone coordinates $u=y$, $z=(t+x)/\sqrt{2}$ and $\bar{z}= (t-x)/\sqrt{2}$ where the field components become $(A_{u}, A, \bar{A})$ and $(B_{u}, B, \bar{B})$. In these new variables the action reads S\_[bf]{}= d u d\^[2]{} z.
The mentioned T symmetry (modulo an exchange of the two gauge fields) in these new variables is (u, z, A, A\_[u]{}, B, B\_[u]{})(u, -|[z]{}, -|[A]{}, A\_[u]{}, |[B]{}, -B\_[u]{});
notice that the $A$ and $B$ fields transform differently under T, which signals the fact that they can be linked to different physical quantities. Indeed, this allows us to interpret $A$ as a charge density and $B$ as a spin density, the fundamental ingredients of the description of TI [@Cho11].
As it is well known, gauge field theories are affected by the presence of redundant degrees of freedom, and a gauge fixing choice is necessary in order to deal with the physical degrees of freedom only. The gauge fixing procedure introduces in the theory unphysical “ghost fields”. Now, while in the Abelian case the ghost fields decouple, and can be integrated out from the Green functions generating functionals, this does not happen in the non-Abelian case, where ghost fields are truly quantum, although unphysical, fields [@Weinberg96]. Once gauge fixed, for the theory partition function and propagators can be defined, and quantum extension can be discussed.
Covariance being already broken by the presence of a boundary, it is convenient to adopt the axial gauge choice $A_{u}=B_{u}=0$, and add to (\[BFaction\]) the corresponding gauge fixing term $S_{gf}$. The coupling with external sources is introduced by means of the term $S_{ext}=\int d u d^{2}z \sum_{\psi} j_{\psi} \psi$, $\psi$ being a generic field of the theory.
Presence of a boundary
======================
We consider now the planar boundary $u=0$. The introduction of a boundary in field theory has been thoroughly discussed long time ago by Symanzik in [@Symanzik81]. The basic idea is quite simple and general, and concerns the propagators of the theory, which only in the gauge fixed theory are well defined: the propagators of the theory between points lying on opposite sides of the boundary, must vanish. This $separability$ condition results in the following general form of the propagator for any field $\psi$ \^[, ’]{}=(u) (u’) \^[, ’]{}\_[+]{}+(-u) (-u’) \^[, ’]{}\_[-]{} where $\pm$ indicates the value of the quantities for $(u\rightarrow 0^{\pm})$ and $\Theta(u)$ is the Heaviside step function. In addition, the usual field theory constraints of locality and power counting are required together with helicity conservation [@Emery91].
The most general boundary action consistent with the previous conditions is S\_[bd]{}=- d u d\^[2]{} z (u) (\_[1]{} A |[A]{}+ \_[2]{} A |[B]{} + \_[3]{} |[A]{} B+ \_[4]{} B |[B]{}), \[Action\_bd\] where $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\alpha_{4}$ are free real parameters. The above boundary term leads to $\delta(u)$-dependent breakings of the equations of motion \_[|[B]{}]{}&=&\_[=]{}(u)(\_[2]{}A\_+\_[4]{}B\_) \[Eq\_breaking\_in\]\
\_[B]{}&=&\_[=]{}(u)(\_[3]{}|[A]{}\_+\_[4]{}|[B]{}\_)\
\_[|[A]{}]{}&=&\_[=]{}(u)(\_[1]{}A\_+\_[3]{}B\_)\
\_[A]{}&=&\_[=]{}(u)(\_[1]{}|[A]{}\_+\_[2]{}|[B]{}\_). \[Eq\_breaking\_fin\] where $\mathcal{F}_{\psi}=\delta S/ \delta \psi$ and $S= S_{bf}+S_{gf}+S_{ext}$.
From now on, we will focus on the $+$ side of the boundary, the $-$ side being obtained by the P symmetry $\left(u, z, A, A_{u}, B, B_{u}\right)\leftrightarrow \left(-u,
\bar{z}, \bar{A}, -A_{u}, \bar{B}, -B_{u}\right)$.
In order to fix the parameters, we integrate (\[Eq\_breaking\_in\]-\[Eq\_breaking\_fin\]) at vanishing sources with respect to $u$ in the infinitesimal interval $(-\epsilon, \epsilon)$. This leads to (1-\_[2]{})A\_[+]{}-\_[4]{}B\_[+]{}&=&0 \[consistency\_in\]\
\_[1]{}A\_[+]{}-(1-\_[3]{})B\_[+]{}&=&0 \[consistency\_2\]\
(1+\_[3]{})|[A]{}\_[+]{}+\_[4]{} |[B]{}\_[+]{}&=&0 \[consistency\_3\]\
\_[1]{}|[A]{}\_[+]{}+(1+\_[2]{}) |[B]{}\_[+]{}&=&0. \[consistency\_fin\]
It is easy to verify that (\[consistency\_in\]-\[consistency\_fin\]) have non trivial T invariant solutions when the determinants vanish, namely \_[1]{} \_[4]{}-(1-\^[2]{}\_[2]{})=0; \_[2]{}=-\_[3]{}. \[contraint\]
The Abelian BF theory with boundary satisfies the following Ward identities (WI) on the generating functional $Z_{c}$:
&&duH\[Z\_[c]{}\]-du (| j\_[|[A]{}]{}+j\_[A]{})=\
&&-\[res1\]\
&&duN\[Z\_[c]{}\]-du (| j\_[|[B]{}]{}+j\_[B]{})=\
&& -. \[res2\]
It is well known that the axial gauge is not a complete gauge fixing [@Bassetto91]: a residual gauge invariance remains, expressed by the two WI (\[res1\]-\[res2\]), one for each gauge field $A$ and $B$. As it is apparent, the presence of the boundary results in $\delta(u)$-dependent $linear$ breaking terms on the r.h.s. of (\[res1\]-\[res2\]). Such linear terms are allowed, since in quantum field theory a non-renormalization theorem assures that linear breakings are present at classical level only, and do not acquire quantum corrections [@Becchi75].
In other words, the WI (\[res1\]-\[res2\]) represent the most general expression of the gauge invariance of the complete theory (bulk and boundary).
Conserved currents and algebra
==============================
We introduce the fields R\_[+]{}&&(1-\_[2]{}) A\_[+]{}+\_[4]{}B\_[+]{}\
S\_[+]{}&&(\_[2]{}-1)A\_[+]{}+\_[4]{}B\_[+]{}, which, according to (\[consistency\_in\]-\[consistency\_fin\]), satisfy the boundary conditions |[R]{}\_[+]{}=S\_[+]{}=0. \[annull\] It is easy to verify that, in terms of these new fields, the WI (\[res1\]-\[res2\]) decouple
d u ( | j\_[|[R]{}]{}+ j\_[R]{})&=& | R\_[+]{} \[resR\]\
d u ( | j\_[|[S]{}]{}+ j\_[S]{})&=& |[S]{}\_[+]{}, \[resS\] from which we read the chirality conditions | R\_[+]{}= |[S]{}\_[+]{}=0. \[chiral\] Notice that $R_{+}$ and $\bar{S}_{+}$ are related by T symmetry and, in virtue of (\[annull\]) and (\[chiral\]), are conserved currents. From the WI (\[resR\]-\[resS\]), we immediately get the following (Abelian limit of a) Kač-Moody algebra
&=& (z-z’)\
&=& | (|[z]{}- |[z]{}’)\
&=&0, \[Abelian\_algebra\] that are connected by T and where the last commutator shows the decoupling of the two currents. We stress that, in the Abelian case, the central charge is not completely fixed since it depends on the boundary parameters. Once required the decoupling of the boundary action (\[Action\_bd\]) in terms of $R_{+}$ and $S_{+}$ we can fix the values of the parameters $\alpha_{2}=0$ and $\alpha_{4}=1$.
The result of our analysis is the presence, on the boundary of the Abelian BF model, of two conserved currents with opposite chiralities, connected by T symmetry. The above picture is in accordance with the phenomenology involved in the helical edge states of the QSH effect [@Bernevig06a; @Konig07; @Qi10] in agreement with the idea that the BF model is a good effective field theoretical description for the $2+1$ D TI [@Cho11]. Note that, due to the presence of the factor $k$, our results could be applied to possible fractional extension as well [@Bernevig06b].
Non-Abelian BF model
====================
The non-Abelian generalization of (\[BFaction\]) is $$S_{BF}=\frac{k}{2\pi}\int\!\! d^3x\ \varepsilon^{\mu\nu\rho}\left\{
F^a_{\mu\nu}B^a_\rho+\frac{1}{3}f^{abc}B^a_\mu
B^b_\nu B^c_\rho\right\}
\ ,
\label{nonabbf}$$ where the fields $A_{\mu}^a$ and $B_{\mu}^a$ belong to the adjoint representation of a compact simple gauge group $G$ whose structure constants are $f^{abc}$. In (\[nonabbf\]), the non-Abelian field strength is defined as $F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu
+ f^{abc} A^b_\mu A^c_\nu$ and the coupling $k$ can be related to the mutual statistics between the quasiparticle sources of the fields. It is worth to note that $k$ can be connected to the “cosmological constant” usually defined in literature [@Maggiore92].
In light–cone coordinates the action in (\[nonabbf\]) reads $$\begin{aligned}
S_{BF}&=&\frac{k}{\pi}\int du d^2z \{ [
B^a(\bar{\partial}A_u^a-\partial_u\bar{A}^a+
f^{abc}\bar{A}^b A_u^c)+\bar{B}^a (\partial_uA^a-\partial A_u^a+f^{abc}
A_u^bA^c) \nonumber\\
&& +B_u^a(\partial\bar{A}^a-\bar{\partial}A^a
+f^{abc}A^b\bar{A}^c)]+ f^{abc}B^a\bar{B}^bB_u^c \}\ .
\label{lcaction}\end{aligned}$$ The axial gauge $A^a_u=B^a_u=0$ is realized, as in the Abelian case, through a suitable gauge fixing term which involves ghost, antighost and lagrange multipliers for both the gauge fields $A_{\mu}^a$ and $B_{\mu}^a$. It is straightforward to verify that the above action satisfies T and P symmetries. The non-Abelian gauge symmetry is realized, as usual in the axial gauge, by means of the local WI $H^a[Z_{c}] = N^a[Z_{c}] = 0$, where $H^a(u,z,\bar{z})$ and $N^a(u,z,\bar{z})$ are local operators whose explicit form is unessential here, and can be found in [@Maggiore92], where the non-Abelian BF theory in the axial gauge with and without boundary is treated in greater detail.
As in the Abelian case, the introduction of the planar boundary $u=0$ is realized through the addition of a term in the action or, equivalently, through $\delta(u)$ - breakings of the fields equations, satisfying the constraints of consistency, power counting and helicity conservation. Consequently, the local WI are broken by the boundary and, once integrated over the $u$-coordinate, read $$\begin{aligned}
\frac{\pi}{k}\int du\,H^a [Z_c]
&=&\Delta^{a}_{H}(z,\bar{z})
\label{brokenward1}\\
\frac{\pi}{k} \int du\,N^a [Z_c]
&=&\Delta^{a}_{N}(z,\bar{z}).
\label{brokenward2}\end{aligned}$$ The above equations are the non-Abelian counterpart of (\[res1\]-\[res2\]), and likewise they describe the residual gauge invariance left on the boundary. By choosing the parameters $\alpha_{1}=\alpha_{4}$ and $\alpha_{2}=\alpha_{3}$ the terms $\Delta^{a}_{H}$ and $\Delta^{a}_{N}$ on their r.h.s. are linear in the quantum fields (hence classical and allowed). Notice that the WI (\[brokenward1\]-\[brokenward2\]) are the most general form of gauge invariance for the non-Abelian theory with boundary [@Becchi75]. This is a central point for what follows. Requiring T symmetry at the boundary and (\[contraint\]), still valid in the non-Abelian case, one fixes the parameters of the boundary action to be $\alpha_{1}=\alpha_{4}=1$ and $\alpha_{2}=\alpha_{3}=0$.
The most general conserved chiral currents on the boundary, together with their algebra, can be fully determined [@Maggiore92]. It turns out that the only T symmetric solution is given by a direct sum of two independent Kač-Moody algebras satisfied by conserved currents of [*[opposite chirality]{}*]{}[^1].
In more detail, the WI corresponding to this solution, read:
$$\begin{aligned}
\frac{\pi}{k}\int du\,H^a [Z_c] &=&
-(\partial\bar{A}^a_{+}+\bar{\partial}A^a_{+})\label{lw1}\\
\frac{\pi}{k}\int du\,N^a [Z_c] &=&
-(\partial\bar{B}^a_{+}+\bar{\partial}B^a_{+}) .
\label{lw2}\end{aligned}$$
The linear combinations of fields R\^a\_[+]{} A\^a\_[+]{}+ B\^a\_[+]{} S\^a\_[+]{} - A\^a\_[+]{} +B\^a\_[+]{} \[redef\]satisfy Dirichlet boundary conditions $S^a_+(z,\bar{z})=\bar{R}^a_+(z,\bar{z})=0$, and the residual WI (\[lw1\]-\[lw2\]) identify two conserved currents with opposite chiralities $\bar{\partial}R^a_{+}(z,\bar{z})=\partial\bar{S}^a_{+}
(z,\bar{z})=0$, for which the following direct sum of Kač-Moody algebras living on the same side of the plane $u=0$ with opposite chiralities, holds: $$\begin{aligned}
&&\left [R^a_{+}(z),R^b_{+}(z')\right]=
f^{abc}\delta(z-z')R^c_{+}(z)+\frac{2\pi}{k}\delta^{ab}\delta'(z-z')\nonumber\\
&&\left [\bar{S}^a_{+} ( \bar{z} ) , \bar{S}^b_{+} ( \bar{z} ' ) \right
] =
f^{abc}\delta(\bar{z}-\bar{z}')\bar{S}^c_{+}(\bar{z})+\frac{2\pi}{k}\delta^{ab}
\delta'(\bar{z}-\bar{z} ')\nonumber\\
&&\left [ R^a_{+}({z}),\bar{S}^b_{+}(\bar{z} ')\right ]=0\end{aligned}$$ where the last commutator shows that $R_{+}$ and $\bar{S}_{+}$ are decoupled. Note that, in the Abelian limit, the above WI and algebras reduce to (\[res1\]-\[res2\]) and (\[Abelian\_algebra\]) respectively in the case of decoupling of the boundary action.
Finally, we notice that, as it has been shown long time ago in [@Guadagnini:1990aw], the redefinition (\[redef\]) allows to rephrase the $2+1$ D non-Abelian BF theory in (\[nonabbf\]) in terms of two CS theories with opposite coupling constants in the bulk. As stated before, in the non-Abelian case, the consistency of the theory also requires the complete decoupling at the boundary that, conversely, is not needed in the Abelian case.
Conclusions
===========
We investigated the $2+1$ D BF model in presence of a boundary both in the Abelian and in the non-Abelian case as a proper effective field theory for the QSH states. The key points of our analysis have been the combined application of the Symanzik’s separability condition and the T symmetry. By means of these conditions we have been able to evaluate the algebraic structure of the current at the edge of the system. In the Abelian case we found two Kač–Moody current algebras with opposite chiralities and with the central charges depending on the coupling constant $k$, together with arbitrary boundary parameters. In the non-Abelian case we also find two Kač–Moody currents with opposite chiralities with the important difference that the central charge is unambigously determined in terms of the coupling constant $k$ of the theory. The appearance of Kač–Moody algebras related by T symmetry, reflects the equivalence between the double CS model and the non-Abelian BF model once the two CS are completely decoupled at the boundary [@Guadagnini:1990aw]. The QSH is characterized by the presence of currents with opposite spin and chiralities on the boundary [@Konig07; @Roth09; @Qi10]. One of the new results of this paper is that we find an algebraic structure which displays this features and, in addition, it respects the T invariance.
The Symanzik’s method of treating boundaries in field theory is widely general and standard, and it is model independent. It is precisely for this reason that we adopted it, having in mind two recently and widely discussed generalizations: the $3+1$ D case and the breaking of the T symmetry, which can be easily treated along the same lines illustrated in this paper.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank M. Sassetti for useful discussions. We acknowledge the support of the EU-FP7 via ITN-2008-234970 NANOCTM.
References {#references .unnumbered}
==========
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[^1]: Note that other solutions exist that break T symmetry at the boundary, but we will not consider them here.
|
---
abstract: 'Hierarchical temporal memory (HTM) is an emerging machine learning algorithm, with the potential to provide a means to perform predictions on spatiotemporal data. The algorithm, inspired by the neocortex, currently does not have a comprehensive mathematical framework. This work brings together all aspects of the spatial pooler (SP), a critical learning component in HTM, under a single unifying framework. The primary learning mechanism is explored, where a maximum likelihood estimator for determining the degree of permanence update is proposed. The boosting mechanisms are studied and found to be a secondary learning mechanism. The SP is demonstrated in both spatial and categorical multi-class classification, where the SP is found to perform exceptionally well on non-spatial data. Observations are made relating HTM to well-known algorithms such as competitive learning and attribute bagging. Methods are provided for using the SP for classification as well as dimensionality reduction. Empirical evidence verifies that given the proper parameterizations, the SP may be used for feature learning.'
author:
- 'James Mnatzaganian, Ernest Fokou[é]{}, and Dhireesha Kudithipudi, [^1] [^2] [^3]'
bibliography:
- 'IEEEabrv.bib'
- 'sp\_math.bib'
title: 'A Mathematical Formalization of Hierarchical Temporal Memory’s Spatial Pooler'
---
[Mnatzaganian : Mathematical Formalization of Hierarchical Temporal Memory Cortical Learning Algorithm’s Spatial Pooler]{}
hierarchical temporal memory, machine learning, neural networks, self-organizing feature maps, unsupervised learning.
Introduction
============
temporal memory (HTM) is a machine learning algorithm that was inspired by the neocortex and designed to learn sequences and make predictions. In its idealized form, it should be able to produce generalized representations for similar inputs. Given time-series data, HTM should be able to use its learned representations to perform a type of time-dependent regression. Such a system would prove to be incredibly useful in many applications utilizing spatiotemporal data. One instance for using HTM with time-series data was recently demonstrated by Cui et al. [@numenta_tp_prediction], where HTM was used to predict taxi passenger counts. The use of HTM in other applications remains unexplored, largely due to the evolving nature of HTM’s algorithmic definition. Additionally, the lack of a formalized mathematical model hampers its prominence in the machine learning community. This work aims to bridge the gap between a neuroscience inspired algorithm and a math-based algorithm by constructing a purely mathematical framework around HTM’s original algorithmic definition.
HTM models, at a high-level, some of the structures and functionality of the neocortex. Its structure follows that of cortical minicolumns, where an HTM region is comprised of many columns, each consisting of multiple cells. One or more regions form a level. Levels are stacked hierarchically in a tree-like structure to form the full network depicted in [Fig. \[fig:htm\]]{}. Within HTM, connections are made via synapses, where both proximal and distal synapses are utilized to form feedforward and neighboring connections, respectively.
The current version of HTM is the predecessor to HTM cortical learning algorithm (CLA) [@cla_whitepaper]. In the current version of HTM the two primary algorithms are the spatial pooler (SP) and the temporal memory (TM). The SP is responsible for taking an input, in the format of a sparse distributed representation (SDR), and producing a new SDR. In this manner, the SP can be viewed as a mapping function from the input domain to a new feature domain. In the feature domain a single SDR should be used to represent similar SDRs from the input domain. The algorithm is a type of unsupervised competitive learning algorithm that uses a form of vector quantization (VQ) resembling self-organizing maps (SOMs). The TM is responsible for learning sequences and making predictions. This algorithm follows Hebb’s rule [@hebb], where connections are formed between cells that were previously active. Through the formation of those connections a sequence may be learned. The TM can then use its learned knowledge of the sequences to form predictions.
![Depiction of HTM, showing the various levels of detail.[]{data-label="fig:htm"}](htm){width="\linewidth"}
HTM originated as an abstraction of the neocortex; as such, it does not have an explicit mathematical formulation. Without a mathematical framework, it is difficult to understand the key characteristics of the algorithm and how it can be improved. In general, very little work exists regarding the mathematics behind HTM. Hawkins et al. [@numneta_tp_math] recently provided a starting mathematical formulation for the TM, but no mentions to the SP were made. Lattner [@lattner] provided an initial insight about the SP, by relating it to VQ. He additionally provided some equations governing computing overlap and performing learning; however, those equations were not generalized to account for local inhibition. Byrne [@byrne] began the use of matrix notation and provided a basis for those equations; however, certain components of the algorithm, such as boosting, were not included. Leake et al. [@leake] provided some insights regarding the initialization of the SP. He also provided further insights into how the initialization may affect the initial calculations within the network; however, his focus was largely on the network initialization. The goal of this work is to provide a complete mathematical framework for HTM’s SP and to demonstrate how it may be used in various machine learning tasks.
The major, novel contributions provided by this work are as follows:
- Creation of a complete mathematical framework for the SP, including boosting and local inhibition.
- Using the SP to perform feature learning.
- Using the SP as a pre-processor for non-spatial data.
- Creation of a possible mathematical explanation for the permanence update amount.
- Insights into the permanence selection.
Spatial Pooler Algorithm
========================
The SP consists of three phases, namely overlap, inhibition, and learning. In this section the three phases will be presented based off their original, algorithmic definition. This algorithm follows an iterative, online approach, where the learning updates occur after the presentation of each input. Before the execution of the algorithm, some initializations must take place.
Within an SP there exist many columns. Each column has a unique set of proximal synapses connected via a proximal dendrite segment. Each proximal synapse tentatively connects to a single column from the input space, where the column’s activity level is used as an input, i.e. an active column is a ‘1’ and an inactive column is a ‘0’.
To determine whether a synapse is connected or not, the synapse’s permanence value is checked. If the permanence value is at least equal to the connected threshold the synapse is connected; otherwise, it is unconnected. The permanence values are scalars in the closed interval \[0, 1\].
$col.overlap \leftarrow 0$ $col.overlap \leftarrow col.overlap + syn.\operatorname{active}()$
$col.overlap \leftarrow 0$ $col.overlap \leftarrow col.overlap * col.boost$
Prior to the first execution of the algorithm, the potential connections of proximal synapses to the input space and the initial permanence values must be determined. Following Numenta’s whitepaper [@cla_whitepaper], each synapse is randomly connected to a unique input bit, i.e. the number of synapses per column and the number of input bits are binomial coefficients. The permanences of the synapses are then randomly initialized to a value close to the connected permanence threshold. A second constraint requires that the permanence value be a function of the distance between the SP column’s position and the input column’s position, such that the closer the input is to the column the larger the value should be. The three phases of the SP are explained in the following subsections.
Phase 1: Overlap
----------------
The first phase of the SP is used to compute the overlap between each column and its respective input, as shown in [Algorithm \[alg:phase1\]]{}. In [Algorithm \[alg:phase1\]]{}, the SP is represented by the object $sp$. The method $col.\operatorname{connected\_synapses}()$ returns an instance to each synapse on $col$’s proximal segment that is connected, i.e. synapses having permanence values greater than the permanence connected threshold, $psyn\_th$. The method $syn.\operatorname{active}()$ returns ‘1’ if $syn$’s input is active and ‘0’ otherwise. $pseg\_th$[^4] is a parameter that determines the activation threshold of a proximal segment, such that there must be at least $pseg\_th$ active connected proximal synapses on a given proximal segment for it to become active. The parameter $col.boost$ is the boost value for $col$, which is initialized to ‘1’ and updated according to [Algorithm \[alg:boost\]]{}.
Phase 2: Inhibition
-------------------
The second phase of the SP is used to compute the set of active columns after they have been inhibited, as shown in [Algorithm \[alg:phase2\]]{}. In [Algorithm \[alg:phase2\]]{}, $\operatorname{kmax\_overlap}(C, k)$ is a function that returns the $k$-th largest overlap of the columns in $C$. The method $sp.\operatorname{neighbors}(col)$ returns the columns that are within $col$’s neighborhood, including $col$, where the size of the neighborhood is determined by the inhibition radius. The parameter $k$ is the desired column activity level. In line 2 in [Algorithm \[alg:phase2\]]{}, the $k$-th largest overlap value out of $col$’s neighborhood is being computed. A column is then said to be active if its overlap value is greater than zero and the computed minimum overlap, $mo$.
$mo \leftarrow \operatorname{kmax\_overlap}(sp.\operatorname{neighbors}(col), k)$
$col.active \leftarrow 1$ $col.active \leftarrow 0$
Phase 3: Learning
-----------------
The third phase of the SP is used to conduct the learning operations, as shown in [Algorithm \[alg:phase3\]]{}. This code contains three parts – permanence adaptation, boosting operations, and the inhibition radius update. In [Algorithm \[alg:phase3\]]{}, $syn.p$ refers to the permanence value of $syn$. The functions $\operatorname{min}$ and $\operatorname{max}$ return the minimum and maximum values of their arguments, respectively, and are used to keep the permanence values bounded in the closed interval \[0, 1\]. The constants $syn.psyn\_inc$ and $syn.psyn\_dec$ are the proximal synapse permanence increment and decrement amounts, respectively.
The function $\operatorname{max\_adc}(C)$ returns the maximum active duty cycle of the columns in $C$, where the active duty cycle is a moving average denoting the frequency of column activation. Similarly, the overlap duty cycle is a moving average denoting the frequency of the column’s overlap value being at least equal to the proximal segment activation threshold. The functions $col.\operatorname{update\_active\_duty\_cycle}()$ and $col.\operatorname{update\_overlap\_duty\_cycle}()$ are used to update the active and overlap duty cycles, respectively, by computing the new moving averages. The parameters $col.odc$, $col.adc$, and $col.mdc$ refer to $col$’s overlap duty cycle, active duty cycle, and minimum duty cycle, respectively. Those duty cycles are used to ensure that columns have a certain degree of activation.
The method $col.\operatorname{update\_boost}()$ is used to update the boost for column, $col$, as shown in [Algorithm \[alg:boost\]]{}, where $maxb$ refers to the maximum boost value. It is important to note that the whitepaper did not explicitly define how the boost should be computed. This boost function was obtained from the source code of Numenta’s implementation of HTM, Numenta platform for intelligent computing (NuPIC) [@nupic].
The method $sp.\operatorname{update\_inhibition\_radius}()$ is used to update the inhibition radius. The inhibition radius is set to the average receptive field size, which is average distance between all connected synapses and their respective columns in the input and the SP.
Adapt permanences
Perform boosting operations
Mathematical Formalization
==========================
The aforementioned operation of the SP lends itself to a vectorized notation. By redefining the operations to work with vectors it is possible not only to create a mathematical representation, but also to greatly improve upon the efficiency of the operations. The notation described in this section will be used as the notation for the remainder of the document.
All vectors will be lowercase, bold-faced letters with an arrow hat. Vectors are assumed to be row vectors, such that the transpose of the vector will produce a column vector. All matrices will be uppercase, bold-faced letters. Subscripts on vectors and matrices are used to denote where elements are being indexed, following a row-column convention, such that $\boldsymbol{X}_{i,j} \in \boldsymbol{X}$ refers to $\boldsymbol{X}$ at row index[^5] $i$ and column index $j$. Element-wise operations between a vector and a matrix are performed column-wise, such that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{x}}} \settoheight{\arght}{\ensuremath{\boldsymbol{x}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{x}}}}}}^T \odot \boldsymbol{Y} = {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{x}}} \settoheight{\arght}{\ensuremath{\boldsymbol{x}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{x}}}}}}_i\boldsymbol{Y}_{i,j}\ \forall i\ \forall j$.
Let $\operatorname{I}(k)$ be defined as the indicator function, such that the function will return 1 if event $k$ is true and 0 otherwise. If the input to this function is a vector of events or a matrix of events, each event will be evaluated independently, with the function returning a vector or matrix of the same size as its input. Any variable with a superscript in parentheses is used to denote the type of that variable. For example, ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{x}}} \settoheight{\arght}{\ensuremath{\boldsymbol{x}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{x}}}}}}^{(y)}$ is used to state that the variable ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{x}}} \settoheight{\arght}{\ensuremath{\boldsymbol{x}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{x}}}}}}$ is of type $y$.
$col.boost \leftarrow maxb$ $col.boost \leftarrow 1$ $col.boost = col.adc * ((1 - maxb) / col.mdc) + maxb$
----------------- ------------------------------------------------
$n$ Number of patterns (samples)
$p$ Number of inputs (features) in a pattern
$m$ Number of columns
$q$ Number of proximal synapses per column
$\phi_+$ Permanence increment amount
$\phi_-$ Permanence decrement amount
$\phi_{\delta}$ Window of permanence initialization
$\rho_d$ Proximal dendrite segment activation threshold
$\rho_s$ Proximal synapse activation threshold
$\rho_c$ Desired column activity level
$\kappa_a$ Minimum activity level scaling factor
$\kappa_b$ Permanence boosting scaling factor
$\beta_0$ Maximum boost
$\tau$ Duty cycle period
----------------- ------------------------------------------------
: User-defined parameters for the SP[]{data-label="tbl:parameters"}
All of the user-defined parameters are defined in [(Table \[tbl:parameters\])]{}[^6]. These are parameters that must be defined before the initialization of the algorithm. All of those parameters are constants, except for parameter $\rho_c$, which is an overloaded parameter. It can either be used as a constant, such that for a column to be active it must be greater than the $\rho_c$-th column’s overlap. It may also be defined to be a density, such that for a column to be active it must be greater than the $\lfloor \rho_c * \operatorname{num\_neighbors}(i) \rfloor$-th column’s overlap, where $\operatorname{num\_neighbors}(i)$ is a function that returns the number of neighbors that column $i$ has. If $\rho_c$ is an integer it is assumed to be a constant, and if it is a scalar in the interval (0, 1\] it is assumed to be used as a density.
Let the terms $s$, $r$, $i$, $j$, and $k$ be defined as integer indices. They are henceforth bounded as follows: $s \in [0, n)$, $r \in [0, p)$, $i \in [0, m)$, $j \in [0, m)$, and $k \in [0, q)$.
Initialization
--------------
Competitive learning networks typically have each node fully connected to each input. The SP; however, follows a different line of logic, posing a new problem concerning the visibility of an input. As previously explained, the inputs connecting to a particular column are determined randomly. Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}} \in \mathbb{Z}^{1 \times m}, {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}} \in [0, m)$ be defined as the set of all columns indices, such that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$ is the column’s index at $i$. Let $\boldsymbol{U} \in \{0, 1\}^{n \times p}$ be defined as the set of inputs for all patterns, such that $\boldsymbol{U}_{s, r}$ is the input for pattern $s$ at index $r$. Let $\boldsymbol{\Lambda} \in \{r\}^{m \times q}$ be the source column indices for each proximal synapse on each column, such that $\boldsymbol{\Lambda}_{i,k}$ is the source column’s index of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$’s proximal synapse at index $k$. In other words, each $\boldsymbol{\Lambda}_{i,k}$ refers to a specific index in $\boldsymbol{U}_s$.
Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ic}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ic}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ic}}}}}}_r\equiv\exists! r\in\boldsymbol{\Lambda}_i\ \forall r$, the event of input $r$ connecting to column $i$, where $\exists!$ is defined to be the uniqueness quantification. Given $q$ and $p$, the probability of a single input, $\boldsymbol{U}_{s, r}$, connecting to a column is calculated by using [(\[eq:p\_connect\])]{}. In [(\[eq:p\_connect\])]{}, the probability of an input not connecting is first determined. That probability is independent for each input; thus, the total probability of a connection not being formed is simply the product of those probabilities. The probability of a connection forming is therefore the complement of the probability of a connection not forming.
$$\begin{split}
\mathbb{P}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ic}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ic}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ic}}}}}}_r) &= 1 - \prod_{k=0}^{q}\left(1 - \frac{1}{p - k}\right)\\
&=\frac{q+1}{p}
\end{split}
\label{eq:p_connect}$$
It is also desired to know the average number of columns an input will connect with. To calculate this, let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\lambda}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\lambda}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\lambda}}}}}}\equiv \sum_{i=0}^{m-1}\sum_{k=0}^{q-1}\operatorname{I}(r=\Lambda_{i,k})\ \forall r$, the random vector governing the count of connections between each input and all columns. Recognizing that the probability of a connection forming in $m$ follows a binomial distribution, the expected number of columns that an input will connect to is simply [(\[eq:e\_columns\])]{}.
$$\mathbb{E}\left[{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\lambda}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\lambda}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\lambda}}}}}}_r\right] = m\mathbb{P}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ic}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ic}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ic}}}}}}_r)
\label{eq:e_columns}$$
Using [(\[eq:p\_connect\])]{} it is possible to calculate the probability of an input never connecting, as shown in [(\[eq:p\_never\_connect\])]{}. Since the probabilities are independent, it simply reduces to the product of the probability of an input not connecting to a column, taken over all columns. Let $\lambda'\equiv \sum_{r=0}^{p-1}\operatorname{I}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\lambda}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\lambda}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\lambda}}}}}}_r=0)$, the random variable governing the number of unconnected inputs. From [(\[eq:p\_never\_connect\])]{}, the expected number of unobserved inputs may then be trivially obtained as [(\[eq:e\_unobserved\])]{}. Using [(\[eq:p\_never\_connect\])]{} and [(\[eq:e\_columns\])]{}, it is possible to obtain a lower bound for $m$ and $q$, by choosing those parameters such that a certain amount of input visibility is obtained. To guarantee observance of all inputs, [(\[eq:p\_never\_connect\])]{} must be zero. Once that is satisfied, the desired number of times an input is observed may be determined by using [(\[eq:e\_columns\])]{}.
$$\mathbb{P}\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\lambda}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\lambda}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\lambda}}}}}}_r=0\right) = (1-\mathbb{P}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ic}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ic}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ic}}}}}}_r))^{m}
\label{eq:p_never_connect}$$
$$\mathbb{E}[\lambda'] = p\mathbb{P}\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\lambda}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\lambda}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\lambda}}}}}}_r=0\right)
\label{eq:e_unobserved}$$
Once each column has its set of inputs, the permanences must be initialized. As previously stated, permanences were defined to be initialized with a random value close to $\rho_s$, but biased based off the distance between the synapse’s source (input column) and destination (SP column). To obtain further clarification, NuPIC’s source code [@nupic] was consulted. It was found that the permanences were randomly initialized, with approximately half of the permanences creating connected proximal synapses and the remaining permanences creating potential (unconnected) proximal synapses. Additionally, to ensure that each column has a fair chance of being selected during inhibition, there are at least $\rho_d$ connected proximal synapses on each column.
Let $\boldsymbol{\Phi} \in \mathbb{R}^{m \times q}$ be defined as the set of permanences for each column, such that $\boldsymbol{\Phi}_i$ is the set of permanences for the proximal synapses for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. Each $\boldsymbol{\Phi}_{i,k}$ is randomly initialized as shown in [(\[eq:perm\_init\])]{}, where $\operatorname{Unif}$ represents the uniform distribution. Using [(\[eq:perm\_init\])]{}, the expected permanence value would be equal to $\rho_s$; thus, $\nicefrac{q}{2}$ proximal synapses would be initialized as connected for each column. To ensure that each column has a fair chance of being selected, $\rho_d$ should be less than $\nicefrac{q}{2}$.
$$\boldsymbol{\Phi}_{i,k} \sim \operatorname{Unif}(\rho_s - \phi_{\delta}, \rho_s + \phi_{\delta})
\label{eq:perm_init}$$
It is possible to predict, before training, the initial response of the SP with a given input. This insight allows parameters to be crafted in a manner that ensures a desired amount of column activity. Let $\boldsymbol{X} \in \{0, 1\}^{m \times q}$ be defined as the set of inputs for each column, such that $\boldsymbol{X}_i$ is the set of inputs for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ai}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ai}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ai}}}}}}_i\equiv\sum_{k=0}^{q-1}\boldsymbol{X}_{i,k}$, the random variable governing the number of active inputs on column $i$. Let $\mathbb{P}(\boldsymbol{X}_{i,k})$ be defined as the probability of the input connected via proximal synapse $k$ to column $i$ being active. $\mathbb{P}(\boldsymbol{X}_i)$ is therefore defined to be the probability of an input connected to column $i$ being active. Similarly, $\mathbb{P}(\boldsymbol{X})$ is defined to be the probability of an input on any column being active. The expected number of active proximal synapses on column $i$ is then given by [(\[eq:e\_active\])]{}. Let $a\equiv\frac{1}{m}\sum_{i=0}^{m-1}\sum_{k=0}^{q-1}\boldsymbol{X}_{i,k}$, the random variable governing the average number of active inputs on a column. Equation [(\[eq:e\_active\])]{} is then generalized to [(\[eq:e\_active\_generalized\])]{}, the expected number of active proximal synapses for each column.
$$\mathbb{E}[{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ai}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ai}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ai}}}}}}_i] = q\mathbb{P}(\boldsymbol{X}_i)
\label{eq:e_active}$$
$$\mathbb{E}[a] = q\mathbb{P}(\boldsymbol{X})
\label{eq:e_active_generalized}$$
Let $\boldsymbol{AC}_{i,k}\equiv\boldsymbol{X}_{i,k}\cap \operatorname{I}\left(\boldsymbol{\Phi}_{i,k}\ge \rho_s\right)$, the event that proximal synapse $k$ is active and connected on column $i$. Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ac}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ac}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ac}}}}}}_i\equiv\sum_{k=0}^{q-1}\boldsymbol{AC}_{i,k}$, the random variable governing the number of active and connected proximal synapses for column $i$. Let $\mathbb{P}(\boldsymbol{AC}_{i,k})\equiv\mathbb{P}(\boldsymbol{X}_{i,k})\rho_s$, the probability that a proximal synapse is active and connected[^7]. Following [(\[eq:e\_active\])]{}, the expected number of active connected proximal synapses on column $i$ is given by [(\[eq:e\_active\_connected\])]{}.
$$\mathbb{E}[{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{ac}}} \settoheight{\arght}{\ensuremath{\boldsymbol{ac}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{ac}}}}}}_i] = q\mathbb{P}(\boldsymbol{AC}_{i,k})
\label{eq:e_active_connected}$$
Let $\operatorname{Bin}(k; n, p)$ be defined as the probability mass function (PMF) of a binomial distribution, where $k$ is the number of successes, $n$ is the number of trials, and $p$ is the success probability in each trial. Let $at\equiv\sum_{i=0}^{m-1}\operatorname{I}\left(\left(\sum_{k=0}^{q-1}\boldsymbol{X}_{i,k}\right) \ge \rho_d\right)$, the random variable governing the number of columns having at least $\rho_d$ active proximal synapses. Let $act\equiv\sum_{i=0}^{m-1}\operatorname{I}\left(\left(\sum_{k=0}^{q-1}\boldsymbol{AC}_{i,k}\right) \ge \rho_d\right)$, the random variable governing the number of columns having at least $\rho_d$ active connected proximal synapses. Let $\pi_x$ and $\pi_{ac}$ be defined as random variables that are equal to the overall mean of $\mathbb{P}(\boldsymbol{X})$ and $\mathbb{P}(\boldsymbol{AC})$, respectively. The expected number of columns with at least $\rho_d$ active proximal synapses and the expected number of columns with at least $\rho_d$ active connected proximal synapses are then given by [(\[eq:e\_active\_threshold\])]{} and [(\[eq:e\_active\_connected\_threshold\])]{}, respectively.
In [(\[eq:e\_active\_threshold\])]{}, the summation computes the probability of having less than $\rho_d$ active connected proximal synapses, where the individual probabilities within the summation follow the PMF of a binomial distribution. To obtain the desired probability, the complement of that probability is taken. It is then clear that the mean is nothing more than that probability multiplied by $m$. For [(\[eq:e\_active\_connected\_threshold\])]{} the logic is similar, with the key difference being that the probability of a success is a function of both $\boldsymbol{X}$ and $\rho_s$, as it was in [(\[eq:e\_active\_connected\])]{}.
![SP phase 1 example where $m = 12$, $q = 5$, and $\rho_d = 2$. It was assumed that the boost for all columns is at the initial value of ‘1’. For simplicity, only the connections for the example column, highlighted in gray, are shown.[]{data-label="fig:phase1"}](sp1){width="\linewidth"}
$$\mathbb{E}[at]=m\left[1-\sum_{t=0}^{\rho_d-1}\operatorname{Bin}(t; q, \pi_x)\right]
\label{eq:e_active_threshold}$$
$$\mathbb{E}[act]=m\left[1-\sum_{t=0}^{\rho_d-1}\operatorname{Bin}(t; q, \pi_{ac})\right]
\label{eq:e_active_connected_threshold}$$
Phase 1: Overlap
----------------
Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{b}}} \settoheight{\arght}{\ensuremath{\boldsymbol{b}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{b}}}}}} \in \mathbb{R}^{1 \times m}$ be defined as the set of boost values for all columns, such that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{b}}} \settoheight{\arght}{\ensuremath{\boldsymbol{b}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{b}}}}}}_i$ is the boost for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. Let $\boldsymbol{Y}\equiv\operatorname{I}(\boldsymbol{\Phi}_i \geq \rho_s)\ \forall i$, the bit-mask for the proximal synapse’s activations. $\boldsymbol{Y}_i$ is therefore a row-vector bit-mask, with each ‘1’ representing a connected synapse and each ‘0’ representing an unconnected synapse. In this manner, the connectivity (or lack thereof) for each synapse on each column is obtained. The overlap for all columns, ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}} \in \{0, 1\}^{1 \times m}$, is then obtained by using [(\[eq:overlap\])]{}, which is a function of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\alpha}}}}}}} \in \mathbb{Z}^{1 \times m}$. ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\alpha}}}}}}}$ is the sum of the active connected proximal synapses for all columns, and is defined in [(\[eq:preoverlap\])]{}.
Comparing these equations with [Algorithm \[alg:phase1\]]{}, it is clear that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\alpha}}}}}}}$ will have the same value as $col.overlap$ before line five, and that the final value of $col.overlap$ will be equal to ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}}$. To help provide further understanding, a simple example demonstrating the functionality of this phase is shown in [Fig. \[fig:phase1\]]{}.
![SP phase 2 example where $\rho_c = 2$ and $\sigma_o = 2$. The overlap values were determined from the SP phase 1 example.[]{data-label="fig:phase2"}](sp2){width="\linewidth"}
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}} \equiv
\begin{cases}
{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\alpha}}}}}}}_i{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{b}}} \settoheight{\arght}{\ensuremath{\boldsymbol{b}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{b}}}}}}_i & {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\alpha}}}}}}}_i \geq \rho_d, \\
0 & \operatorname{otherwise}
\end{cases}
\ \forall i
\label{eq:overlap}$$
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\alpha}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\alpha}}}}}}}_i \equiv \boldsymbol{X}_i \bullet \boldsymbol{Y}_i
\label{eq:preoverlap}$$
Phase 2: Inhibition
-------------------
Let $\boldsymbol{H} \in \{0, 1\}^{m \times m}$ be defined as the neighborhood mask for all columns, such that $\boldsymbol{H}_i$ is the neighborhood for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_j$ is then said to be in ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$’s neighborhood if and only if $\boldsymbol{H}_{i,j}$ is ‘1’. Let $\operatorname{kmax}(S, k)$ be defined as the $k$-th largest element of $S$. Let $\operatorname{max}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{v}}} \settoheight{\arght}{\ensuremath{\boldsymbol{v}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{v}}}}}})$ be defined as a function that will return the maximum value in ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{v}}} \settoheight{\arght}{\ensuremath{\boldsymbol{v}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{v}}}}}}$. The set of active columns, ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}} \in \{0, 1\}^{1 \times m}$, may then be obtained by using [(\[eq:inhibition\])]{}, where ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}}$ is an indicator vector representing the activation (or lack of activation) for each column. The result of the indicator function is determined by ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\gamma}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\gamma}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\gamma}}}}}} \in \mathbb{Z}^{1 \times m}$, which is defined in [(\[eq:activity\_level\])]{} as the $\rho_c$-th largest overlap (lower bounded by one) in the neighborhood of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i\ \forall i$.
Comparing these equations with [Algorithm \[alg:phase2\]]{}, ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\gamma}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\gamma}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\gamma}}}}}}$ is a slightly altered version of $mo$. Instead of just being the $\rho_c$-th largest overlap for each column, it is additionally lower bounded by one. Referring back to [Algorithm \[alg:phase2\]]{}, line 3 is a biconditional statement evaluating to true if the overlap is at least $mo$ and greater than zero. By simply enforcing $mo$ to be at least one, the biconditional is reduced to a single condition. That condition is evaluated within the indicator function; therefore, [(\[eq:inhibition\])]{} carries out the logic in the if statement in [Algorithm \[alg:phase2\]]{}. Continuing with the demonstration shown in [Fig. \[fig:phase1\]]{}, [Fig. \[fig:phase2\]]{} shows an example execution of phase two.
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}} \equiv \operatorname{I}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}}_i \geq {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\gamma}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\gamma}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\gamma}}}}}}_i)\ \forall i
\label{eq:inhibition}$$
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\gamma}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\gamma}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\gamma}}}}}} \equiv \operatorname{max}(\operatorname{kmax}(\boldsymbol{H}_i \odot {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}}, \rho_c), 1)\ \forall i
\label{eq:activity_level}$$
Phase 3: Learning
-----------------
Let $\operatorname{clip}(\boldsymbol{M}, lb, ub)$ be defined as a function that will clip all values in the matrix $\boldsymbol{M}$ outside of the range $[lb, ub]$ to $lb$ if the value is less than $lb$, or to $ub$ if the value is greater than $ub$. $\boldsymbol{\Phi}$ is then recalculated by [(\[eq:permanence\_update\])]{}, where $\boldsymbol{\delta\Phi}$ is the proximal synapse’s permanence update amount given by [(\[eq:permanence\_delta\])]{}[^8].
$$\boldsymbol{\Phi} \equiv \operatorname{clip}\left(\boldsymbol{\Phi} \oplus \boldsymbol{\delta\Phi}, 0, 1\right)
\label{eq:permanence_update}$$
$$\boldsymbol{\delta\Phi} \equiv {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}}^T \odot (\phi_+\boldsymbol{X} - (\phi_-\neg\boldsymbol{X}))
\label{eq:permanence_delta}$$
The result of these two equations is equivalent to the result of executing the first seven lines in [Algorithm \[alg:phase3\]]{}. If a column is active, it will be denoted as such in ; therefore, using that vector as a mask, the result of [(\[eq:permanence\_delta\])]{} will be a zero if the column is inactive, otherwise it will be the update amount. From [Algorithm \[alg:phase3\]]{}, the update amount should be $\phi_+$ if the synapse was active and $\phi_-$ if the synapse was inactive. A synapse is active only if its source column is active. That activation is determined by the corresponding value in $\boldsymbol{X}$. In this manner, $\boldsymbol{X}$ is also being used as a mask, such that active synapses will result in the update equalling $\phi_+$ and inactive synapses (selected by inverting $\boldsymbol{X}$) will result in the update equalling $\phi_-$. By clipping the element-wise sum of $\boldsymbol{\Phi}$ and $\boldsymbol{\delta\Phi}$, the permanences stay bounded between \[0, 1\]. As with the previous two phases, the visual demonstration is continued, with [Fig. \[fig:phase3\]]{} illustrating the primary functionality of this phase. Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta^{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta^{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta^{(a)}}}}}}} \in \mathbb{R}^{1 \times m}$ be defined as the set of active duty cycles for all columns, such that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}}}}}$ is the active duty cycle for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c_i}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c_i}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c_i}}}}}}$. Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta^{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta^{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta^{(min)}}}}}}} \in \mathbb{R}^{1 \times m}$ be defined by [(\[eq:min\_duty\_cycle\])]{} as the set of minimum active duty cycles for all columns, such that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}$ is the minimum active duty cycle for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. This equation is clearly the same as line 9 in [Algorithm \[alg:phase3\]]{}.
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta^{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta^{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta^{(min)}}}}}}} \equiv \kappa_a\operatorname{max}\left(\boldsymbol{H}_i \odot {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta^{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta^{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta^{(a)}}}}}}}\right)\ \forall i
\label{eq:min_duty_cycle}$$
Let $\operatorname{update\_active\_duty\_cycle}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}})$ be defined as a function that updates the moving average duty cycle for the active duty cycle for each ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c_i}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c_i}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c_i}}}}}} \in {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}$. That function should compute the frequency of each column’s activation. After calling $\operatorname{update\_active\_duty\_cycle}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}})$, the boost for each column is updated by using [(\[eq:boost\_update\])]{}. In [(\[eq:boost\_update\])]{}, $\beta\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}}}}}, {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}\right)$ is defined as the boost function, following [(\[eq:boost\_calc\])]{}[^9]. The functionality of [(\[eq:boost\_update\])]{} is therefore shown to be equivalent to [Algorithm \[alg:boost\]]{}.
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{b}}} \settoheight{\arght}{\ensuremath{\boldsymbol{b}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{b}}}}}} \equiv \beta\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}}}}}, {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}\right) \ \forall i
\label{eq:boost_update}$$
$$\resizebox{\hsize}{!}{$ \beta\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}}}}}, {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}\right) \equiv \begin{cases} \beta_0 & {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}} = 0 \\[1ex] 1 & {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}}}}} > {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}} \\[1ex] {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(a)}}}}}}}\frac{1 - \beta_0}{\vphantom{{{{{{a^k}^k}^k}^k}^k}^k}{{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}}} + \beta_0 & \operatorname{otherwise} \end{cases}$ }
\label{eq:boost_calc}$$
![SP phase 3 example, demonstrating the adaptation of the permanences. The gray columns are used denote the active columns, where those activations were determined from the SP phase 2 example.[]{data-label="fig:phase3"}](sp3){width="\linewidth"}
Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta^{(o)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta^{(o)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta^{(o)}}}}}}} \in \mathbb{R}^{1 \times m}$ be defined as the set of overlap duty cycles for all columns, such that ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(o)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(o)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(o)}}}}}}}$ is the overlap duty cycle for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. Let $\operatorname{update\_overlap\_duty\_cycle}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}})$ be defined as a function that updates the moving average duty cycle for the overlap duty cycle for each ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i \in {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}$. That function should compute the frequency of each column’s overlap being at least equal to $\rho_d$. After applying $\operatorname{update\_overlap\_duty\_cycle}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}})$, the permanences are then boosted by using [(\[eq:boost\_permanence\])]{}. This equation is equivalent to lines 13 – 15 in [Algorithm \[alg:phase3\]]{}, where the multiplication with the indicator function is used to accomplish the conditional and clipping is done to ensure the permanences stay within bounds.
$$\boldsymbol{\Phi} \equiv \operatorname{clip}\left(\boldsymbol{\Phi} \oplus \kappa_b\rho_s\operatorname{I}\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(o)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(o)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(o)}}}}}}} < {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}\right), 0, 1\right)
\label{eq:boost_permanence}$$
Let $\operatorname{d}(x, y)$ be defined as the distance function[^10] that computes the distance between $x$ and $y$. To simplify the notation[^11], let $\operatorname{pos}(c, r)$ be defined as a function that will return the position of the column indexed at $c$ located $r$ regions away from the current region. For example, $\operatorname{pos}(0, 0)$ returns the position of the first column located in the SP and $\operatorname{pos}(0, -1)$ returns the position of the first column located in the previous region. The distance between $\operatorname{pos}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i, 0)$ and pos$(\boldsymbol{\Lambda}_{i,k}, -1)$ is then determined by $\operatorname{d}(\operatorname{pos}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i, 0), \operatorname{pos}(\boldsymbol{\Lambda}_{i,k}, -1))$.
Let $\boldsymbol{D} \in \mathbb{R}^{m \times q}$ be defined as the distance between an SP column and its corresponding connected synapses’ source columns, such that $\boldsymbol{D}_{i,k}$ is the distance between ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$ and ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$’s proximal synapse’s input at index $k$. $\boldsymbol{D}$ is computed following [(\[eq:distances\])]{}, where $\boldsymbol{Y}_i$ is used as a mask to ensure that only connected synapses may contribute to the distance calculation. The result of that element-wise multiplication would be the distance between the two columns or zero for connected and unconnected synapses, respectively[^12].
$$\boldsymbol{D} \equiv (\operatorname{d}(\operatorname{pos}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i, 0), \operatorname{pos}(\boldsymbol{\Lambda}_{i,k}, -1)) \odot \boldsymbol{Y}_i\ \forall k)\ \forall i
\label{eq:distances}$$
The inhibition radius, $\sigma_0$, is defined by [(\[eq:inhibition\_radius\])]{}. The division in [(\[eq:inhibition\_radius\])]{} is the sum of the distances divided by the number of connected synapses[^13]. That division represents the average distance between connected synapses’ source and destination columns, and is therefore the average receptive field size. The inhibition radius is then set to the average receptive field size after it has been floored and raised to a minimum value of one, ensuring that the radius is an integer at least equal to one. Comparing [(\[eq:inhibition\_radius\])]{} to line 16 in [Algorithm \[alg:phase3\]]{}, the two are equivalent.
$$\sigma_o \equiv \operatorname{max} \left(1, \left\lfloor\frac{\sum_{i=0}^{m-1}\sum_{k=0}^{q-1}\boldsymbol{D}_{i,k}}{\operatorname{max}(1, \sum_{i=0}^{m-1}\sum_{k=0}^{q-1}\boldsymbol{Y}_{i,k})}\right\rfloor\right)
\label{eq:inhibition_radius}$$
Once the inhibition radius has been computed, the neighborhood for each column must be updated. This is done using the function $\operatorname{h}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i)$, which is dependent upon the type of inhibition being used (global or local) as well as the topology of the system[^14]. This function is shown in [(\[eq:neighborhood\])]{}, where ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\zeta}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\zeta}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\zeta}}}}}}$ represents all of the columns located at the set of integer Cartesian coordinates bounded by an $n$-dimensional shape. Typically the $n$-dimensional shape is a represented by an $n$-dimensional hypercube.
$$\operatorname{h}({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c_i}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c_i}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c_i}}}}}}) \equiv
\begin{cases}
{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}} & global~inhibition \\
{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\zeta}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\zeta}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\zeta}}}}}} & local~inhibition
\end{cases}
\label{eq:neighborhood}$$
Boosting
========
![Demonstration of boost as a function of a column’s minimum active duty cycle and active duty cycle.[]{data-label="fig:boost_value"}](boost_value){width="\linewidth"}
It is important to understand the dynamics of boosting utilized by the SP. The SP’s boosting mechanism is similar to DeSieno’s [@desieno] conscience mechanism. In that work, clusters that were too frequently active were penalized, allowing weak clusters to contribute to learning. The SP’s primary boosting mechanism takes the reverse approach by rewarding infrequently active columns. Clearly, the boosting frequency and amount will impact the SP’s learned representations.
The degree of activation is determined by the boost function, [(\[eq:boost\_calc\])]{}. From that equation, it is clear that a column’s boost is determined by the column’s minimum active duty cycle as well as the column’s active duty cycle. Those two values are coupled, as a column’s minimum active duty cycle is a function of its duty cycle, as shown in [(\[eq:min\_duty\_cycle\])]{}. To study how those two parameters affect a column’s boost, [Fig. \[fig:boost\_value\]]{} was created. From this plot it is found that the non-boundary conditions for a column’s boost follows the shape $1 / {\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\eta}_i^{\boldsymbol{(min)}}}}}}}$. It additionally shows the importance of evaluating the piecewise boost function in order. If the second condition is evaluated before the first condition, the boost will be set to its minimum, instead of its maximum value.
To study the frequency of boosting, the average number of boosted columns was observed by varying the level of sparseness in the input for both types of inhibition, as shown in [Fig. \[fig:boost\_sparseness\]]{}. For the overlap boosting mechanism, [(\[eq:boost\_update\])]{}, very little boosting occurs, with boosting occurring more frequently for denser inputs. This is to be expected, as more bits would be active in the input; thus, causing more competition to occur among the columns.
For the permanence boosting mechanism, [(\[eq:boost\_permanence\])]{}, boosting primarily occurs when the sparsity is between 70 and 76%, with almost no boosting occurring outside of that range. That boosting is a result of the SP’s parameters. In this experiment, $q = 40$ and $\rho_d = 15$. Based off those parameters, there must be at least 25 active inputs on a column for it have a non-zero overlap; i.e. if the sparsity is 75%, a column would have to be connected to each active bit in the input to obtain a non-zero overlap. As such, if the sparsity is greater than 75% it will not be possible for the columns to have a non-zero overlap, resulting in no boosting. For lower amounts of sparsity, boosting was not needed, since adequate input coverage of the active bits is obtained.
Recall that the initialization of the SP is performed randomly. Additionally, the positions of active bits for this dataset are random. That randomness combined with a starved input results in a high degree of volatility. This is observed by the extremely large error bars. Some of the SP’s initializations resulted in more favorable circumstances, since the columns were able to connect to the active bits in the input.
For the SP to adapt to the lack of active bits, it would have to boost its permanence. This would result in a large amount of initial boosting, until the permanences reached a high enough value. Once the permanences reach that value, boosting will only be required occasionally, to ensure the permanences never fully decay. This behavior is observed in [Fig. \[fig:boost\_permanence\]]{}, where the permanence boosting frequency was plotted for a sparsity of 74%. The delayed start occurs because the SP has not yet determined which columns will need to be boosted. Once that set is determined, a large amount of boosting occurs. The trend follows a decaying exponential that falls until its minimum level is reached, at which point the overall degree of boosting remains constant. This trend was common among the sparsities that resulted in a noticeable degree of permanence boosting. The right-skewed decaying exponential was also observed in DeSieno’s work [@desieno].
![Demonstration of frequency of both boosting mechanisms as a function of the sparseness of the input. The top figure shows the results for global inhibition and the bottom figure shows the results for local inhibition.[]{data-label="fig:boost_sparseness"}](boost_sparseness){width="\linewidth"}
These results show that the need for boosting can be eliminated by simply choosing appropriate values for $q$ and $\rho_d$[^15]. It is thus concluded that these boosting mechanisms are secondary learning mechanisms, with the primary learning occurring from the permanence update in [(\[eq:permanence\_update\])]{}. These findings allow resource limited systems (especially hardware designs) to exclude boosting, while still obtaining comparable results; thereby, greatly reducing the complexity of the system.
![Frequency of boosting for the permanence boosting mechanism for a sparsity of 74%. The top figure shows the results for global inhibition and the bottom figure shows the results for local inhibition Only the first 200 iterations were shown, for clarity, as the remaining 800 propagated the trend.[]{data-label="fig:boost_permanence"}](boost_permanence){width="\linewidth"}
Feature learning
================
Probabilistic Feature Mapping
-----------------------------
It is convenient to think of a permanence value as a probability. That probability is used to determine if a synapse is connected or unconnected. It also represents the probability that the synapse’s input bit is important. It is possible for a given input bit to be represented in multiple contexts, where the context for a specific instance is defined to be the set of inputs connected, via proximal synapses, to a column. Due to the initialization of the network, it is apparent that each context represents a random subspace; therefore, each column is learning the probability of importance for its random subset of attributes in the feature space. This is evident in [(\[eq:permanence\_delta\])]{}, as permanences contributing to a column’s activation are positively reinforced and permanences not contributing to a column’s activation are negatively reinforced.
If all contexts for a given input bit are observed, the overall importance of that bit is obtained. Multiple techniques could be conjured for determining how the contexts are combined. The most generous method is simply to observe the maximum. In this manner, if the attribute was important in at least one of the random subspaces, it would be observed. Using those new probabilities the degree of influence of an attribute may be obtained. Let ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}} \in (0, 1)^{1 \times p}$ be defined as the set of learned attribute probabilities. One form of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}}$ is shown in [(\[eq:permanence\_map\])]{}[^16]. In [(\[eq:permanence\_map\])]{}, the indicator function is used to mask the permanence values for $\boldsymbol{U}_{s,r}$. Multiplying that value by every permanence in $\boldsymbol{\Phi}$ obtains all of the permanences for $\boldsymbol{U}_{s,r}$. This process is used to project the SP’s representation of the input back into the input space.
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}} \equiv \operatorname{max}\left(\boldsymbol{\Phi}_{i,k}\operatorname{I}(\boldsymbol{\Lambda}_{i,k}=r)\ \forall i\ \forall k\right)\ \forall r
\label{eq:permanence_map}$$
Dimensionality Reduction
------------------------
The learned attribute probabilities may be used to perform dimensionality reduction. Assuming the form of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}}$ is that in [(\[eq:permanence\_map\])]{}, the probability is stated to be important if it is at least equal to $\rho_s$. This holds true, as ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}}$ is representative of the maximum permanence for each input in $\boldsymbol{U}_s$. For a given $\boldsymbol{U}_{s,r}$ to be observed, it must be connected, which may only happen when its permanence is at least equal to $\rho_s$. Given that, the attribute mask, ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{z}}} \settoheight{\arght}{\ensuremath{\boldsymbol{z}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{z}}}}}}\in \{0, 1\}^{1 \times p}$, is defined to be $\operatorname{I}\left({\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}}\ge \rho_s\right)$. The new set of attributes are those whose corresponding index in the attribute mask are true, i.e. $\boldsymbol{U}_{s,r}$ is a valid attribute if ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{z}}} \settoheight{\arght}{\ensuremath{\boldsymbol{z}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{z}}}}}}_r$ is true.
Input Reconstruction
--------------------
Using a concept similar to the probabilistic feature mapping technique, it is possible to obtain the SP’s learned representation of a specific pattern. To reconstruct the input pattern, the SP’s active columns for that pattern must be captured. This is naturally done during inhibition, where ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}}$ is constructed. ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}}$, a function of $\boldsymbol{U}_s$, is used to represent a specific pattern in the context of the SP.
Determining which permanences caused the activation is as simple as using ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}}$ to mask $\boldsymbol{\Phi}$. Once that representation has been obtained, the process follows that of the probabilistic feature mapping technique, where $\operatorname{I}(\boldsymbol{\Lambda}_{i,k}=r)$ is used as a second mask for the permanences. Those steps will produce a valid probability for each input bit; however, it is likely that there will exist probabilities that are not explicitly in {0, 1}. To account for that, the same technique used for dimensionality reduction is applied, by simply thresholding the probability at $\rho_s$. This process is shown in [(\[eq:input\_reconstruction\])]{}[^17], where ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{u}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{u}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{u}}}}}}}\in\{0, 1\}^{1 \times p}$ is defined to be the reconstructed input.
$${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{u}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{u}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{u}}}}}}} \equiv \operatorname{I}\left(\left[\operatorname{max}\left(\boldsymbol{\Phi}_{i,k}{\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}}}}}}}_i\operatorname{I}(\boldsymbol{\Lambda}_{i,k}=r)\ \forall i\ \forall k\right)\ge \rho_s\right]\right)\ \forall r
\label{eq:input_reconstruction}$$
Experimental Results and Discussion
===================================
To empirically investigate the performance of the SP, a Python implementation of the SP was created, called math HTM (mHTM)[^18]. The SP was tested on both spatial data as well as categorical data. The details of those experiments are explained in the ensuing subsections.
Spatial Data
------------
--------------- -------
column 7.70%
probabilistic 8.98%
reduction 9.03%
--------------- -------
: SP performance on MNIST using global inhibition[]{data-label="tbl:mnist_global"}
The SP is a spatial algorithm, as such, it should perform well with inherently spatial data. To investigate this, the SP was tested with a well-known computer vision task. The SP requires a binary input; as such, it was desired to work with images that were originally black and white or could be readily made black and white without losing too much information. Another benefit of using this type of image is that the encoder[^19] may be de-emphasized, allowing for the primary focus to be on the SP. With those constraints, the modified National Institute of Standards and Technology’s (MNIST’s) database of handwritten digits [@mnist] was chosen as the dataset.
The MNIST images are simple $28\times28$ grayscale images, with the bulk of the pixels being black or white. To convert them to black and white images, each pixel was set to ‘1’ if the value was greater than or equal to $\nicefrac{255}{2}$ and ‘0’ otherwise. Each image was additionally transformed to be one-dimensional by horizontally stacking the rows. The SP has a large number of parameters, making it difficult to optimize the parameter selection. To help with this, $1,000$ independent trials were created, all having a unique set of parameters. The parameters were randomly selected within reasonable limits[^20]. Additionally, parameters were selected such that $\mathbb{E}[\lambda'] = 0$. To reduce the computational load, the size of the MNIST dataset was reduced to $800$ training samples and $200$ testing samples. The samples were chosen from their respective initial sets using a stratified shuffle split with a total of five splits. To ensure repeatability and to allow proper comparisons, care was taken to ensure that both within and across experiments the same random initializations were occurring. To perform the classification, a linear support vector machine (SVM) was utilized. The input to the SVM was the corresponding output of the SP.
Three comparisons were explored for both global and local inhibition: using the set of active columns as the features (denoted as “column”), using ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{\phi}}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{\phi}}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.3\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{\phi}}}}}}}$ as the features (denoted as “probabilistic”), and using the dimensionality reduced version of the input as the features (denoted as “reduction”). For each experiment the average error of the five splits was computed for each method. The top $10$ performers for each method were then retrained on the full MNIST dataset. From those results, the set of parameters with the lowest error across the experiments and folds was selected as the optimal set of parameters.
--------------- -------
column 7.85%
probabilistic 9.07%
reduction 9.07%
--------------- -------
: SP performance on MNIST using local inhibition[]{data-label="tbl:mnist_local"}
The results are shown in [(Table \[tbl:mnist\_global\])]{} and [(Table \[tbl:mnist\_local\])]{} for global and local inhibition, respectively. For reference, the same SVM without the SP resulted in an error of 7.95%. The number of dimensions was reduced by 38.27% and 35.71% for global and local inhibition, respectively. Both the probabilistic and reduction methods only performed marginally worse than the base SVM classifier. Considering that these two techniques are used to modify the raw input, it is likely that the learned features were the face of the numbers (referring to inputs equaling ‘1’). In that case, those methods would almost act as pass through filters, as the SVM is already capable of determining which features are more / less significant. That being said, being able to reduce the number of features by over two thirds, for the local inhibition case, while still performing relatively close to the case where all features are used is quite desirable.
Using the active columns as the learned feature is the default behavior, and it is those activations that would become the feedforward input to the next level (assuming an HTM with multiple SPs and / or TMs). Both global and local inhibition outperformed the SVM, but only by a slight amount. Considering that only one SP region was utilized, that the SP’s primary goal is to map the input into a new domain to be understood by the TM, and that the SP did not hurt the SVM’s ability to classify, the SP’s overall performance is acceptable. It is also possible that given a two-dimensional topology and restricting the initialization of synapses to a localized radius may improve the accuracy of the network. Comparing global to local inhibition, comparable results are obtained. This is likely due to the globalized formation of synaptic connections upon initialization, since that results in a loss of the initial network topology.
To explore the input reconstruction technique, a random instance of each class from MNIST was selected. The input was then reconstructed as shown in [Fig. \[fig:input\_reconstruction\]]{}. The top row shows the original representation of the inputs. The middle row shows the SDR of the inputs. The bottom row shows the reconstructed versions. The representations are by no means perfect, but it is evident that the SP is indeed learning an accurate representation of the input.
![Reconstruction of the input from the context of the SP. Shown are the original input images (top), the SDRs (middle), and the reconstructed version (bottom).[]{data-label="fig:input_reconstruction"}](input_reconstruction){width="\linewidth"}
Categorical Data
----------------
One of the main purposes of the SP is to create a spatial representation of its input through the process of mapping its input to SDRs. To explore this, the SP was tested on Bohanec et al.’s car evaluation dataset [@bohanec1988knowledge], [@Lichman:2013]. This dataset consists of four classes and six attributes. Each attribute has a finite number of states with no missing values. To encode the attributes, a multivariate encoder comprised of categorical encoders was used[^21]. The class labels were also encoded, by using a single category encoder[^22].
The selection of the SP’s parameters was determined through manual experimentation[^23]. Cross validation was used, by partitioning the data using a stratified shuffle split with eight splits. To perform the classification an SVM was utilized, where the output of the SP was the corresponding input the SVM. The SP’s performance was also compared to just using the linear SVM and using a random forest classifier[^24]. For those classifiers, a simple preprocessing step was performed to map the text-based values to integers.
----------------- --------
Linear SVM 26.01%
Random Forest 8.96%
SP + Linear SVM 1.73%
----------------- --------
: Comparison of classifiers on car evaluation dataset[]{data-label="tbl:car_eval"}
The results are shown in [(Table \[tbl:car\_eval\])]{}[^25]. The SVM performed poorly, having an error of 26.01%. Not surprisingly, the random forest classifier performed much better, obtaining an error of 8.96%. The SP was able to far outperform either classifier, obtaining an error of only 1.73%. From literature, the best known error on this dataset was 0.37%, which was obtained from a boosted multilayer perceptron (MLP) [@oza2005online]. Comparatively, the SP with the SVM is a much less complicated system.
This result shows that the SP was able to map the input data into a suitable format for the SVM, thereby drastically improving the SVM’s classification. Based off this, it is determined that the SP produced a suitable encoding.
Extended Discussion
-------------------
Comparing the SP’s performance on spatial data to that of categorical data provides some interesting insights. It was observed that on spatial data the SP effectively acted as a pass through filter. This behavior occurs because the data is inherently spatial. The SP maps the spatial data to a new spatial representation. This mapping allows classifiers, such as an SVM, to be able to classify the data with equal effectiveness.
Preprocessing the categorical data with the SP provided the SVM with a new spatial representation. That spatial representation was understood by the SVM as readily as if it were inherently spatial. This implies that the SP may be used to create a spatial representation from non-spatial data. This would thereby provide other algorithms, such as the TM and traditional spatial classifiers, a means to interpret non-spatial data.
Exploring the Primary Learning Mechanism
========================================
To complete the mathematical formulation it is necessary to define a function governing the primary learning process. Within the SP, there are many learned components: the set of active columns, the neighborhood (through the inhibition radius), and both of the boosting mechanisms. All of those components are a function of the permanence, which serves as the probability of an input bit being active in various contexts.
As previously discussed, the permanence is updated by [(\[eq:permanence\_delta\])]{}. That update equation may be split into two distinct components. The first component is the set of active columns, which is used to determine the set of permanences to update. The second component is the remaining portion of that equation, and is used to determine the permanence update amount.
Plausible Origin for the Permanence Update Amount
-------------------------------------------------
In the permanence update equation, [(\[eq:permanence\_delta\])]{}, it is noted that the second component is an unlearned function of a random variable coming from a prior distribution. That random variable is nothing more than $\boldsymbol{X}$. It is required that $\boldsymbol{X}_{i,k} \sim \operatorname{Ber}(\mathbb{P}(\boldsymbol{X}_{i,k}))$, where $\operatorname{Ber}$ is used to denote the Bernoulli distribution. If it is assumed that each $\boldsymbol{X}_{i,k} \in \boldsymbol{X}$ are independent and identically distributed (i.i.d.), then $\boldsymbol{X} { \savebox{\mybox}{\hbox{$\scriptstylei.i.d.$}} \savebox{\mysim}{\hbox{$\sim$}} \mathbin{\overset{i.i.d.}{\resizebox{\wd\mybox}{\ht\mysim}{$\sim$}}}} \operatorname{Ber}(\theta)$, where $\theta$ is defined to be the probability of an input being active. Using the PMF of the Bernoulli distribution, the likelihood of $\theta$ given $\boldsymbol{X}$ is obtained in [(\[eq:joint\_likelihood\])]{}, where $t\equiv mq$ and $\boldsymbol{\overline{X}} \equiv \frac{1}{t}\sum_{i=0}^{m-1}\sum_{k=0}^{q-1}\boldsymbol{X}_{i,k}$, the overall mean of $\boldsymbol{X}$. The corresponding log-likelihood of $\theta$ given $\boldsymbol{X}$ is given in [(\[eq:joint\_log\_likelihood\])]{}.
$$\begin{split}
\mathcal{L}(\theta; \boldsymbol{X}) &= \prod_{i=0}^{m}\prod_{k=0}^{q}\theta^{\boldsymbol{X}_{i,k}}(1-\theta)^{1-\boldsymbol{X}_{i,k}}\\
&=\theta^{t\boldsymbol{\overline{X}}}(1-\theta)^{t-t\boldsymbol{\overline{X}}}
\end{split}
\label{eq:joint_likelihood}$$
$$\ell(\theta; \boldsymbol{X}) = t\boldsymbol{\overline{X}}\text{log}(\theta) + (t-t\boldsymbol{\overline{X}})\text{log}(1-\theta)
\label{eq:joint_log_likelihood}$$
Taking the gradient of the joint log-likelihood of [(\[eq:joint\_log\_likelihood\])]{} with respect to $\theta$, results in [(\[eq:max\_joint\_log\_likelihood\])]{}. Ascending that gradient results in obtaining the maximum-likelihood estimator (MLE) of $\theta$, $\hat{\theta}_{MLE}$. It can be shown that $\hat{\theta}_{MLE} = \boldsymbol{\overline{X}}$. In this context, $\hat{\theta}_{MLE}$ is used as an estimator for the maximum probability of an input being active.
$$\label{eq:max_joint_log_likelihood}
\boldsymbol{\nabla}\ell(\theta; \boldsymbol{X}) = \frac{t}{\theta}\boldsymbol{\overline{X}} - \frac{t}{1-\theta}(1-\boldsymbol{\overline{X}})$$
Taking the partial derivative of the log-likelihood for a single $\boldsymbol{X}_{i,k}$ results in [(\[eq:max\_log\_likelihood\])]{}. Substituting out $\theta$ for its estimator, $\boldsymbol{\overline{X}}$, and multiplying by $\kappa$, results in [(\[eq:p\_update\_a\])]{}. $\kappa$ is defined to be a scaling parameter and must be defined such that $\frac{\kappa}{\boldsymbol{\overline{X}}} \in [0, 1]$ and $\frac{\kappa}{1-\boldsymbol{\overline{X}}} \in [0, 1]$. Revisiting the permanence update equation, [(\[eq:permanence\_delta\])]{}, the permanence update amount is equivalently rewritten as $\phi_+\boldsymbol{X} - \phi_-(\boldsymbol{J}-\boldsymbol{X})$, where $\boldsymbol{J} \in \{1\}^{m \times q}$. For a single $\boldsymbol{X}_{i,k}$ it is clear that the permanence update amount reduces to $\phi_+\boldsymbol{X}_{i,k} - \phi_-(1-\boldsymbol{X}_{i,k})$. If $\phi_+\equiv\frac{\kappa}{\boldsymbol{\overline{X}}}$ and $\phi_-\equiv\frac{\kappa}{1-\boldsymbol{\overline{X}}}$, then [(\[eq:p\_update\_a\])]{} becomes [(\[eq:p\_update\_b\])]{}. Given this, $\boldsymbol{\delta\Psi}$ is presented as a plausible origin for the permanence update amount. Using the new representations of $\phi_+$ and $\phi_-$, a relationship between the two is obtained, requiring that only one parameter, $\kappa$, be defined. Additionally, it is possible that there exists a $\kappa$ such that $\phi_+$ and $\phi_-$ may be optimally defined for the desired set of parameters.
$$\frac{\partial}{\partial\theta}\ell(\theta; \boldsymbol{X}_{i,k}) = \frac{1}{\theta}\boldsymbol{X}_{i,k} - \frac{1}{1-\theta}(1-\boldsymbol{X}_{i,k})
\label{eq:max_log_likelihood}$$
\[eq:p\_update\] $$\begin{aligned}
\label{eq:p_update_a}
\boldsymbol{\delta\Psi}_{i,k} &\equiv \frac{\kappa}{\boldsymbol{\overline{X}}}\boldsymbol{X}_{i,k} - \frac{\kappa}{1-\boldsymbol{\overline{X}}}(1-\boldsymbol{X}_{i,k})\\
\label{eq:p_update_b}
&\equiv \phi_+\boldsymbol{X}_{i,k} - \phi_-(1-\boldsymbol{X}_{i,k})
\end{aligned}$$
Discussing the Permanence Selection
-----------------------------------
The set of active columns is the learned component in [(\[eq:permanence\_delta\])]{}, obtained through a process similar to competitive learning [@competitive_learning]. In a competitive learning network, each neuron in the competitive learning layer is fully connected to each input neuron. The neurons in the competitive layer then compete, with one neuron winning the competition. The neuron that wins sets its output to ‘1’ while all other neurons set their output to ‘0’. At a global scale, this resembles the SP with two key differences. The SP permits multiple columns to be active at a time and each column is connected to a different subset of the input.
Posit that each column is equivalent to a competitive learning network. This would create a network with one neuron in the competitive layer and $q$ neurons in the input layer. The neuron in the competitive layer may only have the state of ‘1’ or ‘0’; therefore, only one neuron would be active at a time. Given this context, each column is shown to follow the competitive learning rule.
Taking into context the full SP, with each column as a competitive learning network, the SP could be defined to be a bag of competitive learning networks, i.e. an ensemble with a type of competitive learning network as its base learner. Recalling that $\boldsymbol{X} \subseteq \boldsymbol{U}_s$, each $\boldsymbol{X}_i$ is an input for ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{c}}} \settoheight{\arght}{\ensuremath{\boldsymbol{c}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{c}}}}}}_i$. Additionally each $\boldsymbol{X}_i$ is obtained by randomly sampling $\boldsymbol{U}_s$ without replacement. Comparing this ensemble to attribute bagging [@attribute_bagging], the primary difference is that sampling is done without replacement instead of with replacement.
In attribute bagging, a scheme, such as voting, must be used to determine what the result of the ensemble should be. For the SP, a form of voting is performed through the construction of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}}$. Each base learner (column) computes its degree of influence. The max degree of influence is $q$. Since that value is a constant, each ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}}_i$ may be represented as a probability by simply dividing ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\alpha}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\alpha}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{.5\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\alpha}}}}}}_i$ by $q$. In this context, each column is trying to maximize its probability of being selected. During the inhibition phase, a column is chosen to be active if its probability is at least equal to the $\rho_c$-th largest probability in its neighborhood. This process may then be viewed as a form of voting, as all columns within a neighborhood cast their overlap value as their vote. If the column being evaluated has enough votes, it will be placed in the active state.
Conclusion & Future Work
========================
In this work, a mathematical framework for HTM’s SP was presented. Using the framework, it was demonstrated how the SP can be used for feature learning. The primary learning mechanism of the SP was explored. It was shown that the mechanism consists of two distinct components, permanence selection and the degree of permanence update. A plausible estimator was provided for determining the degree of permanence update, and insight was given on the behavior of the permanence selection.
The findings in this work provide a basis for intelligently initializing the SP. Due to the mathematical framework, the provided equations could be used to optimize hardware designs. Such optimizations may include removing the boosting mechanism, limiting support to global inhibition, exploiting the matrix operations to improve performance, reducing power through the reduction of multiplexers, etc…. In the future, it is planned to explore optimized hardware designs. Additionally, it is planned to expand this work to provide the same level of understanding for the TM.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank K. Gomez of Seagate Technology, the staff at RIT’s research computing, and the members of the NanoComputing Research Lab, in particular A. Hartung and C. Merkel, for their support and critical feedback.
[^1]: This work began in August 2014.
[^2]: J. Mnatzaganian and D. Kudithipudi are with the NanoComputing Research Laboratory, Rochester Institute of Technology, Rochester, NY, 14623.
[^3]: E. Fokou[é]{} is with the Data Science Research Group, Rochester Institute of Technology, Rochester, NY, 14623.
[^4]: This parameter was originally referred to as the minimum overlap; however, it is renamed in this work to allow consistency between the SP and the TM.
[^5]: All indices start at 0.
[^6]: The parameters $\kappa_a$ and $\kappa_b$ have default values of 0.01 and 0.1, respectively.
[^7]: $\rho_s$ was used as a probability. Because $\rho_s\in\mathbb{R}, \rho_s\in(0,1)$, permanences are uniformly initialized with a mean of $\rho_s$, and for a proximal synapse to be connected it must have a permanence value at least equal to $\rho_s$, $\rho_s$ may be used to represent the probability that an initialized proximal synapse is connected.
[^8]: Due to $\boldsymbol{X}$ being binary, a bitwise negation is equivalent to the shown logical negation. In a similar manner, the multiplications of ${\text{{{ \settowidth{\argwd}{\ensuremath{\boldsymbol{\hat{c}^T}}} \settoheight{\arght}{\ensuremath{\boldsymbol{\hat{c}^T}}} \addtolength{\argwd}{.1\argwd} \raisebox{\arght}{ \makebox[.04\argwd][l]{ \resizebox{\argwd}{0.4\arght}{$\rightharpoonup$} } } \ensuremath{\boldsymbol{\hat{c}^T}}}}}}$ with $\boldsymbol{X}$ and $\neg\boldsymbol{X}$ can be replaced by an $\operatorname{AND}$ operation (logical or bitwise).
[^9]: The conditions within the piecewise function must be evaluated top-down, such that the first condition takes precedence over the second condition which takes precedence over the third condition.
[^10]: The distance function is typically the Euclidean distance.
[^11]: In an actual system the positions would be explicitly defined.
[^12]: It is assumed that an SP column and an input column do not coincide, i.e. their distance is greater than zero. If this occurs, $\boldsymbol{D}$ will be unstable, as zeros will refer to both valid and invalid distances. This instability is accounted for during the computation of the inhibition radius, such that it will not impact the actual algorithm.
[^13]: The summation of the connected synapses is lower-bounded by one to avoid division by zero.
[^14]: For global inhibition, every value in $\boldsymbol{H}$ would simply be set to one regardless of the topology. This allows for additional optimizations of [(\[eq:activity\_level\])]{} and [(\[eq:min\_duty\_cycle\])]{} and eliminates the need for [(\[eq:inhibition\_radius\])]{} and [(\[eq:neighborhood\])]{}. For simplicity only the generalized forms of the equations were shown.
[^15]: This assumes that there will be enough active bits in the input. If this is not the case, the input may be transformed to have an appropriate level of active bits.
[^16]: The function $\operatorname{max}$ was used as an example. Other functions producing a valid probability are also valid.
[^17]: The function $\operatorname{max}$ was used as an example. If a different function is utilized, it must be ensured that a valid probability is produced. If a sum is used, it could be normalized; however, if caution is not applied, thresholding with respect to $\rho_s$ may be invalid and therefore require a new thresholding technique.
[^18]: This implementation has been released under the MIT license and is available at: https://github.com/tehtechguy/mHTM.
[^19]: An encoder for HTM is any system that takes an arbitrary input and maps it to a new domain (whether by lossy or lossless means) where all values are mapped to the set {0, 1}.
[^20]: The following parameters were kept constant: $\rho_s=0.5$, $30$ training epochs, and synapses were trimmed if their permanence value ever reached or fell below $10^{-4}$.
[^21]: A multivariate encoder is one which combines one or more other encoders. The multivariate encoder’s output concatenates the output of each of the other encoders to form one SDR. A categorical encoder is one which losslessly converts an item to a unique SDR. To perform this conversion, the number of categories must be finite.
For this experiment, each category encoder was set to produce an SDR with a total of 50 bits. The number of categories, for each encoder, was dynamically determined. This value was set to the number of unique instances for each attribute/class. No active bits were allowed to overlap across encodings. The number of active bits, for each encoding, was scaled to be the largest possible value. That scaling would result in utilizing as many of the 50 bits as possible, across all encodings. All encodings have the same number of active bits. In the event that the product of the number of categories and the number of active bits is less than the number of bits, the output was right padded with zeros.
[^22]: This encoding followed the same process as the category encoders used for the attributes.
[^23]: The following parameters were used: $m = 4096$, $q = 25$, $\rho_d = 0$, $\phi_{\delta} = 0.5$, $\rho_c = 819$, $\phi_+ = 0.001$, and $\phi_- = 0.001$. Boosting was disabled and global inhibition was used. Only a single training epoch was utilized. It was found that additional epochs were not required and could result in overfitting. $\rho_c$ was intentionally set to be about 20% of $m$. This deviates from the standard value of $\sim2\%$. A value of 2% resulted in lower accuracies (across many different parameter settings) than 20%. This result is most likely a result of the chosen classifier. For use with the TM or a different classifier, additional experimentation will be required.
[^24]: These classifiers were utilized from scikit-learn [@scikit].
[^25]: The shown error is the median across all splits of the data.
|
---
abstract: 'Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents [*stabilizer states*]{} by keeping track of the operators that preserve them. Such states are obtained by [*stabilizer circuits*]{} (consisting of CNOT, Hadamard and Phase only) and can be represented compactly on conventional computers using $\Omega (n^2)$ bits, where $n$ is the number of qubits [@Gottes98]. Although techniques for the efficient simulation of stabilizer circuits have been studied extensively [@AaronGottes; @Gottes; @Gottes98], techniques for efficient manipulation of stabilizer states are not currently available. To this end, we leverage the theoretical insights from [@AaronGottes] and [@Vanden] to design new algorithms for: ([*i*]{}) obtaining [*canonical generators*]{} for stabilizer states, ([*ii*]{}) obtaining [*canonical stabilizer circuits*]{}, and ([*iii*]{}) computing the inner product between stabilizer states. Our inner-product algorithm takes $O(n^3)$ time in general, but observes quadratic behavior for many practical instances relevant to QECC (e.g., GHZ states). We prove that each $n$-qubit stabilizer state has exactly $4(2^n - 1)$ [*nearest-neighbor stabilizer states*]{}, and verify this claim experimentally using our algorithms. We design techniques for representing arbitrary quantum states using [*stabilizer frames*]{} and generalize our algorithms to compute the inner product between two such frames.'
author:
-
title: |
Efficient Inner-product Algorithm\
for Stabilizer States
---
Introduction {#sec:intro}
============
Gottesman [@Gottes] and Knill showed that for certain types of non-trivial quantum circuits known as [*stabilizer circuits*]{}, efficient simulation on classical computers is possible. Stabilizer circuits are exclusively composed of [*stabilizer gates*]{} – Controlled-NOT, Hadamard and Phase gates (Figure \[fig:chp\_pauli\]a) – followed by measurements in the computational basis. Such circuits are applied to a computational basis state (usually $\ket{00...0}$) and produce output states called [*stabilizer states*]{}. The case of purely unitary stabilizer circuits (without measurement gates) is considered often, e.g., by consolidating measurements at the end. Stabilizer circuits can be simulated in poly-time by keeping track of a set Pauli operators that stabilize[^1] the quantum state. Such [*stabilizer operators*]{} uniquely represent a stabilizer state up to an unobservable global phase. Equation \[eq:stab\_count\] shows that the number of $n$-qubit stabilizer states grows as $2^{n^2/2}$, therefore, describing a generic stabilizer state requires at least $n^2/2$ bits. Despite their compact representation, stabilizer states can exhibit multi-qubit entanglement and are often encountered in many quantum information applications such as Bell states, GHZ states, error-correcting codes and one-way quantum computation. To better understand the role stabilizer states play in such applications, researchers have designed techniques to quantify the amount of entanglement [@Fattal; @Wunder; @Hein] in such states and studied relevant properties such as purification schemes [@Dur], Bell inequalities [@Guehne] and equivalence classes [@Vanden04]. Efficient algorithms for the manipulation of stabilizer states (e.g., computing the angle between them), can help lead to additional insights related to linear-algebraic and geometric properties of stabilizer states.
In this work, we describe in detail algorithms for the efficient computation of the inner product between stabilizer states. We adopt the approach outlined in [@AaronGottes], which requires the synthesis of a unitary stabilizer circuit that maps a stabilizer state to a computational basis state. The work in [@AaronGottes] shows that, for any unitary stabilizer circuit, there exists an equivalent block-structured [*canonical circuit*]{} that applies a block of Hadamard ($H$) gates only, followed by a block of CNOT ($C$) only, then a block of Phase ($P$) gates only, and so on in the $7$-block sequence $H$-$C$-$P$-$C$-$P$-$C$-$H$. Using an alternate representation for stabilizer states, the work in [@Vanden] proves the existence of a ($H$-$C$-$P$-$CZ$)-canonical circuit, where the $CZ$ block consists of Controlled-$Z$ (CPHASE) gates. However, no algorithms are known to synthesize such smaller $4$-block circuits given an arbitrary stabilizer state. In contrast, we describe an algorithm for synthesizing ($H$-$C$-$CZ$-$P$-$H$)-canonical circuits given any input stabilizer state. We prove that any $n$-qubit stabilizer state $\ket{\psi}$ has exactly $4(2^n - 1)$ [*nearest-neighbors*]{} – stabilizer states $\ket{\varphi}$ such that $|{\langle \psi | \varphi \rangle}|$ attains the largest possible value $\neq 1$. Furthermore, we design techniques for representing arbitrary quantum states using [*stabilizer frames*]{} and generalize our algorithms to compute the inner product between two such frames.
This paper is structured as follows. Section \[sec:background\] reviews the stabilizer formalism and relevant algorithms for manipulating stabilizer-based representations of quantum states. Section \[sec:inprod\_stab\] describes our circuit-synthesis and inner-product algorithms. In Section \[sec:validate\], we evaluate the performance of our algorithms. Our findings related to geometric properties of stabilizer states are described in Section \[sec:stabneighbors\]. In Section \[sec:stabframes\], we discuss stabilizer frames and how they can be used to represent arbitrary states and extend our algorithms to compute the inner product between frames. Section \[sec:conclude\] closes with concluding remarks.
----------------------------------------------- --------------------------------------------
$ $ X = \begin{pmatrix}
H = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 \\
1 & 1 \\ 1 & 0 \end{pmatrix} \quad
1 & -1 \end{pmatrix} \quad Y = \begin{pmatrix}
P = \begin{pmatrix} 0 & -i \\
1 & 0 \\ i & 0 \end{pmatrix} \quad
0 & i \end{pmatrix} \quad Z = \begin{pmatrix}
CNOT = \begin{pmatrix} 1 & 0 \\
1 & 0 & 0 & 0 \\ 0 & -1 \end{pmatrix}
0 & 1 & 0 & 0 \\ $
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \end{pmatrix}
$
Fig. 1.(b) The Pauli matrices.
and Controlled-NOT (CNOT).
----------------------------------------------- --------------------------------------------
Background and Previous Work {#sec:background}
============================
Gottesman [@Gottes98] developed a description for quantum states involving the [*Heisenberg representation*]{} often used by physicists to describe atomic phenomena. In this model, one describes quantum states by keeping track of their symmetries rather than explicitly maintaining complex vectors. The symmetries are operators for which these states are $1$-eigenvectors. Algebraically, symmetries form [*group*]{} structures, which can be specified compactly by group generators. It turns out that this approach, also known as the [*stabilizer formalism*]{}, can be used to represent an important class of quantum states.
The stabilizer formalism {#sec:stab}
------------------------
A unitary operator $U$ [*stabilizes*]{} a state $\ket{\psi}$ if $\ket{\psi}$ is a $1$–eigenvector of $U$, i.e., $U\ket{\psi}
= \ket{\psi}$ [@Gottes; @NielChu]. We are interested in operators $U$ derived from the Pauli matrices shown in Figure \[fig:chp\_pauli\]b The following table lists the one-qubit states stabilized by the Pauli matrices.
------- -------------------------------- -- -- -------- --------------------------------
$X$ : $(\ket{0}+\ \ket{1})/\sqrt{2}$ $-X$ : $(\ket{0}-\ \ket{1})/\sqrt{2}$
$Y$ : $(\ket{0}+i\ket{1})/\sqrt{2}$ $-Y$ : $(\ket{0}-i\ket{1})/\sqrt{2}$
$Z$ : $\ket{0}$ $-Z$ : $\ket{1}$
------- -------------------------------- -- -- -------- --------------------------------
Observe that $I$ stabilizes all states and $-I$ does not stabilize any state. As an example, the entangled state $(\ket{00} + \ket{11})/\sqrt{2}$ is stabilized by the Pauli operators $X\otimes X$, $-Y\otimes Y$, $Z\otimes Z$ and $I\otimes I$. As shown in Table \[tab:pauli\_mult\], it turns out that the Pauli matrices along with $I$ and the multiplicative factors $\pm1$, $\pm i$, form a [*closed group*]{} under matrix multiplication [@NielChu]. Formally, the [*Pauli group*]{} $\mathcal{G}_n$ on $n$ qubits consists of the $n$-fold tensor product of Pauli matrices, $P = i^kP_1\otimes\cdot\cdot\cdot\otimes P_n$ such that $P_j\in\{I, X, Y, Z\}$ and $k\in\{0,1,2,3\}$. For brevity, the tensor-product symbol is often omitted so that $P$ is denoted by a string of $I$, $X$, $Y$ and $Z$ characters or [*Pauli literals*]{} and a separate integer value $k$ for the phase $i^k$. This string-integer pair representation allows us to compute the product of Pauli operators without explicitly computing the tensor products,[^2] e.g., $(-IIXI)(iIYII) = -iIYXI$. Since $\mid \mathcal{G}_n\mid= 4^{n+1}$, $\mathcal{G}_n$ can have at most $\log_2 \mid \mathcal{G}_n \mid =
\log_2 4^{n+1} = 2(n + 1)$ irredundant generators [@NielChu]. The key idea behind the stabilizer formalism is to represent an $n$-qubit quantum state $\ket{\psi}$ by its [*stabilizer group*]{} $S(\ket{\psi})$ – the subgroup of $\mathcal{G}_n$ that stabilizes $\ket{\psi}$. As the following theorem shows, if $|S(\ket{\psi})|=2^n$, the group uniquely specifies $\ket{\psi}$.
$I$ $X$ $Y$ $Z$
----- ----- ------- ------- -------
$I$ $I$ $X$ $Y$ $Z$
$X$ $X$ $I$ $iZ$ $-iY$
$Y$ $Y$ $-iZ$ $I$ $iX$
$Z$ $Z$ $iY$ $-iX$ $I$
\[th:gen\_commute\] For an $n$-qubit pure state $\ket{\psi}$ and $k\leq n$, $S(\ket{\psi}) \cong {\mathbb Z}_2^k$. If $k=n$, $\ket{\psi}$ is specified uniquely by $S(\ket{\psi})$ and is called a stabilizer state.
(*i*) To prove that $S(\ket{\psi})$ is commutative, let $P, Q \in S(\ket{\psi})$ such that $PQ\ket{\psi} = \ket{\psi}$. If $P$ and $Q$ anticommute, $-QP\ket{\psi} = -Q(P\ket{\psi}) = -Q\ket{\psi} = -\ket{\psi} \neq \ket{\psi}$. Thus, $P$ and $Q$ cannot both be elements of $S(\ket{\psi})$.
([*ii*]{}) To prove that every element of $S(\ket{\psi})$ is of degree $2$, let $P \in S(\ket{\psi})$ such that $P\ket{\psi} = \ket{\psi}$. Observe that $P^2 = i^lI$ for $l\in\{0,1,2,3\}$. Since $P^2\ket{\psi} = P(P\ket{\psi}) = P\ket{\psi} = \ket{\psi}$, we obtain $i^l = 1$ and $P^2 = I$.
([*iii*]{}) From group theory, a finite Abelian group with $a^2 = a$ for every element must be $\cong {\mathbb Z}_2^k$.
([*iv*]{}) We now prove that $k \leq n$. First note that each independent generator $P \in S(\ket{\psi})$ imposes the linear constraint $P\ket{\psi}=\ket{\psi}$ on the $2^n$-dimensional vector space. The subspace of vectors that satisfy such a constraint has dimension $2^{n-1}$, or half the space. Let $gen(\ket{\psi})$ be the set of generators for $S(\ket{\psi})$. We add independent generators to $gen(\ket{\psi})$ one by one and impose their linear constraints, to limit $\ket{\psi}$ to the shared $1$-eigenvector. Thus the size of $gen(\ket{\psi})$ is at most $n$. In the case $|gen(\ket{\psi})| = n$, the $n$ independent generators reduce the subspace of possible states to dimension one. Thus, $\ket{\psi}$ is uniquely specified.
The proof of Theorem \[th:gen\_commute\] shows that $S(\ket{\psi})$ is specified by only $\log_2 2^{n} = n$ [*irredundant stabilizer generators*]{}. Therefore, an arbitrary $n$-qubit stabilizer state can be represented by a [*stabilizer matrix*]{} ${\mathcal{M}}$ whose rows represent a set of generators $g_1,\ldots,g_n$ for $S(\ket{\psi})$. (Hence we use the terms [*generator set*]{} and [*stabilizer matrix*]{} interchangeably.) Since each $g_i$ is a string of $n$ Pauli literals, the size of the matrix is $n\times n$. The phases of each $g_i$ are stored separately using a vector of $n$ integers. Therefore, the storage cost for ${\mathcal{M}}$ is $\Theta(n^2)$, which is an [*exponential improvement*]{} over the $O(2^n)$ cost often encountered in vector-based representations.
Theorem \[th:gen\_commute\] suggests that Pauli literals can be represented using only two bits, e.g., $00 = I$, $01 = Z$, $10 = X$ and $11 = Y$. Therefore, a stabilizer matrix can be encoded using an $n\times2n$ binary matrix or [*tableau*]{}. The advantage of this approach is that this literal-to-bits mapping induces an isomorphism ${\mathbb Z}_2^{2n} \rightarrow \mathcal{G}_n$ because vector addition in ${\mathbb Z}_2^{2}$ is equivalent to multiplication of Pauli operators up to a global phase. The tableau implementation of the stabilizer formalism is covered in [@AaronGottes; @NielChu].
\[prop:stab\_count\] The number of $n$-qubit pure stabilizer states is given by $$\label{eq:stab_count}
N(n) = 2^n\prod_{k=0}^{n-1}(2^{n-k} + 1) = 2^{(.5 + o(1))n^2}
\vspace{-2pt}$$
The proof of Proposition \[prop:stab\_count\] can be found in [@AaronGottes]. An alternate interpretation of Equation \[eq:stab\_count\] is given by the simple recurrence relation $N(n)=2(2^n+1) N(n-1)$ with base case $N(1) = 6$. For example, for $n=2$ the number of stabilizer states is $60$, and for $n=3$ it is $1080$. This recurrence relation stems from the fact that there are $2^n+1$ ways of combining the generators of $N(n-1)$ with additional Pauli matrices to form valid $n$-qubit generators. The factor of $2$ accounts for the increase in the number of possible sign configurations. Table \[tab:two\_qbssts\] and Appendix \[app:three\_qbssts\] list all two-qubit and three-qubit stabilizer states, respectively.
\[.90\]
\[obs:stabst\_amps\] Consider a stabilizer state $\ket{\psi}$ represented by a set of generators of its stabilizer group $S(\ket{\psi})$. Recall from the proof of Theorem \[th:gen\_commute\] that, since $S(\ket{\psi}) \cong {\mathbb Z}_2^n$, each generator imposes a linear constraint on $\ket{\psi}$. Therefore, the set of generators can be viewed as a system of linear equations whose solution yields the $2^n$ basis amplitudes that make up $\ket{\psi}$. Thus, one needs to perform Gaussian elimination to obtain the basis amplitudes from a generator set.
[**Canonical stabilizer matrices**]{}. Although stabilizer states are uniquely determined by their stabilizer group, the set of generators may be selected in different ways. For example, the state $\ket{\psi} = (\ket{00} + \ket{11})/\sqrt{2}$ is uniquely specified by any of the following stabilizer matrices:
-- ------ -- -- ------- -- -- -------
$XX$ $XX$ -$YY$
$ZZ$ -$YY$ $ZZ$
-- ------ -- -- ------- -- -- -------
-------------------------- --
-------------------------- --
One obtains ${\mathcal{M}}_2$ from ${\mathcal{M}}_1$ by left-multiplying the second row by the first. Similarly, one can also obtain ${\mathcal{M}}_3$ from ${\mathcal{M}}_1$ or ${\mathcal{M}}_2$ via row multiplication. Observe that, multiplying any row by itself yields $II$, which stabilizes $\ket{\psi}$. However, $II$ cannot be used as a stabilizer generator because it is redundant and carries no information about the structure of $\ket{\psi}$. This also holds true in general for ${\mathcal{M}}$ of any size. Any stabilizer matrix can be rearranged by applying sequences of elementary row operations in order to obtain a particular matrix structure. Such operations do not modify the stabilizer state. The elementary row operations that can be performed on a stabilizer matrix are transposition, which swaps two rows of the matrix, and multiplication, which left-multiplies one row with another. Such operations allow one to rearrange the stabilizer matrix in a series of steps that resemble Gauss-Jordan elimination.[^3] Given an $n\times n$ stabilizer matrix, row transpositions are performed in constant time[^4] while row multiplications require $\Theta(n)$ time. Algorithm \[alg:gauss\_min\] rearranges a stabilizer matrix into a [*row-reduced echelon form*]{} that contains: ([*i*]{}) a [*minimum set*]{} of generators with $X$ and $Y$ literals appearing at the top, and ([*ii*]{}) generators containing a [*minimum set*]{} of $Z$ literals only appearing at the bottom of the matrix. This particular stabilizer-matrix structure, shown in Figure \[fig:sminv\], defines a canonical representation for stabilizer states [@Djor; @Gottes98]. The algorithm iteratively determines which row operations to apply based on the Pauli (non-$I$) literals contained in the first row and column of an increasingly smaller submatrix of the full stabilizer matrix. Initially, the submatrix considered is the full stabilizer matrix. After the proper row operations are applied, the dimensions of the submatrix decrease by one until the size of the submatrix reaches one. The algorithm performs this process twice, once to position the rows with $X$($Y$) literals at the top, and then again to position the remaining rows containing $Z$ literals only at the bottom. Let $i\in\{1,\ldots,n\}$ and $j\in\{1,\ldots,n\}$ be the index of the first row and first column, respectively, of submatrix ${\mathcal{A}}$. The steps to construct the upper-triangular portion of the row-echelon form shown in Figure \[fig:sminv\] are as follows.
- Let $k$ be a row in ${\mathcal{A}}$ whose $j^{th}$ literal is $X$($Y$). Swap rows $k$ and $i$ such that $k$ is the first row of ${\mathcal{A}}$. Decrease the height of ${\mathcal{A}}$ by one (i.e., increase $i$).
- For each row $m \in \{0,\ldots,n\}, m \neq i$ that has an $X$($Y$) in column $j$, use row multiplication to set the $j^{th}$ literal in row $m$ to $I$ or $Z$.
- Decrease the width of ${\mathcal{A}}$ by one (i.e., increase $j$).
Stabilizer matrix ${\mathcal{M}}$ for $S(\ket{\psi})$ with rows $R_1,\ldots,R_n$ ${\mathcal{M}}$ is reduced to row-echelon form $\Rightarrow$ ${\mathrm{\tt ROWSWAP}}({\mathcal{M}}, i, j)$ swaps rows $R_i$ and $R_j$ of ${\mathcal{M}}$ $\Rightarrow$ ${\mathrm{\tt ROWMULT}}({\mathcal{M}}, i, j)$ left-multiplies rows $R_i$ and $R_j$, returns updated $R_i$ $i \leftarrow 1$ $k \leftarrow$ index of row $R_{k \in \{i,\ldots, n\}}$ with $j^{th}$ literal set to $X$($Y$) ${\mathrm{\tt ROWSWAP}}({\mathcal{M}}, i, k)$ $R_m = {\mathrm{\tt ROWMULT}}({\mathcal{M}}, R_i, R_m)$ $i \leftarrow i + 1$ $k \leftarrow$ index of row $R_{k \in \{i,\ldots, n\}}$ with $j^{th}$ literal set to $Z$ ${\mathrm{\tt ROWSWAP}}({\mathcal{M}}, i, k)$ $R_m = {\mathrm{\tt ROWMULT}}({\mathcal{M}}, R_i, R_m)$ $i \leftarrow i + 1$
To bring the matrix to its lower-triangular form, one executes the same process with the following difference: ($i$) step 1 looks for rows that have a $Z$ literal (instead of $X$ or $Y$) in column $j$, and ($ii$) step 2 looks for rows that have $Z$ or $Y$ literals (instead of $X$ or $Y$) in column $j$. Observe that Algorithm \[alg:gauss\_min\] ensures that the columns in ${\mathcal{M}}$ have at most two distinct types of non-$I$ literals. Since Algorithm \[alg:gauss\_min\] inspects all $n^2$ entries in the matrix and performs a constant number of row multiplications each time, the runtime of the algorithm is $O(n^3)$. An alternative row-echelon form for stabilizer generators along with relevant algorithms to obtain them were introduced in [@Audenaert]. However, their matrix structure is not canonical as it does not guarantee a minimum set of generators with $X$/$Y$ literals.
\
[**Stabilizer circuit simulation**]{}. The computational basis states are stabilizer states that can be represented using the following stabilizer-matrix structure.
\[def:basis\_form\] A stabilizer matrix is in [*[basis form]{}*]{} if it has the following structure. $$ \begin{array}{r}
\pm \\
\pm \\
\vdots \\
\pm
\end{array}
\left[
\begin{array}{cccc}
Z & I & \cdots & I \\
I & Z & \cdots & I \\
\vdots & \vdots & \ddots & \vdots \\
I & I & \cdots & Z
\end{array}
\right]$$
In this matrix form, the $\pm$ sign of each row along with its corresponding $Z_j$-literal designates whether the state of the $j^{th}$ qubit is $\ket{0}$ ($+$) or $\ket{1}$ ($-$). Suppose we want to simulate circuit ${\mathcal{C}}$. Stabilizer-based simulation first initializes ${\mathcal{M}}$ to specify some basis state $\ket{\psi}$. To simulate the action of each gate $U \in {\mathcal{C}}$, we conjugate each row $g_i$ of ${\mathcal{M}}$ by $U$.[^5] We require that $Ug_iU^\dag$ maps to another string of Pauli literals so that the resulting stabilizer matrix ${\mathcal{M}}'$ is well-formed. It turns out that the Hadamard, Phase and CNOT gates (Figure \[fig:chp\_pauli\]a) have such mappings, i.e., these gates conjugate the Pauli group onto itself [@Gottes98; @NielChu]. Table \[tab:cliff\_mult\] lists the mapping for each of these gates.
For example, suppose we simulate a CNOT operation on $\ket{\psi} = (\ket{00} + \ket{11})/\sqrt{2}$ using ${\mathcal{M}}$,
-- ------ -- -- ------
$XX$ $XI$
$ZZ$ $IZ$
-- ------ -- -- ------
One can verify that the rows of ${\mathcal{M}}'$ stabilize $\ket{\psi}\xrightarrow{CNOT}(\ket{00} + \ket{10})/\sqrt{2}$ as required.
Since Hadamard, Phase and CNOT gates are directly simulated using stabilizers, these gates are commonly called *stabilizer gates*. They are also called *Clifford gates* because they generate the Clifford group of unitary operators. We use these names interchangeably. Any circuit composed exclusively of stabilizer gates is called a *unitary stabilizer circuit*. Table \[tab:cliff\_mult\] shows that at most two columns of ${\mathcal{M}}$ are updated when one simulates a stabilizer gate. Thus, such gates are simulated in $\Theta(n)$ time.
\[th:stabst\] An $n$-qubit stabilizer state $\ket{\psi}$ can be obtained by applying a stabilizer circuit to the $\ket{0}^{\otimes n}$ basis state.
The work in [@AaronGottes] represents the generators using a tableau, and then shows how to construct a unitary stabilizer circuit from the tableau. We refer the reader to [@AaronGottes Theorem 8] for details of the proof.
\[cor:stab\_allzeros\] An $n$-qubit stabilizer state $\ket{\psi}$ can be transformed by stabilizer gates into the $\ket{0}^{\otimes n}$ basis state.
Since every stabilizer state can be produced by applying some unitary stabilizer circuit ${\mathcal{C}}$ to the $\ket{0}^{\otimes n}$ state, it suffices to reverse ${\mathcal{C}}$ to perform the inverse transformation. To reverse a stabilizer circuit, reverse the order of gates and replace every $P$ gate with $PPP$.
[cc]{}
Gate Input Output
------ ------- --------
$X$ $Z$
$H$ $Y$ -$Y$
$Z$ $X$
$X$ $Y$
$P$ $Y$ -$X$
$Z$ $Z$
&
Gate Input Output
------ ---------- ----------
$I_1X_2$ $I_1X_2$
$X_1I_2$ $X_1X_2$
$I_1Y_2$ $Z_1Y_2$
$Y_1I_2$ $Y_1X_2$
$I_1Z_2$ $Z_1Z_2$
$Z_1I_2$ $Z_1I_2$
\
\
The stabilizer formalism also admits one-qubit measurements in the computational basis [@Gottes98]. However, the updates to ${\mathcal{M}}$ for such gates are not as efficient as for stabilizer gates. Note that any qubit $j$ in a stabilizer state is either in a $\ket{0}$ ($\ket{1}$) state or in an unbiased[^6] superposition of both. The former case is called a [*deterministic outcome*]{} and the latter a [*random outcome*]{}. We can tell these cases apart in $\Theta(n)$ time by searching for $X$ or $Y$ literals in the $j^{th}$ column of ${\mathcal{M}}$. If such literals are found, the qubit must be in a superposition and the outcome is random with equal probability ($p(0) = p(1) = .5$); otherwise the outcome is deterministic ($p(0) = 1$ or $p(1) = 1$).
[*Random case*]{}: one flips an unbiased coin to decide the outcome and then updates ${\mathcal{M}}$ to make it consistent with the outcome obtained. This requires at most $n$ row multiplications leading to $O(n^2)$ runtime [@AaronGottes; @NielChu].
[*Deterministic case*]{}: no updates to ${\mathcal{M}}$ are necessary but we need to figure out whether the state of the qubit is $\ket{0}$ or $\ket{1}$, i.e., whether the qubit is stabilized by $Z$ or -$Z$, respectively. One approach is to apply Algorithm \[alg:gauss\_min\] to put ${\mathcal{M}}$ in row-echelon form. This removes redundant literals from ${\mathcal{M}}$ in order to identify the row containing a $Z$ in its $j^{th}$ position and $I$ everywhere else. The $\pm$ phase of this row decides the outcome of the measurement. Since this approach is a form of Gaussian elimination, it takes $O(n^3)$ time in practice.
Aaronson and Gottesman [@AaronGottes] improved the runtime of deterministic measurements by doubling the size of ${\mathcal{M}}$ to include $n$ [*destabilizer generators*]{} in addition to the $n$ stabilizer generators. Such destabilizer generators help identify exactly which row multiplications to compute in order to decide the measurement outcome. This approach avoids Gaussian elimination and thus deterministic measurements are computed in $O(n^2)$ time.
Inner-product and circuit-synthesis algorithms {#sec:inprod_stab}
==============================================
Given ${\langle \psi | \varphi \rangle} = re^{i\alpha}$, we normalize the global phase of $\ket{\psi}$ to ensure, without loss of generality, that ${\langle \psi | \varphi \rangle} \in \mathbb{R}_+$.
\[th:stab\_ortho\] Let $S(\ket{\psi})$ and $S(\ket{\varphi})$ be the stabilizer groups for $\ket{\psi}$ and $\ket{\varphi}$, respectively. If there exist $P \in S(\ket{\psi})$ and $Q \in S(\ket{\varphi})$ such that $P =$ -$Q$, then $\ket{\psi}\perp\ket{\varphi}$.
Since $\ket{\psi}$ is a $1$-eigenvector of $P$ and $\ket{\varphi}$ is a $(-1)$-eigenvector of $P$, they must be orthogonal.
\[th:inprod\_aron\][***[@AaronGottes]***]{} Let $\ket{\psi}$, $\ket{\varphi}$ be non-orthogonal stabilizer states. Let $s$ be the minimum, over all sets of generators $\{P_1,\ldots,P_n\}$ for $S(\ket{\psi})$ and $\{Q_1,\ldots,Q_n\}$ for $S(\ket{\varphi})$, of the number of different $i$ values for which $P_i \neq Q_i$. Then, $|{\langle \psi | \varphi \rangle}|=2^{-s/2}$.
Since ${\langle \psi | \varphi \rangle}$ is not affected by unitary transformations $U$, we choose a stabilizer circuit such that $U\ket{\psi} = \ket{b}$, where $\ket{b}$ is a basis state. For this state, select the stabilizer generators ${\mathcal{M}}$ of the form $I\ldots IZI\ldots I$. Perform Gaussian elimination on ${\mathcal{M}}$ to minimize the incidence of $P_i \neq Q_i$. Consider two cases. If $U\ket{\varphi} \neq \ket{b}$ and its generators contain only $I$/$Z$ literals, then $U\ket{\varphi}\perp U\ket{\psi}$, which contradicts the assumption that $\ket{\psi}$ and $\ket{\varphi}$ are non-orthogonal. Otherwise, each generator of $U\ket{\varphi}$ containing $X$/$Y$ literals contributes a factor of $1/\sqrt{2}$ to the inner product.
\
[**Synthesizing canonical circuits**]{}. A crucial step in the proof of Theorem \[th:inprod\_aron\] is the computation of a stabilizer circuit that brings an $n$-qubit stabilizer state $\ket{\psi}$ to a computational basis state $\ket{b}$. Consider a stabilizer matrix ${\mathcal{M}}$ that uniquely identifies $\ket{\psi}$. ${\mathcal{M}}$ is reduced to [basis form]{} (Definition \[def:basis\_form\]) by applying a series of elementary row and column operations. Recall that row operations (transposition and multiplication) do not modify the state, but column (Clifford) operations do. Thus, the column operations involved in the reduction process constitute a unitary stabilizer circuit ${\mathcal{C}}$ such that ${\mathcal{C}}\ket{\psi} = \ket{b}$, where $\ket{b}$ is a basis state. Algorithm \[alg:inprod\_circ\] reduces an input stabilizer matrix ${\mathcal{M}}$ to [basis form]{} and returns the circuit ${\mathcal{C}}$ that performs such a mapping.
Given a finite sequence of quantum gates, a [*circuit template*]{} describes a segmentation of the circuit into blocks where each block uses only one gate type. The blocks must correspond to the sequence and be concatenated in that order. For example, a circuit satisfying the $H$-$C$-$P$ template starts with a block of Hadamard ($H$) gates, followed by a block of CNOT ($C$) gates, followed by a block of Phase ($P$) gates.
A circuit with a [*template structure*]{} consisting entirely of CNOT, Hadamard and Phase blocks is called a [*canonical stabilizer circuit*]{}.
Canonical forms are useful for synthesizing stabilizer circuits that minimize the number of gates and qubits required to produce a particular computation. This is particularly important in the context of quantum fault-tolerant architectures that are based on stabilizer codes. Given any stabilizer matrix, Algorithm \[alg:inprod\_circ\] synthesizes a $5$-block canonical circuit with template $H$-$C$-$CZ$-$P$-$H$ (Figure \[fig:basiscirc\]-a), where the $CZ$ block consists of Controlled-$Z$ (CPHASE) gates. Such gates are stabilizer gates since CPHASE$_{i,j} = $H$_j$CNOT$_{i,j}$H$_j$ (Figure \[fig:basiscirc\]-b). In our implementation, such gates are simulated directly on the stabilizer. The work in [@AaronGottes] establishes a longer $7$-block[^7] $H$-$C$-$P$-$C$-$P$-$C$-$H$ canonical-circuit template. The existence of a $H$-$C$-$P$-$CZ$ template is proven in [@Vanden] but no algorithms are known for obtaining such $4$-block canonical circuits given an arbitrary stabilizer state.
\[alg:inprod\_circ\]
Stabilizer matrix ${\mathcal{M}}$ for $S(\ket{\psi})$ with rows $R_1,\ldots,R_n$ ([*i*]{}) Unitary stabilizer circuit ${\mathcal{C}}$ such that ${\mathcal{C}}\ket{\psi}$ equals basis state $\ket{b}$, and ([*ii*]{}) reduce ${\mathcal{M}}$ to [basis form]{} $\Rightarrow$ ${\mathrm{\tt GAUSS}}({\mathcal{M}})$ reduces ${\mathcal{M}}$ to canonical form (Figure \[fig:sminv\]) $\Rightarrow$ ${\mathrm{\tt ROWSWAP}}({\mathcal{M}}, i, j)$ swaps rows $R_i$ and $R_j$ of ${\mathcal{M}}$ $\Rightarrow$ ${\mathrm{\tt ROWMULT}}({\mathcal{M}}, i, j)$ left-multiplies rows $R_i$ and $R_j$, returns updated $R_i$ $\Rightarrow$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, \alpha_j)$ conjugates $j^{th}$ column of ${\mathcal{M}}$ by Clifford sequence $\alpha$ ${\mathrm{\tt GAUSS}}({\mathcal{M}})$ ${\mathcal{C}}\leftarrow \emptyset$ $i \leftarrow 1$ $k \leftarrow$ index of row $R_{k \in \{i,\ldots, n\}}$ with $j^{th}$ literal set to $X$ or $Y$ ${\mathrm{\tt ROWSWAP}}({\mathcal{M}}, i, k)$ $k_2 \leftarrow$ index of [*last*]{} row $R_{k_2 \in \{i,\ldots, n\}}$ with $j^{th}$ literal set to $Z$ ${\mathrm{\tt ROWSWAP}}({\mathcal{M}}, i, k_2)$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, \text{H}_j)$ ${\mathcal{C}}\leftarrow {\mathcal{C}}\cup \text{H}_j$ $i \leftarrow i + 1$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, \text{CNOT}_{j, k})$ ${\mathcal{C}}\leftarrow {\mathcal{C}}\cup \text{CNOT}_{j, k}$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, \text{CPHASE}_{j, k})$ ${\mathcal{C}}\leftarrow {\mathcal{C}}\cup \text{CPHASE}_{j, k}$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, \text{P}_j)$ ${\mathcal{C}}\leftarrow {\mathcal{C}}\cup \text{P}_j$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, \text{H}_j)$ ${\mathcal{C}}\leftarrow {\mathcal{C}}\cup \text{H}_j$ $R_k = {\mathrm{\tt ROWMULT}}({\mathcal{M}}, R_j, R_k)$ ${\mathcal{C}}$
We now describe the main steps in Algorithm \[alg:inprod\_circ\]. For simplicity, the updates to the phase array under row and column operations will be left out of our discussion as such updates do not affect the overall execution of the algorithm.
[cc]{} \[.8\]
& $
{\xymatrix @*=<0em>}@C=1.0em @R=1.0em {
& \\
& \ctrl{2} & \qw & & & \qw & \ctrl{2} & \qw & \qw \\
& & & \equiv & & & \\
& \gate{Z} & \qw & & & \gate{H} & \targ & \gate{H} & \qw \\
}$\
\
[**(a)**]{} &
- Reduce ${\mathcal{M}}$ to canonical form.
- Use row transposition to diagonalize ${\mathcal{M}}$. For $j \in \{1, \ldots, n\}$, if the diagonal literal ${\mathcal{M}}_{j, j} = Z$ and there are other Pauli (non-$I$) literals in the row (qubit is entangled), conjugate ${\mathcal{M}}$ by H$_j$. Elements below the diagonal are $Z$/$I$ literals.
- For each above-diagonal element ${\mathcal{M}}_{j, k} = X$/$Y$, conjugate by CNOT$_{j,k}$. Elements above the diagonal are now $I$/$Z$ literals.
- For each above-diagonal element ${\mathcal{M}}_{j, k} = Z$, conjugate by CPHASE$_{j,k}$. Elements above the diagonal are now $I$ literals.
- For each diagonal literal ${\mathcal{M}}_{j, j} = Y$, conjugate by P$_j$.
- For each diagonal literal ${\mathcal{M}}_{j, j} = X$, conjugate by H$_j$.
- Use row multiplication to eliminate trailing $Z$ literals below the diagonal and arrive at [basis form]{}.
\[prop:normform\_circsize\] For an $n\times n$ stabilizer matrix ${\mathcal{M}}$, the number of gates in the circuit ${\mathcal{C}}$ returned by Algorithm \[alg:inprod\_circ\] is $O(n^2)$.
The number of gates in ${\mathcal{C}}$ is dominated by the CNOT and CPHASE blocks, which have $O(n^2)$ gates each. This agrees with previous results regarding the number of gates needed for an $n$-qubit stabilizer circuit in the worst case [@Cleve; @DehaeDemoor].
Observe that, for each gate added to ${\mathcal{C}}$, the corresponding column operation is applied to ${\mathcal{M}}$. Therefore, since column operations run in $\Theta(n)$ time, it follows from Proposition \[prop:normform\_circsize\] that the runtime of Algorithm \[alg:inprod\_circ\] is $O(n^3)$.
Canonical stabilizer circuits that follow the $7$-block template structure from [@AaronGottes] can be optimized to obtain a tighter bound on the number of gates. As in our approach, such circuits are dominated by the size of the CNOT blocks, which contain $O(n^2)$ gates. The work in [@PatelMarkov] shows that that any CNOT circuit has an equivalent CNOT circuit with only $O(n^2/\log n)$ gates. Thus, one simply applies such techniques to each of the CNOT blocks in the canonical circuit. It is an open problem whether one can apply the techniques from [@PatelMarkov] directly to CPHASE blocks, which would facilitate similar optimizations to our proposed $5$-block canonical form.
Stabilizer matrices ([*i*]{}) ${\mathcal{M}}^\psi$ for $\ket{\psi}$ with rows $P_1,\ldots,P_n$, and ([*ii*]{}) ${\mathcal{M}}^\phi$ for $\ket{\phi}$ with rows $Q_1,\ldots,Q_n$ Inner product between $\ket{\psi}$ and $\ket{\phi}$ $\Rightarrow$ ${\mathrm{\tt BASISNORMCIRC}}({\mathcal{M}})$ reduces ${\mathcal{M}}$ to [basis form]{}, i.e, ${\mathcal{C}}\ket{\psi}=\ket{b}$, where $\ket{b}$ is a basis state, and returns ${\mathcal{C}}$ $\Rightarrow$ ${\mathrm{\tt CONJ}}({\mathcal{M}}, {\mathcal{C}})$ conjugates ${\mathcal{M}}$ by Clifford circuit ${\mathcal{C}}$ $\Rightarrow$ ${\mathrm{\tt GAUSS}}({\mathcal{M}})$ reduces ${\mathcal{M}}$ to canonical form (Figure \[fig:sminv\]) $\Rightarrow$ ${\mathrm{\tt LEFTMULT}}(P, Q)$ left-multiplies Pauli operators $P$ and $Q$, and returns the updated $Q$ ${\mathcal{C}}\leftarrow {\mathrm{\tt BASISNORMCIRC}}({\mathcal{M}}^\psi)$ ${\mathrm{\tt CONJ}}({\mathcal{M}}^\phi, {\mathcal{C}})$ ${\mathrm{\tt GAUSS}}({\mathcal{M}}^\phi)$ $k \leftarrow 0$ $k \leftarrow k + 1$ $R \leftarrow I^{\otimes n}$ $R \leftarrow {\mathrm{\tt LEFTMULT}}(P_j, R)$ 0 $2^{-k/2}$
\
[**Inner-product algorithm**]{}. Let $\ket{\psi}$ and $\ket{\phi}$ be two stabilizer states represented by stabilizer matrices ${\mathcal{M}}^\psi$ and ${\mathcal{M}}^\phi$, respectively. Our approach for computing the inner product between these two states is shown in Algorithm \[alg:inprod\]. Following the proof of Theorem \[th:inprod\_aron\], Algorithm \[alg:inprod\_circ\] is applied to ${\mathcal{M}}^\psi$ in order to reduce it to [basis form]{}. The stabilizer circuit generated by Algorithm \[alg:inprod\_circ\] is then applied to ${\mathcal{M}}^\phi$ in order to preserve the inner product. Then, we minimize the number of $X$ and $Y$ literals in ${\mathcal{M}}^\phi$ by applying Algorithm \[alg:gauss\_min\]. Finally, each generator in ${\mathcal{M}}^\phi$ that anticommutes with ${\mathcal{M}}^\psi$ (since ${\mathcal{M}}^\psi$ is in [basis form]{}, we only need to check which generators in ${\mathcal{M}}^\phi$ have $X$ or $Y$ literals) contributes a factor of $1/\sqrt{2}$ to the inner product. If a generator in ${\mathcal{M}}^\phi$, say $Q_i$, commutes with ${\mathcal{M}}^\psi$, then we check orthogonality by determining whether $Q_i$ is in the stabilizer group generated by ${\mathcal{M}}^\psi$. This is accomplished by multiplying the appropriate generators in ${\mathcal{M}}^\psi$ such that we create Pauli operator $R$, which has the same literals as $Q_i$, and check whether $R$ has an opposite sign to $Q_i$. If this is the case, then, by Theorem \[th:stab\_ortho\], the states are orthogonal. Clearly, the most time-consuming step of Algorithm \[alg:inprod\] is the call to Algorithm \[alg:inprod\_circ\], therefore, the overall runtime is $O(n^3)$. However, as we show in Section \[sec:validate\], the performance of our algorithm depends strongly on the stabilizer matrices considered and exhibits quadratic behaviour for certain stabilizer states.
Empirical Validation {#sec:validate}
====================
We implemented our algorithms in C++ and designed a benchmark set to validate the performance of our inner-product algorithm. Recall that the runtime of Algorithm \[alg:inprod\_circ\] is dominated by the two nested for-loops (lines 20-35). The number of times these loops execute depends on the amount of entanglement in the input stabilizer state. In turn, the number of entangled qubits depends on the the number of CNOT gates in the circuit ${\mathcal{C}}$ used to generate the stabilizer state ${\mathcal{C}}\ket{0^{\otimes n}}$ (Theorem \[th:stabst\]). By a simple heuristic argument [@AaronGottes], one generates highly entangled stabilizer states as long as the number of CNOT gates in ${\mathcal{C}}$ is proportional to $n\lg n$. Therefore, we generated random $n$-qubit stabilizer circuits for $n\in\{20, 40, \ldots, 500\}$ as follows: fix a parameter $\beta > 0$; then choose $\beta \lceil n \log_2 n\rceil$ unitary gates (CNOT, Phase or Hadamard) each with probability $1/3$. Then, each random ${\mathcal{C}}$ is applied to the $\ket{00\ldots0}$ basis state to generate random stabilizer matrices (states). The use of randomly generated benchmarks is justified for our experiments because (*i*) our algorithms are not explicitly sensitive to circuit topology and (*ii*) random stabilizer circuits are considered representative [@Knill]. For each $n$, we applied Algorithm \[alg:inprod\] to pairs of random stabilizer matrices and measured the number of seconds needed to compute the inner product. The entire procedure was repeated for increasing degrees of entanglement by ranging $\beta$ from $0.6$ to $1.2$ in increments of $0.1$. Our results are shown in Figure \[fig:stabip\]-a.
[cc]{}  & \
\
& [**(b)**]{}
The runtime of Algorithm \[alg:inprod\] appears to grow quadratically in $n$ when $\beta = 0.6$. However, when the number of unitary gates is doubled ($\beta = 1.2$), the runtime exhibits cubic growth. Therefore, Figure \[fig:stabip\]-a shows that the performance of Algorithm \[alg:inprod\] is highly dependent on the degree of entanglement in the input stabilizer states. Figure \[fig:stabip\]-b shows the average size of the basis-normalization circuit returned by the calls to Algorithm \[alg:inprod\_circ\]. As expected (Proposition \[prop:normform\_circsize\]), the size of the circuit grows quadratically in $n$. Figure \[fig:stabip\_ghz0all\] shows the average runtime for Algorithm \[alg:inprod\] to compute the inner product between: (*i*) the all-zeros basis state and random $n$-qubit stabilizer states, and (*ii*) the $n$-qubit GHZ state[^8] and random stabilizer states. GHZ states are maximally entangled states that have been realized experimentally using several quantum technologies and are often encountered in practical applications such as error-correcting codes and fault-tolerant architectures. Figure \[fig:stabip\_ghz0all\] shows that, for such practical instances, Algorithm \[alg:inprod\] can compute the inner product in roughly $O(n^2)$ time (e.g. ${\langle GHZ | 0 \rangle}$). However, without apriori information about the input stabilizer matrices, one can only say that the performance of Algorithm \[alg:inprod\] will be somewhere between quadratic and cubic in $n$.
Nearest-neighbor stabilizer states {#sec:stabneighbors}
==================================
[cc]{}  & \
\
[**(a)**]{} & [**(b)**]{}
We used Algorithm \[alg:inprod\] to compute the inner product between $\ket{00}$ and all two-qubit stabilizer states. Our results are shown in Table \[tab:two\_qbssts\]. We leveraged these results to formulate the following properties related to the geometry of stabilizer states.
Given an arbitrary state $\ket{\psi}$ with $||\psi|| = 1$, a stabilizer state $\ket{\varphi}$ is a nearest stabilizer state to $\ket{\psi}$ if $|{\langle \psi | \varphi \rangle}|$ attains the largest possible value $\neq 1$.
Consider two orthogonal stabilizer states $\ket{\alpha}$ and $\ket{\beta}$ whose unbiased superposition $\ket{\psi}$ is also a stabilizer state. Then $\ket{\psi}$ is a nearest stabilizer state to $\ket{\alpha}$ and $\ket{\beta}$.
Since stabilizer states are unbiased, $|{\langle \psi | \alpha \rangle}| =
|{\langle \psi | \beta \rangle}| = \frac{1}{\sqrt{2}}$. By Theorem \[th:inprod\_aron\], this is the largest possible value $\neq 1$. Thus, $\ket{\psi}$ is a nearest stabilizer state to $\ket{\alpha}$ and $\ket{\beta}$.
\[lem:num\_stabsts\] For any two stabilizer states, the numbers of nearest-neighbor stabilizer states are equal.
By Corollary \[cor:stab\_allzeros\], any stabilizer state can be mapped to another stabilizer state by a stabilizer circuit. Since the operators effected by these circuits are unitary, inner products are preserved.
\[lem:cross\] Let $|\psi\rangle$ and $|\varphi\rangle$ be orthogonal stabilizer states such that $|\varphi\rangle=P|\psi\rangle$ where $P$ is an element of the Pauli group. Then $\frac{|\psi\rangle+|\varphi\rangle}{\sqrt{2}}$ is a stabilizer state.
Suppose $|\psi\rangle=\langle g_k\rangle_{k=1,2,\dots,n}$ is generated by elements $g_k$ of the $n$-qubit Pauli group. Let $$f(k)=\left\{\begin{array}{cl} 0 & \textrm{if}\ [P,g_k]=0 \\
1 & \textrm{otherwise}\end{array}\right.
\vspace{-4pt}$$
and write $|\varphi\rangle=\langle (-1)^{f(k)}g_k\rangle$. Conjugating each generator $g_k$ by $P$ we see that $\ket{\varphi}$ is stabilized by $\langle (-1)^{f(k)} g_k\rangle$. Let $Z_k$ (respectively $X_k$) denote the Pauli operator $Z$ ($X$) acting on the $k^{th}$ qubit. By Corollary \[cor:stab\_allzeros\], there exists an element $L$ of the $n$-qubit Clifford group such that $L|\psi\rangle=\ket{0}^{\otimes n}$ and $L|\varphi\rangle=(LPL^\dag)L|\psi\rangle=i^t|f(1)f(2)\dots f(n)\rangle$. The second equality follows from the fact that $LPL^\dag$ is an element of the Pauli group and can therefore be written as $i^tX(v)Z(u)$ for some $t \in \{0,1,2,3\}$ and $u,v \in {\mathbb Z}_2^k$. Therefore, $$\frac{|\psi\rangle+|\varphi\rangle}{\sqrt{2}}= \frac{L^\dag(\ket{0}^{\otimes n}
+ i^t\ket{f(1)f(2)\ldots f(n)})}{\sqrt{2}}$$ The state in parenthesis on the right-hand side is the product of an all-zeros state and a GHZ state. Therefore, the sum is stabilized by $S' = L^\dag\langle S_{zero}, S_{ghz}\rangle L$ where $S_{zero} = \langle Z_i, i \in \{k|f(k)=0\}\rangle$ and $S_{ghz}$ is supported on $\{k|f(k)=1\}$ and equals $\langle(-1)^{t/2}XX\ldots X,\forall i\ Z_iZ_{i+1}\rangle$ if $t = 0\mod 2$ or $\langle(-1)^{(t-1)/2}YY\ldots Y,\forall i\ Z_iZ_{i+1}\rangle$ if $t = 1\mod 2$.
For any $n$-qubit stabilizer state $\ket{\psi}$, there are $4(2^n - 1)$ nearest-neighbor stabilizer states, and these states can be produced as described in Lemma \[lem:cross\].
The all-zeros basis amplitude of any stabilizer state $\ket{\psi}$ that is a nearest neighbor to $\ket{0}^{\otimes n}$ must be $\propto 1/\sqrt{2}$. Therefore, $\ket{\psi}$ is an unbiased superposition of $\ket{0}^{\otimes n}$ and one of the other $2^n-1$ basis states, i.e., $\ket{\psi} = \frac{\ket{0}^{\otimes n} + P\ket{0}^{\otimes n}}{\sqrt{2}}$, where $P \in \mathcal{G}_n$ such that $P\ket{0}^{\otimes n} \neq \alpha\ket{0}^{\otimes n}$. As in the proof of Lemma \[lem:cross\], we have $\ket{\psi} = \frac{\ket{0}^{\otimes n} + i^t\ket{\varphi}}{\sqrt{2}}$, where $\ket{\varphi}$ is a basis state and $t \in \{0,1,2,3\}$. Thus, there are $4$ possible unbiased superpositions, and a total of $4(2^n - 1)$ nearest stabilizer states. Since $\ket{0}^{\otimes n}$ is a stabilizer state, all stabilizer states have the same number of nearest stabilizer states by Lemma \[lem:num\_stabsts\].
Table \[tab:two\_qbssts\] shows that $\ket{00}$ has $12$ nearest-neighbor states. We computed inner products between all-pairs of $2$-qubit stabilizer states and confirmed that each had exactly $12$ nearest neighbors. We used the same procedure to verify that all $3$-qubit stabilizer states have $28$ nearest neighbors. We verified the correctness of our algorithm by comparing against inner product computations based on explicit basis amplitudes.
Stabilizer frames {#sec:stabframes}
=================
Given an $n$-qubit stabilizer state $\ket{\psi}$, there exists an orthonormal basis including $\ket{\psi}$ and consisting entirely of stabilizer states. Using Theorem \[th:stab\_ortho\], one can generate such a basis from the stabilizer representation of $\ket{\psi}$. Observe that, one can create a state $\ket{\varphi}$ that is orthogonal to $\ket{\psi}$ by changing the signs of an arbitrary non-empty subset of generators of $S(\ket{\psi})$, i.e., by permuting the phase vector of the stabilizer matrix for $\ket{\psi}$. Moreover, selecting two different subsets will produce two mutually orthogonal states. Thus, one can produce $2^n-1$ additional orthogonal stabilizer states. Such states, together with $\ket{\psi}$, form an orthonormal basis. This is illustrated by Table \[tab:two\_qbssts\] were each row constitutes an orthonormal basis.
\[def:stab\_frame\] A [*stabilizer frame*]{} ${\mathcal{F}}$ is a set of $k\leq 2^n$ stabilizer states that forms an orthonormal basis $\{\ket{\psi_1},\dots, \ket{\psi_k}\}$ and spans a subspace of the $n$-qubit Hilbert space. ${\mathcal{F}}$ is represented by a pair consisting of ([*i*]{}) a stabilizer matrix ${\mathcal{M}}$ and ([*ii*]{}) a set of $k$ distinct phase vectors $\sigma_j({\mathcal{M}}), j \in \{1,\ldots, k\}$. The size of the frame, which we denote by $|{\mathcal{F}}|$, is equal to $k$.
Stabilizer frames are useful for representing arbitrary quantum states and for simulating the action of stabilizer circuits on such states. Let ${\ensuremath\boldsymbol{\alpha}} = (\alpha_1, \ldots, \alpha_k) \in \mathbb{C}^k$ be the decomposition of the arbitrary $n$-qubit state $\ket{\phi}$ onto the basis $\{\ket{\psi_1},\dots, \ket{\psi_k}\}$ defined by ${\mathcal{F}}$, i.e., $\ket{\phi} = \sum_{i=1}^k\alpha_k\ket{\psi_i}$. Furthermore, let $U$ be a stabilizer gate. To simulate $U\ket{\phi}$, one simply [*rotates the basis defined by ${\mathcal{F}}$*]{} to get the new basis $\{U\ket{\psi_1},\dots, U\ket{\psi_k}\}$. This is accomplished with the following two-step process: ([*i*]{}) update the stabilizer matrix ${\mathcal{M}}$ associated with ${\mathcal{F}}$ as per Section \[sec:stab\]; ([*ii*]{}) iterate over the phase vectors in ${\mathcal{F}}$ and update each accordingly (Table \[tab:cliff\_mult\]). The second step is linear in the number of phase vectors as only a constant number of elements in each vector needs to be updated. Also, ${\ensuremath\boldsymbol{\alpha}}$ may need to be updated, which requires the computation of the global phase of each $U\ket{\psi_i}$. Since the stabilizer does not maintain global phases directly, each $\alpha_i$ is updated as follows:
- Use Gaussian elimination to obtain a basis state $\ket{b}$ from ${\mathcal{M}}$ (Observation \[obs:stabst\_amps\]) and store its non-zero amplitude $\beta$. If $U$ is the Hadamard gate, it may be necessary to sample a sum of two non-zero (one real, one imaginary) basis amplitudes.
- Compute $U\beta\ket{b}=\beta'\ket{b'}$ directly using the state-vector representation.
- Obtain $\ket{b'}$ from $U{\mathcal{M}}U^\dag$ and store its non-zero amplitude $\gamma$.
- Compute the global-phase factor generated as $\alpha_i=(\alpha_i\cdot\beta')/\gamma$.
Observe that, all the above processes take time polynomial in $k$, therefore, if $k = poly(n)$, $U\ket{\phi}$ can be simulated [*efficiently*]{} on a classical computer via frame-based simulation.
\
[**Inner product between frames**]{}. We now discuss how to use our algorithms to compute the inner product between arbitrary quantum states. Let $\ket{\phi}$ and $\ket{\varphi}$ be quantum states represented by the pairs $<{\mathcal{F}}^\phi,\ {\ensuremath\boldsymbol{\alpha}}= (\alpha_1, \ldots, \alpha_k)>$ and $<{\mathcal{F}}^\varphi,\ {\ensuremath\boldsymbol{\beta}}= (\beta_1, \ldots, \beta_l)>$, respectively. The following steps compute $|{\langle \phi | \varphi \rangle}|$.
- Apply Algorithm \[alg:inprod\_circ\] to ${\mathcal{M}}^\phi$ (the stabilizer matrix associated with ${\mathcal{F}}^\phi$) to obtain basis-normalization circuit ${\mathcal{C}}$.
- Rotate frames ${\mathcal{F}}^\phi$ and ${\mathcal{F}}^\varphi$ by ${\mathcal{C}}$ as outlined in our previous discussion.
- Reduce ${\mathcal{M}}^\phi$ to canonical form (Algorithm \[alg:gauss\_min\]) and record the row operations applied. Apply the same row operations to each phase vector $\sigma^\phi_i, {i\in \{1,\ldots,k\}}$ in ${\mathcal{F}}^\phi$. Repeat this step for ${\mathcal{M}}^\varphi$ and the phase vectors in ${\mathcal{F}}^\varphi$.
- Let ${\mathcal{M}}^\phi_i$ denote that the leading-phases of the rows in ${\mathcal{M}}^\phi$ are set equal to $\sigma^\phi_i$. Similarly, ${\mathcal{M}}^\varphi_j$ denotes that the phases of ${\mathcal{M}}^\varphi$ are equal to $\sigma^\varphi_j$. Furthermore, let $\delta({\mathcal{M}}^\phi_i, {\mathcal{M}}^\varphi_j)$ be the function that returns $0$ if the orthogonality check from Algorithm \[alg:inprod\] (lines 9–15) returns $0$, and $1$ otherwise. The inner product is computed as, $$|{\langle \phi | \varphi \rangle}| = \frac{1}{2^{s/2}}\sum_{i=1}^k\sum_{j=1}^l
|\alpha_i^*\beta_j|\cdot\delta({\mathcal{M}}^i_\phi, {\mathcal{M}}^j_\varphi)$$ where $s$ is the number of rows in ${\mathcal{M}}_\varphi$ that contain $X$ or $Y$ literals.
Prior work on representation of arbitrary states using the stabilizer formalism can be found in [@AaronGottes]. The authors propose an approach that represents a quantum state as a sum of density-matrix terms. Our frame-based technique offers more compact storage ($|{\mathcal{F}}| \leq 2^n$ whereas a density matrix may have $4^n$ non-zero entries) but requires more sophisticated book-keeping.
Conclusion {#sec:conclude}
==========
The stabilizer formalism facilitates compact representation of stabilizer states and efficient simulation of stabilizer circuits. Stabilizer states appear in many different quantum-information applications, and their efficient manipulation via geometric and linear-algebraic operations may lead to additional insights. To this end, we study algorithms to efficiently compute the inner product between stabilizer states. A crucial step of this computation is the synthesis of a canonical circuit that transforms a stabilizer state into a computational basis state. We designed an algorithm to synthesize such circuits using a $5$-block template structure and showed that these circuits contain $O(n^2)$ stabilizer gates. We analysed the performance of our inner-product algorithm and showed that, although its runtime is $O(n^3)$, there are practical instances in which it runs in linear or quadratic time. Furthermore, we proved that an $n$-qubit stabilizer state has exactly $4(2^n-1)$ nearest-neighbor states and verified this result experimentally. Finally, we designed techniques for representing arbitrary quantum states using stabilizer frames and generalize our algorithms to compute the inner product between two such frames.
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The 1080 three-qubit stabilizer states {#app:three_qbssts}
======================================
Shorthand notation represents a stabilizer state as $\alpha_0, \alpha_1, \alpha_2, \alpha_3$ where $\alpha_i$ are the normalized amplitudes of the basis states. Basis states are emphasized in bold. The $\angle$ column indicates the angle between that state and $\ket{000}$, which has $28$ nearest-neighbor states and $315$ orthogonal states ($\perp$).
\[.5\]
\[.5\]
[^1]: An operator $U$ is said to stabilize a state iff $U\ket{\psi}=\ket{\psi}$.
[^2]: This holds true due to the identity: $(A\otimes B)(C \otimes D)=(AC\otimes BD)$.
[^3]: Since Gaussian elimination essentially inverts the $n\times 2n$ matrix, this could be sped up to $O(n^{2.376})$ time by using fast matrix inversion algorithms. However, $O(n^3)$- time Gaussian elimination seems more practical.
[^4]: Storing pointers to rows facilitates $O(1)$-time row transpositions – one simply swaps relevant pointers.
[^5]: Since $g_i\ket{\psi} = \ket{\psi}$, the resulting state $U\ket{\psi}$ is stabilized by $Ug_iU^\dag$ because $(Ug_iU^\dag) U\ket{\psi} = Ug_i\ket{\psi} = U\ket{\psi}$.
[^6]: An arbitrary state $\ket{\psi}$ with computational basis decomposition $\sum_{k=0}^n \lambda_k\ket{k}$ is said to be [*unbiased*]{} if for all $\lambda_i \neq 0$ and $\lambda_j\neq 0$, $|\lambda_i|^2 = |\lambda_j|^2$. Otherwise, the state is [*biased*]{}. One can verify that none of the stabilizer gates produce biased states.
[^7]: Theorem 8 in [@AaronGottes] actually describes an $11$-step canonical procedure. However, the last four steps pertain to reducing destabilizer rows, which we do not consider in our approach.
[^8]: An $n$-qubit [*GHZ state*]{} is an equal superposition of the all-zeros and all-ones states, i.e., $\frac{\ket{0^{\otimes n}} + \ket{1^{\otimes n}}}{\sqrt{2}}$.
|
---
abstract: 'A large fraction of cool, low-mass stars exhibit brightness fluctuations that arise from a combination of convective granulation, acoustic oscillations, magnetic activity, and stellar rotation. Much of the short-timescale variability takes the form of stochastic noise, whose presence may limit the progress of extrasolar planet detection and characterization. In order to lay the groundwork for extracting useful information from these quasi-random signals, we focus on the origin of the granulation-driven component of the variability. We apply existing theoretical scaling relations to predict the star-integrated variability amplitudes for 508 stars with photometric light curves measured by the [*Kepler*]{} mission. We also derive an empirical correction factor that aims to account for the suppression of convection in F-dwarf stars with magnetic activity and shallow convection zones. So that we can make predictions of specific observational quantities, we performed Monte Carlo simulations of granulation light curves using a Lorentzian power spectrum. These simulations allowed us to reproduce the so-called “flicker floor” (i.e., a lower bound in the relationship between the full light-curve range and power in short-timescale fluctuations) that was found in the [*Kepler*]{} data. The Monte Carlo model also enabled us to convert the modeled fluctuation variance into a flicker amplitude directly comparable with observations. When the magnetic suppression factor described above is applied, the model reproduces the observed correlation between stellar surface gravity and flicker amplitude. Observationally validated models like these provide new and complementary evidence for a possible impact of magnetic activity on the properties of near-surface convection.'
author:
- 'Steven R. Cranmer, Fabienne A. Bastien, Keivan G. Stassun, and Steven H. Saar'
title: 'Stellar Granulation as the Source of High-Frequency Flicker in Kepler Light Curves'
---
Introduction {#sec:intro}
============
The last decade has seen a drastic improvement in the precision of both stellar observations and the consequent determination of accurate fundamental parameters for thousands of nearby stars [e.g., @To10; @CM13]. These improvements have also enabled the discovery and characterization of extrasolar planets with a range of sizes that now extends down to less than the radius of the Earth [@Bc13]. However, Sun-like stars are known to exhibit stochastic variations in both their surface-integrated fluxes (“flicker”) and Doppler radial velocities (“jitter”), and these fluctuations may become a source of noise that limits further progress in planet detection. We aim to improve our understanding of the physical origins of the quasi-random variability associated with stellar activity. This paper focuses on the flicker properties of star-integrated visible continuum light curves; see @Ss09, @Bo11, and @Bv13 for additional studies of radial velocity jitter.
High-resolution observations of the Sun have been important in allowing us to distinguish clearly between a number of distinct signals that appear in stellar light curves. For the Sun, acoustic $p$-mode oscillations dominate at periods of order 3–5 min. Granulation (i.e., the photospheric signature of the upper boundary of the convective zone) dominates at slightly longer periods extending up to 10–30 min. Sunspot variability—in combination with the evolution of bright plage and facular regions— occurs on timescales of days to months, with a definite peak at the solar rotation period of $\sim$27 days. For other stars, the granulation and $p$-mode timescales are expected to scale with the acoustic cutoff period [e.g., @Br91] and most spot-related activity (ignoring impulsive flares associated with magnetic active regions) recurs with the rotation period.
Although stellar $p$-mode oscillations are increasingly being used for asteroseismic determinations of stellar masses and radii, there has been less work done on extracting useful physics from the granulation signals [see, however, @Mi08; @Lu09; @KM10; @Mt11]. In this paper, we follow up on recent observational results from the [*Kepler*]{} mission [@Br10] that reported new correlations between the surface gravities of F, G, and K-type stars and the short-timescale flicker properties of their light curves [@Bf13]. We make use of theoretical granulation models developed by @Sm13a [@Sm13b] to produce improved predictions of the observed light curve properties as a function of the fundamental stellar parameters. We also show that the empirical correlation between gravity and flicker can be interpreted as a direct signature of granulation.
In Section \[sec:model\] of this paper, we present the details of the model and we propose an empirical modification to it that accounts for magnetic suppression of granulation in the hottest stars of the sample. Section \[sec:monte\] describes a Monte Carlo model of the granulation light curves that we constructed in order to better understand how the different variability indices of @Bf13 relate to one another and to the intrinsic granulation power. Section \[sec:flicker\] summarizes the results, including a reproduction of the surface gravity dependence of the short-time light curve “flicker.” Lastly, Section \[sec:conc\] concludes this paper with a brief summary, a discussion of some of the wider implications of these results, and suggestions for future improvements.
The Granulation Model {#sec:model}
=====================
Empirical Scaling Relations {#sec:model:emp}
---------------------------
@Sm13a [@Sm13b] presented theoretical scaling relations for how granular fluctuations in a star’s disk-integrated intensity should vary as a function of its fundamental parameters. We apply a slightly modified version of this model below. The root-mean-square amplitude $\sigma$ of photospheric continuum intensity variations is specified as a function of effective temperature ($T_{\rm eff}$), surface gravity ($\log g$), and stellar mass ($M_{\ast}$). @Sm13b derived the following scaling, $$\sigma \, = \, 0.039 \, \left[
\left( \frac{T_{\rm eff}}{T_{\odot}} \right)^{3/4}
\left( \frac{M_{\odot} \nu_{\odot}}{M_{\ast} \nu_{\rm max}}
\right)^{1/2} \Phi ({\cal M}_{a})^{2} \right]^{1.03}
\label{eq:sigma}$$ where $\sigma$ is given in units of parts per thousand (ppt) and $\Phi$ is a dimensionless temperature fluctuation amplitude that depends on the turbulent Mach number ${\cal M}_{a}$ (see below). The normalizing constants $T_{\odot} = 5777$ K, $\log g_{\odot} = 4.438$, and $\nu_{\odot} = 3.106$ mHz are taken from @Sm13b. The peak frequency $\nu_{\rm max}$ of $p$-mode oscillations is assumed to scale with the acoustic cutoff frequency [e.g., @Br91; @KB95], with $$\nu_{\rm max} \, = \, \nu_{\odot} \frac{g}{g_{\odot}}
\left( \frac{T_{\odot}}{T_{\rm eff}} \right)^{1/2} \,\, .
\label{eq:numax}$$ An additional dependence of $\nu_{\rm max}$ on the Mach number ${\cal M}_{a}$ has been proposed [e.g., @Bk11; @Bk12], but we continue to use Equation (\[eq:numax\]) to retain continuity with other empirical results from asteroseismology.
Equation (\[eq:sigma\]) suggests that $\sigma$ is nearly linearly proportional to the combined quantity $T_{\rm eff}^{3/4} M_{\ast}^{-1/2} \nu_{\rm max}^{-1/2}$. This scaling comes from an assumed dependence on the average number of granules ${\cal N}$ that are distributed across the visible stellar surface. @Lu06 and @Sm13a [@Sm13b] gave the statistical result that $\sigma \propto {\cal N}^{-1/2}$, and ${\cal N}$ depends in turn on the stellar radius $R_{\ast}$ and the characteristic granule size $\Lambda$. Some recent sets of convection models [e.g., @Rf04; @Mg13] show that $\Lambda$ is close to being linearly proportional to the photospheric scale height $H_{\ast} \propto T_{\rm eff}/g$ over a wide range of stellar parameters. @Tp13 found a slightly modified correlation ($\Lambda \propto T_{\rm eff}^{1.321} g^{-1.097}$) when modeling convection for both dwarfs and giants, but we retain the usual assumption of $\Lambda \propto H_{\ast}$.
@Sm13b also investigated the dependence of the granular intensity contrast on the turbulent Mach number ${\cal M}_{a}$ in a series of three-dimensional simulations of photospheric convection. We use the @Sm13b scaling relation for the Mach number, $${\cal M}_{a} \, = \, 0.26
\left( \frac{T_{\rm eff}}{T_{\odot}} \right)^{2.35}
\left( \frac{g_{\odot}}{g} \right)^{0.152} \,\, ,
\label{eq:mach}$$ which is then used as input to their parameterized fit for the root-mean square temperature fluctuation amplitude $$\Phi ({\cal M}_{a}) \, = \,
-0.67 + 8.85 {\cal M}_{a} - 8.73 {\cal M}_{a}^{2} \,\, .
\label{eq:Phi}$$ Note that this fitting formula is strictly applicable only for the range $0.15 < {\cal M}_{a} < 0.45$. The above relations describe an initial, unmodified description of turbulent convection to which we propose a modification in Section \[sec:model:mag\].
The 508 [*Kepler*]{} stars analyzed by @Bf13 all have measured values of $T_{\rm eff}$ and $\log g$ [see @Ch11b; @Pn12]. However, definitive masses for the full set have not yet been determined. For a subset of 322 of these stars, we used masses computed from recent ensemble asteroseismology [@Ch13]. Another 47 of the stars were analyzed in an earlier asteroseismology study [@Ch11b], and we incorporated those masses as well. For the remaining 139 stars, we estimated masses by comparing their measured $T_{\rm eff}$ and $\log g$ values against evolutionary tracks computed by the Cambridge STARS code [@Eg71; @Ed08; @Ed09]. These single-star models assumed classic solar abundances ($Z = 0.02$) and did not include mass loss. Figure \[fig01\](a) compares the tracks to the observationally determined $T_{\rm eff}$ and $\log g$ values of the [*Kepler*]{} stars.
Figure \[fig01\](b) shows the $T_{\rm eff}$ dependence of the derived stellar masses. Both panels in Figure \[fig01\] make it clear that the [*Kepler*]{} sample includes a mix of main-sequence dwarfs, mildly evolved subgiants, and more highly evolved red giants. The masses of @Ch11b and @Ch13 are shown with magenta symbols. The black symbols show masses that we determined by computing a $\chi^{2}$ goodness of fit parameter for each ($T_{\rm eff}$, $\log g$) point along the model tracks and weighting the final mass determination by $1 / \chi^{2}$ for all models satisfying $\chi^{2} \leq 10 \min (\chi^{2})$. We found this to be a less ambiguous process than simple interpolation, since there are places in Figure \[fig01\](a) where the tracks cross over one another. This property of the models leads to the existence of multiple local minima in $\chi^{2}$ space. Thus, we believe that using a weighted average of the low-$\chi^{2}$ models takes better account of the uncertainties than just selecting the single model point corresponding to the global minimum in $\chi^{2}$.
Magnetic Inhibition of Convection {#sec:model:mag}
---------------------------------
Although F-type stars tend to exhibit lower levels of chromospheric emission than their cooler G and K counterparts [@IF10], there has been increasing evidence that many of them have strong enough magnetic activity to suppress the amplitudes of atmospheric oscillations and granular variability [@Ch11a; @Sm13b]. As $T_{\rm eff}$ increases from 6000 to 7000 K and beyond, the convection zone shrinks considerably in thickness. It is suspected that strong-field regions can have a much stronger inhibitive effect on the correspondingly shallower granulation of these stars [e.g., @Ca03; @Mo08]. However, the simulations used by [@Sm13b] did not contain this magnetic suppression effect, and thus Equation (\[eq:sigma\]) was seen to overpredict observed fluctuation amplitudes for the largest values of $T_{\rm eff}$.
In the absence of a complete theory of magnetic suppression, we decided on an empirical approach to fold in an additional $T_{\rm eff}$ dependence to the turbulent Mach number ${\cal M}_a$. The goal was to leave the predictions of ${\cal M}_a$ unmodified for the coolest stars and produce a gradual decrease in the convective velocity field for the hotter, more active stars. We first used Equation (\[eq:mach\]) to estimate the Mach number for each model, then we multiplied it by a dimensionless suppression factor $S$ given by $$S \, = \, \left\{ \begin{array}{ll}
1 \,\, , & T_{\rm eff} \leq 5400 \, \mbox{K} , \\
1 / [ 1 + ( T_{\rm eff} - 5400 ) / 1500 ] \,\, , &
T_{\rm eff} > 5400 \, \mbox{K} . \\
\end{array} \right.
\label{eq:Scorr}$$ As we describe further in Section \[sec:flicker\], this form for $S$ was selected in order to produce optimal agreement between the modeled and measured light curve amplitudes for the full range of F, G, and K stars in the [*Kepler*]{} sample. Equation (\[eq:Scorr\]) is meant to estimate the magnitude of the suppression effect only for stars with $T_{\rm eff} \lesssim 7000$ K. There may be additional dependencies on other stellar parameters for hotter stars with extremely thin surface convection zones [see also @Bc05; @Ct09].
Figure \[fig02\] shows the impact of applying the correction factor defined above. Equations (\[eq:mach\])–(\[eq:Phi\]) alone would have predicted high values of ${\cal M}_{a} \approx 0.4$ and $\Phi \approx 1.5$ for the hottest stars in the sample. Our correction gives rise to roughly a factor of 1.8 decrease in both ${\cal M}_{a}$ and $\Phi$ for the hottest stars. Note that the revised model prediction for the temperature fluctuation amplitude produces values that fall closer to the simple approximation of $\Phi=1$ that was assumed in earlier studies [@KB11; @Mt11].
Equations (\[eq:sigma\])–(\[eq:Scorr\]) allow us to compute the root-mean-square amplitude $\sigma$ for each star in the sample, and for each point along the theoretical evolutionary tracks. However, $\sigma$ does not correspond exactly to observational parameters that are derivable easily from the [*Kepler*]{} data.[^1] Following @Ba11, @Bf13 characterized the light curves with three independent measures of variability:
1. a short-timescale [*flicker*]{} amplitude ($F_8$) that corresponds to fluctuations on timescales of 8 hours or less,
2. the light-curve [*range*]{} ($R_{\rm var}$), defined as the difference between the 5% and 95% percentile intensities, and
3. the number of [*zero crossings*]{} ($Z_{\rm C}$) experienced by the light curve, smoothed with a 10-hour window, over the full 90 days of the data set.
In this paper we attempt to faithfully reproduce these quantities as they were defined by @Ba11 and @Bf13. However, computing these parameters from the granulation model requires additional information about the expected frequency spectrum, which we discuss in the following section.
Statistics of Simulated Light Curves {#sec:monte}
====================================
In order to better understand how the various measures of photometric variability relate to one another, we constructed Monte Carlo models of light curves for a range of representative stellar parameters. We used a Lorentzian form for the granular power spectrum, $${\cal P}(\nu) \, = \,
\frac{4 \tau_{\rm eff} \sigma^2}
{1 + (2\pi \tau_{\rm eff} \nu)^{2}}
\label{eq:Pnu}$$ [e.g., @Hv85], where $\tau_{\rm eff}$ is a characteristic granulation timescale. The integral of ${\cal P}$ over all frequencies $\nu$ gives the fluctuation variance $\sigma^{2}$. However, for these models, we set $\sigma = 1$ in order to focus on the time-domain structure of the light curves.
The granulation timescales to use in the Monte Carlo models were obtained by first estimating $\tau_{\rm eff}$ for each of the observed stars using the scaling relation given by @Sm13b, $$\tau_{\rm eff} \, = \, 300 \, \left(
\frac{\nu_{\odot} {\cal M}_{a \odot}}{\nu_{\rm max} {\cal M}_{a}}
\right)^{0.98} \,\,\, \mbox{s} \,\, ,
\label{eq:taueff}$$ where ${\cal M}_{a \odot} = 0.26$. We used the modified version of ${\cal M}_{a}$ described in Section \[sec:model:mag\] as the input quantity to Equation (\[eq:taueff\]). Thus, the estimated values of $\tau_{\rm eff}$ for the [*Kepler*]{} stars ranged between 300 and 14,000 s, with a median value of 990 s. We then created a set of 500 model light curves with a grid of $\tau_{\rm eff}$ values spread out between 150 and 20,000 s, slightly wider than the observed range. Each Monte Carlo light curve was built up from 500 independent frequency components, where the dimensionless frequency $2\pi \tau_{\rm eff} \nu$ ranged from 0.1 to 100.1 in constant steps of 0.2. Each component was assumed to be a sinusoid with a random phase and a relative amplitude consistent with ${\cal P}(\nu)$.
For each set of random Monte Carlo variables, we constructed a full-resolution light curve with a maximum duration of 90 days (one [*Kepler*]{} quarter) and a point-to-point time step of 2.5 s. A corresponding “observational” light curve was sampled from it with a spacing of 30 min—as in the [*Kepler*]{} low-cadence data—and it was processed in an identical way as the actual data to obtain $F_8$, $R_{\rm var}$, and $Z_{\rm C}$. As an example, Figure \[fig03\](a) shows a full resolution model light curve, its reduced [*Kepler*]{} sampling, and the 8 hr smoothing performed to compute $F_8$. For this model, $\tau_{\rm eff} = 1000$ s, and only a one-day subset of the light curve is shown.
Figure \[fig03\](b) shows the ratio $F_{8}/\sigma$ as a function of $\tau_{\rm eff}$ for the 500 models. For the shortest granular timescales (i.e., $\tau_{\rm eff} \lesssim 1000$ s), the $F_8$ diagnostic seems to capture the full granular fluctuation amplitude, and $F_{8} \approx \sigma$. Figure \[fig03\](b) shows a roughly 10% random scatter around the mean ratio $F_{8}/ \sigma = 1$. We believe this arises from both the random nature of the Monte Carlo models and the 30-min cadence sampling used to obtain the flicker amplitude $F_8$. When the 30-min cadence was replaced by a 10-min cadence, the spread in $F_{8}/ \sigma$ (which we measured by computing the range between the 5% and 95% percentile values of the ratio, for $\tau_{\rm eff} < 1000$ s) is reduced from 9.7% to 6.7%. It may be the case that this scatter contributes to the root mean square deviation of $\sim$0.1 dex that characterizes the spread in the observed correlation between $F_{8}$ and $\log g$ [@Bf13].
For the largest values of $\tau_{\rm eff}$, a significant portion of the low-frequency end of the spectrum is excluded by the 8-hr filtering used to compute $F_8$. Thus, the ratio $F_{8}/ \sigma$ is reduced to values as low as $\sim$0.5 at the largest value of $\tau_{\rm eff} =$ 20,000 s. We can estimate the quantitative impact of the 8-hr filtering by integrating over the power spectrum (Equation (\[eq:Pnu\])) only for frequencies exceeding $\nu_{8} = (8 \, \mbox{hr})^{-1}$. The result, expressed as a ratio of the relevant root-mean-square amplitudes, is $$\frac{F_8}{\sigma} \, = \, \sqrt{1 - \frac{2}{\pi}
\tan^{-1} \left( 4 \tau_{\rm eff} \nu_{8} \right)} \,\, .
\label{eq:F8osig}$$ This expression reproduces the Monte Carlo results shown in Figure \[fig03\](b), and we will use it when converting from $\sigma$ to $F_8$ below.
We also computed the amplitude ratio $R_{\rm var}/F_8$ and the already dimensionless number $Z_{\rm C}$ from the Monte Carlo light curves. @Bf13 noted that the stars that seemed to be least contaminated by rotating spot activity tend to have the lowest values of $R_{\rm var}$. Specifically, when $R_{\rm var}$ is plotted as a function of $F_{8}$, there is a noticeably sharp “flicker floor” below which no stars seem to appear. Stars above the floor were interpreted as spot dominated while those on the floor where interpreted as granulation dominated. This floor is described approximately by $R_{\rm var} \approx 3 F_{8}$. @Bf13 also noted that stars on the flicker floor tend to have the largest values of $Z_{\rm C}$, a “zero-crossing ceiling.”
Figure \[fig03\](c) compares the modeled $R_{\rm var}/F_8$ ratios to the observational data. Indeed, the modeled ratio sits very near the observed flicker floor value of 3. The model ratio also increases slightly for the largest values of $\tau_{\rm eff}$, and the observed cool, low-gravity giants in the sample follow this increase as well. In the Monte Carlo models, the ratio ratio $R_{\rm var}/ \sigma$ remains roughly constant no matter the value of $\tau_{\rm eff}$, so the increase in $R_{\rm var}/F_8$ occurs solely because of the decrease in $F_{8} / \sigma$ as shown in Figure \[fig03\](b).
It is possible to derive an analytic estimate of $R_{\rm var}/F_{8}$ in the small-$\tau_{\rm eff}$ limit. For short-timescale granular fluctuations, we showed that $F_8$ is equivalent to the standard deviation $\sigma$ of the distribution of intensities that appear in the light curve. The 30 min cadence observations draw effectively random samples from this distribution. If we assume a normal distribution, then the 5% percentile is known to be a value that is $1.6449\sigma$ below than mean, and the 95% percentile is $1.6449\sigma$ above the mean. Thus, under these assumptions, $R_{\rm var}/F_{8} = 3.2898.$ For $\tau_{\rm eff} \leq 500$ s, Figure \[fig03\](c) shows that the modeled $R_{\rm var}/F_8$ values are given roughly by $3.34 \pm 0.16$, which overlaps with the analytic prediction.
Figure \[fig03\](d) shows how the Monte Carlo model also seems to reproduce the ceiling in the observed values of $Z_{\rm C}$ from the [*Kepler*]{} sample. As predicted by @Bf13, stars with light curves dominated by granulation exhibit the largest number of zero crossings. Stars with large spots are expected to have light curves dominated by rotational modulation, which produces substantially lower frequency variability than does granulation. For the dwarf stars (i.e., the shortest $\tau_{\rm eff}$ timescales), the Monte Carlo models slightly overestimate the upper range of observed values of $Z_{\rm C}$. The cool giants, with $\tau_{\rm eff} \gtrsim 2000$ s, appear to match the model predictions quite well. The stars that fall along the $Z_{\rm C}$ ceiling are presumably (largely) unspotted and therefore granulation dominated.
A simple prediction for the maximum possible number of zero crossings expected from a noisy, but smoothed, data set would be to assume that $Z_{\rm C} \approx N/M$, where $N$ is the total number of data points and $M$ is the width of the boxcar averaging window used to smooth the data. For the simulated [*Kepler*]{} data, $N = 4320$ (corresponding to 90 days with 30-minute cadence) and $M = 20$ (corresponding to a 10 hr window), and thus $N/M = 216$. However, both the observations and simulations exhibit some data points with $Z_{\rm C}$ greater than this value.[^2] To refine this prediction, we performed a set of separate simulations using a time series of uniformly distributed random numbers between 0 and 1. We smoothed them, subtracted their median values, and counted up the zero crossings in a similar way as was done with the data. We varied both $N$ and $M$ in order to determine a robust scaling relation for $Z_{\rm C}$. These simulations indicated a mean dependence given roughly by $$Z_{\rm C} \, \approx \, \frac{N}{\sqrt{4M}}$$ which for the [*Kepler*]{} data parameters would imply an expectation of $Z_{\rm C} \approx 480$. For these parameters, the simulations also showed an approximate factor of two spread around the mean value. This is in rough agreement with the simulated light curves shown in Figure \[fig03\](d), which for $\tau_{\rm eff} < 1000$ s have a mean value of $Z_{\rm C} = 540$. However, the observed sample of dwarf stars in this part of the diagram exhibits a maximum $Z_{\rm C}$ that is about half of that predicted by the Monte Carlo simulations. This could imply that even the dwarfs with the lowest magnetic activity still possess some low-level starspot or plage coverage that reduces $Z_{\rm C}$ via rotational modulation.
Flicker versus Stellar Gravity {#sec:flicker}
==============================
Lastly, we can use the results of the light curve simulations above to compute model predictions of the flicker amplitude $F_8$ for the 508 [*Kepler*]{} stars. We used Equations (\[eq:sigma\])–(\[eq:Scorr\]) to compute $\sigma$ and Equation (\[eq:taueff\]) to compute $\tau_{\rm eff}$, then we applied Equation (\[eq:F8osig\]) to correct for filtering to obtain $F_8$. Figure \[fig04\] compares the $\log g$ dependence of the observed flicker parameters to those we computed based on the above model. Without including the magnetic suppression effect described in Section \[sec:model:mag\], Figure \[fig04\](b) shows that the flicker amplitudes for stars having $T_{\rm eff} \gtrsim 5700$ K are distinctly larger than observed. We applied the suppression effect, as specified in Equation (\[eq:Scorr\]), to produce the agreement between the models and the observations as shown in Figure \[fig04\](c).
The arrangement of colors in the lower-left part of Figure \[fig04\](c) also suggests there may be a detectable correlation between the flicker amplitude and $T_{\rm eff}$, with hotter stars experiencing lower absolute fluctuation levels. To explore that idea in more detail, Figure \[fig05\] displays how $\sigma$ depends separately on $\log g$ and $T_{\rm eff}$ for the STARS code evolutionary tracks that were shown in Figure \[fig01\]. The computed values of $\sigma$ for the [*Kepler*]{} stars are also shown in both panels of Figure \[fig05\] as symbols. In addition to the tight $\log g$ dependence that is also seen in Figure \[fig04\], there is a slight correlation between $\sigma$ and $T_{\rm eff}$. However, Figure \[fig05\](b) shows that this apparent $T_{\rm eff}$ dependence is likely to be the result of the age spread of the 508 [*Kepler*]{} stars, and the fact that these stars fall along only a relatively narrow set of the evolutionary tracks. As stars with $M_{\ast} \approx 1$–2 $M_{\odot}$ evolve from high to low $\log g$ and from high to low $T_{\rm eff}$, they evolve to higher values of $\sigma$. Thus, the correlation between $\sigma$ and $T_{\rm eff}$ may disappear for a more heterogeneous set of observed stars with a broader range of masses.
Along the zero-age main sequence, the dependence of $\sigma$ on $T_{\rm eff}$ shown in Figure \[fig05\](b) can be fit with an approximate lower-limit amplitude of $$\sigma_{\rm min} \, \approx \,
\frac{0.01 x^{14}}{\sqrt{1 + x^{23}}} \,\, ,
\label{eq:sigmin}$$ where $x = T_{\rm eff}/5000$ K, and we show this relation in Figure \[fig05\](b) with a dashed curve. We predict very low intensity fluctuations for the coolest main sequence dwarfs with $\log g > 4.5$ and $T_{\rm eff} < 4500$ K. For these stars, $\sigma \sim 0.001$ ppt, which would require extremely precise photometry to measure. However, these cool dwarfs are also expected to exhibit shorter $\tau_{\rm eff}$ timescales that could make the flicker signal more easily measurable.
Discussion and Conclusions {#sec:conc}
==========================
The goal of this paper was to apply a model of cool-star granulation to the photometric variability of a sample of stars studied by @Bf13. An existing model of granular intensity fluctuations was supplemented by including an ad hoc, but observationally motivated, correction factor for the magnetic suppression of convection in the hottest stars. We also showed how a theoretical frequency spectrum of granular light curve variability could be used to predict some of the other properties of the flicker, range, and zero-crossing diagnostics that were used to analyze the [*Kepler*]{} data. The correlation found by @Bf13 between surface gravity and the flicker amplitude was reproduced with the empirically derived magnetic suppression factor. The overall ability of this model to reproduce several different aspects of the observed light curve variability provides evidence that short-timescale intensity fluctuations are a good probe of stellar granulation.
A subsequent goal of this work is to find self-consistent and accurate ways to predict the properties of stellar light curve variability, and to use this variability to calibrate against other methods of determining their fundamental parameters. Thus, it may be possible to develop the analysis of granular flicker measurements in a way that augments the results of asteroseismology and improves the accuracy of, e.g., stellar mass and radius measurements. To assist in this process, we provide tabulated data for the 508 [*Kepler*]{} stars analyzed above, which also includes their derived masses and predicted values of ${\cal M}_a$, $\sigma$, and $F_8$. These data are given as online-only supplemental material and are hosted, with updates as needed, on the first author’s Web site.[^3] Packaged with the data is a short code written in the Interactive Data Language (IDL)[^4] that reads the data and reproduces two of the three panels of Figure \[fig04\].
In order to make progress in understanding the properties of granulation light curves, the models need to be improved in several key ways. The scaling relations of @Sm13a [@Sm13b] did not include the effects of varying atmospheric metallicity on $\sigma$. @Mg13 presented a grid of three-dimensional granulation models that were constructed over a range of \[Fe/H\] values from –4 to $+$0.5 [see also @Lu09; @Tb13]. Metallicity appears to have a significant effect on the simulation-averaged granulation intensity contrast, but there is no systematic trend (i.e., in some models, the intensity contrast goes up with increasing \[Fe/H\], and in others it goes down). It would also be useful to combine the granulation model with accompanying predictions of starspot rotational modulation [e.g., @Lz12] and some types of $g$-mode pulsations that can resemble spot variability [@Bn11]. This could lead the way to predicting the full range of flicker, range, and zero-crossing parameters for the more active stars above the flicker floor.
Another source of uncertainty in the models is that our simple treatment of the magnetic suppression of granulation was assumed to depend only on $T_{\rm eff}$. It is clearly desirable to move beyond the empirically guided (i.e., ad hoc) correction factor $S(T_{\rm eff})$ that we derived, but no definite scalings with other stellar properties have emerged. If strong magnetic fields are really at the root of the velocity suppression, then for stars with $T_{\rm eff} \lesssim 6500$ K [@BV02] there should be an additional correlation between velocity suppression and the rotation rate $P_{\rm rot}^{-1}$, in parallel with well-known relationships between rotation and high-energy activity [e.g., @Pz03] and rotation and magnetic flux [e.g., @Rn12]. A correlation between rotation and the suppression effect, if found, might explain the added scatter in $F_8$ seen for the higher gravity stars (Figure \[fig04\](a)). Nevertheless, simulations of the interaction between convection and inhomogeneous magnetic “spots” show that the power in both granular motions and acoustic $p$-mode oscillations is reduced substantially when the spot coverage grows larger [@Mu73; @Ca03; @PK07; @Mo08]. Activity-related changes in the convective Mach number may also affect asteroseismic determinations of quantities like the peak oscillation frequency [@Bk11; @Bk12]. Better models of this interaction may be crucial to improving predictions of the inflated radii of M dwarfs [e.g., @MM13; @FC13], and also to understanding how spots are formed within turbulent convection zones in the first place [@Bd13].
This paper includes data collected by the [*Kepler*]{} mission. Funding for [*Kepler*]{} is provided by the NASA Science Mission directorate. The authors gratefully acknowledge William Chaplin and John Eldridge for making their data available for use in this study. FAB acknowledges support from a NASA Harriet Jenkins graduate fellowship. KGS and FAB acknowledge NSF AST-1009810 and NSF PAARE AST-0849736.
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[^1]: Although it is possible to compute the full frequency spectrum from an observed light curve, and process that spectrum to estimate the values of $\sigma$ and several other granulation properties [e.g., @Mi08; @Mt11], this procedure is much less straightforward than the light curve metrics discussed here.
[^2]: Observed or simulated values of $Z_{\rm C}$ [*less*]{} than this value are easier to understand, since that merely implies a smoother light curve than one that would vary up and down through the median value after every measurement.
[^3]: http://www.cfa.harvard.edu/$\sim$scranmer/
[^4]: IDL is published by ITT Visual Information Solutions. There are also several free implementations with compatible syntax, including the GNU Data Language (GDL) and the Perl Data Language (PDL).
|
---
abstract: 'This paper explores a new natural language processing task, review-driven multi-label music style classification. This task requires the system to identify multiple styles of music based on its reviews on websites. The biggest challenge lies in the complicated relations of music styles. It has brought failure to many multi-label classification methods. To tackle this problem, we propose a novel deep learning approach to automatically learn and exploit style correlations. The proposed method consists of two parts: a label-graph based neural network, and a soft training mechanism with correlation-based continuous label representation. Experimental results show that our approach achieves large improvements over the baselines on the proposed dataset. Especially, the micro F1 is improved from 53.9 to 64.5, and the one-error is reduced from 30.5 to 22.6. Furthermore, the visualized analysis shows that our approach performs well in capturing style correlations.'
author:
- |
Guangxiang Zhao[^1], Jingjing Xu, Qi Zeng, Xuancheng Ren\
[MOE Key Lab of Computational Linguistics, School of EECS, Peking University]{}\
[`{zhaoguangxiang,jingjingxu,pkuzengqi,renxc}@pku.edu.cn `]{}\
bibliography:
- 'emnlp2018.bib'
title: |
Review-Driven Multi-Label Music Style Classification\
by Exploiting Style Correlations
---
[UTF8]{}[gbsn]{}
Introduction
============
As music style (e.g., Jazz, Pop, and Rock) is one of the most frequently used labels for music, music style classification is an important task for applications of music recommendation, music information retrieval, etc. There are several criteria related to the instrumentation and rhythmic structure of music that characterize a particular style. In real life, many pieces of music usually map to more than one style.
Several methods have been proposed for automatic music style classification [@qin2005music; @ZhouZS06; @hyper-graph; @choi2017convolutional]. Although these methods make some progress, they are limited in two aspects. First, their generalization ability partly suffers from the small quantity of available audio data. Due to the limitation of music copyright, it is difficult to obtain all necessary audio materials to classify music styles. Second, for simplification, most of the previous studies make a strong assumption that a piece of music has only one single style, which does not meet the practical needs.
Different from the existing methods, this work focuses on review-driven multi-label music style classification. The motivation of using reviews comes from the fact that, there is a lot of accessible user reviews on relevant websites. First, such reviews provide enough information for effectively identifying the style of music, as shown in Table \[example\]. Second, compared with audio materials, reviews can be obtained much more easily. Taking practical needs into account, we do not follow the traditional single-label assumption. Instead, we categorize music items into fine-grained styles and formulate this task as a multi-label classification problem. For this task, we build a new dataset which contains over 7,000 samples. Each sample includes a music title, a set of human annotated styles, and associated reviews. An example is shown in Table \[example\].
Music Title Mozart: The Great Piano Concertos, Vol.1
------------- ------------------------------------------------------------------------------------------------------------------------
Styles Classical Music, Piano Music
*(1) I’ve been listening to **classical** music all the time.*
*(2) Mozart is always good. There is a reason he is ranked in the top 3 of lists of greatest **classical** composers.*
*(3) The sound of **piano** brings me peace and relaxation.*
*(4) This volume of Mozart concertos is superb.*
The major challenge of this task lies in the complicated correlations of music styles. For example, Soul Music contains elements of R&B and Jazz. These three labels can be used alone or in combination. Many multi-label classification methods fail to capture this correlation, and may mistake the true label \[Soul Music, R&B, Jazz\] for the false label \[R&B, Jazz\]. If well learned, such relations are useful knowledge for improving the performance, e.g., increasing the probability of Soul Music if we find that it is heavily linked with two high probability labels: R&B and Jazz. Therefore, to better exploit style correlations, we propose a novel deep learning approach with two parts: a label-graph based neural network, and a soft training mechanism with correlation based continuous label representation.
First, the label-graph based neural network is responsible for classifying music styles based on reviews and style correlations. A hierarchical attention layer collects style-related information from reviews based on a two-level attention mechanism, and a label graph explicitly models the relations of styles. Two information flows are combined together to output the final label probability distribution.
Second, we propose a soft training mechanism by introducing a new loss function with continuous label representation that reflects style correlations. Without style relation information, the traditional discrete label representation sometimes over-distinguishes correlated styles, which does not encourage the model to learn style correlations and limits the performance. Suppose a sample has a true label set \[Soul Music\], and currently the output probability for Soul Music is 0.8, and the probability for R&B is 0.3. It is good enough to make a correct prediction of \[Soul Music\]. However, the discrete label representation suggests the further modification to the parameters, until the probability of Soul Music becomes 1 and the probability of R&B becomes 0. Because Soul Music and R&B are related as mentioned above, over-distinguishing is harmful for the model to learn the relation between Soul Music and R&B. To avoid this problem, we introduce the continuous label representation as the supervisory signal by taking style correlations into account. Therefore, the model is no longer required to distinguish styles completely because a soft classification boundary is allowed.
Our contributions are the followings:
- To the best of our knowledge, this work is the first to explore review-driven multi-label music style classification.
- To learn the relations among music styles, we propose a novel deep learning approach with two parts: a label-graph based neural network, and a soft training mechanism with correlation-based continuous label representation.
- Experimental results on the proposed dataset show that our approach achieves significant improvements over the baselines in terms of all evaluation metrics.
Related works
=============
Music Style Classification
--------------------------
Previous works mainly focus on using audio information to identify music styles. Traditional machine learning algorithms are adopted in this task, such as Support Vector Machine (SVM) [@xu2003musical], Hidden Markov Model (HMM) [@chai2001folk; @DBLP:journals/taslp/PikrakisTK06], and Decision Tree (DT) [@ZhouZS06]. Furthermore, several studies explore different hand-craft feature templates [@tzanetakis2002musical; @qin2005music; @review]. Recently, neural networks have freed researchers from cumbersome feature engineering. For example, @choi2017convolutional introduced a convolutional recurrent neural network for music classification. @medhat2017automatic designed a masked conditional neural network for multidimensional music classification.
Motivated by the fact that many pieces of music usually have different styles, several studies aim at multi-label musical style classification. For example, @hyper-graph proposed to solve multi-label music genre classification with a hyper-graph based SVM. explored how representation learning approaches for multi-label audio classification outperformed traditional handcrafted feature based approaches.
The previous studies have two limitations. First, they are in shortage of available audio data, which limits the generalization ability. Second, their studies are based on a strong assumption that a piece of music should be assigned with only one style. Different from these studies, we focus on using easily obtained reviews in conjunction with multi-label music style classification.
Multi-Label Classification
--------------------------
In contrast to traditional supervised learning, in multi-label learning, each music item is associated with a set of labels. Multi-label learning has gradually attracted attention, and has been widely applied to diverse problems, including image classification [@DBLP:conf/mm/QiHRTMZ07; @DBLP:conf/civr/WangZC08], audio classification [@DBLP:journals/pr/BoutellLSB04; @DBLP:conf/sigir/SandenZ11], web mining [@DBLP:conf/nips/KazawaITM04], information retrieval [@DBLP:conf/sigir/ZhuJXG05; @DBLP:conf/sigir/GopalY10], etc. Compared to the existing multi-label learning methods [@DBLP:journals/corr/abs-1805-04033; @DBLP:journals/corr/abs-1801-09030; @DBLP:journals/corr/abs-1808-05437; @DBLP:conf/coling/YangSLMWW18], our method has novelties: a label graph that explicitly models the relations of styles; a soft training mechanism that introduces correlation-based continuous label representation. To our knowledge, most of the existing studies of learning label representation only focus on single-label classification [@DBLP:journals/corr/HintonVD15; @label], and there is few research on multi-label learning.
Review-Driven Multi-Label Music Style Classification
====================================================
Task Definition
---------------
Given several reviews from a piece of music, the task requires the model to predict a set of music styles. Assume that $X = \{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{i}, \ldots, \boldsymbol{x}_{K}\}$ denotes the input $K$ reviews, and $\boldsymbol{x}_{i}={x_{i,1},\ldots, x_{i,J}}$ represents the $i^{th}$ review with $J$ words. The term $Y=\{y_{1}, y_{2}, \ldots, y_{M}\}$ denotes the gold set with $M$ labels, and $M$ varies in different samples. The target of review-driven multi-label music style classification is to learn the mapping from input reviews to style labels.
Dataset
-------
We construct a dataset consisting of 7172 samples. The dataset is collected from a popular Chinese music review website, where registered users are allowed to comment on all released music albums.
The dataset contains 5020, 646, and 1506 samples for training, validation, and testing respectively. We define an album as a data sample in the dataset, the dataset contains over 287K reviews and over 3.6M words. 22 styles are found in the dataset. Each sample is labeled with 2 to 5 style types. Each sample includes the title of an album, a set of human annotated styles, and associated user reviews sorted by time. An example is shown in Table \[example\]. On average, each sample contains 2.2 styles and 40 reviews, each review has 12.4 words.
{width="0.95\linewidth"}
Proposed Approach
=================
In this section, we introduce our proposed approach in detail. An overview is presented in Section \[overview\]. The details are explained in Section \[model\] and Section \[soft\].
Overview
--------
The proposed approach contains two parts: a label-graph based neural network and a soft training mechanism with continuous label representation. An illustration of the proposed method is shown in Figure \[fig:hie\].
The label-graph based neural network outputs a label probability distribution $e$ based on two kinds of information: reviews and label correlations. First, a hierarchical attention layer produces a music representation $\boldsymbol{z}$ by using a two-level attention mechanism to extract style-related information from reviews. Second, we transforms $\boldsymbol{z}$ into a “raw” label probability distribution $\boldsymbol{z}'$ via a sigmoid function. Third, a label graph layer outputs the final label probability distribution $\boldsymbol{e}$ by multiplying the “raw” label representation with a label graph that explicitly models the relations of labels. Due to noisy reviews, the model sometimes cannot extract all necessary information needed for a correct prediction. The label correlations can be viewed as supplementary information to refine the label probability distribution. For example, the low probability of a true label will be increased if the label is heavily linked with other high probability labels. With the label correlation information, the model can better handle multi-label music style classification, where there are complicated correlations among music styles.
Typically, the model is trained with the cross entropy between the discrete label representation $\boldsymbol{y}$ and the predicted label probability distribution $\boldsymbol{e}$. However, we find it hard for the model to learn style correlations because the discrete label representation does not explicitly contain style relations. For example, for a true label set \[Soul Music\], the discrete label representation assigns Soul Music with the value of 1 while its related styles, R&B and Jazz, get the value of 0. Such discrete distribution does not encourage the model to learn the relation between Soul Music and its related styles. To better learn label correlations, a continuous label representation $\boldsymbol{y}'$ that involves label relations is desired as training target. Therefore, we propose a soft training method that combines the traditional discrete label representation $\boldsymbol{y}$ (e.g., $[1, 1, 0]$) and the continuous label representation $\boldsymbol{y}'$ (e.g., $[0.80, 0.75, 0.40]$).
We first propose to use the learned label graph $\mathcal{G}$ to transform the discrete representation $\boldsymbol{y}$ into a continuous form. The motivation comes from that in a well-trained label graph, the values should reflect label relations to a certain extent. Two highly related labels should get a high relation value, and two independent labels should get a low relation value. However, in practice, we find that for each label, the relation value with itself is too large and the relation value with other labels is too small, e.g., \[0.95, 0.017, 0.003\]. It causes the generated label representation lacking sufficient label correlation information. Therefore, to enlarge the label correlation information in the generated label representation, we propose a smoothing method that punishes the high relation values and rewards the low relation values in $\mathcal{G}$. The method applies a softmax function with a temperature $\tau$ on $\mathcal{G}$ to get a softer label graph $\mathcal{G}'$, and uses $\mathcal{G}'$ to transform $\boldsymbol{y}$ into a softer label representation.
For ease of understanding, we introduce our approach from the following two aspects: one for extracting music representation from reviews, the other for exploiting label correlations.
Hierarchical Attention Layer for Extracting Music Representation {#model}
----------------------------------------------------------------
This layer takes a set of reviews $X$ from the same sample as input, and outputs a music representation $\boldsymbol{z}$. Considering that the dataset is built upon a hierarchical structure where each sample has multiple reviews and each review contains multiple words, we propose a hierarchical network to collect style-related information from reviews.
We first build review representations via a Bi-directional Long-short Term Memory Network (Bi-LSTM) and then aggregate these review representations into the music representation. The aggregation process also adopts a Bi-LSTM structure that takes the sequence of review representations as input. Second, it is observed that different words and reviews are differently informative. Motivated by this fact, we introduce a two level of attention mechanism [@DBLP:journals/corr/BahdanauCB14]: one at the word level and the other at the review level. It lets the model to pay more or less attention to individual words and sentences when constructing the music representation $\boldsymbol{z}$.
Label Correlation Mechanism {#soft}
---------------------------
### Label Graph Layer
To explicitly take advantage of the label correlations when classifying music styles, we add a label graph layer to the network. This layer takes a music representation $\boldsymbol{z}$ as input and generates a label probability distribution $\boldsymbol{e}$.
First, given an input $\boldsymbol{z}$, we use a sigmoid function to produce a “raw” label probability distribution $\boldsymbol{z}'$ as $$\boldsymbol{z}' = sigmoid(f(\boldsymbol{z})) = \frac{1}{1+e ^{-f(\boldsymbol{z})}}$$ where $f()$ is a feed-forward network.
Formally, we denote $\mathcal{G} \in R_{m \times m}$ as the label graph, where $m$ is the number of labels in the dataset, $\mathcal{G}$ is initialized by an identity matrix. An element $\mathcal{G}[l_{i}, l_{j}]$ is a real-value score indicating how likely the label $l_{i}$ and the label $l_{j}$ are related in the training data. The graph $\mathcal{G}$ is a part of parameters and can be learned by back-propagation.
Then, given the “raw” label probability distribution $ \boldsymbol{z}'$ and the label graph $\mathcal{G}$, the output of this layer is: $$\boldsymbol{e} = \boldsymbol{z}' \cdot \mathcal{G}$$
Therefore, the probability of each label is determined not only by the current reviews, but also by its relations with all other labels. The label correlations can be viewed as supplementary information to refine the label probability distribution.
### Soft Training
Given a predicted label probability distribution $\boldsymbol{e}$ and a target discrete label representation $\boldsymbol{y}$, the typical loss function is computed as $$\label{loss_ye}
L(\theta) = \mathcal{H}(\boldsymbol{y}, \boldsymbol{e}) = -\sum_{i=1}^{m}{y_{i}\log e_{i}}$$ where $\theta$ denotes all parameters, and $m$ is the number of the labels. The function $ \mathcal{H}(,)$ denotes the cross entropy between two distributions.
However, the widely used discrete label representation does not apply to the task of music style classification, because the music styles are not mutually exclusive and highly related to each other. The discrete distribution without label relations makes the model over-distinguish the related labels. Therefore, it is hard for the model to learn the label correlations that are useful knowledge.
Instead, we propose a soft training method by combining a discrete label representation $\boldsymbol{y}$ with a correlated-based continuous label representation $\boldsymbol{y}'$. The probability values of $\boldsymbol{y}'$ should be able to tell which labels are correct, and the probability gap between two similar labels in $\boldsymbol{y}'$ should not be large. With the combination between $\boldsymbol{y}'$ and $\boldsymbol{y}$ as training target, the classification model is no longer required to distinguish styles completely and can have a soft classification boundary.
A straight-forward approach to produce the continuous label representation is to use the label graph matrix $\mathcal{G}$ to transform the discrete representation $\boldsymbol{y}$ into a continuous form: $$\boldsymbol{y_c} = \boldsymbol{y} \cdot \mathcal{G}$$ We expect that the values in a well-learned label graph should reflect the degree of label correlations. However, in practice, we find that for each label, the relation value with itself is too large and the relation value with other labels is too small. It causes the generated label representation $\boldsymbol{y_c}$ lacking sufficient label correlation information. Therefore, to enlarge the label correlation information in $\boldsymbol{y_c}$, we propose a smoothing method that punishes the high relation values and rewards the low relation values in $\mathcal{G}$. We apply a softmax function with a temperature $\tau$ on $\mathcal{G}$ to get a softer $\mathcal{G}'$ as $$(\mathcal{G}')_{ij}= \frac{\exp{[(\mathcal{G})_{ij}/\tau]}}{\sum_{i=1}^{N}\exp{[(\mathcal{G})_{ij}/\tau]}}$$ where $N$ is the dimension of each column in $\mathcal{G}$. This transformation keeps the relative ordering of relation values unchanged, but with much smaller range. The higher temperature $\tau$ makes the steep distribution softer. Then, the desired continuous representation $\boldsymbol{y}'$ is defined as $$\boldsymbol{y}' = \boldsymbol{y} \cdot \mathcal{G}'$$
Finally, we define the loss function as $$Loss(\theta) = \mathcal{H}(\boldsymbol{e}, \boldsymbol{y}) + \mathcal{H}(\boldsymbol{e}, \boldsymbol{y}')$$ where the loss $\mathcal{H}(\boldsymbol{e}, \boldsymbol{y})$ aims to correctly classify labels, and the loss $\mathcal{H}(\boldsymbol{e}, \boldsymbol{y}')$ aims to avoid the over-distinguishing problem and to better learn label correlations.
With the new objective, the model understands not only which labels are correct, but also the correlations of labels. With such soft training, the model is no longer required to distinguish the labels completely because a soft classification boundary is allowed.
Experiment
==========
In this section, we evaluate our approach on the proposed dataset. We first introduce the baselines, the training details, and the evaluation metrics. Then, we show the experimental results and provide the detailed analysis.
Baselines
---------
We first implement the following widely-used multi-label classification methods for comparison. Their inputs are the music representations which are produced by averaging word embeddings and review representations at the word level and review level respectively.
- ML-KNN [@mlknn]: It is a multi-label learning approach derived from the traditional K-Nearest Neighbor (KNN) algorithm.
- Binary Relevance [@DBLP:reference/dmkdh/TsoumakasKV10]: It decomposes a multi-label learning task into a number of independent binary learning tasks (one per class label). It learns several single binary models without considering the dependences among labels.
- Classifier Chains [@read2011classifier]: It takes label dependencies into account and keeps the computational efficiency of the binary relevance method.
- Label Powerset [@DBLP:conf/ecml/TsoumakasV07]: All classes assigned to an example are combined into a new and unique class in this method.
- MLP: It feed the music representations into a multilayer perceptron, and generate the probability of music styles through a sigmoid layer.
Different from the above baselines, the following two directly process word embeddings. Similar to MLP, they produce label probability distribution by a feed-forward network and a sigmoid function.
- CNN: It consists of two layers of CNN which has multiple convolution kernels, then feed the word embeddings to get the music representations.
- LSTM: It consists of two layers of LSTM, which processes words and sentences separately to get the music representations.
Training Details
----------------
The features we use for the baselines and the proposed method are the pre-trained word embeddings of reviews. For evaluation, we introduce a hyper-parameter $p$, and a label will be considered a music style of the song if its probability is greater than $p$. We tune hyper-parameters based on the performance on the validation set. We set the temperature $\tau$ in soft training to 3, $p$ to 0.2, hidden size to 128, embedding size to 128, vocabulary size to 135K, learning rate to 0.001, and batch size to 128. The optimizer Adam [@DBLP:journals/corr/KingmaB14] and the maximum training epoch is set to 100. We choose parameters with the best performance on the validation set and then use the selected parameters to predict results on the test set.
Evaluation Metrics
------------------
Multi-label classification requires different evaluation metrics from traditional single-label classification. In this paper, we use the following widely-used evaluation metrics.
- F1-score: We calculate the micro F1 and macro F1, respectively. Macro F1 computes the metric independently for each label and then takes the average, whereas micro F1 aggregates the contributions of all labels to compute the average metric.
- One-Error: One-error evaluates the fraction of examples whose top-ranked label is not in the gold label set.
- Hamming Loss: Hamming loss counts the fraction of the wrong labels to the total number of labels.
Experimental Results
--------------------
--------------------- ---------- ----------- ---------- ----------
ML-KNN 77.3 0.094 23.6 38.1
Binary Relevance 74.4 0.083 24.7 41.8
Classifier Chains 67.5 0.107 29.9 44.3
Label Powerset 56.2 0.096 37.7 50.3
MLP 71.5 0.081 29.8 45.8
CNN 37.9 0.099 32.5 49.3
LSTM 30.5 0.089 33.0 53.9
**HAN (Proposal)** 25.9 0.079 52.1 61.0
**+LCM (Proposal)** **22.6** **0.074** **54.4** **64.5**
--------------------- ---------- ----------- ---------- ----------
: The comparisons between our approach and the baselines on the test set. The OE and HL denotes one-error and hamming loss respectively, the implemented approach HAN and LCM denotes the hierarchical attention network and the label correlation mechanism respectively. “+” represents that higher scores are better and “-” represents that lower scores are better. It can be seen that the proposed approach significantly outperforms the baselines. []{data-label="table:state-of-the-art"}
We evaluate our approach and the baselines on the test set. The results are summarized in Table \[table:state-of-the-art\]. It is obvious that the proposed approach significantly outperforms the baselines, with micro F1 of 64.5, macro F1 of 54.4, and one-error of 22.6, improving the metrics by 10.6, 21.4, and 7.9 respectively. The improvement is attributed to two parts, a hierarchical attention network and a label correlation mechanism. Only using the hierarchical attention network outperforms the baselines, which shows the effectiveness of hierarchically paying attention to different words and sentences. The greater F1-score is achieved by adding the proposed label correlation mechanism, which shows the contribution of exploiting label correlations. Especially, the micro F1 is improved from 61.0 to 64.5, and the macro F1 is improved from 52.1 to 54.4. The results of baselines also reveal the usefulness of label correlations for improving the performance. ML-KNN and Binary Relevance, which over-simplify multi-label classification and neglect the label correlations, achieve the worst results. In contrast, Classifier Chains and Label Powerset, which take label correlations into account, get much better results. Though without explicitly taking advantage of label correlations, the neural baselines, MLP, CNN, and LSTM, still achieve better results, due to the strong learning ability of neural networks.
Incremental Analysis
--------------------
---------- ---------- ----------- ---------- ----------
HAN 25.9 0.079 52.1 61.0
+**LG** 23.4 0.077 54.2 62.8
+ **ST** **22.6** **0.074** **54.4** **64.5**
---------- ---------- ----------- ---------- ----------
: Performance of key components in the proposed approach. LG and ST denote the label graph layer and the soft training.[]{data-label="Tab:module"}
In this section, we conduct a series of experiments to evaluate the contributions of our key components. The results are shown in Table \[Tab:module\].
The method with the label graph does not achieve the expected improvements. It indicates that though with explicitly modeling the label correlations, the label graph does not play the expected role. It verifies our assumption that the traditional training method with discrete label representation makes the model over-distinguish the related labels, and thus does not learn label correlations well. To solve this problem, we propose a soft training method with a continuous label representation $\boldsymbol{y}'$ that takes label correlations into account. It can be clearly seen that with the help of soft training, the proposed method achieves the best performance. Especially, the micro F-score is improved from 62.8 to 64.5, and the one-error is reduced from 23.4 to 22.6. With the new loss function, the model not only knows how to distinguish the right labels from the wrong ones, but also can learn the label correlations that are useful knowledge, especially when the input data contains too much style unrelated words for the model to extract all necessary information.
**Ground Truth** **Without LCM** **With LCM**
--------------------------------------------- ------------------- --------------
Britpop, Rock ,
Hip-Hop, Pop, R&B Electronic Music, ,
Pop, R&B , Rock, Britpop ,
Country Music, Folk, Pop , , ,
Classical Music, New-Age Music, Piano Music , , ,
: Examples generated by the methods with and without the label correlation mechanism. The labels correctly predicted by two methods are shown in blue. The labels correctly predicted by the method with the label correlation mechanism are shown in orange. We can see that the method with the label correlation mechanism classifies music styles more precisely.[]{data-label="Tab:casestudy"}
For clearer understanding, we compare several examples generated with and without the label correlation mechanism in Table \[Tab:casestudy\]. By comparing gold labels and predicted labels generated by different methods, we find that the proposed label correlation mechanism identifies the related styles more precisely. This is mainly attributed to the learned label correlations. For example, the correct prediction in the first example shows that, the label correlation mechanism captures the close relation between “Britpop” and “Rock”, which helps the model to generate a more appropriate prediction.
Visualization Analysis {#visualization}
----------------------
Since we do not have enough space to show the whole heatmap of all 22 labels, we randomly select part of the heatmap to visualize the learned label graph. Figure \[fig:LG\] shows that some obvious music style relations are well captured. For “Country Music”, the most related label is “Folk Music”. In reality, these two music styles are highly similar and the boundary between them is not well-defined. For three kinds of rock music, “Heavy Metal Music”, “Britpop Music”, and “Alternative Music”, the label graph correctly captures that the most related label for them is “Rock”. For a more complicated relation where “Soul Music” is highly linked with two different labels, “R&B” and “Jazz”, the label graph also correctly capture such relation. These examples demonstrate that the proposed approach performs well in capturing relations among music styles.
![The heatmap generated by the learned label graph. The deeper color represents the closer relation. For space, we abbreviate some music style names. We can see that some obvious relations are well captured by the model, e.g., “Heavy Metal Music (Metal)” and “Rock”, “Country Music (Country)” and “Folk”. []{data-label="fig:LG"}](LG-8.eps){width="0.8\linewidth"}
Error Analysis {#error}
--------------
Although the proposed method has achieved significant improvements, we also notice that there are some failure cases. In this section, we give the detailed error analysis.
First, the proposed method performs worse on the styles with low frequency in the training set. Table \[Tab:label unbalance\] compares the performance on the top 5 music styles of highest and lowest frequencies. As we can see, the top 5 fewest music styles get much worse results than top 5 most music styles. This is because the label distribution is highly imbalanced where unpopular music styles have too little training data. For future work, we plan to explore various methods to handle this problem. For example, re-sample original data to provide balanced labels. Second, we find that some music items are wrongly classified into the styles that are similar with the gold styles. For example, a sample with a gold set \[Country Music\] is wrongly classified into \[Folk\] by the model. The reason is that some music styles share many common elements and only subtly differ from each other. It poses a great challenge for the model to distinguish them. For future work, we would like to research how to effectively address this problem.
------------------- ------------------ --------
Rock 30.4 75.8
Independent Music 30.0 64.8
Pop 26.2 67.1
Folk Music 21.9 73.7
Electronic Music 13.9 61.8
**Least styles** **% of Samples** **F1**
Jazz 4.3 37.5
Heavy Metal Music 3.9 55.6
Hip-Hop 3.1 7.5
Post-punk 2.5 17.1
Dark Wave 1.3 17.4
------------------- ------------------ --------
: The performance of the proposed method on most and fewest styles.[]{data-label="Tab:label unbalance"}
Conclusions
===========
In this paper, we focus on classifying multi-label music styles with user reviews. To meet the challenge of complicated style relations, we propose a label-graph based neural network and a soft training mechanism. Experiment results show that our proposed approach significantly outperforms the baselines. Especially, the micro F1 is improved from 53.9 to 64.5, and the one-error is reduced from 30.5 to 22.6. Furthermore, the visualization of label graph also shows that our method performs well in capturing label correlations.\
[^1]: Equal Contribution
|
---
abstract: 'A significant part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over physical states that can be prepared. This is termed an “epistemic restriction” because it implies a fundamental limit on the amount of knowledge that any observer can have about the physical state of a classical system. This article provides an overview of [*epistricted theories*]{}, that is, theories that start from a classical statistical theory and apply an epistemic restriction. We consider both continuous and discrete degrees of freedom, and show that a particular epistemic restriction called [*classical complementarity*]{} provides the beginning of a unification of all known epistricted theories. This restriction appeals to the symplectic structure of the underlying classical theory and consequently can be applied to an arbitrary classical degree of freedom. As such, it can be considered as a kind of *quasi-quantization* scheme; “quasi” because it generally only yields a theory describing a subset of the preparations, transformations and measurements allowed in the full quantum theory for that degree of freedom, and because in some cases, such as for binary variables, it yields a theory that is a distortion of such a subset. Finally, we propose to classify quantum phenomena as weakly or strongly nonclassical by whether or not they can arise in an epistricted theory.'
author:
- 'Robert W. Spekkens'
bibliography:
- 'EpistrictedTheoriesReview.bib'
date: 'Sept. 16, 2014'
title: 'Quasi-quantization: classical statistical theories with an epistemic restriction'
---
Introduction
============
Epistricted theories
--------------------
Start with a classical theory for some degree of freedom and consider the statistical theory associated with it. This is the theory that describes the statistical distributions over the space of physical states and how they change over time. If one then postulates, as a fundamental principle, that there is a restriction on what kinds of statistical distributions can be prepared, then the resulting theory reproduces a large part of quantum theory, in the sense of reproducing precisely its operational predictions. This article reviews recent work on such theories and their relevance for notions of nonclassicality, for the interpretation of the quantum state, and for the program of deriving the formalism of quantum theory from axioms. Some clarifications are in order regarding statistical theories. Given a system whose physical state is drawn from some ensemble of possibilities, the statistical distribution associated to this ensemble can be taken either to describe relative frequencies of physical properties within the virtual ensemble or it can be taken to describe the knowledge that an agent has about an individual system when she knows that it was drawn from that ensemble. The latter sort of language is preferred by those who take a Bayesian approach to statistics, and we shall adopt it here. The distinction between the physical state of a system and an agent’s state of knowledge of that physical state will be critical in what follows. As such, we will make use of some jargon to clearly distinguish the two sorts of states. Recalling the Greek terms for reality and for knowledge, *ontos* and *epistēmē*, we will henceforth refer to physical states as *ontic states* and states of knowledge as *epistemic states* [@spekkens2007evidence]. The theory that governs the evolution of ontic states is an *ontological theory*, while the [*statistical theory*]{} describes the evolution of epistemic states. A restriction on knowledge is an *epistemic restriction*. The theories we are considering, therefore, are epistemically-restricted statistical theories of classical systems. Given that this is a rather unwieldy descriptor, we introduce the term *epistricted theory* as an abbreviation to it. It is worth considering in a bit more detail the scheme by which one infers from a given classical theory the epistricted version thereof. One starts with a particular classical ontological theory (first column of Table \[Table:epistrictedtheories\]). We are here considering the usual notion of an ontological theory: one which provides a kinematics and a dynamics, that is, a hypothesis about the possible physical states that a system can occupy at a given time and a law describing how each state can evolve over time. One then constructs the statistical theory for the classical ontological theory under consideration (second column of Table \[Table:epistrictedtheories\]). The fundamental object here is a statistical distribution over the physical state space rather than a point in the physical state space, that is, an epistemic state rather than an ontic state. The statistical theory answers questions such as: If the physical state of the system undergoes deterministic dynamics, how does the statistical distribution change over time? Or, more precisely, if an agent assigns a statistical distribution over physical states at one time and she knows the dynamics, what statistical distribution should she assign at a later time? If an agent implements a measurement on the system and takes note of the outcome, how should she update her statistical distribution?
In the final and most significant step of the theory-construction scheme, one postulates a fundamental restriction on the sorts of statistical distributions that can describe an agent’s knowledge of the system (third column of Table \[Table:epistrictedtheories\]). As a first example, consider the classical ontological theory of particle mechanics. The associated statistical theory is what is sometimes called *Liouville mechanics*. If one then postulates a classical version of the uncertainty principle as the epistemic restriction [@bartlett2012reconstruction], then one obtains a theory which we shall refer to here as *Gaussian epistricted mechanics* (it was called *epistemically-restricted Liouville mechanics* in Ref. [@bartlett2012reconstruction]). This theory is equivalent to a subtheory of quantum mechanics, the Gaussian subtheory, which is defined in Ref. [@bartlett2012reconstruction].
The case of optics is a straightforward extension of the case of mechanics because each optical mode is a scalar field and the phase spaces of a field mode and of a particle are both Euclidean. The canonically conjugate variables, which are position and momentum in the mechanical case, are field quadratures in the optical case. The statistical theory of optics is well-known [@born1999principles]. Upon postulating an epistemic restriction in the form of an uncertainty principle, one obtains the optical analogue of the Gaussian subtheory of quantum mechanics, namely, the Gaussian subtheory of quantum optics, which is sometimes referred to as *linear quantum optics*. The latter theory includes a wide variety of quantum optical experiments.
-------------------------------------- ---------------------------------------------- ------------------------------------------------------
[**Statistical theory**]{} [**Epistemically-restricted statistical theory** ]{}
[**Classical ontological theory**]{} [**for the classical ontological theory**]{} [**for the classical ontological theory**]{}
Mechanics Liouville mechanics Gaussian epistricted mechanics
=Gaussian subtheory of quantum mechanics
Quadrature epistricted mechanics
=Quadrature subtheory of quantum mechanics
trits Statistical Theory of trits Quadrature epistricted theory of trits
= Quadrature/Stabilizer subtheory for qutrits
Bits Statistical Theory of bits Quadrature epistricted theory of bits
$\simeq$ Quadrature/Stabilizer subtheory for qubits
Optics Statistical optics Gaussian epistricted optics
= Gaussian subtheory of quantum optics
Qudrature epistricted optics
= quadrature subtheory of quantum optics
-------------------------------------- ---------------------------------------------- ------------------------------------------------------
One can apply the same strategy for a classical ontological theory wherein the fundamental degrees of freedom are discrete, so that every system has an integer number $d$ of ontic states. It is unusual for physicists to discuss discrete degrees of freedom in a classical context. Nonetheless, this is done when considering the possibility of models that are cellular automata. It is also common when describing the physics of digital computers. The language of computation, therefore, is a natural one for describing such a theory.
The simplest case to consider is $d=2$, in which case the fundamental degree of freedom is a bit. A collection of such fundamental degrees of freedom corresponds to a string of bits. An interaction between two distinct degrees of freedom can be understood as a gate acting on two bits. Similarly for interactions between $n$ systems. General dynamics, which corresponds to an arbitrary sequence of interactions, can be understood as a circuit. The statistical theory of bits is just a theory of the statistical distributions over the possible bit-strings, how these evolve under gates, and how these are updated as a result of registering the outcome of measurements performed on the bits. One then imposes a restriction on what kinds of statistical distributions can characterize an agent’s knowledge of the value of the bit-string.
This is the arena in which the first epistricted theory was constructed [@spekkens2007evidence]. The restriction on knowledge was implemented through a principle that asserted that any agent could have the answers to at most half of a set of questions that would specify the ontic state of the system. Consequently, when one has maximal knowledge, then the number of independent questions that are answered is equal to the number of independent questions that are unanswered; in this case, one’s measure of knowledge is equal to one’s measure of ignorance. This epistemic restriction was dubbed the *knowledge-balance principle*, and the epistricted theory of bits that resulted was called a *toy theory* in Ref. [@spekkens2007evidence]. This theory mirrors very closely a subtheory of the quantum theory of qubits, namely, the one which is known to quantum information theorists as the *stabilizer formalism* and which we will term the *stabilizer subtheory* of qubits. It will be presented in Sec. \[quadraturesubtheories\]. Stabilizer states are defined to be the eigenstates of products of Pauli operators, stabilizer measurements are measurements of commuting sets of products of Pauli operators, and stabilizer transformations are unitary transformations which take stabilizer states to stabilizer states. Although the toy theory is not operationally equivalent to the stabilizer subtheory, it reproduces qualitatively the same phenomenology. Also, the toy theory can be cast in the same sort of language as the stabilizer theory, as noted in Ref. [@pusey2012stabilizer].
Subsequent work sought to develop an epistricted theory for discrete systems with $d$ ontic states, where $d > 2$. There were two natural avenues to pursue: generalize the knowledge-balance principle used in Ref. [@spekkens2007evidence] or devise a discrete version of the classical uncertainty principle used in Ref. [@bartlett2012reconstruction]. The former approach was pursued by van Enk [@van2007toy].[^1] However, some important work by Gross [@gross2006hudson] established that it is possible to define a discrete phase space for a $d$-level systems where $d$ is an odd prime and it is possible to define a Wigner representation based on this phase space such that the stabilizer theory for these qudits admits of a nonnegative Wigner representation. Gross’s Wigner representation can be understood as a hidden variable model for the stabilizer subtheory. This suggests that one should be able to define a classical theory of $d$-level systems using this discrete phase space and then to find an epistemic restriction that yields precisely this hidden variable model. In other words, Gross’s work strongly suggests that one should look for an epistemic restriction that appeals to the phase-space structure, analogously to the epistemic restriction that was used in the Gaussian epistricted mechanics [@bartlett2012reconstruction]. Such an epistemic restriction was subsequently identified [@Sch08]. Using the phase-space structure, one can define *quadrature variables* for the classical system. The epistemic restriction then asserts that one can have joint knowledge of a set of quadrature variables if and only if they commute relative to a discrete analogue of the Poisson bracket. The epistemic restriction is dubbed *classical complementarity* and the theory that results is called the *quadrature epistricted theory* of $d$-level classical systems. If we apply the complementarity-based epistemic restriction in the case of $d=2$, the resulting theory—the qudrature epistricted theory of bits—turns out to be equivalent to the toy theory of Ref. [@spekkens2007evidence], and as mentioned previously, this is operationally a close phenomenological cousin of the stabilizer theory of qubits.
On the other hand, for $d$ an *odd* prime, i.e., any prime besides 2, the quadrature epistricted theory reproduces *precisely* the stabilizer theory for qudits. For such values of $d$, the epistemic restriction of classical complementarity turns out to be inequivalent to the knowledge-balance principle. The latter specifies only that at most half of the full set of variables can be known, whereas the former picks out particular halves of the full set of variables, namely, the halves wherein all the variables Poisson-commute. Because the restriction of classical complementarity actually reproduces the stabilizer theory for qudits while the knowledge-balance principle does not [@Sch08], epistemic restrictions based on the symplectic structure seem to be preferable to those based on a principle of knowledge balance.
We will also show that on the quantum side, one can define the notion of a quadrature *observable*, a quantum analogue of a classical quadrature variable. In $d=2$, the Pauli operators are both unitary and Hermitian; as unitaries, they constitute the quantum analogue of classical phase-space displacements, while as observables, they correspond to our quadrature observables. In $d>2$, on the other hand, the generalized Pauli operators are unitary but not always Hermitian and therefore cannot always be interpreted as observables. Consequently, the stabilizer of a state in $d>2$ specifies the unitaries that leave the state invariant, not the observables for which the state is an eigenstate. In $d>2$, the quadrature observables are the ones that are defined in terms of the eigenbases of the generalized Pauli operators. They provide a means for acheiving a characterization of stabilizer states for any $d$ as joint eigenstates of a commuting set of quadrature observables. This characterization is more analogous to our characterization, in epistricted theories, of the valid epistemic states as states wherein one has joint knowledge of a Poisson-commuting set of quadrature variables. Finally, the epistemic restriction of classical complementarity can also be applied to particle mechanics, where it is different from the restriction based on the uncertainty principle that is used in Ref. [@bartlett2012reconstruction]. In particular, a smaller set of statistical distributions are considered valid epistemic states. Using the principle of classical complementarity, one obtains a different theory at the end, which we call *quadrature epistricted mechanics*. We prove that this is equivalent to a subtheory of quantum mechanics that we will call the *quadrature subtheory of quantum mechanics* and which we will describe in detail in Sec. \[quadraturesubtheories\]. The latter stands to the Gaussian subtheory of quantum mechanics as the quadrature epistricted theory of mechanics stands to the Gaussian epistricted theory of mechanics. One can similarly define analogous theories for optics.
It follows that the epistemic restriction of classical complementarity provides the beginning of a unification of all known epistricted theories. It can be applied for both continuous and discrete degrees of freedom, and the formalism can be made to look precisely the same in each case.
It remains an open question whether one can find a form of the epistemic restriction that is applicable to an arbitrary degree of freedom and that when applied in the case of a $d$-level system yields the Stabilizer/quadrature subtheory of qudits while when applied in the case of continuous variable systems yields the Gaussian subtheory of quantum mechanics/optics rather than merely the quadrature subtheory.
Guided by the bridge between the epistricted theories and the quantum subtheories, we present the formalism of the associated quantum subtheories in a unified manner for continuous and discrete degrees of freedom. This presentation focusses on quadrature observables rather than stabilizer groups and helps to reveal the analogies between the subtheories for the different degrees of freedom.
For any epistemic restriction that is applicable to many different degrees of freedom, such as the principle of classical complementarity described here, one can think of the process of applying this restriction to the corresponding classical statistical theories as a kind of quantization scheme, or more precisely, a *quasi-quantization* scheme. It is “quasi” because it does not succeed at obtaining the full quantum theory from its classical counterpart and because in certain cases, such as binary variables, it does not even yield a subtheory of quantum theory.[^2] Unlike normal quantization schemes, which are mathematically inspired, the quasi-quantization scheme of this approach is [*conceptually*]{} inspired. There is no ambiguity about how to interpret the formalism that results.
Although our quasi-quantization scheme has already been applied to a few different sorts of degrees of freedom, it is clear that one could apply it to others. Vector fields are a good example, one which promises the possibility of a quasi-quantization of classical electrodynamics. By finding the appropriate epistemic restriction on a statistical theory of electrodynamics, one can imagine deriving a theory that might be equivalent to—or perhaps, as for the case of bits, merely analogous to—some subtheory of quantum electrodynamics[^3]. At present, it is not obvious how to do this because the epistemic restrictions that have worked best for the degrees of freedom considered thus far have made reference to canonically conjugate degrees of freedom. One therefore expects to encounter precisely the same difficulties that were faced by those who attempted a canonical quantization of classical electrodynamics. Presumably, therefore, it would be useful to develop a Lagrangian, or least-action quasi-quantization scheme in addition to the canonical one. If one could succeed at devising an epistricted theory of electrodynamics, then it would of course be very interesting to attempt to apply quasi-quantization to classical theories of gravity. This would not yield a full quantum theory of gravity, but it might reconstruct some subtheory, or a distorted version of such a subtheory.
The rest of the introduction makes explicit what can and cannot be explained in epistricted theories, together with their significance for interpretation and axiomatization. We have put this material up front rather than at the end of the paper for the benefit of those readers who are reluctant to engage with the detailed development until they have had certain questions answered, in particular, questions about the precise explanatory scope of these epistricted theories, and the question of why one should care about a quantization scheme that does not recover the full quantum theory.
Explanatory scope
-----------------
We return now to the claim that epistricted theories reproduce a “large part” of quantum theory. At this stage, a sceptic might be unconvinced on the grounds that for each classical ontological theory, the subtheory of the corresponding quantum theory that has been derived via this quantization scheme is *far* from the full quantum theory. For instance, Gaussian epistricted mechanics yields a part of quantum mechanics wherein the dynamics include only those Hamiltonians that are at most quadratic in position and momentum observables [@bartlett2012reconstruction]. Clearly, this is a small subset of all possible Hamiltonians. Nonetheless, we argue that the relative size of the space of Hamiltonians is not the correct metric by which to assess this project. The primary object of the exercise is to achieve conceptual clarity on the principles that might underly quantum theory. As such, it is better to ask: how many distinctively quantum phenomena are reproduced within these subtheories? In particular, how many of the phenomena that are usually taken to defy classical explanation? In terms of the phenomena they include, the subtheories of quantum theory one obtains by an epistemic restriction *do* subsume a large part of the full theory. In support of this claim, Table \[tbl:categorization\] provides a categorization of some prominent quantum phenomena into those that arise in epistricted theories (on the left), and those that do not (on the right). As one can easily see, for this particular list, the lion’s share are found on the left, and this set includes many of the phenomena that are typically taken to provide the greatest challenge to the classical worldview.[^4]
[**Phenomena arising in epistricted theories**]{} [**Phenomena not arising in epistricted theories**]{}
--------------------------------------------------- -------------------------------------------------------
Noncommutativity Bell inequality violations
Coherent superposition Noncontextuality inequality violations
Collapse Computational speed-up (if it exists)
Complementarity Certain aspects of items on the left
No-cloning
No-broadcasting
Interference
Teleportation
Remote steering
Key distribution
Dense coding
Entanglement
Monogamy of entanglement
Choi-Jamiolkowski isomorphism
Naimark extension
Stinespring dilation
Ambiguity of mixtures
Locally immeasurable product bases
Unextendible product bases
Pre and post-selection effects
Quantum eraser
And many others...
Note that it is typically the case that if one looks hard enough at a given quantum phenomenon that appears on the left list, one can usually find *some* feature of it that cannot be explained within an epistricted theory. When we place a given phenomenon on the left, therefore, what we are claiming is that an epistricted theory can reproduce *the features of this phenomenon that are most frequently cited as making it difficult to understand classically*. Consider the example of quantum teleportation. What is most frequently taken to be mysterious about teleportation from a classical perspective is that the amount of information that is required to describe the quantum state exceeds the amount of information that is communicated in the protocol. This is just as true, however, if one seeks to teleport a quantum state within the stabilizer theory of qubits: for a single qubit, this subtheory includes only *six* distinct quantum states (rather than an infinite number), but the teleportation protocol still succeeds while communicating only *two* bits of classical information, which is less than $\log_2{6}$ and hence not enough to describe a state drawn from this set. As such, we judge the teleportation protocol in the stabilizer formalism to include the essential mystery of teleportation, and, because an epistricted theory can reproduce this notion of teleportation, we put teleportation on the left-hand list. One can always point to features of the teleportation protocol in the full quantum theory that *do not* arise in the stabilizer theory of qubits, for instance, the fact that it works for an infinite number of quantum states rather than just six. However, this is not the feature of teleportation that is typically cited as “the mystery”. Such incidental features are the sorts of things that we mean to include on the right-hand side of our classification under “certain aspects of items on the left”. Of course, one of the lessons of this categorization exercise is that the usual story about what is mysterious about a given quantum phenomenon should be supplanted by one that highlights these more subtle features, and these should henceforth be our focus when puzzling about quantum theory. The left-hand list includes basic quantum phenomena, such as non-commutativity of observables, interference, coherent superposition, collapse of the wave function, the existence of complementary bases, and a no-cloning theorem [@wootters1982single]. It also includes many quantum information processing tasks, such as teleportation [@bennett1993teleporting] and key distribution [@bennett1984quantum]. A large part of entanglement theory [@horodecki2009quantum] is there, as are more exotic phenomena, such as locally indistinguishable product bases [@bennett1999quantum] and unextendible product bases [@bennett1999unextendible]. One also gets many of the distinctive relations that hold between (and within) the sets of quantum states, quantum measurements, and quantum transformations, such as the Choi-Jamiołkowski isomorphism between bipartite states and unipartite operations [@Choi1975; @Jamiolkowski1972], the Naimark extension of positive-operator valued measures into projector-valued measures [@Naimark1940], the Stinespring dilation of irreversible operations into reversible (unitary) operations [@stinespring1955positive], and the fact that there are many convex decompositions and many purifications of a mixed state.
On the right-hand side, we find Bell-inequality violations [@Bell1964] and noncontextuality-inequality violations [@kochen1967problem; @spekkens2005contextuality; @liang2011specker]. This is expected, as these phenomena are the operational signatures of the impossiblity of locally causal and noncontextual ontological models respectively, whereas the particular sorts of epistemic restrictions that have been considered to date yield theories that satisfy local causality and noncontextuality (even the generalized sense of noncontextuality of Ref. [@spekkens2005contextuality]). The right-hand side also includes quantum computational speedup, with the caveat that the claim of an exponential speed-up is predicated on certain unproven conjectures, such as the factoring problem being outside the complexity class P.
There are also some phenomena that have not yet been conclusively categorized. Two examples are: the quantization of quantities such as energy and angular momentum and the statistics of indistinguishable particles.
There is always some satisfaction in adding a quantum phenomenon to the left-hand list: it suggests that the idea of an epistemic restriction captures much of the innovation of quantum theory, that the phenomena in question is not so mysterious after all. However, the right-hand list is the one that we would most like to see grow, because the phenomena that appear there are the ones that still seem surprising, and it is by focusing on these that one can best develop the research program wherein quantum states are understood as states of knowledge.
Our quasi-quantization scheme sheds light on the old question “what is the conceptual innovation of quantum theory relative to classical theories?” In particular, it implies that the frontier between what *can* and what *cannot* be explained classically extends much deeper into quantum territory than previously thought. This is because in order to pronounce a phenomenon nonclassical, one should be maximally permissive in *how* a classical theory manages to reproduce the phenomenon. For instance, when considering whether certain operational statistics admit of a local or a noncontextual hidden variable model, one must allow an arbitrary space of ontic states, i.e., arbitrary hidden variables. With the benefit of hindsight, one sees that previous assessments of the scope of classical explanations were overly pessimistic because they did not consider the possibility that some phenomenon exhibited by quantum states was reproduced by classical *epistemic states* rather than by classical *ontic states.*
Phenomena arising in an epistricted theory might still be considered to exhibit a type of nonclassicality insofar as an epistemic restriction is, strictly speaking, an assumption that goes beyond classical physics. But it is a weak type of nonclassicality, as it ultimately can be understood via a relatively modest addendum to the classical worldview. By contrast, quantum phenomena that do not arise in epistricted theories constitute a strong type of nonclassicality, one which marks a significant departure from the classical worldview. Table \[tbl:categorization\] may therefore be understood as sorting quantum phenomena into categories of weak and strong nonclassicality.
Interpretational significance
-----------------------------
Epistricted theories serve to highlight the existence (and the appeal) of a type of ontological model that has previously received almost no attention. With the exception of a model of a qubit proposed by Kochen and Specker in 1967, previous ontological models have been such that the ontic state included a description of the quantum state, and therefore any two distinct pure quantum states necessarily described different ontic states. This was true whether the model considered the space of ontic states to be precisely the space of pure quantum states, or whether the quantum state was supplemented by additional variables, such as occurs, for instance, in Bohmian mechanics. By contrast, in epistricted theories, two distinct pure quantum states that are nonorthogonal correspond to two probability distributions that overlap on one or more ontic states. Indeed, it was the work on epistricted theories that led to the articulation of the distinction between a $\psi$-ontic model, wherein the ontic state encodes the quantum state, and a $\psi$-epistemic model, wherein it does not [@spekkens2007evidence; @spekkens2005contextuality; @Harrigan2010].
For the subtheories of quantum theory described above (Gaussian and quadrature quantum mechanics and the stabilizer theory of qudits for $d$ an odd prime), $\psi$-epistemic models exist and provide a compelling causal explanation of the operational predictions of those theories. The breadth of quantum phenomenology that is reproduced within epistricted theories suggests that something about the principles underlying these theories must be correct. The assumption that quantum states should be interpreted as epistemic states rather than ontic states seems a good candidate.
This in no way implies, however, that the innovation of quantum theory is *merely* to impose an epistemic restriction on some underlying classical physics. This is clearly *not* the only innovation of quantum theory. As we have noted, epistricted theories, considered as ontological models, are by construction both local and noncontextual, and because of Bell’s theorem and the Kochen-Specker theorem, we know that the full quantum theory cannot be explained by such models. If quantum computers really do allow an exponential speed-up over their classical counterparts, then this too cannot be reproduced by such models. What the success of epistricted theories suggests, rather, is that pure quantum states and statistical distributions over classical ontic states are *the same category of thing*, namely, an epistemic thing. This is a point of view that has been central also to the “QBist” research program [@caves2002quantum; @Fuchs2003a; @fuchs2013quantum].
A question that naturally arises is whether one can construct a $\psi$-epistemic model of the *full* quantum theory, or whether one can find natural assumptions under which such models are ruled out, This question was first posed by Lucien Hardy and was formalized in Ref. [@Harrigan2010]. It has become the subject of much debate in recent years [@pusey2012reality; @lewis2012distinct; @colbeck2012system]. It is worth noting, however, that the standard framework for such models contains many implicit assumptions, including the idea that the correct formalism for describing epistemic states is classical probability theory. This assumption can be questioned, and indeed, the nonlocality and contextuality of quantum theory already suggest that it should be abandoned, as argued in [@leifer2013towards; @wood2012lesson].
The investigation of epistricted theories, therefore, need not—and indeed *should not*—be considered as the first step in a research program that seeks to find a $\psi$-epistemic model of the full quantum theory. Even though such a model could always circumvent any no-go theorems by violating their assumptions, it would be just as unsatisfying as a $\psi$-ontic model insofar as it would need to be explicitly nonlocal and contextual. Rather, the investigation of epistricted theories is best considered as a first step in a larger research program wherein the framework of ontological models—in particular the use of classical probability theory for describing an agent’s incomplete knowledge—is ultimately rejected, but where one holds fast to the notion that a quantum state is epistemic.
Significance for the axiomatic program
--------------------------------------
Most reconstruction efforts are focussed on recovering the formalism of the full quantum theory. However, it may be that there are particularly elegant axiomatic schemes that are not currently in our reach and that the road to progress involves temporarily setting one’s sights a bit lower. The quasi-quantization scheme described here only recovers certain subtheories of the full quantum theory, which include only a subset of the preparations, transformations and measurements that the latter allows. However, such derivations may provide clues for more ambitious axiomatization schemes. In particular, it provides further evidence for the usefulness of foundational principles asserting a fundamental restriction on what can be known [@caves2002quantum; @zeilinger1999foundational; @paterek2010theories]. It also seems to highlight the importance of symplectic structure, which is not currently a feature of any reconstruction program.
Furthermore, epistricted theories constitute a foil to the full quantum theory in two senses. When they are operationally equivalent to a subtheory of the full quantum theory, they are still a foil in the sense that the universe might have been governed by this subtheory rather than the full theory. Why did nature choose the full theory rather than the subtheory? When the epistricted theory is not operationally equivalent to the corresponding quantum subtheory, as in the case of bits, it is a foil not only to the full quantum theory, but to the quantum subtheory as well. In this case, the epistricted theory describes a set of operational predictions that are not instantiated in our universe and again the question is: why nature did not avail itself of this option? Because epistricted theories share so much of the operational phenomenology of quantum theory, they constitute points in the landscape of possible operational theories that are particularly close to quantum theory. They are therefore particularly helpful in the project of determining what is unique about quantum theory. For any purported attempt to derive the formalism of quantum theory from axioms, it is useful to ask which of the axioms rule out epistricted theories. Axiom sets that may seem promising at first can often be ruled out immediately if they fail to pass this simple test.
Finally, epistricted theories have significance for the problem of developing mathematical frameworks for describing the landscape of operational theories. In particular, they provide a test of the *scope* of any given framework. Broadness of scope is the key virtue of any framework because an axiomatic derivation of quantum theory is only as impressive as the size of the landscape within which it is derived. For instance, the formalism of $C^{*}$-algebras essentially includes only operational theories that are fully quantum or fully classical, or that are quantum within each of a set of superselection sectors and classical between these. This is a rather limited scope, and consequently axiomatizations within this framework are less impressive than those formulated within broader frameworks. On this front, epistricted theories serve to highlight two deficiencies in the prevailing framework of convex operational theories.
First, quadrature epistemic theories are an example of a *possibilistic* or *modal* theory, wherein one does not specify the *probabilities* of measurement outcomes, but only which outcomes are *possible* and which are *impossible*. This perspective on the toy theory of Ref. [@spekkens2007evidence], for instance, is emphasized in Ref. [@coecke2011phase]. Possibilistic theories have recently been receiving renewed attention [@mansfield2012hardy; @abramsky2012logical; @schumacher2012modal] because in the context of discussions of Bell’s theorem they highlight the fact that quantum theory cannot merely imply an innovation to probability theory but must also imply an innovation to logic.
Second, these epistricted theories have operational state spaces that are not convex. They only allow certain mixtures of operational states. As such, they cannot be captured by the prevailing framework of *convex* operational theories [@barrett2007information; @hardy2001quantum] because these assume from the outset a convex state space. On the other hand, epistricted theories can be captured by the category-theoretic framework for process theories [@Coecke2009], as shown in [@coecke2011toy], or by the framework of general probabilistic theories described in Ref. [@chiribella2010probabilistic]. Epistricted theories therefore provide a concrete example of how the category-theoretic framework necessarily describes some real estate in the landscape of foil theories that is not on the map of the convex operational framework.
Quadrature epistricted theories
===============================
Classical complementarity as an epistemic restriction {#complementarity}
-----------------------------------------------------
The criterion on the joint knowability of classical variables that is used here is inspired by the criterion on the *joint measurability* of quantum observables.
> [**Guiding analogy**]{}:\
> A set of observables is *jointly measurable* if and only if it is commuting relative to the matrix commutator.\
> A set of variables is *jointly knowable* if and only if it is commuting relative to the Poisson bracket.\
The full epistemic restriction that we adopt is a combination of this notion of joint knowability together with the restriction that the only variables that can be known by the agent are *linear* combinations of the position and momentum variables. We refer to such variables as *quadrature variables*.[^5] We term the full epistemic restriction *classical complementarity*.
> [**Classical complementarity**]{}: The valid epistemic states are those wherein an agent knows the values of a set of quadrature variables that commute relative to the Poisson bracket and is maximally ignorant otherwise.
It is presumed that maximal ignorance corresponds to a probability distribution that is uniform over the region of phase-space consistent with the known values of the quadrature variables. In the case of a phase-space associated to a continuous field, uniformity is evaluated relative to the measure that is invariant under phase-space displacements. Hence a valid epistemic state is a uniform distribution over the ontic states that is consistent with a given valuation of some Poisson-commuting set of quadrature variables. It is because of the uniformity of these distributions that they can be understood as merely specifying, for the given constraints, which ontic states are possible and which are impossible. Consequently, the epistemic state in this case is aptly described as a *possibilistic* state. There is a subtlety here. The epistemic restriction is assumed to apply *only* to what an agent can know about a set of variables based on information acquired entirely to the past or entirely to the future of those variables. It is not assumed to apply to what an agent can know about a set of variables based on pre- and post-selection. The same caveats on applicability hold for the quantum uncertainty principle, so this constraint on applicability is not unexpected. To describe the epistemic restriction in more detail, we introduce some formalism. The continuous and discrete cases are considered in turn.
**Continuous degrees of freedom.** Assume $n$ classical continuous degrees of freedom. The configuration space is $\mathbb{R}^{n}$ and a particular configuration is denoted $$({\tt q}_{1},{\tt q}_{2},\dots,{\tt q}_{n})\in \mathbb{R}^{n}.$$ These could describe the positions of $n$ particles in a 1-dimensional space, or the positions of $n/3$ particles in a 3-dimensional space, or the amplitudes of $n$ scalar fields, etcetera. The associated phase space is$$\Omega\equiv\mathbb{R}^{2n}
$$ and we denote a point in this space by $$\textbf{m} \equiv ({\tt q}_{1},{\tt p}_{1},{\tt q}_{2},{\tt p}_{2},\dots,{\tt q}_{n},{\tt p}_{n})\in \Omega.$$
We consider real-valued functionals over this phase space $$f:\Omega\rightarrow\mathbb{R}\text{.}$$ In particular, the functionals associated with the position and momentum of the $i$th degree of freedom are defined respectively by$$q_{i}({\bf m})={\tt q}_{i},\text{ }p_{i}({\bf m})={\tt p}_{i}.$$ The Poisson bracket is a binary operation on a pair of functionals, defined by$$\left[ f,g\right]_{\rm PB} \left( {\bf m}\right) \equiv\sum_{i=1}^{n}\left(
\frac{\partial f}{\partial q_{i}}\frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q_{i}}\right)
\left( {\bf m}\right) .$$ In particular, we have $$\label{eq:CCRcontinuous}
\left[ q_i,p_j\right]_{\rm PB} \left( {\bf m}\right) = \delta_{i,j}.$$
The assumption of classical complementarity incorporates a restriction on the sorts of functionals that an agent can know. Specifically, an agent can only know the value of a functional that is *linear* in the position and momentum functionals, that is, those of the form$$f={\tt a}_{1}q_{1}+{\tt b}_{1}p_{1}+\dots+{\tt a}_{n}q_{n}+{\tt b}_{n}p_{n} +{\tt c},
\label{eq:linearfunctionals}$$ where ${\tt a}_{1},{\tt b}_{1},\dots,{\tt a}_{n},{\tt b}_{n},{\tt c} \in\mathbb{R}$. (Note that functionals that differ only by addition of a scalar or by a multiplicative factor ultimately describe the same property.) We will call these *quadrature functionals* or *quadrature variables*. The vector of coefficients of the position and momentum functionals for a given quadrature functional will be denoted by the boldface of the notation used for the functional itself. The vector ${\bf f}$ specifying the position and momentum dependence of the quadrature functional $f$ defined in Eq. is $${\bf f} \equiv ({\tt a}_{1},{\tt b}_{1},\dots,{\tt a}_{n},{\tt b}_{n}),$$ such that if we define the vector of position and momentum functionals $$\label{vectorqpfunctionals}
{\bf z} \equiv (q_{1},p_{1},\dots,q_{n},p_{n}),$$ we can express $f$ as $$\label{eq:f}
f= {\bf f}^T {\bf z} + {\tt c}.$$ Similarly, the action of the functional $f$ on a phase space vector ${\bf m}$ is given by $$f({\bf m}) = {\bf f}^T {\bf m}+ {\tt c}.$$ In other words, the space of quadrature functionals is the dual of the phase space $\Omega$, but each functional $f$ is associated with a vector in the phase space, ${\bf f} \in \Omega$. Note that the vectors associated with the position and momentum functionals $q_i$ and $p_i$ are ${\bf q}_i\equiv (0,0,\dots,1,0,\dots,0,0)$ where the only nonzero component is ${\tt a}_i$ and ${\bf p}_i \equiv (0,0,\dots,0,1,\dots,0,0)$, where the only nonzero component is ${\tt b}_i$. It is not difficult to see that the Poisson bracket of two quadrature functionals always evaluates to a functional that is uniform over the phase space. Its value is equal to the symplectic inner product of the associated vectors, $$\left[ f,g\right] _{PB}({\bf m})=\langle {\bf f}, {\bf g} \rangle, \label{eq:PoissonSymplecticIP}$$ where $$\langle {\bf f}, {\bf g} \rangle \equiv {\bf f}^{T}J{\bf g},$$ with $T$ denoting transpose and $J$ denoting the skew-symmetric $2n\times 2n$ matrix with components $J_{ij} \equiv\delta _{i,j+1}-\delta _{i+1,j}$, that is, $$\label{eq:SymplecticForm}
J \equiv
\begin{pmatrix}
0 & 1 & 0 & 0 & \dots \\
-1 & 0 & 0 & 0 & \\
0 & 0 & 0 & 1 & \\
0 & 0 & -1 & 0 & \\
\vdots & & & & \ddots
\end{pmatrix}.$$ (Note that $J$ squares to the negative of the $2n \times 2n$ identity matrix, $J^2 = -I$, it is an orthogonal matrix, $J^T J =I$, it has determinant +1, and it has an inverse given by $J^{-1} = J^T =-J$.) For instance, for $\Omega =\mathbb{R}^2$, if ${\bf f}=\left( {\tt a},{\tt b}\right) $ and ${\bf g}=\left( {\tt a}^{\prime},{\tt b}^{\prime}\right) ,$ then $\langle {\bf f}, {\bf g} \rangle =
{\tt a}{\tt b}^{\prime}-{\tt b}{\tt a}^{\prime}.$ The symplectic inner product on a phase space $\Omega$ is a bilinear form $\langle \cdot ,\cdot \rangle :\Omega \times \Omega \rightarrow \mathbb{R}$ that is skew-symmetric ($\langle {\bf f}, {\bf g} \rangle = -\langle {\bf g}, {\bf f} \rangle $ for all ${\bf f}, {\bf g} \in \Omega$) and non-degenerate (if $\langle {\bf f}, {\bf g} \rangle =0$ for all ${\bf g} \in \Omega,$ then ${\bf f}=0$). By equipping the vector space $\Omega$ with the symplectic inner product, it becomes a symplectic vector space. This connection to symplectic geometry allows us to provide a simple geometric interpretation of the Poisson-commuting sets of quadrature functionals, which we will present in Sec. \[validepistemicstates\].
**Discrete degrees of freedom.** For discrete degrees of freedom, the formalism is precisely the same, except that variables are no longer valued in the real field $\mathbb{R}$, but a finite field instead. Recall that all finite fields have order equal to the power of a prime. We shall consider here only the case where the order is itself a prime, denoted $d$, in which case the field is isomorphic to the integers modulo $d$, which we will denote by $\mathbb{Z}_{d}$. Therefore, the configuration space is $\left( \mathbb{Z}_{d}\right)^n$, the associated phase space is $$\Omega\equiv\left( \mathbb{Z}_{d}\right) ^{2n},$$ the functionals have the form $$f:\Omega\rightarrow \mathbb{Z}_{d} \text{,}$$ and the linear functionals are of the form of Eq. (\[eq:linearfunctionals\]), $$f={\tt a}_{1}q_{1}+{\tt b}_{1}p_{1}+\dots+{\tt a}_{n}q_{n}+{\tt b}_{n}p_{n} + {\tt c},$$ but where ${\tt a}_{1},{\tt b}_{1},\dots,{\tt a}_{n},{\tt b}_{n},{\tt c} \in\mathbb{Z}_d$ and the sum denotes addition modulo $d$. It follows that the vector ${\bf f} \equiv ({\tt a}_1, {\tt b}_1, \dots, {\tt a}_n, {\tt b}_n)$ associated with the functional $f$ lives in the phase space $\Omega \equiv\left( \mathbb{Z}_{d}\right)^n$ as well. The Poisson bracket, however, cannot be defined in the conventional way, because without continuous variables we do not have a notion of derivative. Nonetheless, one can define a discrete version of the Poisson bracket in terms of finite differences. For any functionals $f: \Omega \to \mathbb{Z}_d$ and $g: \Omega \to \mathbb{Z}_d$, their Poisson bracket, denoted $[f,g]_{PB}$, is also such a functional, the one defined by $$\left[ f,g\right] _{PB}({\bf m})\equiv\sum_{i=1}^{n}\left[
\begin{array}[c]{c}
\left( f\left( {\bf m}+{\bf q}_{i}\right) -f\left( {\bf m}\right) \right) \left(
g\left( {\bf m} +{\bf p}_{i} \right) -g\left( {\bf m}\right) \right) \\
-\left( f\left( {\bf m} +{\bf p}_{i}\right) -f\left( {\bf m}\right) \right) \left(
g\left( {\bf m}+{\bf q}_{i}\right) -g\left( {\bf m}\right) \right)
\end{array}
\right],$$ where the differences in this expression are evaluated with modular arithmetic. The requirement that $$\label{eq:CCRdiscrete}
\left[ q_i,p_j\right]_{\rm PB} \left( {\bf m}\right) = \delta_{i,j},$$ is clearly satisfied. Furthermore, it is straightforward to verify that Eq. (\[eq:PoissonSymplecticIP\]) also holds under this definition, so that one can relate the Poisson bracket in the discrete setting to the symplectic inner product on the discrete phase space, $\langle \cdot ,\cdot \rangle :\Omega \times \Omega \rightarrow \mathbb{Z}_d$.
**Simple examples.** It is useful to consider some simple examples of commuting pairs of quadrature variables, that is, some examples of 2-element sets $\left\{
f,g\right\} $ such that $\left[ f,g\right] _{PB}=0.$ Any quadrature variable defined for system 1 commutes with any quadrature variable for system 2, e.g., the pair$${\tt a}_{1}q_{1}+{\tt b}_{1}p_{1},\;\;\;{\tt a}_{2}q_{2}+{\tt b}_{2}p_{2}
$$ is a commuting pair for any values of ${\tt a}_{1},{\tt b}_{1},{\tt a}_{2},{\tt b}_{2}\in \mathbb{R}$ (or ${\tt a}_{1},{\tt b}_{1},{\tt a}_{2},{\tt b}_{2}\in\mathbb{Z}_d$). Additionally, there are commuting pairs of quadrature variables describing joint properties of the two systems, for instance$$q_{1}-q_{2},\;\;\;p_{1}+p_{2}
$$ (when the field is $\mathbb{Z}_d$, the coefficient $-1$ is equivalent to $d-1$, so that $q_{1}-q_{2}=q_{1}+(d-1)q_{2}$).
Another useful concept in the following will be that of *canonically conjugate* variables. A pair of variables are said to be canonically conjugate if $\left[ f,g\right] _{PB} = 1.$ On a single system, the pair of quadrature variables$$ {\tt a}q+{\tt b}p,\;\;\;-{\tt b}q+{\tt a}p
$$ are canonically conjugate for any values ${\tt a},{\tt b} \in \mathbb{R}$ (or ${\tt a},{\tt b} \in \mathbb{Z}_d$) such that ${\tt a}^2 + {\tt b}^2 =1$; in particular $\{q,p\}$ is such a pair.
Note that we were able to present these examples without specifying the nature of the field. We will follow this convention of presenting results in a unified field-independent manner for the next few sections.
Characterization of quadrature epistricted theories
---------------------------------------------------
### The set of valid epistemic states {#validepistemicstates}
Using the connection between the Poisson bracket for quadrature functionals and the symplectic inner product, one obtains a geometric interpretation of the epistemic restriction and the valid epistemic states.
To specify an epistemic state one must specify: (i) the set of quadrature variables that are known to that agent and (ii) the values of these variables. We will consider each aspect in turn.
The epistemic restriction asserts that the only sets of variables that are jointly knowable are those that are Poisson-commuting (which is to say that every pair of elements in the set Poisson-commutes). Note, however, that if every variable in a set has a known value, then any function of those variables also has a known value, in particular any linear combinations of those variables has a known value. It follows that for any Poisson-commuting set of variables, we can close the set under linear combination and preserve the property of being Poisson-commuting. In terms of the vectors representing these variables, this implies that we can take their *linear span* while preserving the property of having vanishing symplectic inner product for every pair of vectors. In terms of the symplectic geometry, a subspace all of whose vectors have vanishing symplectic inner product with one another is called an *isotropic* subspace of the phase space. Formally, a subspace $V\subseteq \Omega$ is isotropic if $$\forall {\bf f},{\bf g}\in V: \langle {\bf f}, {\bf g} \rangle =0.$$ It follows that we can parametrize the different possible sets of known variables in terms of the isotropic subspaces of the phase space $\Omega$.
For a $2n$-dimensional phase space, the maximum possible dimension of an isotropic subspace is $n$. These are called *maximally isotropic* or *Lagrangian* subspaces. This case corresponds to the maximal possible knowledge an agent can have according to the epistemic restiction. The agent then knows a *complete* set of Poisson-commuting variables, which is the analogue of measuring a complete set of commuting observables in quantum theory.
For a given Poisson-commuting set of variables, define a basis of that set to be any subset containing linearly independent elements and from which the entire set can be obtained by linear combinations. In the symplectic geometry, this corresponds to a vector basis for the associated isotropic subspace. There are, of course, many choices of bases for a given isotropic subspace or Poisson-commuting set.
Next, we must characterize the possible value assignments to a Poisson-commuting set of quadrature variables. That is, we must specify a linear functional $v$ acting on a quadrature functional $f$ and taking values in the appropriate field (continuous or discrete) such that $v(f)$ is the value assigned to $f$. Denote the isotropic subspace of $\Omega$ that is associated to this Poisson-commuting set by $V$, such that ${\bf f} \in V$ is the vector associated with the quadrature function $f$. The set of value assignments corresponds precisely to the set of vectors in $V$. In other words, for every vector ${\bf v} \in V$, which we call a *valuation vector*, we obtain a distinct value assignment $v$, via $$v(f) = {\bf f}^T {\bf v}.$$ To see this, it suffices to note that the ontic state of the system determines the values of all functionals and therefore the set of possible value assignments is given by the set of possible ontic states. Specifically, each ontic state ${\bf m} \in \Omega$ defines the value assignment $$v_{\bf m}( f) = {\bf f}^T {\bf m}.$$ However, many different ontic states yield the same value assignment. Denoting the projector onto $V$ by $P_{V}$, we can express the relevant equivalence relation thus: the ontic states ${\bf m}$ and ${\bf m'}$ yield the same value assignment to the quadrature functionals associated to $V$ if and only if $P_{V} {\bf m} = P_{V} {\bf m'}$. It follows that the set of possible value assignments to the associated to $V$ can be parametrized by the set of projections of all ontic states ${\bf m}\in \Omega$ into $V$, which is simply the set of ontic states in $V$. This establishes what we set out to prove. As an example, consider the case where we have two degrees of freedom, so that $\Omega$ is 4-dimensional, and suppose that the set of quadrature variables that are jointly known are the position variables, $\{ q_1, q_2\}$, and that these are known to each take the value $1$. In this case, the associated isotropic subspace $V \subseteq \Omega$, and the valuation vector ${\bf v}\in V$ are, respectively, $$\begin{aligned}
V & =\mathrm{span}\{{\bf q}_{1},{\bf q}_{2}\}\\
&= \mathrm{span}\{ (1,0,0,0), (0,0,1,0)\}\\
&=\{ ({\tt s},0,{\tt t},0): {\tt s},{\tt t} \in \mathbb{R}/\mathbb{Z}_d \}\\
{\bf v} & =(1,0,1,0).\end{aligned}$$ These are depicted in green in Fig. \[fig:Green1\].
Next, we consider a given epistemic state, where the known quadrature variables are specified by an isotropic subspace $V \subseteq \Omega$, and their values are specified by ${\bf v} \in V$, and ask: what probability distribution over the phase space $\Omega$ does it correspond to? Recalling that this probability distribution should be maximally uninformative relative to the given constraint, the answer is simply a uniform distribution on the set of all ontic states that yield this value assignment, that is, on the set $$\begin{aligned}
\label{setofonticstates}
&\{ {\bf m}\in \Omega : {\bf f}^T {\bf m} = {\bf f}^T {\bf v} \;\forall {\bf f} \in V \} \nonumber \\
& =\left\{ {\bf m}\in\Omega:P_{V}{\bf m}={\bf v}\right\}.\end{aligned}$$ If we denote the subspace of $\Omega$ that is orthogonal to $V$ (relative to the Euclidean inner product) by $V^{\perp},$ and we denote the translation of a subspace $W$ by a vector ${\bf v}$ as $W+{\bf v}\equiv\left\{ {\bf m}:{\bf m}={\bf w}+{\bf v},\text{ } {\bf w}\in W\right\} ,$ then it is clear that the set of ontic states of is simply $$V^{\perp}+{\bf v}.$$
For instance, in the example described above, $$\begin{aligned}
V^{\perp}
&= \{ (0,{\tt s},0,{\tt t}): {\tt s},{\tt t} \in \mathbb{R}/\mathbb{Z}_d \},\end{aligned}$$ and consequently the set of ontic states consistent with the agent’s knowledge is $$V^{\perp}+{\bf v} \equiv \left\{ \left( 1,{\tt s},1,{\tt t}\right) :{\tt s},{\tt t} \in\mathbb{R}/\mathbb{Z}
_{d}\right\},$$ which is depicted in blue in Fig. \[fig:Green1\].
As a probability distribution over $\Omega$, the epistemic state associated to $\left( V,{\bf v}\right)$ has the following form $$\label{epiprod}
\mu_{V,{\bf v}}\left( {\bf m}\right) = \frac{1}{\mathcal{N}_V } \delta_{V^{\perp}+{\bf v}}({\bf m})
$$ where we have introduced the notation $$\label{defndelta}
\delta_{V^{\perp}+{\bf v}}({\bf m}) \equiv \prod_{{\bf f}^{(i)}: {\rm span}\{ {\bf f}^{(i)} \} =V} \delta ( {\bf f}^{(i)T} {\bf m} - {\bf f}^{(i)T}{\bf v}),$$ where in the discrete case $\delta ({\tt c})=1$ if ${\tt c}=0$ and $\delta ({\tt c})=0$ otherwise, while in the continuous case $\delta$ denotes a Dirac delta function. In this expression, $\{ {\bf f}^{(i)} \}$ can be any basis of $V$.
Geometrically, $\mu_{V,{\bf v}}$ is simply the uniform distribution over the ontic states in $V^{\perp}+{\bf v}$.
Some epistemic states are seen to be mixtures of others in this theory. A valid epistemic state is termed *pure* if it is convexly extremal among valid epistemic states, that is, if it cannot be formed as a convex combination of other valid opistemic states. Non-extremal epistemic states are termed *mixed*. Note that we are judging extremality relative to the set of valid epistemic states, not relative to the set of all epistemic states. In our approach, the pure epistemic states are those corresponding to maximal knowledge, that is, knowledge associated to a complete set of Poisson-commuting quadrature variables. Note, however, that because of the epistemic restriction, maximal knowledge is always incomplete knowledge.
### The set of valid transformations
In addition to specifying the valid epistemic states, we must also specify what transformations of the epistemic states are allowed in our theory. To begin with, we consider the reversible transformations on an isolated system.
Suppose an agent knows the precise ontological dynamics of a system over some period of time. This transformation is represented by a bijective map on the ontic state space, and this induces a bijective map on the space of epistemic states. Because we assume that the underlying ontological theory has symplectic structure, it follows that the allowed transformations must be within the set of *symplectic transformations* (sometimes called *symplectomorphisms*). The requirement that the epistemic restriction must be preserved under the transformation implies that the valid transformations are a subset of the symplectic transformations, namely, those that map the set of quadrature variables to itself. Each such transformation can be represented in terms of its action on the phase space vector ${\bf m}\in\Omega$ as $$\label{symplecticaffine}
{\bf m}\mapsto S{\bf m}+{\bf a}$$ where ${\bf a} \in\Omega$ is a phase-space displacement vector and where $S$ is a $2n \times 2n$ *symplectic matrix*, that is, one which preserves the symplectic form $J$ defined in Eq. , $$S^{T}JS=J,$$ or equivalently, one which preserves symplectic inner products, i.e., $\left(
S{\bf m}\right)^{T}J\left( S{\bf m}^{\prime}\right) ={\bf m}^{T}J{\bf m}^{\prime}\;\forall {\bf m},{\bf m}' \in \Omega$. These are combinations of phase-space rotations and phase-space displacements.
Equation describes an affine transformation, but it does not include all such transformations because $S$ is not a general linear matrix. Following [@gross2006hudson], we call transformations of the form of Eq. *symplectic affine transformations*. Two such transformations, $S \cdot +{\bf a}$ and $S' \cdot +{\bf a}'$ compose as $$\label{sympaffine}
S \left( S' \cdot +{\bf b'} \right) +{\bf b} = SS' \cdot + (S{\bf b}' + {\bf b}).$$ The inverse of a symplectic matrix $S$ is $S^{-1} = J^T S^T J$, and the inverse of the phase-space displacement ${\bf a}$ is of course $-{\bf a}$. We call the resulting group of transformations the *symplectic affine group*.
If the epistemic state is described by a probability distribution/density over ontic states, $\mu : \Omega \to \mathbb{R}_+$, then under the ontological transformation ${\bf m} \mapsto S{\bf m} + {\bf a}$, the transformation induced on the epistemic state is $$\mu ({\bf m}) \mapsto \mu'({\bf m}) = \mu (S^{-1}{\bf m} -{\bf a}),$$
We can equivalently represent this transformation by a conditional probability distribution $\Gamma_{S,{\bf a}}: \Omega \times \Omega \to \mathbb{R}_+$, that is, $$\mu'({\bf m}) = \int {\rm d}{\bf m}' \Gamma_{S,{\bf a}}({\bf m}|{\bf m}') \mu({\bf m}'),$$ where $$\label{Cliffordcond}
\Gamma_{S,{\bf a}}({\bf m}|{\bf m}') = \delta({\bf m} - (S{\bf m}' + {\bf a})).$$
There is a subtlety worth noting at this point. The map $\mu \mapsto \mu'$ on the space of probability distributions, which is induced by the map ${\bf m} \mapsto S{\bf m} + {\bf a}$ on the space of ontic states, has the following property: it maps the set of valid epistemic states (those satisfying the classical complementarity principle) to itself. However, not every map from the set of valid epistemic states to itself can be induced by some map on the space of ontic states. A simple counterexample is provided by the map corresponding to time reversal. For a single degree of freedom, time reversal is represented by the map ${\bf m} = ({\tt q},{\tt p}) \mapsto {\bf m}' = ({\tt q},-{\tt p})$, which obviously fails to preserve the symplectic form. In terms of symplectic geometry, it is a reflection rather than a rotation in the phase space. Nonetheless, it maps isotropic subspaces to isotropic subspaces and therefore it also maps valid epistemic states to valid epistemic states. Therefore, in considering a given map on the space of distributions over phase space, it is not sufficient to ensure that it takes valid epistemic states to valid epistemic states, one must also ensure that it arises from a possible ontological dynamics. We say that the map must *supervene* upon a valid ontological transformation [@bartlett2012reconstruction].
Note that if the phase space is over a [*discrete*]{} field, then the transformations must be discrete in time. Only in the case of continuous variables can the transformations be continuous in time and only in this case can they be generated by a Hamiltonian. In addition to transformations corresponding to reversible maps over the epistemic states, there are also transformations corresponding to irreversible maps. These correspond to the case where information about the system is lost. The most general such transformation corresponds to adjoining the system to an ancilla that is prepared in a quadrature state, evolving the pair by some symplectic affine transformation that involves a nontrivial coupling of the two, and finally marginalizing over the ancilla. The reason this leads to a loss of information about the ontic state of the system is that the transformation of the system depends on the initial ontic state of the ancilla, and the latter is never completely known, by virtue of the epistemic restriction.
### The set of valid measurements
We must finally address the question of which *measurements* are consistent with our epistemic restriction. We will distinguish sharp and unsharp measurements. The sharp measurements are the analogues of those associated with projector-valued measures in quantum theory and can be defined as those for which the outcome is deterministic given the ontic state. The unsharp measurements are the analogues of those in quantum theory that cannot be represented by a projector-valued measure but instead require a positive operator-valued measure; they can be defined as those for which the outcome is not deterministic given the ontic state. We begin by considering the valid *sharp* measurements. Without the epistemic restriction, one could imagine the possibility of a sharp measurement that would determine the values of *all* quadrature variables, and hence also determine what the ontic state of the system was prior to the measurement. Given classical complementarity, however, one can only jointly retrodict the values of a set of quadrature variables if these are a Poisson-commuting set, and therefore the only sets of quadrature variables that can be jointly measured are the Poisson-commuting sets.
Given that every Poisson-commuting set of quadrature variables defines an isotropic subspace, the valid sharp measurements are parametrized by the isotropic subspaces. Furthermore, the possible joint value-assignments to a Poisson-commuting set of variables associated with isotropic subspace $V$ are parametrized by the vectors in $V$, so that the outcomes of the measurement associated with $V$ are indexed by ${\bf v} \in V$.
Such measurements can be represented as a conditional probability, specifying the probability of each outcome ${\bf v}$ given the ontic state ${\bf m}$, namely, $$\label{responsefns}
\xi_{V}\left({\bf v}| {\bf m}\right) =
\delta_{V^{\perp} + {\bf v}}({\bf m}),$$ where $\delta_{V^{\perp}+{\bf v}}({\bf m})$ is defined in Eq. . We refer to the set $\{ \xi_V ({\bf v} | {\bf m}): {\bf v}\in V\}$, considered as functions over $\Omega$, as the *response functions* associated with the measurement.
The set of all valid [*unsharp*]{} measurements can then be defined in terms of the valid sharp measurements as follows. An unsharp measurement on a system is valid if it can be implemented by adjoining to the system an ancilla that is described by a valid epistemic state, coupling the two by a symplectic affine transformation, and finally implementing a valid sharp measurement on the system+ancilla. Note that this construction of unsharp measurements from sharp measurements on a larger system is the analogue of the Naimark dilation in quantum theory. A full treatment of measurements would include a discussion of how the epistemic state is updated when the system survives the measurement procedure, but we will not discuss the transformative aspect of measurements in this article.
### Operational statistics
Suppose that one prepares a system with phase space $\Omega$ in the epistemic state $\mu_{V,{\bf v}}({\bf m})$ associated with isotropic subspace $V$ and valuation vector ${\bf v}$, and one subsequently implements the sharp measurement associated with the isotropic subspace $V'$. What is the probability of obtaining a given outcome ${\bf v}' \in V'$? The answer follows from an application of the law of total probability. The probability is simply $$\begin{aligned}
\label{opstat1}
&&\sum_{{\bf m} \in \Omega} \xi_{V'}({\bf v}' | {\bf m}) \mu_{V,{\bf v}}({\bf m}).\nonumber\\
$$ If a symplectic affine transformation ${\bf m} \mapsto S{\bf m} +{\bf a}$ is applied between the preparation and the measurement, the probability of outcome ${\bf v}'$ becomes $$\begin{aligned}
\label{opstat2}
&&\sum_{{\bf m} \in \Omega} \xi_{V'}({\bf v}' | {\bf m}) \sum_{{\bf m}' \in \Omega} \Gamma_{S,{\bf a}}({\bf m}|{\bf m}') \mu_{V,{\bf v}}({\bf m}').
\nonumber\\
$$
These statistics constitute the operational content of the quadrature epistricted theory.
Quadrature epistricted theory of continuous variables
-----------------------------------------------------
We now turn to concrete examples of quadrature epistricted theories for particular choices of the field. In this section, we consider the case of a phase space of $n$ [*real*]{} degrees of freedom, $\Omega=\mathbb{R}^{2n}.$ We begin by discussing the valid epistemic states for a single degree of freedom, $n=1$. In this case, the phase space is 2-dimensional and the isotropic subspaces are the set of 1-dimensional subspaces. We have depicted a few examples in Fig. \[fig:CVsingledofs\]. The isotropic subspace $V$ is depicted in light green, the valuation vector ${\bf v}$ is depicted as a dark green arrow, and the set $V^{\perp}+{\bf v}$ of ontic states in the support of the associated epistemic state is depicted in blue. Fig. \[fig:CVsingledofs\](a) depicts a state of knowledge wherein position is known (and hence momentum is unknown). Fig. \[fig:CVsingledofs\](b) depicts the vice-versa. Fig. \[fig:CVsingledofs\](c) corresponds to knowing the value of a quadrature $\left( \cos\theta\right) q+\left( \sin\theta\right) p$ (and hence having no knowledge of the canonically conjugate quadrature $-\left(
\sin\theta\right) q+\left( \cos\theta\right) p$). Finally, an agent could know nothing at all, in which case the epistemic state is just the uniform distribution over the whole phase space, as depicted in Fig. \[fig:CVsingledofs\](d).
If one considers a pair of continuous degrees of freedom, then it becomes harder to visualize the epistemic states because the phase space is 4-dimensional. Nonetheless, we present 3-dimensional projections as a visualization tool. We know that for every pair of isotropic subspace and valuation vector, $(V,{\bf v})$, there is a distinct epistemic state. In Fig. \[fig:CVtwodofs\](a), we depict the example where $q_1$ and $q_2$ are the known variables and both take the value 1, so that $V=\mathrm{span}\left\{ {\bf q}_{1},{\bf q}_{2}\right\}=\mathrm{span}\left\{ (1,0,0,0), (0,0,1,0)\right\} $ and ${\bf v}=(1,0,1,0)$, while in Fig. \[fig:CVtwodofs\](b), it is $q_{1}-q_{2}$ and $p_{1}
+p_2$ that are the known variables and both take the value 1, so that $V=\mathrm{span}\left\{ {\bf q}_{1}-{\bf q}_{2},{\bf p}_{1}+{\bf p}_{2}\right\}=\mathrm{span}\left\{ (1,0,-1,0), (0,1,0,1)\right\} $ and ${\bf v}=\left( \tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2}\right) .$ In the example of Fig. \[fig:CVtwodofs\](c), only a single variable, $q_1$, is known and takes the value 1, so that $V=\mathrm{span}\{{\bf q}_1 \}=\mathrm{span}\left\{ (1,0,0,0) \right\} $ and ${\bf v}=(1,0,0,0)$.
Quadrature epistricted theory of trits
--------------------------------------
We turn now to discrete systems. We begin with the case where the configuration space of every degree of freedom is three-valued, i.e., a *trit*, and represented therefore by $\mathbb{Z}_{3}$, the integers modulo 3. The configuration space of $n$ degrees of freedom is $\mathbb{Z}_{3}^n$ and the phase space is $\Omega=\left( \mathbb{Z}_{3}\right) ^{2n}.$
For a single system ($n=1)$, we can depict $\Omega$ as a 3$\times3$ grid. Consider all of the quadrature functionals that can be defined on such a system. They are of the form $f={\tt a}q+{\tt b}p+{\tt c}$ where ${\tt a},{\tt b},{\tt c} \in \mathbb{Z}_3$. Some of these functionals partition the phase-space in equivalent ways. It suffices to look at the inequivalent quadrature functionals. There are four of these: $$q,\;p,\;q+p,\;q+2p.$$ Note that because addition is modulo 3, $q+2p$ could equally well be written $q-p.$
Because no two of these functionals Poisson-commute, the principle of classical complementarity implies that an agent can know the value of at most one of these variables. It follows that there are twelve pure epistemic states, depicted in Fig. \[fig:epistemicstates1toytrit\]. The only mixed state is the state of complete ignorance. Here we depict in blue the ontic states in the support of the epistemic state. We have not explicitly depicted the isotropic subspace and valuation vector, but these are analogous to what we had in the continuous variable case.
Next, we can consider *pairs* of trits $(n=2)$. The quadrature variables are linear combinations of the positions and momentum of each, with coefficients drawn from $\mathbb{Z}_3$. Just as in the continuous case, one now has quadrature variables that describe joint properties of the pair of systems. The complete sets of Poisson-commuting variables now contain a pair of variables. Rather than attempting to portray the 4-dimensional phase space, as we did in the continous case, we can depict each 2-dimensional symplectic subspace along a line, as in Fig. \[fig:Sudokupuzzle\]. This is the Sudoku puzzle depiction of the two-trit phase space.
Figs. \[fig:twotoytrits\](a), \[fig:twotoytrits\](b), \[fig:twotoytritsentangled\](a), and \[fig:twotoytritsentangled\](b) each depict a mixed epistemic state, wherein the value of a single quadrature variable is known. Figs. \[fig:twotoytrits\](c) and \[fig:twotoytritsentangled\](c) depict pure epistemic states, wherein the values of a pair of Poisson-commuting variables are known. If one of the pair of known variables refers to the first subsystem and the other refers to the second subsystem, as in Fig. \[fig:twotoytrits\](c), the epistemic state corresponds to a product state in quantum theory. If both of the known variables describe joint properties of the pair of trits, as in Fig. \[fig:twotoytritsentangled\](c), the epistemic state corresponds to an entangled state.
The valid reversible transformations are the affine symplectic maps on the phase-space. These correspond to a particular subset of the permutations. Some examples are depicted in Fig. \[fig:twotrittransf\].
Just as in the continuous case, the valid measurements are those that determine the values of a set of Poisson-commuting quadrature variables. For instance, for a single trit, there are only four inequivalent measurements: of $q$, of $p$, of $q+p$ and of $q+2p$, depicted in Fig. \[fig:mmtstrits\](a), with different colours denoting different outcomes. Fig. \[fig:mmtstrits\](b) depicts some valid measurements on a pair of trits. The left depicts a joint measurement of $q_1$ and $q_2$, which corresponds to a product basis in quantum theory. The right depicts a joint measurement of $q_1-q_2$ and $p_1 + p_2$, which corresponds to a basis of entangled states.
Quadrature epistricted theory of bits
-------------------------------------
The epistricted theory of bits is very similar to that of trits, except with $\mathbb{Z}_2$ rather than $\mathbb{Z}_3$ describing the configuration space of a single degree of freedom. For a single system ($n=1)$, we can depict the phase space $\Omega$ as a 2$\times2$ grid. There are only three inequivalent linear functionals: $$q,\;p,\;q+p.$$ Unlike the case of trits, $q-p$ is not a distinct functional because in arithmetic modulo 2, $q-p=q+p$.
It follows that the valid epistemic states for a single system are those depicted in Fig. \[fig:epistemicstates1toybit\]. There are six pure states and one mixed state. We adopt a similar graphical convention to depict the 4-dimensional phase space of a pair of bits as we did for a pair of trits, presented in Fig. \[fig:4by4sudoku\]. Because the combinatorics are not so bad for the case of bits, we depict *all* of the valid epistemic states for a pair of bits in Fig. \[fig:allepistemicstatestwobits\]. We categorize these into those for which two variables are known (the pure states) and those for which only one or no variable is known (the mixed states). We also categorize these according to whether they exhibit correlation between the two subsystems or not. The pure correlated states correspond to the entangled states.
The reversible transformations for the case of a single system $(n=1)$ are particularly simple. In this case, $\Omega = (\mathbb{Z}_d)^2$, and the symplectic form is simply $J=
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}.$ because $-1=1$ in arithmetic modulo 2. As such, the symplectic matrices in this case are those with elements in $\mathbb{Z}_2$ and satisfying $S^T
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
S = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$. These are all the $2\times 2$ matrices having at least one column containing a $0$, that is, $$\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},\;\;
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},\;\;
\begin{pmatrix}
1 & 0 \\
1 & 1
\end{pmatrix},\;\;
\begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix},\;\;
\begin{pmatrix}
0 & 1 \\
1 & 1
\end{pmatrix},\;\;
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix},$$ corresponding respectively to the tranformations $$\begin{matrix}
{\tt q \mapsto q} \\
{\tt p \mapsto p}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto p} \\
{\tt p \mapsto q}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto q} \\
{\tt p \mapsto q+p}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto q+p} \\
{\tt p \mapsto p}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto p} \\
{\tt p \mapsto q+p}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto q+p} \\
{\tt p \mapsto q}
\end{matrix}.$$ Each of these symplectic transformations can be composed with the four possible phase-space displacements, $$\begin{matrix}
{\tt q \mapsto q} \\
{\tt p \mapsto p}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto q+1} \\
{\tt p \mapsto p}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto q} \\
{\tt p \mapsto p+1}
\end{matrix}\;\;,\;\;
\begin{matrix}
{\tt q \mapsto q+1} \\
{\tt p \mapsto p+1}
\end{matrix}.$$ In all, this leads to 24 reversible symplectic affine transformations, which are depicted in Fig. \[fig:transf1toybit\]. Given that there are only 24 permutations on the discrete phase space, we see that [*every*]{} reversible ontic transformation is physically allowed in this case.
On the other hand, for a pair of systems $(n=2)$, only a subset of the permutations of the ontic states correspond to valid sympectic affine transformations. In Fig. \[fig:mmtsbits\](a), we present the valid reproducible measurements on a single bit, and in Fig. \[fig:mmtsbits\](b) we present some examples of such measurements on a pair of bits, one corresponding to a product basis and the other an entangled basis.
Quadrature quantum subtheories {#quadraturesubtheories}
==============================
We now shift our attention to quantum theory, and build up to a definition of the subtheories of quantum theory that our epistricted theories will ultimately be shown to reproduce.
Quadrature observables
----------------------
We are interested in describing collections of elementary systems that each describe some continuous or discrete degree of freedom. If the elementary system is a continuous degree of freedom, it is associated with the Hilbert space $\mathcal{L}^2(\mathbb{R})$, the space of square-integrable functions on $\mathbb{R}$. For the case of $n$ such systems, the Hilbert space is $\mathcal{L}^2(\mathbb{R})^{\otimes n}=\mathcal{L}^2(\mathbb{R}^n)$. The sorts of discrete degrees of freedom we consider are those wherein all the elementary systems have $d$ levels where $d$ is a prime. These are described by the Hilbert space $\mathbb{C}^d$. For $n$ such systems, the Hilbert space is $(\mathbb{C}^{d})^{\otimes n} = \mathbb{C}^{dn}$.
We seek to describe both discrete and continuous systems in the language of symplectic structure. For a scalar field, for instance, we describe each mode of the field in terms of a pair of field quadratures. In the example of a 2-level system, even though the physical degree of freedom in question may be spin or polarization, we seek to understand it in terms of a configuration variable and its canonically conjugate momentum. In all of these cases, we will conventionally refer to the pair of conjugate variables, regardless of the degrees of freedom they describe, as ‘position’ and ‘momentum’.
We wish to present the quadrature subtheories for the continuous and discrete cases in a unified manner. Towards this end, we will avoid using a Hermitian operator to represent the quantum measurement associated to a quadrature variable. The reason is that although this works well for the continuous case, it fails to make sense in the discrete case. Recall that in the continuous case, we can define Hermitian operators on $\mathcal{L}^2(\mathbb{R})$, denoted $\hat{q}$ and $\hat{p}$, and satisfying the commutation relation $[ \hat{q},\hat{p} ]= \hat{\mathbb{1}},$ where $[ \cdot, \cdot]$ denotes the matrix commutator and $\hat{\mathbb{1}}$ is the identity operator on $\mathcal{L}^2(\mathbb{R})$. In the discrete setting, however, we would expect the operators associated to the discrete position and momentum variables to have eigenvalues in the finite field $\mathbb{Z}_d$, whereas the eigenvalues of Hermitian operators are necessarily real. Even if we did pick a pair of Hermitian operators to serve as discrete position and momentum observables, these would necessarily fail to provide an analogue of the commutation relation $[ \hat{q},\hat{p} ]= \hat{\mathbb{1}},$ because in a finite-dimensional Hilbert space, the commutator of any two Hermitian operators has vanishing trace and therefore cannot be proportional to the identity operator on that space.
In any case, within the fields of quantum foundations and quantum information, there has been a move away from representing measurements by Hermitian operators because the eigenvalues of these operators are merely arbitrary labels of the measurement outcomes and have no operational significance. It is only the projectors in the spectral resolution of such a Hermitian operator that appear in the Born rule and hence only these that are relevant to the operational statistics. Therefore, a measurement with outcome set $K$ is associated with a set of projectors $\{ \Pi_k : k \in K\}$ such that $\Pi_k^2 = \Pi_k,\; \forall k\in K$ and $\sum_{k\in K} \Pi_k = \mathbb{1}$ (integral in the case of a continuum of outcomes). Such a set is called a *projector-valued measure* (PVM).
In the continuous variable case, we define the position observable, denoted $\mathcal{O}_q$, to be the PVM consisting of projectors onto position eigenstates, $$\mathcal{O}_q \equiv \{ \hat{\Pi}_q( {\tt q}) : {\tt q}\in \mathbb{R} \},$$ where $$\hat{\Pi}_q( {\tt q}) \equiv |{\tt q}\rangle_q \langle {\tt q}|.$$ The momentum observable, denoted $\mathcal{O}_p$, is defined to be the PVM of projectors onto momentum eigenstates $$\mathcal{O}_p \equiv \{\hat{\Pi}_p( {\tt p}) : {\tt p} \in \mathbb{R} \},$$ where $$\hat{\Pi}_p( {\tt p}) \equiv |{\tt p}\rangle_p \langle {\tt p}|,$$ and the momentum eigenstates are related to the position eigenstates by a Fourier transform, $$| {\tt p} \rangle_p \equiv \frac{1}{2 \pi \hbar} \int_{\mathbb{R}} {\rm d}{\tt q} e^{i \frac{ {\tt qp}}{\hbar}} | {\tt q}\rangle_q.$$ Strictly speaking, one needs to make use of rigged Hilbert space to define position and momentum eigenstates rigorously but we will adopt the standard informal treatment of such states here.
In the discrete case, we can also define position and momentum observables in this way. A discrete position basis for $\mathbb{C}^d$ (which one can think of as the [*computational basis*]{} in a quantum information setting) can be chosen arbitrarily. Denoting this basis by $\{ |{\tt q}\rangle_q : {\tt q}\in \mathbb{Z}_d \}$, the PVM defining the position observable, denoted $\mathcal{O}_q$, is $$\mathcal{O}_q \equiv \{ \hat{\Pi}_q( {\tt q}) : {\tt q}\in \mathbb{Z}_d \},$$ where $\hat{\Pi}_q( {\tt q}) \equiv |{\tt q}\rangle_q \langle {\tt q}|$. We can define a discrete momentum basis, denoted $\{ |{\tt p}\rangle_p : {\tt p} \in \mathbb{Z}_d \}$, via a discrete Fourier transform, $$| {\tt p}\rangle_p \equiv \frac{1}{\sqrt {d}} \sum_{{\tt q} \in \mathbb{Z}_d} e^{i 2\pi \frac{ {\tt qp}}{d}} | {\tt q}\rangle.$$ and in terms of it, the PVM defining the momentum observable, $$\mathcal{O}_p \equiv \{ \hat{\Pi}_p( {\tt p}) : {\tt p} \in \mathbb{Z}_d \},$$ where $\hat{\Pi}_p( {\tt p}) \equiv |{\tt p}\rangle_p \langle {\tt p}|$. If one does not associate a Hermitian operator to each observable, then joint measurability of two observables can no longer be decided by the commutation of the associated Hermitian operators. Rather, it is determined by whether the associated PVMs commute or not, where two PVMs are said to commute if every projector in one commutes with every projector in the other.
To define the rest of the quadrature observables (and the commuting sets of these), we must first define a unitary representation of the symplectic affine transformations. We begin by specifying the unitaries that correspond to phase-space displacements. To do this in a uniform manner for discrete and continuous degrees of freedom, we define functions $\chi: \mathbb{R} \to \mathbb{C}$ and $\chi: \mathbb{Z}_d \to \mathbb{C}$ as $$\begin{aligned}
\chi({\tt c} ) &=& e^{i \frac{{\tt c}}{\hbar}} \textrm{ for } {\tt c}\in \mathbb{R}\nonumber \\
\chi({\tt c} ) &=& e^{i \frac{2\pi}{d} {\tt c}} \textrm{ for } {\tt c}\in \mathbb{Z}_d, \textrm{ when $d$ is an odd prime}\nonumber \\
\chi({\tt c} ) &=& e^{i \frac{\pi}{2} {\tt c}} \textrm{ for } {\tt c}\in \mathbb{Z}_d, \textrm{ when $d=2$.}\end{aligned}$$ In the continuous case, this is the standard exponential function; in the discrete case where $d$ is an odd prime, $\chi({\tt a})$ is the ${\tt a}$th power of the $d$th root of unity; in the discrete case where $d=2$, $\chi({\tt a})$ is the ${\tt a}$th power of the fourth (not the second) root of unity. In terms of this function, we can define a unitary that shifts the position by ${\tt q}$, where ${\tt q} \in \mathbb{R}$ in the continuous case and ${\tt q} \in \mathbb{Z}_d$ in the discrete case, as $$\begin{aligned}
\hat{S}({\tt q}) &= \sum_{{\tt p} \in \mathbb{R}/\mathbb{Z}_d} \chi({\tt q p}) |{\tt p}\rangle_{p} \langle {\tt p}|\nonumber\\
&= \sum_{{\tt q}' \in \mathbb{R}/\mathbb{Z}_d} |{\tt q}'-{\tt q}\rangle_q \langle {\tt q}'|
$$ and a unitary that boosts the momentum by ${\tt p}$, where ${\tt p} \in \mathbb{R}$ in the continuous case and ${\tt p} \in \mathbb{Z}_d$ in the discrete case, as $$\begin{aligned}
\hat{B}({\tt p}) &= \sum_{{\tt q} \in \mathbb{R}/\mathbb{Z}_d} \chi({\tt q p}) |{\tt q}\rangle_q \langle {\tt q}|\nonumber\\
&= \sum_{{\tt p}' \in \mathbb{R}/\mathbb{Z}_d} |{\tt p}'-{\tt p}\rangle_{p} \langle {\tt p}'|
$$ Note that the shift unitaries do not commute with the boost unitaries. The unitaries corresponding to phase-space displacements—typically called the *Weyl operators*—are proportional to products of these. In particular, the Weyl operator associated with the phase-space displacement vector ${\bf a} = ({\tt q},{\tt p}) \in \mathbb{R}^2/(\mathbb{Z}_d)^2$ is defined to be $$\label{def:Weyl}
{\hat W}({\bf a}) = \chi(2 {\tt pq}) \hat{S}({\tt q}) \hat{B}({\tt p}).$$ This is easily generalized to the case of a phase-space displacement for $n$ degrees of freedom, ${\bf a} = ({\tt q}_1,{\tt p}_1,\dots,{\tt q}_n,{\tt p}_n) \in \mathbb{R}^{2n}/(\mathbb{Z}_d)^{2n}$ via the tensor product, $$\label{generalWeyl}
{\hat W}({\bf a}) = \bigotimes_{i=1}^n \chi(2 {\tt p}_i {\tt q}_i) \hat{S}({\tt q}_i) \hat{B}({\tt p}_i).$$ For ${\bf a},{\bf a}' \in \Omega$, the product of the corresponding Weyl operators is $$\label{productWeyl}
{\hat W}({\bf a}){\hat W}({\bf a}')= \chi(2 \langle {\bf a}, {\bf a}'\rangle) {\hat W} ( {\bf a} + {\bf a}').$$ Thus it is clear that the Weyl operators constitute a projective unitary representation of the group of phase-space displacements ${\bf m}\to {\bf m}+{\bf a}$, where the composition law is $$\label{phasespacedispls}
(\cdot + {\bf a}) + {\bf a}' = \cdot + ({\bf a} +{\bf a}').$$
Next, we define a projective unitary representation $\hat{V}$ of the symplectic group acting on a $2n$-dimensional phase space $\Omega$. For every $2n \times 2n$ symplectic matrix $S : \Omega \to \Omega$, there is a unitary $\hat{V}(S)$ acting on the Hilbert space $\mathcal{L}(\mathbb{R}^n)/\mathbb{C}^{dn}$, such that $$\label{productsymp}
\hat{V}(S)\hat{V}(S')=e^{i\phi} \hat{V}(SS')$$ for some phase factor $e^{i \phi}$. These can be defined via their action on the Weyl operators. Specifically, $\forall {\bf a} \in \Omega$, $$\label{SympactiononWeyl}
\hat{V}(S) \hat{W}({\bf a}) \hat{V}^{\dag}(S) \propto \hat{W}(S {\bf a}).$$
In the following, we will often consider the action of these unitaries under conjugation, therefore, we define the superoperators associated to phase-space displacement ${\bf a}$ and symplectic matrix $S$, $$\begin{aligned}
\mathcal{W}({\bf a}) (\cdot) &\equiv \hat{W}({\bf a}) \cdot \hat{W}({\bf a})^{\dag},\nonumber \\
\mathcal{V}(S) (\cdot) &\equiv \hat{V}(S) \cdot \hat{V}(S)^{\dag}.\end{aligned}$$ Note that Eq. implies that $$\begin{aligned}
\mathcal{W}({\bf a}) \circ \mathcal{V}(S) (\cdot) =\mathcal{V}(S) \circ \mathcal{W}(S^{-1}{\bf a}) (\cdot).\end{aligned}$$ In the classical theory, every Poisson-commuting set of quadrature functionals $\{ f^{(1)}, f^{(2)}, \dots, f^{(k)} \}$ can be obtained from every other such set by a symplectic linear transformation (here, $k\le n$). The proof is as follows. If $f^{(i)}={\bf f^{(i)}}^T{\bf z}$ is a quadrature functional, then so is $\tilde{f}^{(i)}= (S{\bf f}^{(i)})^T {\bf z}$ for all $i \in \{ 1, \dots ,k\}$ when $S$ is a symplectic matrix. Furthermore, if the initial set is Poisson-commuting, then $\langle {\bf f}^{(i)}, {\bf f}^{(j)} \rangle =0$ for all $i \ne j \in \{ 1, \dots, k\}$, and then because $$\begin{aligned}
\langle {\bf \tilde{f}}^{(i)}, {\bf \tilde{f}}^{(j)} \rangle &= \langle S{\bf f}^{(i)}, S{\bf f}^{(j)} \rangle \nonumber\\
&= \langle {\bf f}^{(i)}, {\bf f}^{(j)} \rangle,\end{aligned}$$ it follows that $\langle {\bf \tilde{f}}^{(i)}, {\bf \tilde{f}}^{(j)} \rangle =0$ for all $i \ne j \in \{ 1, \dots, k\}$ so the final set is Poisson-commuting as well. Here, we have used the fact that the symplectic inner product is invariant under the action of a symplectic matrix.
We can define commuting sets of quantum quadrature *observables* similarly. Consider a single degree of freedom, $\Omega = \mathbb{R}^2/\mathbb{Z}_d^2$. Denote by $S_f$ the symplectic matrix that takes the position functional $q$ to a quadrature functional $f$, so that $S_f {\bf q} = {\bf f}$. (Given that ${\bf q} \equiv (1,0)$, we see that ${\bf f}$ is the first column of $S_f$.) We define the quadrature *observable* associated with $f$, denoted $\mathcal{O}_f$, to be the image under the action of the unitary $\hat{V}(S_f)$ of the position observable, that is, $$\mathcal{O}_f \equiv \{ \hat{\Pi}_f ({\tt f}): {\tt f} \in \mathbb{Z}_d \},$$ where $$\begin{aligned}
\hat{\Pi}_f ({\tt f}) &\equiv \mathcal{V}(S_f) (\hat{\Pi}_q ({\tt f}) ).
$$ It is useful to note how these projectors transform under phase-space displacements and symplectic matrices. By definition of the quadrature observables, we infer that for a symplectic matrix $S$, $$\label{effectsymp}
\mathcal{V}(S) (\hat{\Pi}_{f}({\tt f}) ) = \hat{\Pi}_{Sf}({\tt f}),$$ where $S f$ denotes the quadrature functional associated to the vector $S {\bf f}\in \Omega$. Now consider the action of a Weyl superoperator. First note that the projectors onto position eigenstates transform as $$\mathcal{W}({\bf a}) (\hat{\Pi}_q({\tt q}))=\hat{\Pi}_q({\tt q}+q({\bf a})).$$ It follows that if $f = S_f q$, then $$\begin{aligned}
\label{effectaffine}
\mathcal{W}({\bf a}) (\hat{\Pi}_f({\tt f}))&=\mathcal{W}({\bf a}) ( \hat{\Pi}_{S_f q}({\tt f}) ),\nonumber\\
&=\mathcal{W}({\bf a}) \mathcal{V}(S_f ) (\hat{\Pi}_q ({\tt f}) ),\nonumber\\
&=\mathcal{V}(S) \mathcal{W}(S_f^{-1}{\bf a}) (\hat{\Pi}_q ({\tt f}) ),\nonumber \\
&=\mathcal{V}(S) (\hat{\Pi}_q ({\tt f}+ q(S_f^{-1}{\bf a})) ),\nonumber \\
&=\hat{\Pi}_f({\tt f}+f({\bf a})).\end{aligned}$$
In all, $$\begin{aligned}
\label{effectaffine}
\mathcal{V}(S) \mathcal{W}({\bf a}) (\hat{\Pi}_f({\tt f}))&= \hat{\Pi}_{S f}({\tt f} + f({\bf a})).\end{aligned}$$
The case of $n$ degrees of freedom, $\Omega = \mathbb{R}^2/\mathbb{Z}_d^2$, is treated similarly. In this case, our quadrature observables need not be rank-1. Our fiducial quadrature can be taken to be $q_1$, the position functional for system 1. The associated quadrature observable is $$\mathcal{O}_{q_1} \equiv \{ \hat{\Pi}_{q_1} ({\tt q}_1) \otimes \mathbb{1}_2 \otimes \cdots \otimes \mathbb{1}_n : {\tt q}_1 \in \mathbb{R}/\mathbb{Z}_d \}.$$ For an arbitrary functional on the $n$ systems, $f:\Omega \to \mathbb{R}/\mathbb{Z}_d$, we find the symplectic matrix $S_f$ such that $S_f {\bf q}_1 = {\bf f}$, and we define the quadrature observable associated with $f$ to be $$\mathcal{O}_{f} \equiv \{ \hat{\Pi}_{f}({\tt f}) : {\tt f} \in \mathbb{R}/\mathbb{Z}_d \}.$$ where $$\hat{\Pi}_{f}({\tt f}) \equiv \hat{V}(S_f) \left( \hat{\Pi}_{q_1} ({\tt f}) \otimes \mathbb{1}_2 \otimes \cdots \otimes \mathbb{1}_n \right) \hat{V}(S_f)^{\dag}.$$ It follows that for every classical quadrature *functional* $f$, there is a corresponding quadrature *observable* $\mathcal{O}_f$, which stands in relation to the position and momentum observables as $f$ stands to the position and momentum functionals.
As an aside, one may note that in the continuous variable case, the quadrature observables are simply the spectral resolutions of those Hermitian operators that are linear combinations of position and momentum operators. In particular, for a quadrature observable $\mathcal{O}_f$ associated to a vector ${\bf f} \in \Omega$, the associated Hermitian operator is simply $$\hat{f} = {\bf f}^T {\bf \hat{z}},$$ where $$\label{vectorqpobservables}
{\bf \hat{z}} \equiv (\hat{q}_1 ,\hat{p}_1, \dots,\hat{q}_n,\hat{p}_n),$$ is the vector of position and momentum operators. Hence for every classical quadrature *variable* $f = {\bf f}^T {\bf z}$, as defined in Eq. , there is a corresponding quadrature *operator* $\hat{f}={\bf f}^T {\bf \hat{z}}$, where we have simply replaced the position and momentum functionals with their corresponding Hermitian operators.
We are now in a position to describe the commuting sets of quadrature observables. A set of quadrature observables $\{ \mathcal{O}_{f^{(1)}}, \dots, \mathcal{O}_{f^{(k)}} \}$ is a commuting set if and only if the corresponding quadrature functionals $\{ f^{(1)}, \dots, f^{(k)} \}$ are Poisson-commuting. The proof is as follows. The functionals $\{ f^{(1)}, \dots, f^{(k)} \}$ are Poisson-commuting if and only if they can be obtained by some symplectic transformation from any other such set, in particular, the set of position functionals for the first $k$ systems, $\{ q_1, \dots, q_k \}$. In other words, $\{ f^{(1)}, \dots, f^{(k)} \}$ are Poisson-commuting if and only if there is a sympectic matrix $S$ such that ${\bf f}^{(i)} = S{\bf q}_i$ for all $i \in \{1,\dots, k\}$ (which implies that the vectors ${\bf f}^{(i)}$ are the first $k$ columns of $S$). Given the definition of quadrature observables, this condition is equivalent to the statement that there exists a symplectic matrix $S$ such that $\mathcal{O}_{f^{(i)}} = \hat{V}(S) \mathcal{O}_{q_i} \hat{V}(S)^{\dag}$ for all $i \in \{1,\dots, k\}$. But given that the elements of the set $\{ \mathcal{O}_{q_1}, \dots, \mathcal{O}_{q_k} \}$ (the position observables for the first $k$ systems) commute, and commutation relations are preserved under a unitary, it follows that the elements of the set $\{ \mathcal{O}_{f^{(1)}}, \dots, \mathcal{O}_{f^{(k)}} \}$ commute if and only if there exists such an $S$, hence they commute if and only if the corresponding quadrature functionals $\{ f^{(1)}, \dots, f^{(k)} \}$ Poisson-commute.
Again, this has a simple interpretation in the continuous variable case. There, it is easy to verify that the matrix commutator of two quadratures operators is equal to the symplectic inner product of the corresponding vectors, that is, $[\hat{f},\hat{g}]= \langle {\bf f},{\bf g}\rangle.$ In particular, it follows that $[\hat{f},\hat{g}]=0$ if and only if $\langle {\bf f},{\bf g}\rangle=0$, which provides another proof of the fact that a commuting set of quadrature observables is associated with an isotropic subspace of the phase space.
As described in Sec. \[validepistemicstates\], every set of Poisson-commuting quadrature functionals defines an istropic subspace $V\subseteq \Omega$ and therefore the sets of commuting quadrature observables are also parameterized by the isotropic subspaces of $\Omega$. If a commuting set of quadrature observables is such that the corresponding quadrature functionals are associated with an isotropic subspace $V$, then this set defines a single quadrature observable, denoted $\mathcal{O}_V$, by $$\mathcal{O}_V = \{ \hat{\Pi}_{V}({\bf v}): {\bf v} \in V\}$$ where $$\label{pivv}
\hat{\Pi}_{V}({\bf v}) \equiv \prod_{{\bf f}^{(i)}: {\rm span}({\bf f}^{(i)})=V} \hat{\Pi}_{f^{(i)}}\left(f^{(i)}({\bf v})\right).$$ For instance, the quadrature functionals $f = q_1 - q_2$ and $g = p_1 +p_2$ are Poisson-commuting and therefore the associated quadrature observables, $\mathcal{O}_f$ and $\mathcal{O}_g$, commute, which is to say that the projectors $\{\hat{\Pi}_{f}({\tt f}) : {\tt f}\in \mathbb{R}/\mathbb{Z}_d \}$ all commute with the projectors $\{\hat{\Pi}_{g}({\tt g}) : {\tt g}\in \mathbb{R}/\mathbb{Z}_d \}$. If $V = {\rm span}\{{\bf f},{\bf g}\}$, then the possible pairs of values for the two observables can be expressed as the possible components of a vector ${\bf v} \in V$ along the basis vectors ${\bf f}$ and ${\bf g}$ respectively. These are the pairs $\{ ({\tt f}, {\tt g})\}$ such that ${\tt f} = {\bf f}^T{\bf v} = f({\bf v}) $ and ${\tt g} = {\bf g}^T{\bf v} = g({\bf v}) $ for some ${\bf v} \in V$. It follows that we can parametrize the possible values of this commuting set by vectors ${\bf v}\in V$.
In the continuous variable case, $\hat{\Pi}_f({\tt f})$ is the projector onto the eigenspace of $\hat{q}_1 - \hat{q}_2$ with eigenvalue ${\tt f}$, $\hat{\Pi}_g({\tt g})$ is the projector onto the eigenspace of $\hat{p}_1 + \hat{p}_2$ with eigenvalue ${\tt g}$, and $\hat{\Pi}_{V}({\bf v})$ is the projector onto the joint eigenspace of $\hat{q}_1 - \hat{q}_2$ and $\hat{p}_1 + \hat{p}_2$ with eigenvalues $({\tt f},{\tt g})$, which corresponds to an Einstein-Podolsky-Rosen entangled state.
With this background established, we are in a position to define the quadrature quantum subtheories.
Characterization of quadrature quantum subtheories
--------------------------------------------------
In this section, we define a quadrature subtheory of the quantum theory for a given system (discrete or continuous). In the discrete case, this subtheory is closely connected to the stabilizer formalism, a connection that we make precise in the appendix.
### The set of valid quantum states
In order to define the valid quantum states in the quadrature quantum subtheory, we use the guiding analogy of Sec. \[complementarity\], together with the isomorphism between quadrature functionals and quadrature observables noted above.
As we have just seen, for both the discrete and continuous cases, every commuting set of quadrature observables is associated to an isotropic subspace $V \subset \Omega$. Furthermore, every set of values that these observables can jointly take is associated to a vector ${\bf v} \in V$. We have denoted the projector that yields these values by $\hat{\Pi}_V({\bf v})$. The quantum states that are part of the quadrature subtheory, termed *quadrature states*, are simply the density operators that are proportional to such projectors. It follows that the quadrature states are parameterized by pairs consisting of an isotropic subspace $V$ and a valuation vector ${\bf v} \in V$ (in precisely the same way as one parametrizes the set of valid epistemic states in the epistricted classical theory). Specifically, it is the set of states of the form $$\label{qdelt}
\rho_{V,{\bf v}} = \frac{1}{\mathcal{N}_V } \hat{\Pi}_{V}({\bf v}),$$ where $V\subseteq \Omega$ is isotropic, ${\bf v} \in V$, and $\mathcal{N}_V$ is a normalization factor. Equivalently, if $\{ {\bf f}^{(i)} \}$ is a basis of $V$, then $$\label{qprod}
\rho_{V,{\bf v}} \equiv \frac{1}{\mathcal{N}_V } \prod_{\{ {\bf f}^{(i)} : {\rm span}\{ {\bf f}^{(i)} \} =V \} } \hat{\Pi}_{f^{(i)}} \left( f^{(i)}({\bf v}) \right).
$$
### The set of valid transformations
Because the overall phase of a Hilbert space vector is physically irrelevant, physical states are properly represented by density operators, and consequently a reversible physical transformation is not represented by a unitary operator but rather by the superoperator corresponding to conjugation by that unitary.
When a Weyl operator $\hat{W}({\bf a})$ acts by conjugation, it defines what we will call the *Weyl superoperator*, $$\mathcal{W}({\bf a})(\cdot) \equiv \hat{W}({\bf a}) (\cdot) \hat{W}({\bf a})^{\dag}.$$ Unlike the Weyl operators of two phase-space displacements, which, by Eq. , commute if and only if the corresponding phase-space displacement vectors have vanishing symplectic inner product, $$[ \hat{W}({\bf a}) , \hat{W}({\bf a}') ]=0 \;\textrm{ if and only if }\; \langle {\bf a}, {\bf a}'\rangle=0,$$ the Weyl superoperators of any two phase-space displacements necessarily commute, $$[\mathcal{W}({\bf a}) ,\mathcal{W}({\bf a}') ]=0\; \forall {\bf a},{\bf a}' \in \Omega.$$ This follows from Eq. and the skew-symmetry of the symplectic inner product. It follows that $$\mathcal{W}({\bf a}) \mathcal{W}({\bf a}') = \mathcal{W}({\bf a} + {\bf a}')\; \forall {\bf a},{\bf a}' \in \Omega.$$ As such, the Weyl superoperators constitute a nonprojective representation of the group of phase-space displacements, Eq. .
Next, we consider the projective unitary representation $\hat{V}$ of the symplectic group acting by conjugation. This defines a superoperator representation of the symplectic group which is nonprojective, that is, for $$\mathcal{V}(S)(\cdot) \equiv \hat{V}(S) (\cdot) \hat{V}(S)^{\dag},$$ we have $$\mathcal{V}(S)\mathcal{V}(S')=\mathcal{V}(SS').$$
The Clifford group of unitaries is defined as those which, when acting by conjugation, take the set of Weyl operators to itself. That is, a unitary $\hat{U}$ is in the Clifford group if $\forall {\bf b} \in \Omega$, $$\label{CliffordactiononWeyl}
\hat{U} \hat{W}({\bf b}) \hat{U}^{\dag} = c({\bf b}) \hat{W}(S {\bf b}),$$ for some maps $c : \Omega \to \mathbb{C}$ and $S: \Omega \to\Omega$.
It turns out that every such unitary can be written as a product of a Weyl operator and an element of the unitary projective representation of the symplectic group, that is, $$\hat{U}(S,{\bf a}) = \hat{W}({\bf a}) \hat{V}(S),$$ for some symplectic matrix $S: \Omega \to \Omega$ and phase-space vector ${\bf a} \in \Omega$. From Eqs. and we infer that a product of such unitaries is $$\label{Ucomposition}
\hat{U}(S,{\bf a})\hat{U}(S',{\bf a}')= e^{i\phi} \hat{U}(SS', S{\bf a}' + {\bf a} ).$$ for some phase factor $\phi$. Recalling Eq. , it is clear that the Clifford group of unitaries $\hat{U}(S,{\bf a})$ constitutes a projective representation of the symplectic affine group.
When a Clifford unitary $\hat{U}(S,{\bf a})$ acts by conjugation, it defines what we will call a *Clifford superoperator* $\mathcal{U}(S,{\bf a})(\cdot) \equiv \hat{U}(S,{\bf a}) (\cdot) \hat{U}(S,{\bf a})^{\dag}$. It follows that $$\mathcal{U}(S,{\bf b})\mathcal{U}(S',{\bf b}')= \mathcal{U}(SS', {\bf b} + S{\bf b}'),$$ and therefore, recalling Eq. , these form a nonprojective representation of the symplectic affine group.
The reversible transformations that are included in quadrature quantum mechanics are precisely those associated with Clifford superoperators. These map every quadrature state to another quadrature state.
The valid *irreversible* transformations in the quadrature subtheory are those that admit of a Stinespring dilation of the following form: the system is coupled to an ancilla of arbitrary dimension that is prepared in a quadrature state, the system and ancilla undergo a reversible transformation associated with a Clifford superoperator, and a partial trace operation is performed on the ancilla.
### The set of valid measurements
Finally, the reproducible measurements included in quadrature quantum mechanics are simply those associated with a commuting set of quadrature observables. Recall that these are parametrized by the isotropic subspaces $V \subset \Omega$, and correspond to PVMs of the form $\{ \hat{\Pi}_V({\bf v}) : {\bf v}\in V\}$, as defined in Eq. .
The most general measurement allowed is one whose Naimark extension can be achieved by preparing an ancilla in a quadrature state, coupling to the system via a Clifford superoperator, and finally measuring a commuting set of quadrature observables on the ancilla.
Comparing quantum subtheories to epistricted theories
=====================================================
Equivalence for continuous and odd-prime discrete cases
-------------------------------------------------------
The operational equivalence result is proven using the Wigner representation. The latter is a quasi-probability representation of quantum mechanics, wherein Hermitian operators on the Hilbert space are represented by real-valued functions on the corresponding classical phase space. For the case of $n$ continuous degrees of freedom, where the Hilbert space is $\mathcal{L}^2(\mathbb{R}^{n})$ and the phase space is $\mathbb{R}^{2n}$, the Wigner representation is a well-known formulation of quantum theory, particularly in the field of quantum optics [@wigner1932quantum; @gardiner2004quantum]. For the case of *discrete* degrees of freedom, there are many proposals for how to define a quasi-probability representation that is analogous to Wigner’s but for a discrete phase space. We here make use of a proposal due to Gross [@gross2006hudson], which is built on (but distinct from) a proposal by Wootters [@gibbons2004discrete]. For $n$ $d$-level systems (qudits), where $d$ is a prime, the phase space is taken to be $(\mathbb{Z}_d)^{2n}$.
We shall attempt to present the proof for the continuous case and for the odd-prime discrete case in a unified notation. Towards this end, we will provide a definition of the Wigner representation that is independent of the nature of the phase space. In the case of $\Omega = \mathbb{R}^{2n}$ and $\Omega = (\mathbb{Z}_d)^{2n}$ for $d$ an odd prime, our definition will reduce, respectively, to the standard Wigner representation and the discrete Wigner representation proposed by Gross [@gross2006hudson]. Marginalizing over the entire phase space $\Omega$ will be denoted by a sum over $\Omega$ in all of our expressions, which will be taken to represent a discrete sum in the discrete case and an integral with a phase-space invariant measure in the continuous case.
### Wigner representation of quantum theory
The Wigner representation of an operator $\hat{O}$, denoted $\hat{W}_{\hat{O}}({\bf m})$, can be understood as the components of that operator in a particular basis for the vector space of Hermitian operators where the inner product is the Hilbert-Schmidt inner product, $\langle \hat{O}, \hat{O}' \rangle \equiv {\rm tr}(\hat{O} \hat{O}')$. The elements of this operator basis are indexed by the elements of the phase space and termed the *phase-space point operators*. Denoting this operator basis by $\{ \hat{A}({\bf m}) : {\bf m}\in \Omega\}$, we have $$\hat{W}_{\hat{O}}({\bf m})=\mathrm{Tr}[\hat{O} \hat{A}({\bf m})]\,.$$
The phase-space point operators can be defined as the symplectic Fourier transform of the Weyl operators (which in turn are defined for both continuous and discrete degrees of freedom in Eq. ), $$\label{defnpointoperators}
\hat{A}({\bf m}) \equiv \frac{1}{\mathcal{N}_{\Omega}} \sum_{{\bf m}' \in \Omega} \chi(\langle {\bf m}, {\bf m}' \rangle) \hat{W}({\bf m}').$$ where $\mathcal{N}_{\Omega}$ is a normalization factor chosen to ensure that $${\rm Tr}[\hat{A}({\bf m})]=1.$$ The key property of the phase-space point operators is that they transform covariantly under symplectic affine transformations, $$\label{pointcovariance}
\mathcal{U}(S,{\bf a}) \left[ \hat{A}({\bf m}) \right] \propto \hat{A}(S{\bf m}+{\bf a}),$$ which can be inferred from Eq. and the manner in which the Weyl operators transform under the action of the Clifford superoperators, Eq. . This in turn implies that the Wigner representation of an operator also transforms covariantly under symmplectic affine transformations, $$\begin{aligned}
\hat{W}_{\mathcal{U}(S,{\bf a})(\hat{O})}({\bf m})&= {\rm tr}\left( \mathcal{U}(S,{\bf a})(\hat{O}) \hat{A}({\bf m}) \right)\nonumber\\
&= {\rm tr}\left(\hat{O}\; \mathcal{U}(S^{-1}, -{\bf a}) ( \hat{A}({\bf m}) ) \right)\nonumber \\
&= \hat{W}_{\hat{O}}(S^{-1}{\bf m}-{\bf a}).\end{aligned}$$
In both the discrete and continuous cases, we have $$\frac{1}{\mathcal{N}_{\Omega}} \sum_{{\bf m}\in \Omega} \chi(\langle {\bf m}, {\bf m}' \rangle) = \delta_{\bf 0}({\bf m}'),$$ where $\delta_{\bf 0}({\bf m}') = \prod_{i=1}^n \delta({\tt q}'_i)\delta({\tt p}'_i) $ for ${\bf m}' \equiv ({\tt q}'_1,{\tt p}'_1, \dots, {\tt q}'_n,{\tt p}'_n)$ and where $\delta$ denotes the Dirac-delta function in the continuous case and a Kronecker-delta in the discrete case. It then follows from Eq. that $$\begin{aligned}
\sum_{{\bf m}\in \Omega} \hat{A}({\bf m}) &=& \sum_{{\bf m}' \in \Omega} \delta({\bf m}') \hat{W}({\bf m}'),\nonumber\\
&=& \hat{W}({\bf 0}),\nonumber\\
&=& \mathbb{1}.\end{aligned}$$ Consequently the trace of an arbitrary operator is given by the normalization of the corresponding Wigner representation on the phase-space, $${\rm Tr}(\hat{O})= \sum_{{\bf m}\in \Omega} W_{\hat{O}}({\bf m}).$$
The phase-space point operators are Hermitian, and therefore the Wigner representation of any Hermitian operator is real-valued. They are orthogonal, $$\label{OrthogPoint}
{\rm Tr}\left( \hat{A}({\bf m}) \hat{A}({\bf m}') \right) \propto \delta({\bf m} - {\bf m}'),$$ and form a complete basis for the operator space relative to the Hilbert-Schmidt inner product, that is, for arbitrary $\hat{O}$, $$\sum_{{\bf m}\in \Omega} \hat{A}({\bf m}) {\rm Tr}\left( \hat{A}({\bf m}) \hat{O} \right) = \hat{O}.$$ It follows from this completeness that for any pair of Hermitian operators $\hat{O}$ and $\hat{O}'$, $$\label{eq:HSinnerproductinWigrep}
\mathrm{Tr}\left( \hat{O} \hat{O}'\right) = \sum_{{\bf m}\in \Omega} W_{\hat{O}}({\bf m}) W_{\hat{O}'} ({\bf m})\,.$$
The Wigner representation of a quantum state $\rho$ is the function $W_{\rho}: \Omega \to \mathbb{R}$ defined by $$W_{\rho}({\bf m}) = {\rm Tr}[\rho \hat{A}({\bf m})],$$ where the fact that ${\rm Tr}(\rho)=1$ implies $$\sum_{{\bf m} \in \Omega} W_{\rho}({\bf m}) = 1.$$
A superoperator $\mathcal{E}$ corresponding to the transformation $\rho \mapsto \mathcal{E}(\rho)$ can be modelled in the Wigner representation by a conditional quasiprobability function $W_{\mathcal{E}}({\bf m}'| {\bf m})$ such that $$W_{\rho}({\bf m}) \mapsto \sum_{{\bf m}'\in \Omega} W_{\mathcal{E}}({\bf m}| {\bf m}') W_{\rho}({\bf m}').$$ Specifically, the function $W_{\mathcal{E}}: \Omega \times \Omega \to \mathbb{R}$ is defined as $$\label{WignerSuperoperator}
W_{\mathcal{E}}({\bf m}| {\bf m}') = {\rm Tr}\left[ \hat{A}({\bf m}) \mathcal{E}\left(\hat{A}({\bf m}')\right) \right]$$ If $\mathcal{E}$ is trace-preserving, then $$\sum_{{\bf m}\in \Omega} W_{\mathcal{E}}({\bf m}| {\bf m}') = 1.$$
A sharp measurement with outcome set $K$, associated with a projector-valued measure $\mathcal{O} \equiv \{\hat{\Pi}_{\bf k}: {\bf k} \in K\}$ is represented by a conditional quasi-probability function $W_{\mathcal{O}} : K \times \Omega \to \mathbb{R}$ defined by $$\begin{aligned}
W_{\mathcal{O}}({\bf k}| {\bf m}) &= W_{\hat{\Pi}_{{\bf k}}}({\bf m}),\nonumber\\
&= {\rm Tr}[\hat{\Pi}_{{\bf k}} \hat{A}({\bf m})],
$$ where the fact that $\sum_{{\bf k} \in K} \hat{\Pi}_{\bf k} =\hat{\mathbb{1}}$ implies that $$\sum_{{\bf k} \in K} W_{\mathcal{O}}({\bf k}| {\bf m}) = 1.$$ Finally, we can infer from Eq. that the probability of obtaining outcome ${\bf k}$ in a measurement of $\{\hat{\Pi}_{\bf k}: {\bf k} \in K\}$ on the state $\rho$ is expressed in the Wigner representation as $$\label{kkk}
\mathrm{Tr}\left( \hat{\Pi}_{\bf k} \rho \right) = \sum_{{\bf m}\in \Omega} W_{\mathcal{O}}({\bf k}| {\bf m}) W_{\rho}({\bf m}) \,.$$ Similarly, if a transformation associated with the completely-positive trace-preserving map $\mathcal{E}$ acts between the preparation and the measurement, then the probability of obtaining outcome $k$ is expressed in the Wigner representation as $$\label{lll}
\mathrm{Tr}\left( \hat{\Pi}_{\bf k} \mathcal{E} (\rho) \right) = \sum_{{\bf m}\in \Omega} W_{\mathcal{O}}({\bf k}| {\bf m}) \sum_{{\bf m}'\in \Omega} W_{\mathcal{E}}({\bf m}| {\bf m}') W_{\rho}({\bf m}') \,.$$
Note that if $W_{\rho}({\bf m})$ is nonnegative, it can be interpreted as a probability distribution on phase-space. Similarly, if $W_{\mathcal{O}}({\bf k}| {\bf m}) $ and $W_{\mathcal{E}}({\bf m}| {\bf m}')$ are nonnegative, then can be interpreted as conditional probability distributions. In this case, Eqs. and for the probability of a measurement outcome can be understood as an application of the law of total probability, in analogy with Eqs. and . This sort of interpretation is indeed possible for the quadrature quantum subtheories and yields precisely the operational predictions of the quadrature epistricted theory. To show this, it remains only to show that the Wigner representation of the preparations, transformations and measurements of the quadrature subtheory are precisely equal to those of the quadrature epistricted theory.
### Wigner representation of the quadrature quantum subtheory
Our proof of equivalence relies on two of the defining features of the Wigner representation. First, the fact that the Wigner representation transforms covariantly under the symplectic affine transformations, Second, the fact that the Wigner representation of the projectors defining the position and momentum observables are the response functions associated to the position and momentum functionals in the classical theory, that is, $$\begin{aligned}
W_{\hat{\Pi}_{q_i}({\tt q}_i)}({\bf m}) &=& \delta(q_i({\bf m})-{\tt q}_i),\nonumber \\
W_{\hat{\Pi}_{p_j}({\tt p}_j)}({\bf m}) &=& \delta(p_j({\bf m})-{\tt p}_j).\end{aligned}$$
It follows from these facts that the Wigner representation of the projectors in the quadrature observable $\mathcal{O}_f$ are equal to the response functions associated with the corresponding quadrature functional $f$ in the classical theory, $$\begin{aligned}
\label{quadWig}
W_{\hat{\Pi}_{f}({\tt f})}({\bf m})
&=& W_{\hat{\Pi}_{S_f q_1}({\tt f})}({\bf m}),\nonumber\\
&=& W_{\mathcal{V}(S_f)(\hat{\Pi}_{q_1}({\tt f}))}({\bf m}),\nonumber\\
&=& W_{\hat{\Pi}_{q_1}({\tt f})}(S_f^{-1}{\bf m}),\nonumber\\
&=& \delta(q_1(S_f^{-1}{\bf m})-{\tt f}),\nonumber\\
&=& \delta((S_f q_1)({\bf m})-{\tt f}),\nonumber\\
&=& \delta(f({\bf m})-{\tt f})\nonumber\\
&=& \delta({\bf f}^T {\bf m}-{\tt f}).\end{aligned}$$
As noted previously, the sharp measurements that are included in the quadrature quantum subtheory are those associated to a set of commuting quadrature observables, $\{ \mathcal{O}_{f^{(i)}} \}$ which in turn is associated with a PVM $\mathcal{O}_{V'}\equiv \{ \Pi_{V',{\bf v'}}: {\bf v}' \in V'\}$ where $V' = {\rm span}\{ {\bf f}^{(i)} \}$. Given that $\hat{\Pi}_{V}({\bf v}) \equiv \prod_{\{{\bf f}^{(i)}: {\rm span}({\bf f}^{(i)})=V\}} \hat{\Pi}_{f^{(i)}}\left({\bf f}^{(i)T}{\bf v} \right)$ (Eq. ), and using Eq. , we conclude that $$\begin{aligned}
\label{Piexp}
W_{\hat{\Pi}_{V'}({\bf v}')}({\bf m})
&= \prod_{\{{\bf f}^{(i)}: {\rm span}({\bf f}^{(i)})=V\}} \delta({\bf f}^{(i)T}{\bf m}- {\bf f}^{(i)T}{\bf v}).\end{aligned}$$ Recalling Eq. , we see that the Wigner representation of the projector valued measure associated with $(V',{\bf v}')$ is the set of response functions associated with $(V',{\bf v}')$ in the quadrature epistricted theory, that is, $$W_{\mathcal{O}_{V'}}({\bf v}'|{\bf m})= \xi_{V'}({\bf v}'|{\bf m}).$$
The Wigner representation of the quadrature state associated with $(V,{\bf v})$ is $$\begin{aligned}
\label{mmm}
W_{\rho_{V,{\bf v}}}({\bf m}) &=&
{\rm Tr}\left(\rho_{V,{\bf v}} \hat{A}({\bf m}) \right) \nonumber\\
&=& \frac{1}{\mathcal{N}_V } \prod_{{\bf f}^{(i)} : {\rm span}\{{\bf f}^{(i)} \} =V} \delta ({\bf f}^{(i)T} {\bf m} - {\bf f}^{(i)T}{\bf v})
$$ where we have used Eqs. and . Recalling Eqs. and , we conclude that the Wigner representation of the quadrature state associated with $(V,{\bf v})$ is the epistemic state associated with $(V,{\bf v})$ in the quadrature epistricted theory, that is, $$W_{\rho_{V,{\bf v}}}({\bf m}) = \mu_{V,{\bf v}}({\bf m}).$$
The Wigner representation of the Clifford superoperator $\mathcal{U}(S,{\bf a})$ is $$\begin{aligned}
\label{proofequivtransf}
W_{\mathcal{U}(S,{\bf a})}({\bf m}|{\bf m}') &=&
{\rm Tr}\left[ \hat{A}({\bf m})\mathcal{U}(S,{\bf a})\left(\hat{A}({\bf m}')\right) \right]
\nonumber\\
&=& {\rm Tr}\left[ \hat{A}({\bf m})\hat{A}( S{\bf m}' + {\bf a}) \right]
\nonumber\\
&=&
\delta( {\bf m} -(S{\bf m}'+{\bf a})).
$$ Here, the first equality follows from the form of the Wigner representation of superoperators, Eq. . The second equality follows from the fact that the phase-space point operators transform covariantly under the action of the Clifford superoperators, Eq. . The third equality in Eq. follows from the orthogonality of the phase-space point operators, Eq. .
Recalling Eq. , we see that this is precisely the transition probability associated with the symplectic affine transformation, $\Gamma_{S,{\bf a}}({\bf m}|{\bf m}')$, in the quadrature epistricted theory, $$\begin{aligned}
\label{proofequivtransf}
W_{\mathcal{U}(S,{\bf a})}({\bf m}|{\bf m}')&=&
\Gamma_{S,{\bf a}}({\bf m}|{\bf m}').\end{aligned}$$
This concludes the proof of equivalence.
Inequivalence for bits/qubits
-----------------------------
In the case where $d=2$, the only even prime, the situation is more complicated. We have shown that in *both* the quadrature epistricted theory of bits and in the quadrature subtheory of qubits, we have: (i) the set of possible operational states is isomorphic to the set of pairs $(V,{\bf v})$ where $V$ is an isotropic subspace of the phase-space $\Omega = (\mathbb{Z}_2)^{2n}$ and ${\bf v}\in V$; (ii) the set of possible sharp measurements is isomorphic to the set of isotropic subspaces $V'$ (with the different outcomes associated to the elements ${\bf v}' \in V'$); (iii) the set of possible reversible transformations is isomorphic to the elements $(S,{\bf a})$ of the symplectic affine group acting on $\Omega = (\mathbb{Z}_2)^{2n}$. It follows that the operational states, measurements and transformations of one theory are respectively isomorphic to those of the other. The valid *unsharp* measurements and *irreversible* transformations are defined in terms of the sharp and reversible ones respectively, and they are defined *in the same way* in the quadrature epistricted theory and the quadrature quantum subtheory. It follows that we also have the unsharp measurements and irreversible transformations of one theory isomorphic to those of the other.
Despite this strong structural similiarly, the two theories nonetheless make different predictions. The particular algorithm that takes as input a triple of preparation, measurement and transformation and yields as output a probability distribution over measurement outcomes, is not equivalent in the two theories. More precisely, $${\rm Tr}\left(\hat{\Pi}_{V'}({\bf v}') \mathcal{U}_{S,{\bf a}} (\rho_{V,{\bf v}})\right) \nonumber\\
\ne \sum_{{\bf m} \in \Omega} \xi_{V'}({\bf v}' | {\bf m}) \sum_{{\bf m}' \in \Omega} \Gamma_{S,{\bf a}}({\bf m}|{\bf m}') \mu_{V,{\bf v}}({\bf m}').$$
Gross’s Wigner representation for discrete systems only works for systems of dimension $d$ for $d$ a power of an odd prime. It therefore does not work for $d=2$. Nonetheless, a Wigner representation can be constructed for the quadrature subtheory of qubits. One can define it in terms of tensor products of the phase-space point operators for a single qubit as proposed by Gibbons, Hoffman and Wootters [@gibbons2004discrete]. In this representation, we have $${\rm Tr}\left(\hat{\Pi}_{V'}({\bf v}')\; \mathcal{U}({S,{\bf a}}) \left(\rho_{V,{\bf v}}\right)\right) \nonumber\\
= \sum_{{\bf m} \in \Omega} W_{\mathcal{O}_{V'}}({\bf v}' | {\bf m}) \sum_{{\bf m}' \in \Omega} W_{\mathcal{U}(S,{\bf a})}({\bf m}|{\bf m}') W_{\rho_{V,{\bf v}}}({\bf m}').$$
So why can’t we identify the Wigner representations with the corresponding objects in the epistricted theory, just as we did for $d$ an odd prime and in the continuous case? The problem is that in the qubit case, the Wigner functions representing quadrature states sometimes go negative. It follows that these cannot be interpreted as probability distributions over the phase space. Similarly, the Wigner representations of quadrature observables and Clifford superoperators also sometimes go negative and hence cannot always be interpreted as conditional probability distributions.
It is also straightforward to prove that no alternative definition of the Wigner representation can achieve positivity. First, we make use of a fact shown in Ref. [@spekkens2008negativity], that if a set of preparations and measurements supports a proof of contextuality in the sense of Ref. [@spekkens2005contextuality], then *all* quasiprobability representations must necessarily involve negativity. It then suffices to note that the quadrature subtheory is contextual. There are many ways of seeing this. For instance, Mermin’s magic square proof of contextuality using two qubits [@mermin1993hidden] uses only the resources of the stabilizer theory of qubits. The same is true of the Greenberger-Horne-Zeilinger proof of nonlocality using three qubits [@greenberger1989going], which is also a proof of contextuality.
The quadrature subtheory of qubits simply makes different operational predictions than the quadrature epistricted theory of bits. It admits of contextuality and nonlocality proofs while the quadrature epistricted theory is local and noncontextual by construction.[^6]
By contrast, the quadrature subtheory of odd-prime qudits and the quadrature subtheory of mechanics make precisely the same predictions as the corresponding epistricted theories. They are consequently devoid of any contextuality or nonlocality because they admit of hidden variable models that are both local and noncontextual—the quadrature epistricted theory *is* the hidden variable model. Other differences between the two theories are discussed in Ref. [@spekkens2007evidence].
The conceptual significance of the difference between the quadrature subtheory of qubits and the epistricted theory of bits remains a puzzle, despite various formalizations of the difference [@pusey2012stabilizer; @coecke2011phase]. This puzzle is perhaps the most interesting product of these investigations.
It shows in particular that whatever conceptual innovation over the classical worldview is required to achieve the phenomenology of contextuality and nonlocality, it must be possible to make sense of this innovation even in the thin air of the quadrature subtheory of qubits. This is an advantage because the latter theory uses a more meagre palette of concepts than full quantum theory. For instance, we can conclude that it must be possible to describe the innovation of quantum over classical in terms of possibilistic inferences rather than probabilistic inferences.
I acknowledge Stephen Bartlett and Terry Rudolph for discussions on the quadrature subtheory of quantum mechanics, Jonathan Barrett for suggesting to define the Poisson bracket in the discrete case in terms of finite differences, and Giulio Chiribella and Joel Wallman for comments on a draft of this article. Much of the work presented here summarizes unpublished results obtained in collaboration with Olaf Schreiber. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
Quadrature quantum subtheories and the Stabilizer formalism
===========================================================
In quantum information theory, there has been a great deal of work on a particular quantum subtheory for discrete systems of prime dimension (qubits and qutrits in particular) which is known as the *stabilizer formalism* [@gottesman1998heisenberg; @gross2006hudson].
A stabilizer state is defined as a joint eigenstate of a set of commuting Weyl operators. By Eq. , two Weyl operators commute if and only if the corresponding phase-space displacement vectors have vanishing symplectic inner product, $$[ \hat{W}({\bf a}) , \hat{W}({\bf a}') ]=0 \;\textrm{ if and only if }\; \langle {\bf a}, {\bf a}'\rangle=0.$$ Consequently, the sets of commuting Weyl operators, and therefore the stabilizer states, are parametrized by the isotropic subspaces of $\Omega$. Specifically, for each isotropic subspace $M$ of $\Omega$ and each vector ${\bf v}\in JM\equiv \{ J{\bf u} : {\bf u} \in M \}$, we can define a stabilizer state $\rho^{(\rm stab)}_{M,{\bf v}}$ as the projector onto the joint eigenspace of $\{ \hat{W}({\bf a}): {\bf a} \in M\}$ where $\hat{W}({\bf a})$ has eigenvalue $\chi(\langle {\bf v}, {\bf a} \rangle)$.
We will show here that the set of stabilizer states is precisely equivalent to the set of quadrature states.
To describe the connection, it is convenient to introduce some additional notions from symplectic geometry. The *symplectic complement* of a subspace $V$, which we will denote as $V^C$, is the set of vectors that have vanishing symplectic inner product with every vector in $V$, $$V^{C}\equiv \{ {\bf m}' \in \Omega : {\bf m'}^T J {\bf m} =0, \; \forall {\bf m} \in V \},$$ where $J$ is the symplectic form, defined in Eq. . This is not equivalent to the Euclidean complement of a subspace $V$, which is the set of vectors that have vanishing Euclidean inner product with every vector in $V$, $$V^{\perp}\equiv \{ {\bf m}' \in \Omega : {\bf m'}^T {\bf m} =0, \; \forall {\bf m} \in V \},$$ The composition of the two complements will be relevant in what follows. It turns out that the latter is related to $V$ by an isomorphism; it is simply the image of $V$ under left-multiplication by the symplectic form $J$, $$(V^{\perp})^{C}\equiv JV = \{ J{\bf u} : {\bf u} \in V \}.$$ Note that if $V$ is isotropic, then $(V^{\perp})^{C}$ is as well.
\[equivstabquad\] Consider the quadrature state $\rho_{V,{\bf v}}$, with $V$ an isotropic subspace of $\Omega$ and ${\bf v}\in V$ a valuation vector, which is the joint eigenstate of the commuting set of quadrature observables $\{ \mathcal{O}_{f} : {\bf f}\in V \}$, where the eigenvalue of $\mathcal{O}_f$ is $f({\bf v})$. This is equivalent to the stabilizer state $\rho^{({\rm stab})}_{M,{\bf v}}$, which is the joint eigenstate of the commuting set of Weyl operators $\{ \hat{W}({\bf a}) : {\bf a} \in M\}$ where $M \equiv (V^{\perp})^C$ is the isotropic subspace that is the symplectic complement of the Euclidean complement of $V$, and where the eigenvalue of $\hat{W}({\bf a})$ is $\chi(\langle {\bf v},{\bf a} \rangle)$.
Consider first a single degree of freedom. Every quadrature observable $\mathcal{O}_{f}$ can be expressed in terms of the position observable $\mathcal{O}_q$ as follows: if $S_f$ is the symplectic matrix such that ${\bf f} = S_f {\bf q}$, then $\mathcal{O}_{f} = \hat{V}(S_f) \mathcal{O}_{q} \hat{V}(S_f)^{\dag}$. Now note that the position basis can equally well be characterized as the eigenstates of the boost operators. Specifically, $\hat{B}({\tt p}) |{\tt q}\rangle_q = \chi({\tt q p}) |{\tt q}\rangle_q$, that is, an element $|{\tt q}\rangle_q$ of the position basis is an eigenstate of the set of operators $\{ \hat{B}({\tt p}): {\tt p}\in \mathbb{R}/\mathbb{Z}_d \}$ where the eigenvalue of $\hat{B}({\tt p})$ is $\chi({\tt qp})$. The element $|{\tt f}\rangle_f$ of the basis associated to the quadrature operator $\mathcal{O}_f$ is defined as $|{\tt f}\rangle_f \equiv \hat{V}(S_f)|{\tt f}\rangle_q$ and consequently can be characterized as an eigenstate of the set of operators $\{ \hat{V}(S_f) \hat{B}({\tt g}) \hat{V}(S_f)^{\dag}: {\tt g}\in \mathbb{R}/\mathbb{Z}_d \}$ where the eigenvalue of $\hat{V}(S_f) \hat{B}({\tt g}) \hat{V}(S_f)^{\dag}$ is $\chi({\tt fg})$. This can be stated equivalently as follows: the element $|{\tt f}\rangle_f$ of the basis associated to the quadrature operator $\mathcal{O}_f$ is the eigenstate of the set of Weyl operators $\{ \hat{W}({\bf a}): {\bf a}\in {\rm span}(S_f{\bf p}) \}$ where the eigenvalue of $\hat{W}({\bf a})$ is $\chi( {\tt f}\langle {\bf f},{\bf a}\rangle)$. Noting that $$\begin{aligned}
{\rm span}(S_f{\bf p})&={\rm span}(S_f J{\bf q}),\nonumber\\
&= {\rm span}(J S_f {\bf q}),\nonumber\\
&= {\rm span}(J {\bf f}),\end{aligned}$$ we can just as well characterize $\hat{\Pi}_f({\tt f})$ as the projector onto the joint eigenspace of the Weyl operators $\{ \hat{W}({\bf a}): {\bf a}\in {\rm span}(J {\bf f}) \}$.
Now consider $n$ degrees of freedom. The quadrature state associated with $(V,{\bf v})$ has the form $$\label{pivv}
\rho_{V,{\bf v}}
= \frac{1}{\mathcal{N}} \prod_{{\bf f}^{(i)}: {\rm span}({\bf f}^{(i)})=V} \hat{\Pi}_{f^{(i)}}\left(f^{(i)}({\bf v})\right).$$ By an argument similar to that used for a single degree of freedom, this is an eigenstate of the Weyl operators $\{ \hat{W}({\bf a}): {\bf a}\in {\rm span}(J {\bf f}^{(i)}) \}$ where the eigenvalue of $\hat{W}({\bf a})$ is $\chi( \langle {\bf v}, {\bf a} \rangle)$. Noting that ${\rm span}(J {\bf f}^{(i)}) = J V = (V^{\perp})^C$, we have our desired isomorphism.
The stabilizer formalism allows all and only the Clifford superoperators as reversible transformations. The sharp measurements that are included in the stabilizer formalism are the ones associated with PVMs corresponding to the joint eigenspaces of a set of commuting Weyl operators, which, by proposition \[equivstabquad\], are precisely those corresponding to the joint eigenspaces of a set of commuting quadrature observables. It follows that the stabilizer formalism coincides precisely with the quadrature subtheory.
Gross has argued that the discrete analogue of the Gaussian quantum subtheory for continuous variable systems is the stabilizer formalism [@gross2006hudson]. Our results show that the connection between the discrete and continuous variable cases is a bit more subtle than this. In the continuous variable case, there is a distinction between the Gaussian subtheory and the quadrature subtheory, with the latter being contained within the former. In the discrete case, there is no distinction, so the stabilizer formalism can be usefully viewed as either the discrete analogue of the Gaussian subtheory or as the discrete analogue of the quadrature subtheory. While Gross’s work showed that the stabilizer formalism for discrete systems could be defined similarly to how one defines Gaussian quantum mechanics, our work has shown that it can also be defined in the same way that one defines quadrature quantum mechanics.
To our knowledge, quadrature quantum mechanics has not previously received much attention. However, given that it is a natural continuous variable analogue of the stabilizer formalism for discrete systems, it may provide an interesting paradigm for exploring quantum information processing with continuous variable systems.
[^1]: We are here refering to the first part of Ref. [@van2007toy]. In the second part, the author proposes a theory wherein there is a restriction on what can be known about the outcome of measurements, rather than a restriction on what can be known about some underlying ontic state. As such, the latter theory is not an epistricted theory.
[^2]: Note that for the purposes of this article, the term quantum theory refers to a theory schema that can be applied to many different degrees of freedom: particles, fields and discrete systems.
[^3]: Note that the theory of *stochastic electrodynamics* has some significant similarities to an epistricted theory of electrodynamics, but there are also significant differences. Many authors who describe themselves as working on stochastic electrodynamics posit a *nondeterministic* dynamical law for the fields, whereas an epistricted theory of electrodynamics is one wherein agents merely lack knowledge of the electrodynamic fields, which continue to evolve deterministically. That being said, Boyer’s version of stochastic electrodynamics [@Boy80] does not posit any modification of the dynamical law and so is closer to what we are imagining here. A second difference is that in stochastic electrodynamics, there is no epistemic restriction on the matter degrees of freedom. However, if one degree of freedom can interact with another, then to enforce an epistemic restriction on one, it is necessary to enforce a similar epistemic restriction on the other. In other words, the assumptions of stochastic electrodynamics were inconsistent. The sort of epistricted theory of electrodynamics we propose here is one that would apply the epistemic restriction to the matter and to the fields.
[^4]: It should be noted that many researchers had previously recognized the possibility of recovering many of these quantum phenomena if one compared quantum states to probability distributions in a classical statistical theory [@caves9601025quantum; @emerson; @hardy1999disentangling; @kirkpatrick2003quantal]
[^5]: This terminology comes from optics, where it was originally used to describe a pair of variables that are canonically conjugate to one another. It was inherited from the use of the expression in astronomy, where it applies to a pair of celestial bodies and describes the configuration in which they have an angular separation of 90 degrees as seen from the earth.
[^6]: It seems that the quadrature epistricted theory of bits is about as close as one can get to the stabilizer theory for qubits while still being local and noncontextual.
|
---
abstract: 'This article proposes a novel collective decision making scheme to solve the multi-agent drift-diffusion-model problem with the help of spiking neural networks. The exponential integrate-and-fire model is used here to capture the individual dynamics of each agent in the system, and we name this new model as Exponential Decision Making (EDM) model. We demonstrate analytically and experimentally that the gating variable for instantaneous activation follows Boltzmann probability distribution, and the collective system reaches meta-stable critical states under the Markov chain premises. With mean field analysis, we derive the global criticality from local dynamics and achieve a power law distribution. Critical behavior of EDM exhibits the convergence dynamics of Boltzmann distribution, and we conclude that the EDM model inherits the property of self-organized criticality, that the system will eventually evolve toward criticality.'
author:
- 'Yanlin Zhou, Chen Peng, and Qing Hui[^1][^2]'
bibliography:
- 'IEEEabrv.bib'
- 'library.bib'
title: '**A Spiking Neural Dynamical Drift-Diffusion Model on Collective Decision Making with Self-Organized Criticality** '
---
Introduction {#sec:intro}
============
Self-Organized Criticality (SOC), a ground breaking achievement of statistical physics, has gained growing interest in neural firing and brain activity in recent years [@soc_fundm]. Bak’s hypothesis [@bak_1996] and recent studies [@soc_fundm; @Neurobiologically_SOC_Rubinov] suggest that criticality is evolutionarily chosen for optimal computational and fast reactionary purposes, and that the brain is always balanced precariously at the critical point.
Such critical dynamics emerge during the phase transition between randomness (sub-critical) and order (super-critical), and usually follow power-law distributed spatial and temporal properties [@Neurobiologically_SOC_Rubinov]. A dynamic network system with the SOC behavior then has the potential to be spatial and/or temporal scale free [@Watkins2016] and to fast switch between phases and attain optimal computational capability. This offers a possible approach to model a decision making process.
The systems that exhibit SOC behavior are usually high dimensional and slowly driven, with nonlinearity properties [@bak_1996; @Watkins2016]. To this end, Brochini *et al* [@phase_2016] have discussed the phase transitions and SOC in stochastic spiking neural networks. Also, Bogacz *et al* [@formal_ddm] demonstrated that standard Drift Diffusion Model (DDM) can be used for stochastic spiking dynamics and they relate DDM to a highly interactive “pooled inhibition" model.
However, to authors’ best knowledge, although the SOC has been recognized as a fundamental property of neural systems [@soc_fundm], there has yet to be a decision making model capitalizing the SOC property.
In this paper, we incorporate the nonlinearity of Exponential Integrate-and-Fire (EIF) model to replace the stochastic spiking scheme in DDM proposed by [@formal_ddm]. Introduced by Fourcaud-Trocme *et al* [@eif_fourcaud], the nonlinear EIF model is experimentally verified to be able to accurately capture the response properties. It will be demonstrated that neural sampling and mean field branching can be derived with the Boltzmann distribution. The proposed Exponential Decision Making (EDM) model therefore reaches a set of absorbing states, and the corresponding global criticality follows power law distribution, thus attaining the SOC behavior.
The contributions of this paper are summarized as follows.
1. To the best of authors’ knowledge, this paper is the first published work on modeling decision making processes with the SOC property.
2. A collective decision making model, i.e., EDM, is proposed to implement the DDM methodology on EIF spiking neurons.
3. A probability inference scheme on EIF sampling is proposed, which extends an existing leaky integrate-and-fire sampling method.
4. Mean field analysis of the connectivity of EDM is given, which exhibits global criticality.
5. Detailed analysis is given to reveal SOC behavior of the EDM model under criticality conditions.
This paper is organized as follows. In Section \[sec:model\], literature review is given and the necessary background concepts are introduced. Then, in Section \[sec:dm\_dynam\], the dynamics of each agent as well as the EIF neural sampling are discussed. In Section \[sec:coll\_beh\], the collective behavior of the network system is analyzed. Section \[sec:converge\] provides convergence analysis and simulation results. Section \[sec:conclusions\] concludes the paper and looks into future work.
{width="48.00000%"}
[\[fig:relation\]]{}
Notation and Preliminaries {#notation-and-preliminaries .unnumbered}
--------------------------
Here we use a classical directed graph representation $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ with nonempty finite number of nodes and edges. Specifically, $\mathcal{V}$ is the set of nodes, $\mathcal{E}$ is the set of directed edges, and $\mathcal{A}=[a_{ij}]$ is an adjacency matrix with weights $a_{ij} > 0$ if, $\left<i,j\right>\in\mathcal{E}$, an edge from node $i$ to node $j$. Note that the assumed graph is simple, i.e., $\left<i,i\right>\notin\mathcal{E}$, $\forall i$, with no multiple edges going in the same direction between the same pair of nodes and no self loop. In this case, the diagonal elements of $\mathcal{A}$ are zero. In addition, the Laplacian matrix of $\mathcal{G}$ is denoted by $L$.
Backgrounds and Related Information {#sec:model}
===================================
Self-Organized Criticality {#ssec:socintro}
--------------------------
Self-Organized Criticality describes a self-tuned internal interactions that show critical dynamics in complex systems [@bak_1996]. The interacting node groups are called active sites while the nodes that are less sensitive to the input are called inactive sites. Usually in the sand pile model, each agent has their own steep slope, which represents the membrane potential of the spiking neurons. When a certain threshold is hit and the sand in that specific area is steep enough $Z_{local} > Z_{critical}$, the avalanches will be triggered, which follow a power-law distribution of $1/f$ noise. Plenz and Beggs [@Beggs11167] observed a similar pattern of avalanches in the cortical neural electrical activity, which was the first evidence that the brain functions at criticality.
For the spiking neural network sense, if an agents activates too many neighboring neurons (super-critical), it leads to the massive activation of the entire network, while if too few neurons are activated (sub-critical), propagation dies out too fast [@formal_ddm].
In our case, the local rigidity level is expressed by two terms: the firing threshold of each agent, and the correlation between each two agents in the same local active sites.
Boltzmann Machine {#ssec:bmintro}
-----------------
Boltzmann Machines (BM) is a special type of stochastic recurrent neural networks based on non-stochastic Hopfield nets. In recent years, BM’s property of binary output attracts more attention in both the theoretical neuroscience and the high dimensional parallel stochastic computation [@bm_basic; @neural_sampling_2011].
BM is proven to be efficient for the models with connectivity properly constrained, to be specific, machine learning and probability inference are two major applications. In this paper, we apply neural sampling and show that the probability function of the gating current variable for the activation term follows the Boltzmann functions.
The global energy function of the Boltzmann Machine is defined as: $
E = - \sum_{i<j}h_{ij} s_i s_j - \sum_{i}b_i s_i
$ , where $E$ is the global energy, $h_{ij}$ is the connection strength between Unit $j$ and Unit $i$, $s_i\in \{0,1\}$ is the state of Unit $i$, and $b_i$ is the bias of Unit $i$.
As for the Boltzmann distribution, the probability that the $i$th unit is on is $$\label{eq:bmprobdist}
P_{i=on} = \frac{1}{1+\exp(-\triangle E_i/T)},$$ where $T$ is the temperature of the system.
The probability is calculated using only the information of the energy difference from the initial state, and the temperature of the current time. In our framework, we consider the term $-\triangle E_i/T$ as a logistic activation function as in [@Barranca2014; @richardson_2009]. It has already been shown in [@richardson_2009] that some stochastic neurons sample from a Boltzmann distribution. Ideally, after a long running period and without further inputs, the probability of a global state will not be affected by other terms, i.e., time constants and conductance values in the EIF model. At this stage, the system is at its “thermal equilibrium", that converges to a low temperature distribution where the energy level hovers around a global minimum.
This feature presents a similar behavior as SOC if we consider the criticality as this thermal equilibrium that energy level fluctuates around. Also, the log-probabilities eventually becomes a linear term, which helps us simplify the exponential term in the EIF model. Further discussion will be given in later sections.
Moreover, the neural sampling technique in the later section incorporates the Boltzmann machine according to some local switching, with conditional probability integrated, the multivariable Boltzmann joint distribution has the form [@noise_resource] $$\label{eq:boltzmenergyt}
P_m=\dfrac{e^{-\epsilon_m/kT}}{\sum_{n=1}^M e^{-\epsilon_j/kT}},$$ where $P_m$ stands for the probability of state $m$, $\epsilon_m$ is the energy at state $m$, $k$ is a constant, and $M$ is the total number of the states.
DDM {#ssec:ddm}
---
Drift Diffusion Model has been applied on Two-Alternative Forced-Choice (TAFC) task in an extensive amount of work (see [@formal_ddm; @ddm_naomi] for instance). The fact that DDM integrates the difference between two choices according to one or two threshold makes it possible to describe a decision making process in a spiking neural network.
In the pure DDM, the accumulation of the unbiased evidence has the form $$\label{eq:pureddm}
dx = g dt + \beta dw , ~x(0) = 0,$$ where $dx$ represents the changes in difference over the time interval $dt$, $g$ is the increase in evidence supporting the correct choice each time, $w$ is the independent, identically distributed (i.i.d.) Wiener processes, and $\beta$ is the standard deviation. The probability density $P(x,t)$ is normally distributed with mean $g t$ and standard deviation $\beta\sqrt{t}$.
Since the second term in (\[eq:pureddm\]) is represented by a standard Wiener process that describes the noise, it is common to consider $dx$ in DDM to be the change in membrane potential within a certain amount of time [@stochastic_diffusion].
Here we consider a network system with $N$ agents. Each agent relates the DDM model to the non-stationary dynamics of the firing of an EIF spiking neural model. While the forced-response protocol is usually considered, we follow the free-response protocol, that each consecutive fires determine the range of the time interval. The common assumption made for this equation usually considers $g>0$ to support the first choice, and $g<0$ for the other [@formal_ddm]. The term $g$ can either be a constant for inactive nodes, or a function for active nodes that depends on membrane potential.
While (\[eq:pureddm\]) only describes the dynamics of a single DDM system, we need extra terms to capture the impact from neighbors. We have the following stochastic diffusion process with an initial condition $x_0$: $$\label{eq:voltage_ddm}
dx = \big( \alpha(x(t),t)(x(t)-x_0) +g(x(t),t) \big) dt + \beta(x(t),t) dw,$$ where $\alpha(t)$ is a measurable gain function that models the external input to accelerate the potential increment, and linear drifting term $g(t)$ represents the dynamic drifting variable of the node itself. In this regime, we have transferred our model to a Ornstein-Uhlenbeck (OU) process, which is known to be the solution to the famous Fokker-Planck equation. Here, to further simplify the model, we may eliminate the afterhyperpolarization, that is, let $x_0 = 0$.
For the model proposed above, it is possible to receive the spike generations with arbitrary shape, i.e., different spiking time intervals and different incremental speed of membrane potential. With a proper defined activation function, which will be discussed in Section \[sec:dm\_dynam\], the behavior of each single stochastic diffusion process can be bounded. Before further discussing into the individual dynamics and their boundedness properties, we need to look into the specific local dynamics by applying a most commonly used neuron model.
Generalized Exponential Integrate-and-Fire (EIF) Model {#ssec:EIF}
------------------------------------------------------
Exponential integrate-and-fire (EIF) is a well developed biological neuron model introduced by Fourcaud-Trocme *et al* [@eif_fourcaud] as an extension of the standard leaky integrate-and-fire model. As concluded in several studies [@Barranca2014; @richardson_2009], EIF is a suitable simple model for very large scale network simulations. For the generalized EIF [@richardson_2009], arbitrary spike shapes are allowed and gated currents usually reach a steady-state with nonlinear voltage activation function.
The EIF model holds a nonlinearity property consisting of a linear leakage term combined with an exponential activation term, which follows a simple RC-circuit dynamics before $V$, the membrane potential, reaches a set threshold $V_T$. After reaching the threshold, it can be considered that the neuron has fired, and its membrane potential is then set to a resting voltage, $V_R$, approximately $-60$ mV [@Barranca2014; @richardson_2009] or $0$ mV [@stochastic_diffusion] by different assumptions.
The dynamics of the membrane potential is given by $$\label{eq:eif}
C \dfrac{dV}{dt} = -\varrho_L(V-V_L)+ \varrho_T\Delta_T \exp \bigg( \dfrac{V-V_T}{\Delta——T} \bigg)+I_{ion}.$$ In this equation, $C$ is the membrane capacitance, $V_T$ is the membrane potential threshold, $\Delta_T$ is the sharpness of action potential initiation, or slope factor, $V_L$ is the leak reversal potential, $\varrho$ is the conductance, and $I_{ion}$ is input current. While $I_{ion}$ only represents synaptic current in Fourcaud’s model, here we have extended the ionic current by summing up input current $I_{neib}$ from neighbors with connectivity, external noise current $I_{noise}$ that integrates i.i.d. Wiener process, and synaptic input $I_{syn}$ that incorporates the drifting term, serving as a bias. We have $I_{ion}=I_{neib}+ I_{noise}+I_{syn}$, where the term $I_{neib}$ takes identical form as leakage current in the first term of (\[eq:eif\]), while $I_{syn}=\varrho_{syn}\Gamma(V-V_{syn})$ represents the slow voltage activated current with a gating variable $\Gamma=\Gamma(t)$. The term $\Gamma_\infty=\lim_{t\to\infty}\Gamma(t)$ can be used to describe the instantaneous activation.
Here, we alter the usually constant conductance $\varrho$, and change it to a function of $V$ and $t$. Multiplying $dt$ to both sides of the (\[eq:eif\]), we now have $$\begin{aligned}
\label{eq:eif_xdt}
C dV =& -\varrho_L(V-V_L){dt}+ \varrho_L\Delta_T \exp \bigg( \dfrac{V-V_T}{\Delta——T} \bigg){dt} \nonumber\\
&+ \varrho_{syn}\Gamma(V,t)(V-V_{syn}){dt} + \varrho_{neib}(V-V_{noise}){dt} \nonumber\\
&+ I_{noise}dt.
\end{aligned}$$
It is clear that most terms in (\[eq:eif\_xdt\]) have very similar forms to those in (\[eq:voltage\_ddm\]). As most current terms do not need to be altered to fit in (\[eq:voltage\_ddm\]), the conductance term can change over time and become a function. Functions $\varrho$ and $\alpha$ are sometimes interchangeable. However, the exponential term can be tricky to work around, and we will talk about it in the later part after done discussing absorbing state.
Henceforth, for simplicity purpose, we call this EIF and DDM combined model as Exponential Decision Model, or EDM for short.
Decision Making Dynamics {#sec:dm_dynam}
========================
The commonly discussed decision making process is an adaptive behavior that makes use of a series of external input variables and then leads to an optimal or sub-optimal choice of action over other competing alternatives.
We begin by discussing this optimal decision rule. There are two thresholds $z_i$ in DDM model, with the same magnitude but different signs to represent different choices. In EDM, we consider this choice to be optimal if the threshold of correct choice is reached, or sub-optimal, if our expectation value $\mathbb{E}$ ends up with the same sign as correct threshold but with smaller magnitude.
Now we start solving the OU process described in (\[eq:voltage\_ddm\]).
\[thm:sol\_ou\] The solution of the collective decision making system in (\[eq:voltage\_ddm\]) is given by $$\label{eq:xsol_ref}
x = e^{\alpha(t)}\bigg(c+\int^t_{0}e^{-\alpha(t-\eta)}g(\eta)d\eta+\int^t_{0}e^{-\alpha(t-\eta)}\beta(\eta)dw_{\eta} \bigg)$$ which has a similar form as in [@arnold1974stochastic], with the updated expectation $$\label{eq:xsol_mean}
\mathbb{E}(x(t)) = \int^t_{0}e^{-\alpha(t-\eta)}g(s)ds.$$
Let $\phi(t)$ be a fundamental solution matrix.
Also, let $Y$ be $$c+\int^t_{0}\phi(\eta)^{-1}g(\eta)d\eta + \int^t_{0}\phi(\eta)^{-1}\beta(\eta)dw_{\eta}.$$ Then $Y$ has the stochastic differential equation $$dY = \phi(t)^{-1} \big( g(t)dt+\beta(t)dw_t \big),$$ which implies $$\begin{aligned}
x &= \phi(t) Y \\
&= \phi(t)\bigg(c+\int^t_{0}\phi(\eta)^{-1}g(\eta)d\eta + \int^t_{0}\phi(\eta)^{-1}\beta(\eta)dw_{\eta} \bigg).
\end{aligned}$$ Furthermore, combining all above together, it is straightforward to conclude (\[eq:xsol\_ref\]).
Behaviors {#ssec:each_behav}
---------
Here we consider a very special but common network system.
\[brownagents\] For a high dimensional dynamical network system with $N$ agents described by (\[eq:voltage\_ddm\]), each unit integrates an inward stimulus $\alpha_i$ and receives signals $I_{neib}$ and noise $\beta_idw_i^j$ from local $j$ neighbors. The dynamics of $w^j$ follows Correlated Brownian motions, with standard correlation $col \in (-1,1)$.
In Reference [@tao_li], the authors proposed a Itô consensus S.D.E formalized by $N^2$-dimensional standard white noise, and expanded the right hand side terms in (\[eq:pureddm\]) to a matrix form with graph theory, i.e., $\alpha(t)Lx(t)$. Different from their encoded gain function, we now assume that all the state information is available to others.
For the considered network system with $N$ agents, the state of each agent is observable by others.
Extending (\[eq:voltage\_ddm\]), then we have the dynamic equation for each agent with neighbor’s dynamics added $$\begin{gathered}
\label{eq:single_net_ddm}
dx_i = \bigg( \sum_{j=1}^K \alpha(x_i(t),t)l_{ij}(y_{ji}-x_i(t))+ g(x_i(t),t) \bigg) dt \\ + \beta(x_i(t),t) dw,
\end{gathered}$$ where $K$ is the total number of agents connecting the agent $i$, $l_{ij}$ are elements in the Laplacian matrix $L$, $y_{ji}$ denotes the observed membrane potential of $j$th agent by the $i$th agent.
In Reference [@girsanov], Charalambos *et al* have shown a method of achieving optimization problems under a reference probability measure by transferring continuous and discrete-time stochastic dynamic decision systems, via Girsanov’s Measure Transformation. To this end, we have the following claim.
\[prop:equi\] The collective stochastic dynamic decision system with common team optimality can be transformed to the equivalent static optimization problem with independent distributed sequences. And under the reference probability space, states and observations are independent Brownian motions.
Consider a series of $d$ inputs, $\textbf{X}_i(t)=[x_{i1}(t),...,x_{id}(t)]$, for $N$ agents, each of which has 2 states $$s_i =\begin{cases}
1, & \text{if $t\in(t-\tau_{ref},t] $},\\
0, & \text{otherwise}.
\end{cases}$$ The firing during $\tau_{ref}$ is sometimes called absolute refractory period.
Such series of inputs can be modeled as discrete decision making scheme, which we have applied the free-response paradigm upon, with adaptive prescribed time interval $\tau_{x}$ for each agent in the network system. Equation (\[eq:single\_net\_ddm\]) then becomes the following distributed protocol $$\label{eq:measureable_ddm}
d\textbf{X} = \bigg(L\cdot\alpha(\textbf{X}(t),t)\cdot\big(\textbf{Y}(t)-\textbf{X}(t)\big)+g(t)\bigg)dt + \beta(\textbf{X}(t),t) d\textbf{w}.$$
Sampling with EIF {#ssec:sample}
-----------------
Recall from Proposition \[prop:equi\] that, since the Brownian motions are indpendent of all other team decisions, it opens up the possibility for Markov chain related methods. For efficiency and flexibility purpose, Markov Chain Monte Carlo (MCMC) has been applied in sampling the spiking network of neurons [@neural_sampling_2011].
In Reference [@richardson_2009], Richardson has shown the equilibrium value of a slow driven voltage-activated current gating variable with the form $ \tau_\Gamma \dfrac{d\Gamma}{dt} = \Gamma_{\infty} - \Gamma$, where $\tau_\Gamma(V)$ is an adaptive time constant characterized by different voltage values, and $\Gamma_{\infty}$ is the equilibrium value. Then we have $\Gamma=\Phi(\left(V_\Gamma-V\right)/\Delta_\Gamma)$, where $\Phi$ is the sigmoid function that digests a membrane potential function into a probability density function with range $[0,1]$. $$\label{eq:gateequi}
\Gamma_{\infty}=\dfrac{1}{1+e^{-(V-V_\Gamma)/\Delta_\Gamma}}.$$ Here, the equilibrium term $\Gamma_{\infty}$ holds a very similar form to Boltzmann probability distribution as in (\[eq:bmprobdist\]).
It is clear that (\[eq:gateequi\]) is a slowly varying function, which can be proved simply by applying the definition. Now we can use the property of the slow varying function to deal with the exponential term in (\[eq:eif\_xdt\]). The Karamata representation theorem is one of the mostly used property of slow varying functions that transfers a function into a general exponential form. In our case, $\Gamma_\infty$ is expressed as: $\Gamma_\infty=\exp \left( \hslash(\Gamma) + \int_B^\Gamma \frac{\varepsilon(t)}{t} \,dt \right) $, for some $B>0$, where $\hslash(\Gamma)$ is a bounded measurable function converging to a finite number, and $\varepsilon(t)$ is a bounded measurable function converging to $0$.
Here, since the exponential term is only a property of membrane potential increment that adds to nonlinearity, the potential accumulation of each agent does not affect collective network decision as much during the firing period $(t-\tau_{ref},t]$. And for the absolute refractory period, the neuron model is guaranteed not to fire. For such a piece wise continuous function, if we only consider the time interval to be one firing, then the variable $V$ can be bounded. Taking out the exponential term in (\[eq:eif\_xdt\]), we have $$\dfrac{CdV_{e}}{\varrho_T(V_e,t)\Delta_T dt} = \exp \bigg( \dfrac{V_e-V_T}{\Delta——T} \bigg),$$
$$\begin{aligned}
\lim_{t\to t_{end}}\exp\left(\dfrac{V_e-V_T}{\Delta_T}\right)&=\lim_{t\to t_{end}}\exp(\hslash(V_e)) \\
&=\exp\left(\dfrac{V_{end}-V_T}{\Delta_T}\right) = \mathcal{C} \\
&= \tau_\Gamma \dfrac{d\Gamma}{dt}+\Gamma.
\end{aligned}$$ where $t_{end}$ is the end of the refractory time, $V_{end}$ is the membrane potenial at end of the refractory period, and $V_e$ is the membrane potential incremented by exponential term only. Since $\hslash$ converges to a finite number and $\varepsilon(t)$ converges to 0, the limit of the exponential term converges to a constant $\mathcal{C}$ during the refractory period.
It can be thought of as entering an absorbing state where its behavior at infinity is very similar to the behavior of converging to infinity. Therefore, at the absolute refractory period, we can safely ignore this exponential term in (\[eq:eif\_xdt\]), and treat it just as other linear terms.
{width="48.00000%"}
[\[fig:boltzmann\]]{}
For simulation, we have used the same parameters as the work done by Barranca *et al* [@Barranca2014]. As shown in Fig. \[fig:boltzmann\], this monotonically increasing activation function has a sigmoidal general form, with an upper bound 1 and lower bound 0. In some cases, as mentioned above, we may set the initial condition to be $0$ mV to eliminate the afterhyperpolarization. For our simulation, we have removed such constraints to show that the dynamics of the membrane potential can be bounded by the activation function with arbitrary initial conditions. This is because the higher threshold in the EIF model always requires higher voltage to be breached, and the logistic activation function describes such positive correlation well enough.
In fact, very similar dynamics of activation function has been capitalized in the neural sampling frame work [@richardson_2009; @neural_sampling_2011; @sto_infere]. Thus, our collective decision making model can be thought of as a network consisting of $N$ agents (or neurons) sampling from a probability distribution $p$ using the stochastic dynamics carried from the Drift-Diffusion model.
The firing activity of the generalized EIF model, which represents each agent in a collective DDM network system, follows a Markov chain process.
With the information-coded signal from each DDM agent of the system, in other words, the firing information within the time interval $(t-\tau_{ref},t]$, the neural sampling follows conditional probability distribution, and most of time is Boltzmann distributions.
For each agent, we treat the collective behavior of connected nodes as a accelerator/damper. For instance, if the agent is surrounded by nodes with higher membrane potential, it receives more current than normal drifting, and vice versa.
$ p(s_i=1|X_i(t-1))= $
$$\label{eq:prob_each}
\dfrac{X_i(t-1)+g(t)+\sum_{j=1}^K\alpha(t)l_{ij}\big(y_{ji}-x_i(t)\big)}{V_T+\dfrac{\sum_{j=1}^K\alpha(t)l_{ji}\big(y_{ij}-x_j(t)\big)}{K}}.$$
Collective Behavior {#sec:coll_beh}
===================
For most multi-agent dynamic network systems, it is common that new communication links can be established over two agents with no previous connection. In the previous sections, we have assumed that the connection is known at each state. Now we define the coupling and connection behavior among the agents in the system.
Coupling and Connectivity
-------------------------
\[asp:coup\_neib\] For the system described in this paper, agent $i$ forms at most $K$ outward links randomly at $t=0$. When $s_i=1$, Agent $i$ tries to establish new connections with new neighbors, for instance, connecting to Agent $j$ with the coupling probability $\mathcal{P}_{ij}$ depends on the voltage difference. When successful, the equal number of previous connections are lost according to a decoupling probability function $\mathcal{Q}$. When $s_i=0$, Agent $i$ will not actively modify its neighboring connections.
It is worth pointing out that both $\mathcal{P}$ and $\mathcal{Q}$ follow a sigmoid (or reverse sigmoid) relation $\Phi$ (or $1-\Phi$). For function $\mathcal{P}$, the greater the difference in membrane potentials has, the higher the probability is. The situation is reversed for function $\mathcal{Q}$, the greater the difference in membrane potentials has, the lower the probability is.
Then we have the following equations $$\mathcal{P}_{ij} = \dfrac{X_j-X_i}{V_{Ti}+V_{Tj}}\dfrac{K-\mathcal{K}_i}{K},$$ $$\mathcal{Q}_{ij} = \left(1-\dfrac{X_i-X_j}{V_{Ti}+V_{Tj}}\right)\dfrac{\mathcal{K}_i}{K},$$ where $\mathcal{K}_i$ is the number of current established connections of node $i$. In the case of probability values that are less than zero or greater than one, we simply set them to $0$ and $1$ respectively for mean field analysis. The negative probability is also provided in Fig. \[fig:globalbranch\] . Different from the common equation for branching probability, our process is not a tree-like process and the maximal connectivity is defined by $K$. Therefore they are Markovian processes with respect to the number of connected nodes.
{width="48.00000%"}
[\[fig:powerlaw\]]{}
Mean Field Analysis
-------------------
In the mean field analysis, the individual drifting variable usually follows a distribution with the average. In our case, we assume that the expanded term follows the average distribution of $g_i/W$, where $W$ can be considered as the sum of the total drift. Here $g_{i}$ is still a gain function representing the external input exerted by neighbors. However, due to the free-response property, $g_{i}$ is essentially a piece-wise continuous gain function representing synaptic strength.
Recalling the term “active site" described in Section [\[ssec:socintro\]]{}, the local active site for each agent can be thought as the local field: $\mathcal{F}_i=-\sum_j^Na_{ji}g_{j}/W$, where $a_{ji}$ is the elements in an adjacency matrix, then the global field has the average $\bar{\mathcal{F}}=\sum^N_i \left(\mathcal{F}_i/K\right)/N $.
Therefore, with regard to a sequence of $d$ number of inputs, the probability of a single DDM having the state $s_i=1$, represented by a single generalized EIF neuron, has approximately the following probability with respect to the field. $$\label{eq:eifboltz}
\begin{aligned}
p_i(s) = &S^{-1} \exp\bigg(-\gamma \bigg(\sum_{j=1}^{N \setminus K}\bar{\mathcal{F}}\mathcal{P}_{ij}-\\
&~~~~~~~~~~~\sum_{j=1}^K\mathcal{F}_j \mathcal{Q}_{ij}+\sum_0^d b_i \bigg)\bigg),
\end{aligned}$$ where $S$ is some partition function, $b_i$ is the bias term that supports the correct choice, $\gamma$ is a thermodynamic beta in Boltzmann factor with the form $\gamma = 1/\left(k_B \mathcal{F}_i\right)$, and $k_B$ is a Boltzmann constant.
Putting aside the coupling strength, the mean activity of this network is measured as $\sum^N_{i=1}s_i/N$. However, for such a spiking neural model, it is not always practical to have normally distributed probability density function, and in fact, most of these processes are stochastic with certain thresholds or even highly constrained. That being said, we would have a stochastic Itô based integral for probability density for the mean activity: $\mathcal{H}=\int^{\infty}_{-\infty}\Phi(X)p(s_i|X)dX$.
Convergence Analysis {#sec:converge}
====================
There are two main evidences for systems presenting the SOC behavior [@bak_1996; @Watkins2016], the first is the power law distribution, and the second is the critical dynamics, which we consider as converging to absorbing states in the EDM model. In this section, we expand the results from Sections \[sec:dm\_dynam\] and \[sec:coll\_beh\], and examine the global convergence behavior of the collective EDM model. We then provide both pieces of evidence to show the EDM system has the SOC behavior.
Global Criticality from Local Dynamics {#ssec:globfloc}
--------------------------------------
Recall that in SOC, active nodes trigger self avalanches when a threshold is reached, and update the information of all connecting active nodes. Also, nodes in nearby inactive sites will be communicated to establish more connections if needed.
Moreover, each local active site has their own dynamics of reaching out to other active or inactive sites. Inspired by the work of Harris on the theory of branching processes [@harris2002theory], we use a branching parameter $\sigma$ that captures the subsequent activity of connectivity triggering or dying-out [@soc_branching]. The *local branching ratio* and *global branching ratio* have the form
$$\sigma_j(t)= \sum^K_{i=1} \mathcal{P}_{ij}(t), \quad\tilde{\sigma}(t)=\dfrac{1}{N-1}\sum^N_{j=1}\sigma_j(t),$$ respectively. As discussed in [@harris2002theory; @soc_branching], the system exhibits criticality at $\sigma = 1$, and is sub-critical (super-critical) for $\sigma < 1$ ($\sigma > 1$).
[0.48]{} ![In Fig. (\[fig:gbmp\]), as the firing probability increases with the membrane potential, $\sigma$ converges to 1 with proper connectivity constraints. Cases with different numbers of active and connected neighbors are shown in a network system with $N=10$. Fig (\[fig:gbnn\]) removes the constraint that cumulation of local branching ratio in each iteration caps at 1. It is clear that as the number of connected neighbors increases, the network system enters the active/loading phase first, and then evolves to the dissipation/absorbing phase. The system clearly shows SOC dynamics, that is $\sigma = 1$ at both minimal and maximal connectivity.[]{data-label="fig:globalbranch"}](figs/globalbran.jpg "fig:"){width="\textwidth"}
[0.48]{} ![In Fig. (\[fig:gbmp\]), as the firing probability increases with the membrane potential, $\sigma$ converges to 1 with proper connectivity constraints. Cases with different numbers of active and connected neighbors are shown in a network system with $N=10$. Fig (\[fig:gbnn\]) removes the constraint that cumulation of local branching ratio in each iteration caps at 1. It is clear that as the number of connected neighbors increases, the network system enters the active/loading phase first, and then evolves to the dissipation/absorbing phase. The system clearly shows SOC dynamics, that is $\sigma = 1$ at both minimal and maximal connectivity.[]{data-label="fig:globalbranch"}](figs/globalbranminmax.jpg "fig:"){width="\textwidth"}
In our simulation, sub-critical dynamics are characterized by low potential and rapidly decaying agent neuron firing distributions, while super-critical dynamics are characterized by high potential and slowly decaying firing activities. Critical dynamics are characterized by firing activity that follows power-law distributions.
As shown in Fig. \[fig:powerlaw\], it can be easily recognized that the collective behaviors of the firing density in the EDM model follow a power law distribution, $\mathcal{D}(z) \sim z^{\Lambda}$ with different cluster size $z$ and scaling factor $\Lambda<0$, which is a primary evidence supporting the SOC behavior [@Watkins2016].
Absorbing States {#ssec:absorb}
----------------
Recalling the avalanches described in self-organized criticality, the system keeps tuning itself to one of many meta-stable states, which commonly have lifespans shorter than ground state and longer than excited states [@bak_1996]. And without further inputs, the distribution reaches meta-stability, a very special energy well that is able to temporarily trap the system for a limited number of states.
This can be modeled as Boltzmann distribution with global energy level in a simulated annealing system from any initial conditions. With the results from (\[eq:prob\_each\]) and (\[eq:eifboltz\]), as well as the exponential property of the generalized EIF system, the Lyapunov based semistability [@HHB:TAC:2008] can be achieved with great potential, but due to the length restriction, this concept will not be discussed in this paper. Nevertheless, we are going to show that the system described in this paper converges to some absorbing states.
In mean field theory, as being discussed in [@soc_branching], absorbing state becomes unstable when the probability of a node creating connection with neighbors is greater than $1/2$. In our case, this can be thought as the coupling probability $\mathcal{P}>\mathcal{P}_{critical} = 1/2$.
\[lem:multi\_absorb\] The attractor of the system is a set of discrete states.
If a non-conserving system, such as the drift and diffusion based EDM model described in this paper, has shown a temporary stable configuration after the avalanches, then the system is at least at a critical point. The critical and super-critical session are usually slow driving [@soc_branching]. So there must be a drift load and a diffusion dissipation fluctuating to keep all the nodes in the system from either forming active sites, or staying quiescent completely.
Thus, if the system presents the thermodynamic behavior as Boltzmann distribution with simulated annealing, there can exist infinite numbers of infinitesimally varied absorbing states in thermodynamic limit. As for a finite number of total states, this absorbing phase becomes a set of discrete states.
This obeys another property of SOC, that is, the dynamical system with a critical point as an attractor, is able to keep itself at the critical point between two phases, which in our case, are the active phase and the absorbing phase.
Meta-stable states generally hold more energy than the ground states, and less energy than the excited states. Therefore, SOC is the process of the EDM model losing global energy, and falling into a certain set of absorbing states, regardless of guaranteed stability.
Each agent in the EDM model is essentially drift diffusion terms taking input variable from EIF markup. It is clear that when the individual thresholds are reached, the agents will initiate the spike and send information-coded signals (current) to connected nodes or nodes with great probability of establishing connectivity. The exited states usually carry higher membrane potential than the incremental states, and then the fired node resets its membrane potential to $V_R$ and enters a refractory period. Therefore, the states during this absolute refractory period $\tau_{ref}$ can be considered as comparable meta-stable states that trap the dynamics of each node for $\tau_{ref}$. And since we have proved Lemma \[lem:multi\_absorb\], the system self-organizes itself, instead of fine-tuning, to a small region of absorbing criticality, or in another word, metastability.
Since the system is slow varying at the absorbing state, the fundamental solution is independent of $t$, that is, $\alpha(X(t),t)) = \alpha(X)$. And (\[eq:xsol\_ref\]) becomes $$\label{eq:xsol_ref_inv}
X = e^{\alpha(t-\tau_{ref})}c+\int^t_{t_0}e^{-L\alpha(t-\eta)} \big(g(\eta)d\eta+\beta(\eta)dw_{\eta} \big) .$$
At such slow varying, time independent absorbing states, it is natural to assume that the dimension of $\beta$ is only 1. Using the corollary in [@arnold1974stochastic], $ \phi(t) = \exp\bigg( \int^t_{(t_0)} \alpha(\eta)d\eta \bigg)$.
Then we can further turn Theorem \[thm:sol\_ou\] to be $$\label{eq:sol_ou_gen}
\begin{aligned}
&X(t) = \exp\bigg( \int^t_{(t_0)} \alpha(\eta)d\eta \bigg)\bigg(c + \\
&~~~~\int^t_{t_0}\exp \bigg( \int_{t_0}^s A(u)du \bigg) \big(g(\eta)d\eta+\beta(\eta)dw_{\eta} \big) \bigg) .
\end{aligned}$$
The threshold presents the local rigidity level. Also, the convergence dynamics of these absorbing states show Boltzmann distribution as well, i.e., $
P_i(V_x|s=0)= e^{-X_i/k_B\bar{\mathcal{F}}}/\left(\sum_{j=1}^N e^{-X_i/k_B\bar{\mathcal{F}}}\right).
$
As shown in Fig. \[fig:globalbranch\], the branching pattern of the EDM model across multiple fires follows the SOC behavior and will eventually evolve to a certain set of absorbing states, known as the recurrence sets. Also, these stationary distributions of network states can be convergent from any initial state.
Therefore, without the presence of a proper controller, the system fine-tunes itself, and then converges to $
\bar{u} = \lim_{t \to \infty} \sum_0^{t-1} \mathbb{E}(X(t))/t.
$
Conclusion and Future Work {#sec:conclusions}
==========================
Conclusion
----------
In this work, we have proposed a collective decision making model with a specific type of spiking neurons, exponential integrate-and-fire (EIF). Our method is based on the well-known Two-Alternative Forced Choice Task solver–Drift-Diffusion model. We recognize that DDM and EIF share very common terms in their dynamic equations, and the exponential term can be ignored during the absolute refractory period. We have derived the probability of each agent’s firing based on a Markov chain conditional premise. Then the mean field theory is used to approximate the global criticality from local dynamics.
Both analytically and experimentally, we have found out that the global branching ratio follows a power law distribution and the EDM system eventually evolves to a set of absorbing states, which are two main evidences suggesting the Self-Organized Criticality behavior. The activation function follows the Boltzmann state probability and the convergence dynamics of absorbing states follow Boltzmann distribution as well.
Future Work
-----------
At this point, we have set up a detailed model that is ready to be expanded from different aspects. For instance, since theoretical and experimental studies have demonstrated that critical systems are often optimizing computational capability, it is promising to suggest that the system with the SOC behavior is both robust and flexible to ensure homeostatic stability.
In fact, due to the nature of absorbing states and criticality property, any initial conditions of a spiking network decision making system can converge and/or fluctuate around a set of states, potentially semistability [@HHB:TAC:2008]. Therefore, the convergence property of such a model can be useful for fault pre-screening and is in a way, robust to quantified uncertainties.
Besides the theoretical analysis, there are potential applications as well. Bak has demonstrated that both traffic dynamics and brain dynamics exhibit similar criticality [@bak_1996]. This opens up the way to extend our model to formulate real-world applications. With specific problem solving scenarios, it is natural to extend the proposed multi-agent spiking neural decision making system to a parallel distributed process, and eventually leads to a neuromorphic chip development.
Since the nature of SOC is to organize the system between two phases, the fast switching capability is useful for sensitivity analysis. Furthermore, we will also look into fully stochastic neurons such as *Galves-Löcherbach* (GL) model to incorporate different activation profiles and to increase the candidate choices as well.
[^1]: This work was supported by the Defense Threat Reduction Agency, Basic Research Award \#HDTRA1-15-1-0070.
[^2]: The authors are with the Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0511, USA, [[email protected]; [email protected]; [email protected]]{}.
|
KEK-TH-1171\
August 2007
[**Yasuhiro Okada**]{}$^{(a,b)}$ [^1] 0.2in [*$^{(a)}$[Theory Group, KEK, Oho 1-1 Tsukuba, 305-0801, Japan]{}\
$^{(b)}$[The Graduate University for Advanced Studies (Sokendai),\
Oho 1-1 Tsukuba, 305-0801, Japan]{}\
*]{}
Introduction
============
Current understanding of the elementary particle physics is based on two important concepts, gauge invariance and spontaneous symmetry breaking. Out of four fundamental interactions of Nature, namely strong, weak and electromagnetic and gravity interactions, three of them except for gravity are described on the same footing in terms of gauge theory. The gauge group corresponding to the strong interaction is $SU(3)$, and the weak and the electromagnetic interactions arise from $SU(2)$ and $U(1)$ groups and are called the electroweak interaction. Once quarks and leptons are assigned in proper representations of the three gauge groups, all properties of the three fundamental interactions are determined from the requirement of gauge invariance.
For more than thirty years, high energy experiments have been testing various aspects of gauge symmetry and have established gauge invariance as a fundamental principle of Nature. We have discovered gauge bosons mediating the three interactions, namely gluon for the strong interaction, W and Z bosons for the electroweak interaction. The couplings between quarks/leptons and gauge bosons have been precisely measured at the CERN LEP and SLAC SLC experiments, and we have confirmed the assignments of the gauge representations for quarks and leptons.
The gauge principle alone, however, cannot describe the known structure of the elementary particle physics. In the Standard Model of the elementary particle physics, all quarks, leptons and gauge bosons are first introduced as massless fields. In order to generate masses for these particles, the $SU(2) \times U(1)$ symmetries have to be broken spontaneously.
Spontaneous symmetry breaking itself is not new for particle physics [@Nambu:1961tp; @Goldstone:1961eq]. The theory of the strong interaction, QCD, possesses an approximate symmetry among three light quarks called chiral symmetry. The vacuum of QCD corresponds to a state where quark and anti-quark pair is condensed, and the chiral symmetry is broken spontaneously. As a consequence, pseudo scalar mesons such as pions and kaons are light compared to the typical energy scale of the strong interaction since they behave approximately as Nambu-Goldstone bosons, a characteristic signature of spontaneous symmetry breaking.
In the case of the electroweak symmetry, it is shown theoretically that the Nambu-Goldstone bosons associated with spontaneous breakdown are absorbed by gauge bosons, providing the mass generation mechanism for gauge bosons (Higgs mechanism) [@Higgs:1964ia]. Although we are now quite sure that this is the mechanism for gauge boson mass generation, we know little about how the symmetry breaking occurs, or what is dynamics behind the Higgs mechanism. Clearly, we need a new interaction other than four known fundamental forces, but we do not know what it is. The goal of the Higgs physics is to answer this question.
In this article, I would like to explain what are theoretical issues of the Higgs sector, what is expected at the future collider experiments, LHC and ILC, and what would be impacts of the Higgs physics on a deeper understanding of the particle physics.
Higgs boson in the Standard Model
=================================
In the Standard Model, a single Higgs doublet field is included for the symmetry breaking of the $SU(2)\times U(1)$ gauge groups. This was introduced in S. Weinberg’s 1967 paper “A Model of Leptons” [@Weinberg:1967tq], and is the simplest possibility for generating the gauge boson masses.
The Higgs potential is given by $$V(\Phi)=-\mu^2|\Phi|^2+\lambda |\Phi|^4,
\label{eq1}$$ where the two component complex field is defined as $$\Phi(x)=\left(
\begin{array}{c}
\phi(x)^+\\
\phi(x)^0
\end{array}
\right).$$ In order for the stability of the vacuum the parameter $\lambda$ must be positive. The coefficient of the quadratic term, on the other hand, can be either sign. In fact, if the sign is negative, namely $\mu^2>0$, the origin of the potential is unstable, and the vacuum state corresponds to a non-zero value of the $\Phi$ field. The states satisfying $|\phi^+|^2+|\phi^0|^2=\frac{\mu^2}{2\lambda}\equiv\frac{v^2}{2}$ are degenerate minimum of the potential. We can choose the vacuum expectation value in the $\phi^0$ direction, $<\phi^0>=\frac{v}{\sqrt{2}}$, and then there are three massless modes corresponding to the flat directions of the potential (Nambu-Goldstone modes). When the symmetry is a gauge symmetry, these massless particles disappear from the physical spectrum, and become longitudinal components of massive gauge bosons. This is seen most clearly if we take the “Unitary gauge” where the Nambu-Goldstone modes are removed by an appropriate gauge transformation. The kinetic term of the scalar field is defined as $|D_{\mu}\Phi|^2=|(\partial_{\mu}+g\frac{\tau^a}{2}W_{\mu}^a+\frac{g'}{2}B_{\mu})\Phi|^2$, and the gauge boson mass terms are obtained by substituting the vacuum expectation value into $\Phi(x)$.
Mass terms of quarks and leptons are also generated through interactions with the Higgs field. This follows from the chiral structure of quarks and leptons. Since the discovery of the parity violation in the weak interaction [@Lee:1956qn], chiral projected fermions (Weyl fermions) instead of Dirac fermions have been considered as building blocks of a particle physics model. In particular, only left-handed quarks and leptons are assigned as $SU(2)$ doublets because the weak interaction has a V(vector)-A(axial vector) current structure. Right-handed counter parts are singlet under the $SU(2)$ gauge group. This unbalance in the $SU(2)$ quantum number assignment forbids us to write direct mass terms for quarks and leptons: the only possible way to generate mass terms is to introduce Yukawa couplings with help of the $\Phi$ field such as $y_d\Phi^{\dagger}\bar{d}_R q_L$ where $q_L=(u_L,d_L)^T$. After replacing $\Phi(x)$ by its vacuum expectation value, this term generates a mass of $y_d v/\sqrt{2}$ for down-type quarks. Similar mechanism works for up-type quarks and charged leptons.
There is one important prediction of this model. Since we introduce a two-component complex field and three real degrees of freedom are absorbed by gauge bosons, one scalar particle appears in the physical spectrum, which is called the Higgs particle ($\equiv$ Higgs boson). In the Unitary gauge, interactions related to the Higgs boson can be obtained by replacing $v$ with $v+H(x)$ in the Lagrangian where $H(x)$ represents the Higgs boson. The mass of the Higgs boson is given by $m_h=\sqrt{2\lambda}v$, which means that the Higgs boson becomes heavier if the Higgs self-coupling gets larger. In fact, this is a general property of the particle mass generation mechanism due to the Higgs field: A stronger interaction leads to a heaver particle. The mass formula for the W, Z bosons, quarks, leptons and the Higgs boson at the lowest order approximation with respect to coupling perturbation (i.e. tree-revel) are summarized in table \[t1\].
W boson Z boson quarks, leptons Higgs boson
---------------- ------------------------------- -------------------------- --------------------
$\frac{g}{2}v$ $\frac{\sqrt{g^2+g'^2}}{2} v$ $y_f \frac{v}{\sqrt{2}}$ $\sqrt{2\lambda}v$
: Mass formula for elementary particles. $g$, $g'$, $y_f$, and $\lambda$ are the $SU(2)$ and the $U(1)$ gauge coupling constants, the Yukawa coupling constant for a fermion $f$, and the Higgs self-coupling constant. []{data-label="t1"}
Naturalness and Physics beyond the Standard Model
=================================================
Although the Higgs potential in Eq.\[eq1\] is very simple and sufficient to describe a realistic model of mass generation, we think that this is not the final form of the theory but rather an effective description of a more fundamental theory. It is therefore important to know what is limitation of this description of the Higgs sector.
In renormalizable quantum field theories, the form of Lagrangian is specified by requirement for renormalizability. In the case of the Higgs potential, quadratic and quartic terms are only renormalizable interactions. We can then consider two kinds of corrections to the potential. One is a calculable higher order correction within the Standard Model. For instance the correction from the top Yukawa coupling constant can be evaluated up to a desired accuracy applying renormalization procedure of field theory. Another type of corrections comes from outside of the present model, presumably from physics at some high energy scale. We cannot really compute these corrections until we know the more fundamental theory. In this sense, the present theory is considered as an effective theory below some cutoff energy scale $\Lambda$.
Although the effective theory cannot include all physical effects, it is still useful because unknown correction is expected to be suppressed by $(E/\Lambda)^2$ where $E$ is a typical energy scale under consideration. Therefore, as long as the cutoff scale is somewhat larger than $E$, the theory can make fairly accurate predictions. For example, the correction is $0(10^{-4})$ when the cutoff scale is around 10 TeV for physical processes in the 100 GeV range. If the theory is valid up to the Planck scale $(\sim 10^{19}$ GeV) where the gravity interaction becomes as strong as the other gauge interactions, the correction becomes extremely small. In this way, an effective theory is useful description as long as we restrict ourself to the energy regime below the cutoff scale.
Once we take a point of view that the Higgs sector of the Standard Model is an effective description of a more complete theory below $\Lambda$, naturalness with regard to parameter fine-tuning becomes a serious problem. In particular, the quadratic divergence of the Higgs mass radiative correction is problematic, and this has been one of main motivations to introduce various models beyond the Standard Model.
In the Higgs potential in Eq.\[eq1\] the only mass parameter is $\mu^2$. At the tree level, this parameter is related to the vacuum expectation value $v$ by $\mu^2=\lambda v^2$ where $v$ is known to be about 246 GeV. ($v=(\sqrt{2} G_F)^{-\frac{1}{2}}$, where $G_F$ is the Fermi constant representing the coupling constant of the weak interaction.) If we include the radiative correction, $\mu^2$ becomes a sum of two contributions $\mu_0^2 + \delta \mu^2$ where $\mu_0^2$ is a bare mass term and $\delta \mu^2$ is the radiative correction. In the Standard Model, the top quark and gauge boson loop corrections are important and $\delta \mu^2$ from these sources are represented by a sum of terms of a form $C_i \frac{g_i^2}{(4\pi)^2}\Lambda^2$ where $g_i$ is the top Yukawa coupling constant, or $U(1)$ or $SU(2)$ gauge coupling constant and $C_i$ are $O(1)$ coefficients. Since the radiative correction depends on the cutoff scale quadratically, the fine-tuning between the bare mass term and the radiative correction is necessary if the cutoff scale is much larger than 1 TeV. Roughly speaking, the fine-tuning at 1% level is necessary for $\Lambda=10$ TeV. If the cutoff scale is close to the Planck scale, the degree of the fine-tuning is enormous: A tuning of one out of $10^{32}$ is required. This is the naturalness problem of the Standard Model, and sometime also called the hierarchy problem. This problem suggests that the description of the Higgs sector by the simple potential in Eq. \[eq1\] is not very satisfactory, and probably will be replaced by a more fundamental form at a higher energy scale.
Since the problem arises from the quadratic divergence in the renormalization of the Higgs mass terms, proposed solutions involve cancellation mechanism of such divergence. Supersymmetry[@Wess:1974tw] is a unique symmetry that guarantees complete cancellation of the quadratic divergence in scalar field mass terms. This is a new symmetry between bosons and fermions and the cancellation occurs between loop diagrams of bosons and fermions. Particle physics models based on supersymmetry such as supersymmetric grand unified theory (SUSY GUT) have been proposed and studied since early 1980’s as a possible way out of the hierarchy problem [@Sakai:1981gr]. In a realistic model of a supersymmetric extension of the Standard Model, we need to introduce new particles connected by supersymmetry to ordinary quarks, leptons, gauge bosons, and Higgs particles. In 1990’s, precision studies on the Z boson were preformed at LEP and SLC experiments, and it was pointed out that three precisely measured coupling constants are consistent with the prediction of SUSY GUT, although the gauge coupling unification fails badly without supersymmetric partner particles [@Langacker:1991an]. Supersymmetry is also an essential ingredient of the superstring theory, a potential unified theory including gravity and gauge interactions. In this way, the supersymmetric model has become a promising candidate beyond the Standard Model. If supersymmetry realizes at or just above the TeV scale, it can provide a consistent and unified picture of the particle physics from the weak scale to the Planck scale.
An opposite idea for solution of the naturalness problem is considering that the cutoff scale is close to the electroweak scale. In particular, the Higgs field is considered to be a composite state of more fundamental objects at a relatively low energy scale. The simplest form of this model is called the technicolor model [@Susskind:1978ms] proposed in late 1970’s, in which the cutoff scale is about 1 TeV. The technicolor model is however strongly constrained from precision tests of electroweak theory later at the LEP and SLC experiments [@Peskin:1990zt], but there have been continuous attempts to construct a phenomenologically viable model of a composite Higgs field. Little Higgs models[@Arkani-Hamed:2002qx] are a recent proposal on this line, where the physical Higgs boson is dynamically formed by a new strong interaction around 10 TeV. An interesting feature of this model is that the quadratic divergence of the Higgs boson mass term is canceled by loop corrections due to new gauge bosons and a heavy partner of the top quark at one loop level. In this way the hierarchy problem between the electroweak scale and 10 TeV is nicely solved.
In addition to supersymmetry and little Higgs models, there have been many proposals for TeV scale physics. Motivations for many of them are solving the naturalness problem of the Standard Model or explaining the large (apparent) hierarchy between the weak scale and the gravity scale. Examples are models with large extra dimensions [@Arkani-Hamed:1998rs], models with warped extra-dimensions [@Randall:1999ee], the Higgsless model[@Csaki:2003dt], the twin Higgs model[@Chacko:2005pe], and the inert Higgs model[@Barbieri:2006dq], the split-supersymmetry model[@Arkani-Hamed:2004fb], etc. All of these proposals involve some characteristic signals around a TeV region. These signals are important to choose a correct model at the TeV scale and clarify the mechanism of the electroweak symmetry breaking.
Experimental Prospects of Higgs Physics
=======================================
Higgs physics is expected to be the center of the particle physics in coming years starting from the commissioning of the CERN LHC experiment. The first step will be a discovery of a new particle which is a candidate of the Higgs boson. We then study its properties in detail and compare them with the prediction of the Standard Model Higgs boson. We may be able to confirm that the discovered particle is the Higgs boson responsible for the mass generation for elementary particles. Another possibility would be to find some deviation from the Standard Model Higgs boson. Deviation could be something like small difference of production cross section and decay branching ratios from the Standard Model predictions, or more drastic new signals such as discovery of several Higgs states. At the same time, we may also find other new particles predicted in extensions of the Standard Model, for example supersymmetric particles in the supersymmetric model or the heavy gauge bosons and the top partner in the little Higgs model. In order to accomplish these goals we probably need several steps in collider experiments including LHC and ILC experiments and possible upgrades for these facilities.
If we restrict ourselves to the Higgs boson in the Standard Model, all physical properties are determined by one parameter, the Higgs boson mass. Present experimental lower bound for the mass of the Standard Model Higgs boson is 114.4 GeV at the 95% confidence level, set by the direct Higgs boson search at LEP [@Barate:2003sz]. It is remarkable that we can also draw an upper bound from a global fit of electroweak precision data. Although a heavy Higgs boson means a large self-coupling $\lambda$, we have not seen any evidence of such a large coupling in physical observables related to $Z$ and $W$ gauge boson processes. The upper limit of the Standard Model Higgs boson is 166 GeV at the 95% confidence level [@Alcaraz:2006mx]. This implies that a relatively light Higgs boson is favored. If the Higgs boson turns out to be heaver than 200 GeV, we would expect some additional new particles that have significant couplings to gauge bosons.
The decay branching ratios of the Higgs boson depends strongly on the Higgs boson mass, and therefore the discovery strategy for the Higgs boson at LHC differs for light and heavy Higgs bosons. The branching ratios for the Standard Model Higgs boson is shown in figure \[f1\]. Since the Higgs boson couples more strongly to a heaver particle, it tends to decay to heaver particles as long as kinematically allowed. For instance, the Higgs boson mostly decays into two gauge bosons if the Higgs boson mass is larger than 200 GeV, whereas the bottom and anti-bottom pair is the main decay mode for its mass less than 140 GeV. For this mass range, the Higgs boson search at LHC relies on other decay modes such as the loop-induced two photon decay mode, because two bottom modes are hidden by overwhelming QCD background processes. Detail simulation studies on the Higgs discovery at LHC have been performed, and it is shown that the Higgs boson can be found at LHC experiments within a few years for the entire mass region as long as the production and decay properties are similar to the Standard Model Higgs boson[@atlas:1999; @cms:2006]. Furthermore, information on the Higgs couplings is obtained with a higher luminosity. Estimated precision for coupling ratios are typically 0(10)% [@Duhrssen:2004cv].
![Decay branching ratio of the Standard Model Higgs boson as a function of its mass. $c$, $b$, $t$, $\tau$ represent charm, bottom, top quarks and tau lepton. $\gamma$, $g$ , $W$, $Z$ are photon, gluon, $W$ and $Z$ bosons. []{data-label="f1"}](80516Fig1.eps){width="8cm"}
ILC is a future electron-positron linear collider project proposed in the international framework [@ILC]. One aspect of this facility is a Higgs factory. For instance, the number of produced Higgs bosons can be $0(10^5)$ in the first stage of experiments with the center-of-mass collider energy of 500 GeV. Under clean environment of the $e^+e^-$ collider, precise determinations on the mass, quantum numbers, and coupling constants of the Higgs boson are possible. Typical production and decay processes are shown figures \[f2\]. Precision of the coupling constant determination reaches a few % level for Higgs-$WW$, Higgs-$ZZ$, and Higgs-$b \bar{b}$ couplings for the case of a relatively light Higgs boson. We can also measure the Higgs self-coupling from the double Higgs boson production process and the top Yukawa coupling from the Higgs-$t \bar{t}$ production. Figure \[f3\] shows precision of the Higgs coupling constant determination for various particles at ILC. The proportionality between coupling constants and particle masses is a characteristic feature of the one Higgs doublet model where the particle mass formulas involve only one vacuum expectation value. An important feature of ILC experiments is that absolute values of these coupling constants can be determined in a model-independent way. This is crucial in establishing the mass generation mechanism for elementary particles.
![Production process of the Higgs boson at ILC (left) and Higgs boson decay to fermions (right).[]{data-label="f2"}](80516Fig2a.eps "fig:") ![Production process of the Higgs boson at ILC (left) and Higgs boson decay to fermions (right).[]{data-label="f2"}](80516Fig2b.eps "fig:")
![Precision of the coupling-constant determination for various particles at ILC with the integrated luminosity of 500 $fb^{-1}$. The Higgs boson mass is taken to be 120 GeV. For charm, tau, bottom, W, and Z coupling measurement, $\sqrt{s}$=300 GeV is assumed. $\sqrt{s}=500$ GeV (700 GeV) is taken for the triple Higgs $(t\bar{t} H)$ coupling measurement[@:2003mg].[]{data-label="f3"}](80516Fig3.eps){width="8cm"}
The precise determination of the Higgs coupling constants is also useful to explore physics beyond the Standard Model. In some case, the Higgs boson coupling is modified from the Standard Model.
- The Higgs sector of supersymmetric models is different from the Standard Model. In any realistic supersymmetric model, the Higgs sector contains at lease two sets of doublet fields. In the minimal supersymmetric standard model (MSSM), in particular, the Higgs sector is a two Higgs doublet model. Furthermore, there is a rather strict theoretical upper bound for the the lightest neutral Higgs boson[@Okada:1990vk], which is about 130 GeV. Since this light boson plays a role of the usual Higgs particle, this particle may be the only Higgs particle discovered at LHC. In such case, the branching ratio measurement for the lightest neutral Higgs boson is useful to obtain information on the masses of heavy Higgs bosons [@Abe:2001gc; @Aguilar-Saavedra:2001rg; @Weiglein:2004hn]. In particular, the tau and bottom coupling constants show sizable enhancement if the heavy Higgs boson exists below 600 GeV. The ratio like $B(H \to WW)/B(H \to \tau \tau)$ is useful to determine the heavy Higgs mass scale indirectly.
- In models with extra dimensions, there appears a scalar field called Radion, corresponding to the size of the extra space dimension. Since Radion is a neutral scalar field, it can mix with the Higgs field. It is pointed out that Radion-Higgs mixing in the warped extra dimension model could reduce the magnitude of Yukawa coupling constants and $WWH$ and $ZZH$ constant in a universal way [@Hewett:2002nk]. In order to observe such effects, absolute coupling measurements at ILC are necessary.
- The two-gluon width of the Higgs boson is generated by loop diagrams, so that it can be a probe to virtual effects of new particles. The same is true for the two-photon width, the measurement of which is improved at the photon-photon collider option of ILC [@Badelek:2001xb]. There are many new physics models where such loop effects are sizable.
- Explaining the baryon number of the Universe is one of most outstanding questions for particle physics in connection with cosmology [@Riotto:1999yt]. One possibility is the electroweak baryogenesis scenario, in which the baryon number was generated at the electroweak phase transition. For a successful electroweak baryogenesis, the Higgs sector has to be extended from that of the minimal Standard Model to realize a strong first-order phase transition. The change of the Higgs potential can lead to observable effects in the triple Higgs coupling measurement [@Grojean:2004xa; @Kanemura:2004ch].
As we can see above examples, observations of new physics effects require precise determination of coupling constants. This will be an important goal of the future ILC experiment.
Conclusions
===========
The Higgs sector is an unknown part of the particle physics model. Although a simple potential is assumed in the Standard Model, this description is supposed to be valid below some cutoff scale, beyond which the theory of the electroweak symmetry breaking takes in a more fundamental form. If the cutoff scale is as low as 1 TeV, some direct signals on new physics is likely to appear at LHC. If the cutoff scale is much larger, the fine-tuning of the Higgs boson mass term becomes a serious problem. Proposed solutions to this problem such as supersymmetry or little Higgs models also predict new physics signals at the TeV scale. These signals are targets of future collider experiments starting from LHC.
Experimental prospects for the Higgs physics are quite bright. The Higgs particle can be found and studied at LHC. At the proposed ILC, precise information on coupling constants between the Higgs boson and other particles will be obtained. These measurements are an essential step to establish the mass generation mechanism. At the same time, the precision measurement may reveal evidence of new force and/or new symmetry because these new physics is most probably related to the physics of electroweak symmetry breaking, i.e. the Higgs sector. In this way, the Higgs particle will play a special role in determining the future direction of the particle physics.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported in part by the Grant-in-Aid for Science Research, Ministry of Education, Culture, Sports, Science and Technology, Nos. 16081211 and 17540286.
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[^1]: E-mail: [email protected]
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abstract: 'We demonstrate that at long times the rate of passive scalar decay in a turbulent, or simply chaotic, flow is dominated by regions (in real space or in inverse space) where mixing is less efficient. We examine two situations. The first is of a spatially homogeneous stationary turbulent flow with both viscous and inertial scales present. It is shown that at large times scalar fluctuations decay algebraically in time at all spatial scales (particularly in the viscous range, where the velocity is smooth). The second example explains chaotic stationary flow in a disk/pipe. The boundary region of the flow controls the long-time decay, which is algebraic at some transient times, but becomes exponential, with the decay rate dependent on the scalar diffusion coefficient, at longer times.'
author:
- 'M. Chertkov$^a$ and V. Lebedev$^{a,b}$'
title: Decay of scalar turbulence revisited
---
Study of advection of a steadily supplied scalar in smooth flow, pioneered by Batchelor [@59Bat] for the case of a slowly changing strain and extended by Kraichnan to the opposite limit of short-correlated flow [@67Kra], has developed into a universal theory, applicable to any kind of statistics and temporal correlations of the smooth chaotic flow [@94SS; @95CFKLa]. (See also the reviews [@00SS; @01FGV].) Extension of this theory to the decay problem [@99Son; @99BF] suggested a fast (exponential) decay of the passive scalar in the case of spatially smooth (that is approximated by linear velocity profiles at relevant scales) and unbounded random flow. (We will call this case the “pure Batchelor" one.) In parallel, laboratory experiments [@96MD; @97WMG; @00JCT; @01GSb] and numerical simulations [@93Pie; @94HS; @95AFO; @97BDY] were conducted to verify the Batchelor-Kraichnan theory. However, a comparison between theory, experiment and numerical simulations was not conclusive. Different experiments and simulations showed different results, inconsistent with each other and with the theory. In this letter we explain a possible source of the discrepancy. Our major point is that the Batchelor-Kraichnan theory does not apply to the late-time stage of the scalar decay measured in the decay laboratory and numerical experiments [@93Pie; @96MD; @97WMG; @01GSb]. This is because the long-time decay is dominated by the regions of the flow, where the mixing is not as efficient as in the pure Batchelor case.
Before constructing a quantitative theory, let us draw a qualitative physical picture. We start from the first example of a multi-scale turbulent flow with the following hierarchy of scales assumed: $L_v\gg\eta\gg r_d$, where $L_v$ is the energy containing scale of the flow, $\eta$ is the Kolmogorov (viscous) scale and $r_d$ is the diffusive scale of the scalar. It is known that the passive scalar mixing in the inertial range of turbulent flow is slower at larger scales, since the turnover time grows with scale. This explains the algebraic-in-time decay of the scalar for the inertial range of scales, $L_v\gg r\gg\eta$ [@00EX; @01CEFV]. If we now take into account that turbulent velocity is smooth at scales smaller than $\eta$, it becomes important to understand what happens with the scalar fluctuations at these small scales. We show in the letter that scalar decay is accompanied by both upscale and downscale transport, however, the situation is asymmetric. The decay of scalar fluctuations in time is due to inertial interval physics, while the spatial correlations of the scalar at the smallest scales are mainly controlled by the viscous part of the flow. (Here, the spatial correlations of the scalar are logarithmic, like in the stationary case [@95CFKLa].) An analogous situation is realized if the scalar is advected by a chaotic (spatially smooth) flow in a box or a pipe. Advection of the scalar slows down in the boundary region, while mixing continues to be efficient in the bulk, i.e. far from the boundaries. The result is that the boundary region supplies the scalar into the bulk. Plumes (blobs) of the scalar, injected from the boundary domain into the box center, cascade in a fast way down to $r_d$ where diffusion smears it out. Therefore, the temporal decay of the scalar is mainly controlled by the rate of scalar injection from the boundary layer. This causes an essential slow down of the decay in comparison with the pure Batchelor case, since in this case there are no stagnation (boundary) regions. The decay of the scalar is algebraic during the transient stage, when the scalar is supplied to the bulk from the boundary region which is wider than the diffusive boundary layer. The boundary region width decreases with time. Whenever the boundary layer width becomes of the order of the diffusive layer width the decrease stops and the algebraic temporal decay turns into an exponential one. The exponential decay rate appears to be parametrically smaller than in the pure Batchelor case.
In all the cases discussed, we consider decay of a passive scalar, $\theta$, described by the equation $$\begin{aligned}
\partial_t\theta+({\bm v}\nabla)\theta
=\kappa\nabla^2\theta,
\label{theta} \end{aligned}$$ where $\bm v$ is the flow velocity and $\kappa$ is the diffusion coefficient. Our goal is, starting from the equation (\[theta\]) and assuming some statistical properties of the velocity field $\bm v$, to describe statistics of the passive scalar decay. In order to understand the qualitative features of the passive scalar decay, we choose (in the spirit of Kraichnan [@94Kra]) the simplest possible case of a short-correlated in time velocity, possessing Gaussian statistics. This model enables us to obtain analytically a time evolution of the passive scalar correlation functions. Two situations are discussed. The first is of a statistically homogeneous flow (infinite volume) with a multi-scale velocity which is smooth at small scales, and possesses a scaling behavior at larger scales. The Reynolds number $Re=(L_v/\eta)^{4/3}$ and the Prandtl number $Pr=(\eta/r_d)^2$ are both assumed to be large. The second situation is $2d$ chaotic flow confined within a disk. We also discuss an extension to the case of a more realistic pipe flow. The flow is assumed to be smooth and obeying the standard viscous behavior in the neighborhood of the boundary: the longitudinal and transverse components of the velocities tend to zero quadratically and linearly (respectively) with the distance to the boundary.
We begin with an analysis of the homogeneous unbounded case. The statistics of a short-correlated in time Gaussian velocity is completely characterized by its pair correlation function $$\begin{aligned}
\langle v_\alpha(t_1,{\bm r}) v_\beta(t_2,0)\rangle
=\delta(t_1-t_2) \left[V_0\delta_{\alpha\beta}
-{\cal K}_{\alpha\beta}(\bm r)\right],
\label{BK1} \\
{\cal K}_{\alpha\beta}\equiv \frac{r}{2}
W'(r)\left(\delta_{\alpha\beta}
-\frac{r_\alpha r_\beta}{r^2}\right)
+\frac{d-1}{2}W(r)\delta_{\alpha\beta},
\label{BK2} \end{aligned}$$ where $d$ is dimensionality of space, $V_0$ is a constant characterizing fluctuations of the velocity at the integral scale, $L_v$, and $W(r)$ is a function, which determines the scale-dependence of velocity fluctuations. To describe both viscous and inertial ranges, we introduce $$\begin{aligned}
W=Dr^2/[1+(r/\eta)^\gamma] \,,
\label{BK3} \end{aligned}$$ where $D$ stands for the amplitude of the velocity fluctuations and $\gamma$ characterizes the velocity scaling in the inertial interval. The Kolmogorov scale $\eta$ separates the inertial and viscous intervals. The diffusive length is now expressed as $r_d=\sqrt{\kappa/D}$. Measuring time, $t$, in the units of $1/D$, one replaces $D$ by unity in all the forthcoming formulas. The object we examine here is the simultaneous pair correlation function of the passive scalar, $F(t,{\bm r})= \langle\theta(t,\bm
r)\theta(t,0)\rangle$, which is the simplest non-zero correlation function in the homogeneous case. Under the assumption of isotropy one derives from Eqs. (\[theta\]-\[BK2\]) $$\begin{aligned}
-r^{1-d}\partial_r\left\{r^{d-1}\left[W(r)+\kappa\right]
\partial_r F_\lambda\right\}=\lambda F_\lambda,
\label{Flambda} \end{aligned}$$ where $F_\lambda(r)$ is the Laplace transform of $F(t,r)$ with respect to $t$. The operator on the left-hand side of Eq. (\[Flambda\]) is self-adjoint and non-negative (with respect to the measure $\int\mbox dr\, r^{d-1}$ and under condition that $F(r)$ is smooth at the origin and $\partial_r F(r=0)=0$).
We assume that $\int\mbox d\bm r\, \theta=0$ (the Corrsin invariant is zero), and also, that initially the scalar is correlated at the scale $r_0$ from the viscous-convective range, $r_d\ll r_0\ll \eta$, i.e. $F(0,r)$ is $\approx F(0,0)$ at $r<r_0$ and decays fast enough at $r>r_0$. Then some part of the initial evolution of $F(t,r)$ is not distinguishable from the decay in the pure Batchelor case, described by the limit $\eta\to\infty$ in Eq.(\[BK3\]). Let us briefly describe this initial stage of the dacay. The solutions of Eq. (\[Flambda\]) are power-like, $F_\lambda\propto
r^{\pm\sqrt{d^2/4-\lambda}-d/2}$, outside the diffusive range, $r\gg r_d$. The functions are normalizable only if $\lambda>d^2/4$. Therefore there is a gap in the spectrum ($0<\lambda<d^2/4$ are forbidden). The existence of this gap guarantees an exponential decay of $F(t,r)$ with time. This feature is, of course, in agreement with the previous analysis of the pure Batchelor case [@99Son; @99BF]. A complete set of functions at $r\gg r_d$ is $F_\lambda=r^{-d/2}\cos[\alpha_k+k\ln(r/r_d)]$, where $k$ is a positive number related to $\lambda$ via $\lambda=d^2/4+k^2$, and $\alpha_k$ are phases, determined by matching at $r\sim r_d$. Taking into account the orthogonality condition for $F_\lambda$, $\int\mbox dr\,r^{d-1}F_\lambda(r)
F_{\lambda'}(r)=\pi\delta(k-k')/2$, one can express $F(t,r)$ via $F(0,r)$. Analysis of this expression leads to the following picture. The correlation function, $F(t,r)$, does not change at $r<r_-=r_0\exp(- td)$. At the scales larger than $r_-$, $F$ decays exponentially, $\propto\exp[-t d^2/4]$. Note, that $r_-$, also, marks the position of the maximum of the spatial derivative of $F(t,r)$, $\partial_r F(t,r)$. Therefore, $r_-(t)$ describes the front running downscale (from $r_0$). The front reaches $r_d$ at $t_d=\ln(r_0/r_d)/d$, so that at $t>t_d$, $F(t,r)$ decays exponentially at all the small scales. Complimentary to the front running downscale, there exists another front running upscale. Indeed, the integral quantity, $\int_0^s\mbox dr\,r^{d-1}F(t,r)$, which determines the overall amount of the scalar at all the scales smaller than $s$, achieves its maximum around $r_+=r_0\exp(td)$.
At $t>t_\eta\sim\ln[\eta/r_0]/D$ the pure Batchelor description ceases to be valid and one does need to account for complete multiscale form of eddy-diffusivity function $W(r)$ given by Eq. (\[BK3\]). Our further analysis is devoted to this general case. The spatial decay of the functions $F_\lambda$ in the multi-scale case is not as steep as in its pure Batchelor one. It is convenient to change to a new field, $\Phi_\lambda$, $F_\lambda=r^{1-d}\partial_r[r^{d/2} \Phi_\lambda(z)]$, where $z=2\sqrt{\lambda}r^{\gamma/2}/\gamma$. The general solution of the resulting equation for $\Phi_\lambda(z)$ is a linear combination of the Bessel functions, $\Phi_\lambda= J_{\pm\nu}(z)$, where $\nu=2\sqrt{d^2/4 -\lambda}/\gamma$. If $\lambda<d^2/4$, the positive root for $\Phi_\lambda$ has to be chosen to satisfy the matching conditions at $r_d$. The eigen function is normalizable for any positive $\lambda$, so that even the smallest positive $\lambda$ are not forbidden. The lack of the gap in the spectrum of the multi-scale model means an algebraic in time decay of $F(t,r)$, i.e. much slower decay than the one found for the Batchelor-Kraichnan model. We present here the expression for the long-time, $t\gg t_\eta$, asymptotic behavior of $F(t,r)$: $$\begin{aligned}
F(t,r)\!\propto \!\left\{
\begin{array}{cc}
\left[t d+\ln(r/\eta)
\!+\!(r/\eta)^\gamma/\gamma\right]^{-1-d/\gamma},
& r\ll r_+; \\
t^{-5/4-d/2\gamma}r^{-d/2+\gamma/4}, & r\gg r_+
\end{array}\right.
\label{F2multi} \end{aligned}$$ Here $r_+=(\gamma t)^{1/\gamma}$ stands for position of the front running upscale, $r_+$ lies in the scaling region $r_+\gg \eta$, where the multi-scale model turns into the so-called Kraichnan model [@94Kra], $W\to r^{2-\gamma}$. (Therefore, it is not surprising, that our result (\[F2multi\]) is consistent at $r\gtrsim r_+$ with the expression for the pair correlation function derived before for the Kraichnan model [@00EX; @01CEFV] in the regime of zero Corrsin invariant.) Note, that when $t-t_\eta$ is yet moderate in value, the small-scale part of the asymptotic expression (\[F2multi\]) is correct only at $r\gg R_-(t)=\eta e^{-td}$, where $R_-$ is the position of the front initiated at $t\sim t_\eta$ and running downscale from $\eta$. At $t\sim t_\eta+\ln(\eta/r_d)/d$, the downscale front reaches the dissipative scale, and afterwards the expression (\[F2multi\]) is correct for any scales $r\gg r_d$. Eq. (\[F2multi\]) shows that just like in the pumped case [@59Bat; @67Kra; @95CFKLa] the scalar structure function is logarithmic, $\langle[\theta(t,\bm r)
-\theta(t,0)]^2\rangle \sim t^{-2-d/\gamma}\ln[r/r_d]$, at $r_d\ll r\ll \eta$ and large times, where, therefore, the time-dependent factor at the logarithm can be interpreted as a scalar flux from the inertial range, $r\gtrsim \eta$, downscale to the viscous range, $r\lesssim \eta$.
To conclude, the existence of the inertial region in velocity affects drastically the spatio-temporal distribution of the scalar in the viscous range (where the velocity can be approximated by linear profiles). The inertial interval, where the scalar is mainly concentrated, serves as a kind of reservoir for the smaller scales. This explains why the decay of scalar correlation function in the viscous is much slower (algebraic) than in the pure Batchelor case (when it would be exponential).
Now we proceed to the situation of spatially bounded flows. Aiming to demonstrate the main qualitative features of the confined geometry, that is strong sensitivity of the scalar decay to the peripheral (close to the boundary) part of the flow, we examine a model case of $2d$ chaotic (i.e. consisting of only few spatial harmonics) flow inside a disk. (It is, actually, clear that the picture of temporal decay and spatial distribution of the scalar, described below, is universal, i.e. it applies to a typical chaotic flow confined in a close box, particularly in $3d$.) Incompressible flow in $2d$ can be characterized by a stream function $\psi$, then the radial and azimutal components of the velocity are $v_r=-\partial_\varphi\psi/r$ and $v_\varphi
=\partial_r\psi$. Our model of $\psi$ is $$\begin{aligned}
\psi=-\frac{\xi_1}{2}r^2 U(r)\sin(2\varphi)
+\frac{\xi_2}{2}r^2U(r)\cos(2\varphi) \,,
\label{streamU} \\
\langle\xi_i(t_1)\xi_j(t_2)\rangle
=2D\delta_{ij}\delta(t_1-t_2) \,,
\label{xi} \end{aligned}$$ where $U(r)$ is a function of $r$, finite at the origin, $U(0)=1$, and becoming zero, together with its first derivative, at the disk boundary, $r=1$, and $\xi_{1,2}(t)$ are zero mean short correlated random Gaussian functions. The value of the passive scalar, $\theta$, averaged over the statistics of $\xi$, $\langle\theta(t,\bm r)\rangle$, is the object of our interest here. One examines how the average concentration of the scalar evolves with time at different locations ${\bf r}$ within the disk. The short-correlated feature of the velocity field allows a closed description for $\langle\theta\rangle$ in terms of a partial differential equation. Considering the spherically symmetric part of $\langle\theta\rangle$ only (asymptotically, at large times only this $varphi$-independent part remains essential), and passing from $\langle\theta\rangle$ and $r$ to $\Upsilon$ and $q$, respectively, where $\langle\theta\rangle= r^{-1}\partial_r
[r^2\Upsilon(t,q)]$ and $q=-\ln r$, one finds that the Laplace transform of $\Upsilon$ with respect to $t$ (measured, again, in $D^{-1}$ units) satisfies $$\begin{aligned}
&&
(U^2+\kappa e^{-2q})
(\partial_q^2\Upsilon_\lambda -2\partial_q\Upsilon_\lambda)
+\lambda\Upsilon_\lambda=0.
\label{Philambda} \end{aligned}$$ The diffusion term is important only at the center of the disk and near the boundaries. Analysis of Eq. (\[Philambda\]) is straightforward but bulky. Below we will present only selected details of the analysis, aiming to describe the general picture of the phenomenon. (The complete account for the derivation details will be published elsewhere [@02CLT].)
In the bounded flow the decay of passive scalar splits into three distinct stages. The major effect dominating the first stage (just as in the pure Batchelor case, explained by Eq. (\[Philambda\]) with $U\to1$) is formation of elongated structures (stripes) of the scalar in the bulk region of the flow. The stripes are getting thinner with time, i.e. inhomogeneities of smaller and smaller scales are produced. Once the width of the stripes decreases down to the dissipative scale, $r_d\sim\sqrt{\kappa}$, the stripes are smeared out by diffusion. The stretching-contraction process is exponential in time, so that the initial stage when the stripes are formed lasts for $\tau_1\sim\ln[1/r_d]$. By the end of this first stage the scalar is exhausted in the central region of the flow. The stretching rate, however, is smaller in the peripheral domain than in the bulk. Thus the scalar literally remains longer in the peripheral domain. This defines the second, transient, stage. At $\kappa^{1/4}\ll q\approx 1-r
\ll\lambda^{1/4}\ll 1$, the solution of Eq. (\[Philambda\]) is $\Upsilon_\lambda\propto q \sin\left(\alpha_\lambda +\sqrt\lambda q/U\right)$. In the other asympotitc region at $q\gg\lambda^{1/4}$ one can drop the second derivative (with respect to $q$) and diffusive terms in Eq. (\[Philambda\]). One finds that, $\partial_q\ln\Upsilon_\lambda\sim 1$ at $q\sim\lambda^{1/4}$. Matching those two asymptotic regions at $q\sim \lambda^{1/4}$ we come to the $\alpha_\lambda=0$ condition. Once the asymptotic form of the eigen-function $\Upsilon_\lambda$ is known, it is straightforward to restore the behavior of $\Upsilon(t,q)$. One finds that $\Upsilon(t,q)$ (and, therefore, $\langle\theta\rangle$) is concentrated in the $\delta(t)\sim 1/\sqrt{t}$-small (and shrinking with time) vicinity of the boundary. In the domain outside of the shrinking layer the decay is algebraic, $\langle\theta\rangle\propto
t^{-3/2}q^{-3}$. The second stage lasts for $\tau_2\sim\kappa^{-1/2}$, i.e. until $\delta(t)$ shrinks down to $r_{bl}=\kappa^{1/4}$, which is the width of the diffusive boundary layer. Account for diffusivity in the $q\ll 1$ analysis gives yet another matching condition for $\Upsilon_\lambda$, at $r\sim r_{bl}$, resulting in formation of a discrete spectrum for $\lambda$. The smallest $\lambda$ (and therefore the value of the level spacing in the discrete spectrum) is estimated by $\sqrt\kappa$. (Using the periodic character of the $\sin$-function, one finds that the $n$-th level eigenvalue $\lambda_n$ is estimated by $\sim n^2\sqrt\kappa$ at large $n$.) Therefore, at $t\gg\tau_2$, $\langle\theta\rangle$ decays exponentially, $\propto\exp(-t/t_d)$, where $t_d\sim\kappa^{-1/2}$. The smallest $\lambda$ eigen-function is localized at $q\sim r_{bl}$. Thus the third (final) stage of evolution is characterized by the majority of the scalar remaining in the diffusive boundary layer.
To conclude, the temporal behavior of the passive scalar correlations, predicted by the bounded flow theory, is complicated. It involves two different exponential regimes separated by an algebraic one, so that a special accuracy is required in order to quantify the theory against various experiments in the field.
The bounded flow theory can be applied to the experiment of Groisman and Steinberg [@01GSb], where the passive scalar is advected through a pipe by dilute-polymer-solution flow. The polymer-related elastic instability makes the flow chaotic. The mean flow along the pipe is also essential, so that according to the standard Taylor hypothesis, measurements of scalar concentration at different positions along the pipe are interpreted as correspondent to different times in an artificial decay problem. Exponential decay of the scalar was reported in [@01GSb] at long times (long pipes). The theoretical picture explaining the experiment is similar to the one proposed above for the disk. The width of the diffusive boundary layer near the pipe boundaries is estimated by $R\cdot{Pe}^{-1/4}$, where $R$ is the pipe radius and $Pe$ is the Peclet number, $Pe=R^2 \sigma/\kappa$ and $\sigma$ is a typical value of the velocity gradient (${Pe}$ is $\sim 10^4$ in the conditions of [@01GSb]). The characteristic decay time (fixed by the diffusive boundary layer width) is estimated by $t_d\sim\sigma^{-1}{Pe}^{1/2}$. However, one should be careful when converting this time to the decrement of the passive scalar decay along the direction of the flow (pipe). The average velocity of the flow tends to zero near the boundary of the pipe, making the advection near the boundary less efficient than in the bulk. Assuming a linear profile of the average velocity near the boundary, one gets the following estimate for the law of scalar fluctuation decay along the pipe, $\sim\exp(-\gamma z/u_0)$, where $z$ is the coordinate in the direction of the pipe, $u_0$ is the average velocity at the center of the pipe, and $\gamma\sim\sigma\,{Pe}^{-1/4}$. The factor $Pe^{-1/4}$ here is a manifestation of a slow down of the passive scalar decay in comparison with the pure Batchelor case. The same factor characterizes decay of higher order correlation functions of the scalar as well. For example, one can consider the passive scalar pair correlation function $F(r)$ near the center of the disk. One finds that $F(r)\propto r^{-\alpha}$, where $\alpha$ is small and can be estimated as $\alpha\sim\,\mbox{Pe}^{-1/4}$. (Note, that this behavior is almost indistinguishable from the logarithmic one, found for the stationary case [@59Bat; @67Kra; @95CFKLa].) This estimations agree with the experimental data of [@01GSb].
Let us now briefly discuss two other available numerical and experimental results. Pierrehumbert [@93Pie] reported exponential decay of the scalar correlations in his numerical experiment with chaotic map velocity in a periodic box. This observation is consistent with our results, since the periodic boundary conditions are less restrictive than the zero condition for velocity at the box boundary, and, therefore, should leave a finite gap in the $\lambda$-spectrum even in the limit $\kappa\to0$. In the experiment of Jullien, Castiglione and Tabeling on $2d$ stationary flow steered by magnets in a finite volume beneath the $2d$ layer [@00JCT], the decay rate of the scalar seems slower than exponential (Fig. 2 of [@00JCT]). This observation is in agreement with the absence of the gap in the $\lambda$-spectrum (at $\kappa\to0$) we found for the finite box model.
The brevity of this letter does not allow us to discuss effects of intermittency (which manifests in high-order moments of the scalar), of higher order angular harmonics (in the disk or pipe geometry), and details of the spatial distribution of the scalar in a variety of other inhomogeneous cases (e.g. periodic flow). This detailed discussion is postponed for a longer paper to be published elsewhere [@02CLT] (where we will also present results of direct numerical simulations of the passive scalar decay for these cases).
We thank B. Daniel, R. Ecke, G. Eyink, G. Falkovich, A. Fouxon, A. Groisman, I. Kolokolov, M. Riviera, V. Steinberg, P. Tabeling and Z. Toroczkai for helpful discussions.
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---
abstract: 'Brightest Cluster Galaxies (BCGs) in massive dark matter halos are shaped by complex merging processes. We present a detailed stellar population analysis in the central region of Abell 3827 at $z\sim0.1$, including five-nucleus galaxies involved in a BCG assembly. Based on deep spectroscopy from Multi Unit Spectroscopic Explorer (MUSE), we fit the optical spectra of 13 early-type galaxies (ETGs) in the central $70$ kpc of the cluster. The stellar populations in the central $R=1$ kpc of these ETGs are old ($>6$ Gyr). Their \[Fe/H\] increases with $\sigma_{\star}$ and stellar mass. More importantly, \[$\alpha$/Fe\] of galaxies close to the cluster center do not seem to depend on $\sigma_{\star}$ or stellar mass, indicating that the cluster center shapes the \[$\alpha$/Fe\]–$\sigma_{\star}$ and \[$\alpha$/Fe\]–$M_{\star}$ relations differently than other environments where \[$\alpha$/Fe\] is observed to increase with increasing $\sigma_{\star}$ or $M_{\star}$. Our results reveal the coordinated assembly of BCGs: their building blocks are different from the general low mass populations by their high \[$\alpha$/Fe\] and old ages. Massive galaxies thus grow by accreting preferentially high \[$\alpha$/Fe\] and old stellar systems. The radial profiles also bear the imprint of the coordinated assembly. Their declining \[Fe/H\] and flat \[$\alpha$/Fe\] radial profiles confirm that the accreted systems have low metallicity and high \[$\alpha$/Fe\] stellar contents.'
author:
- 'Meng Gu, Charlie Conroy and Gabriel Brammer'
title: Coordinated Assembly of Brightest Cluster Galaxies
---
Introduction
============
According to the $\Lambda$–Cold Dark Matter model, galaxy assembly is closely linked to the hierarchical growth of dark matter structures. Local massive early-type galaxies (ETGs) are considered to have evolved from the compact “red nuggets” at $z\approx2$ by doubling their stellar masses and increasing their effective radii by a factor of $3-5$ [e.g. @vanDokkum2010; @Patel2013]. Recent simulations describe this transformation by the two-phase scenario [e.g. @Naab2009; @Oser2010; @Oser2012], in which massive ETGs experience strong dissipational processes that lead to rapid and concentrated mass growth at high redshifts, and accrete low mass systems at later times to build up the outer envelopes. Brightest cluster galaxies (BCGs) are a special class of ETGs at the extreme high-mass end of the stellar mass function and in the densest environments. They have diffuse and extended envelops [e.g. @Schombert1988] that can be explained by a series of merging events [e.g., @Ostriker1975; @Hausman1978; @Dubinski1998].
It is still ambiguous whether the low mass galaxies we observe today are intrinsically different from the building blocks of massive galaxies, or their surviving counterparts. One useful approach is to compare the abundance trends of their stellar contents, especially the ratio between $\alpha-$elements to iron. \[$\alpha$/Fe\] is used to indicate the star formation timescales due to its sensitivity to the time delay between SNe II and SNe Ia [e.g. @Tinsley1979]. SNe II from massive stars yield both $\alpha$–elements and Fe, while SNe Ia from low mass binary systems contribute mostly Fe on longer timescales. High \[$\alpha$/Fe\] in old stellar population suggests short star formation timescales in the past. Stellar population analysis has revealed correlations between stellar population properties and stellar mass or stellar velocity dispersion [e.g. @Trager2000; @Worthey2003; @Thomas2005; @Schiavon2007; @Thomas2010; @Conroy2014; @McDermid2015] in a way that massive galaxies are older, more metal rich and more $\alpha$–enhanced compared to low mass galaxies. If these trends are universal for all environments and epochs, an apparent tension under the hierarchical assembly paradigm would emerge: the building blocks of massive galaxies, especially the BCGs would have “diluted” the \[$\alpha$/Fe\] at the high mass end, and/or would produce steep \[$\alpha$/Fe\] gradients. This would make it difficult to reconcile with the general observational facts that more massive galaxies are more $\alpha$–enhanced.
0.1cm
The Milky Way satellite dwarf galaxies were once thought to be the surviving counterparts of Galactic building blocks until studies revealed their stellar populations occupy different locations from Milky Way halo stars on the \[$\alpha$/Fe\] vs. \[Fe/H\] diagram [e.g. @Tolstoy2009]. Looking beyond the Milky Way, @Liu2016 showed that the \[$\alpha$/Fe\] of low mass ETGs in the Virgo cluster depend on the distance to the cluster center and the \[$\alpha$/Fe\]–$\sigma_{\star}$ relation in this cluster has larger scatter, indicating that the densest environments quench low mass galaxies earlier than other environments. From these studies it seems that the assembly of massive galaxies are coordinated in a way that their building blocks have early truncated star formation histories, making them a particular sample with high \[$\alpha$/Fe\] among the low mass systems.
In this Letter, we present stellar population analysis on 13 ETGs in Abell 3827. Five of them are involved in a rarely observed BCG assembly from multiple mergers, therefore we are fortunate to directly analyze the building blocks of a prospective BCG. \[Mg/Fe\] is used as a tracer of \[$\alpha$/Fe\]. We compare the stellar population scaling relations in this special environment to the general samples in previous work. We assume a flat $\Lambda$CDM cosmology with $h=0.73$, $\Omega_m=0.27$, $\Omega_{\Lambda}=0.73$. The redshift of Abell 3827 is $cz=29500$ [@Struble1999]. The distance is assumed to be 433 Mpc. This corresponds to a distance modulus of 38.18 mag and a scale of 1.74 kpc arcsec$^{-1}$. All magnitudes in this paper are in the AB system. The mass center of Abell 3827 is assumed to be at RA $=22^h 01^m 52.90^s$, DEC $=-59^d56^m44.89^s$ [@Massey2015]. We assume the $r$ band solar absolute magnitude to be 4.76 mag [@Blanton2003].
Data and methods
================
We use the datacube obtained by the Multi-Unit Spectroscopic Explorer (MUSE) Integral Field Unit (IFU) spectrograph [@Bacon2010; @LeFevre2013] on the European Southern Observatory (ESO) Very Large Telescope (VLT). Abell 3827 was observed in 2015 (ESO programme 295.A-5018(A), PI: Richard Massey, @Massey2015 [@Massey2017]), centered at RA $=22^h 03^m 14.65^s$, DEC $=-59^d56^m43.19^s$. We reduce and combine the data using MUSE Pipeline [muse-2.2]{} [@Weilbacher2014; @Weilbacher2016]. According to @Massey2017, observations were taken in dark time with seeing around $0.7^{\prime\prime}$. The total integration time in the final datacube is $3.2$ hr. The field of view (FoV) is $1$$\times1$. The wavelength coverage is $475-935$ nm, sampled at 1.25Å/pixel, with mean spectral resolution $\sim 3000$ at the optical wavelength range. The spatial pixel size is $0.2$$\times0.2$. Due to the small FoV we limit the sky regions during the data reduction, by setting [skymodel‘\_fraction]{} to $0.01$. As a result, the surface brightness profiles in of the MUSE datacube are consistent with the /ACS image [^1] out to $\mu_{\rm{F606W}}=24$mag arcsec$^{-2}$, with $\Delta\mu_{\rm{F606W}}\approx0.22$mag arcsec$^{-2}$ at $\mu_{\rm{F606W}}=24$mag arcsec$^{-2}$.
0.15cm
Galaxies in Abell 3827 and foreground stars are identified using SExtractor [@Bertin1996]. 13 ETGs in Abell 3827 are selected for further analysis. They are all confirmed spectroscopically and are shown in Figure 1. We mask out foreground stars and background lensed galaxies (whited hatched region). To study the radial profiles of stellar populations, we analyze the spectra by binning spaxels in radial directions for N1–N4 and N9. The binning scheme is shown in the right panel of Figure 1.
To model galaxies spectra we use the absorption line fitter [[alf]{}, @Conroy2012; @Conroy2014; @Conroy2018]. [alf]{} enables stellar population modeling of the full spectrum for stellar ages $>1$Gyr and for metallicities from $\sim-2.0$ to $+0.25$. Parameter space is explored using a Markov Chain Monte Carlo algorithm [[emcee]{}, @ForemanMackey2013]. [alf]{} adopts the MIST stellar isochrones [@Choi2016] and uses a new spectral library [@Villaume2017] that includes continuous wavelength coverage from $0.35-2.4\mu m$ over a wide range in metallicity, which taken from new IRTF NIR spectra for stars in the MILES optical spectral library [@SanchezBlazquez2006]. Theoretical elemental response functions were computed with the ATLAS and SYNTHE programs [@Kurucz1970; @Kurucz1993]. They tabulate the effect on the spectrum of enhancing each of the individual elements. With [alf]{} in “full” mode we fit for parameters including a two burst star formation history, the redshift, velocity dispersion, overall metallicity, 18 individual element abundances, several IMF parameters [@Conroy2018]. Throughout this paper, we use [alf]{} with the IMF fixed to the @Kroupa2001 form. We use flat priors within these ranges: $-10^3$–$10^5$ km/s for recession velocity, $100$–$1000$ km/s for velocity dispersion, $1.0$–$14$ Gyr for age and $-1.8-+0.3$ for metallicities. For each spectrum we fit a continuum in the form of a polynomial to the ratio between model and data. The order of polynomial is $(\lambda_{max} - \lambda_{min})/100$Å. During each likelihood call the polynomial divided input spectrum and model are matched. The continuum normalization occurs in three separate wavelength intervals, $4300-5080$Å , $5080-5700$Å and $5700-6700$Å .
Five galaxies have central velocity dispersion smaller than the resolution of the models ($100$ km/s): N5, N6, N11, N12, and N15. Their spectra are smoothed by convolving a wavelength dependent Gaussian kernel with $\sigma=\sqrt{100^2-{\sigma_i}^2}$ prior to the modeling, where $\sigma_i$ is the wavelength dependent instrumental resolution.
To study the spatial distribution of stellar population parameters, spectra in adjacent spaxels are binned by Voronoi tessellation [@Cappellari2003]. The mean S/N in observed frame $4500$–$5000$Å of this binning scheme is shown in the top left panel of Figure 2. The S/N is between $40$Å$^{-1}$ and $110$Å$^{-1}$. In addition to foreground stars and lensed galaxies, We also exclude bins where foreground and background galaxies overlap with each other, e.g., some spaxels between N4 and N2. Only bins where the best-fit spectra and residuals are visually consistent with data and data uncertainties are shown here. The low S/N at the outskirts ($R>3$ kpc) of low mass ETGs makes it hard to derive reliable mass to light ratio (M/L). Therefore, we assume constant M/L as a function of radius and adopt the M/L measured within $R=1$ kpc. The stellar masses within $R=5$ kpc of the 13 ETGs are derived by multiplying the rest-frame $r$ band total integrated luminosity within $R=5$ kpc, by the best-fit M/L in $r$ band within $R=1$ kpc.
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Results and Discussion
======================
We present results in this section. Objects N1, N2, N3, N4 and N9 have the closest projected distances to the mass center ($\leq18$kpc). Their recession velocities relative to the cluster center ($cz=29500$ km/s) are $cz=-45^{+4}_{-3}$ km/s, $167^{+3}_{-3}$ km/s, $359^{+3}_{-3}$ km/s, $-830^{+4}_{-3}$ km/s and $452^{+4}_{-4}$ km/s, respectively. Given their small projected distances to the cluster center and very similar velocities, they are on their way to form a massive BCG in the near future. If we estimate the time it would take based the dynamical friction time [@Binney1987], the 5 ETGs will merge within $1$ Gyr. Therefore we assume N1–N4 and N9 are building blocks of a prospective BCG with $\log{(M_{\star}/M_{\odot})}\approx11.7$.
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Spatial distribution of the following parameters are shown in Figure 2: $cz$, $\sigma_{\star}$, stellar age, \[Fe/H\] and \[Mg/Fe\], focusing on $D\approx\pm25$ kpc around the cluster center. The $cz$ distribution shows that the five ETGs are all members of the cluster. The $\sigma_{\star}$ map shows that $\sigma_{\star}$ declines outwards from the center of N1–N4 within $R\sim2$ kpc, and rise towards the outskirts, reaching to $\sim400-500$ km/s. This shows that the stars in the outskirts are tracing the gravitational potential of the cluster, instead of any individual galaxy, and describes the formation of the cD envelope, or the intracluster light component that have been measured in other galaxy clusters [e.g. @Kelson2002]. Due to the large $\sigma_{\star}$, the typical errors of $\log{(\mathrm{age/Gyr})}$, \[Fe/H\] and \[Mg/Fe\] are $\sim0.1$ dex in bins with S/N$\approx50$Å$^{-1}$. The stellar age distribution shows this region is uniformly old. From the \[Fe/H\] distribution we see declining \[Fe/H\] from the galaxy centers outwards, indicating that the inner regions of the galaxies are more metal rich than the cD envelop where the stellar content is likely from disrupted low mass systems. The distribution of \[Mg/Fe\] shows that the $\alpha$–abundance in this region are generally high, with no particular pattern of differences between the galaxy centers and the cD envelope, indicating that the stellar content have short star formation timescales in general.
Figure 3 shows the main result. We compare the median stellar population parameters within $R=1$ kpc, including \[Fe/H\], stellar ages and \[Mg/Fe\] as a function of galaxy velocity dispersion $\sigma_{\star}$ and galaxy stellar mass. Colors indicate the projected distance to the cluster mass center. Previous studies found scaling relations between stellar population properties and the central $\sigma_{\star}$: the stellar components in galaxies with higher central $\sigma_{\star}$ are older, more metal rich and more $\alpha$–enhanced [e.g. @Trager2000; @Thomas2005; @Schiavon2007; @Graves2009; @Conroy2014]. We compare our results to previous studies in Figure 3, including a large sample of morphologically selected SDSS ETGs in @Thomas2005 (red dash–dotted line), ETGs in ATLAS$^{3D}$ by @McDermid2015 (blue circles), 12 ETGs in the Coma cluster by @Trager2008 (green triangles). We estimate \[Fe/H\] in the above literatures using Eq 4 in @Thomas2003 assuming the factor $A=0.94$ [@Trager2000]. We also compare to stacked SDSS ETGs that are binned in $\sigma_{\star}$ and stellar mass in @Conroy2014, except that they are fit with the up-to-date [alf]{} with upgraded response functions.
For stellar age and \[Fe/H\], our results agree with the trends found in the general ETG populations: galaxies with higher central stellar velocity dispersion or larger stellar mass are older and more metal rich. For \[Mg/Fe\] our results are distinctly different from ETGs in the field. The \[Mg/Fe\] of galaxies within $\sim 40$ kpc to the cluster center have high \[Mg/Fe\], even indicating a trend that smaller galaxies are more \[Mg/Fe\] enhanced. If we assume the variation of \[Mg/Fe\] is due to the differences of star formation timescale [e.g. @Thomas2005], Figure 3 illustrates that compared to galaxies in all environment, the ETGs with lower $\sigma_{\star}$ or stellar masses within $\sim 40$ kpc in Abell 3827 are quenched as early (if not more) as more massive galaxies. Our results highlight the effect of “environmental quenching” [e.g. @Peng2010] on low mass galaxies, possibly due to the complex interplay between processes such as ram pressure stripping and strangulation. The high \[$\alpha$/Fe\] and old stellar ages make the building blocks very different from low mass galaxies in general, and are possibly due to early quenching by the dense environment.
As described earlier N1–N4 and N9 are very likely building blocks of a BCG in the future. Although multiple nucleus are rarely observed, it has been predicted by recent simulations that major merger plays an important role on buildup of massive galaxies [e.g. @RodriguezGomez2016]. We estimate the stellar population properties of the prospective BCG through weighting the properties of the five ETGs by luminosity. The lower limit of its stellar mass is estimated using the total stellar mass of N1–N4 and N9 within $R=5$ kpc: $\log{(M_{\star}/M_{\odot})}\approx11.7$. The diffuse component is not included due to the contamination from the lensed galaxies and the foreground stars. The predictions are shown as black stars in Figure 3. This prospective BCG would fall on all the empirical trends between stellar population parameters and stellar mass.
0.15cm
The coordinated assembly picture can be described by the schematic diagrams in Figure 5: the building blocks of BCGs are low mass galaxies in a special environment–cluster centers. They follow a relatively flat relation between \[$\alpha$/Fe\] or stellar age and stellar mass (red box), which are distinct from the relations followed by galaxies in the fields (blue box). The right panel shows the expected radial profiles of the massive ETGs: As the low mass systems in the cluster center accrete onto the outskirts of massive ETGs, the coordinated assembly produces flat radial profiles of stellar age and \[$\alpha$/Fe\] (red), whereas if the accreted systems are random draws from the low mass galaxy sample in all environments the profiles of stellar age and \[$\alpha$/Fe\] would decline with radius.
The radial profiles of stellar population properties confirm the coordinated assembly picture. Figure 4 shows the radial profiles of velocity dispersion $\sigma_{\star}$, stellar ages, \[Fe/H\], and \[Mg/Fe\] for N1–N4 and N9. The binning schemes are shown in the right panel of Figure 1. These profiles are compared to that of 6 ETGs in @vanDokkum2017. Note that we only show the combined profiles from both sides of the 6 ETGs. Due to the contamination of the foreground stars and background lensed galaxies, we are not able to extend the radial trends beyond $14$ kpc. However, we can already see rising velocity dispersion profiles built up for N1–N3, which is consistent with the expected cD galaxy profiles [e.g. @Kelson2002]. The stellar ages profiles indicate that the ages of the 5 ETGs are generally old from the center to the outskirts. \[Fe/H\] declines with the radius, and seems to depend on the central velocity dispersions within $R=2$ kpc. This is consistent with the two phase formation scenario [e.g. @Naab2009; @vanderWel2014] that the inner regions are mostly build up by the dissipational process at high redshifts, thus strongly depend on the central stellar velocity dispersions, and the outskirts are dominated by accretion of small stellar systems. \[Mg/Fe\] profiles are generally flat. Flat \[$\alpha$/Fe\] profiles have also been observed previously [e.g. @Sanchezblazquez2007; @Greene2015]. This is consistent with the coordinated assembly picture.
Summary
=======
We have presented detailed stellar population studies for 13 ETGs in Abell 3827 using the optical spectra from MUSE on the VLT. The sample includes five ETGs that are involved in an ongoing assembly of a BCG. Our conclusions are summarized as follows:
- The 13 ETGs are spectroscopically confirmed members of Abell 3827. Their stellar age and \[Fe/H\] fall on empirical trends that galaxies with higher $\sigma_{\star}$ or stellar mass are older and more metal rich. However, ETGs within $40$ kpc from the cluster center show higher \[Mg/Fe\] compared to the \[Mg/Fe\]–$\sigma_{\star}$ and \[Mg/Fe\]–$M_{\star}$ relations in the field.
- From the spatial distribution in the central region of the cluster, the stellar populations in the diffuse stellar light of Abell 3827 are generally old and $\alpha$–enhanced.
- We show the radial profiles of $\sigma_{\star}$, stellar age, \[Fe/H\] and \[Mg/Fe\] that are consistent with previous studies. The flat stellar age and \[Mg/Fe\] profiles confirm the coordinated assembly picture.
- Our results highlight the effect of “environmental quenching”, and reveal the coordinated assembly of BCGs: the building blocks of the prospective BCG in Abell 3827 are distinct from the general low mass systems by high \[$\alpha$/Fe\] due to early quenching by the dense environment.
Future spectroscopic observations of cluster centers will place constraints on the formation history of massive galaxies, and will shed additional light on the general picture of coordinated assembly.
M.G. acknowledges support from the National Science Foundation Graduate Research Fellowship. C.C. acknowledges support from NASA grant NNX15AK14G, NSF grant AST-1313280, and the Packard Foundation.
This research has made use of the services of the ESO Science Archive Facility, based on observations collected at the European Organization for Astronomical Research in the Southern Hemisphere under ESO programme 295.A-5018(A), PI: Richard Massey, and based on data obtained from the ESO Science Archive Facility under request number 345883. The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
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[^1]: Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Mikulski Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. These observations are associated with program 12817.
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abstract: 'This is the first nontrivial construction to date of instantons over a compact manifold with ${\textnormal {Hol\,}}=G_2$. The HYM connections on asymptotically stable bundles over Kovalev’s noncompact Calabi-Yau 3-folds, obtained in the first article [@G2I], are glued compatibly with a twisted connected sum, to produce a $G_2-$instanton over the resulting compact $7-$manifold [@kovalevzinho; @kovalevzao]. This is accomplished under a nondegeneracy *acyclic* assumption on the bundle ‘at infinity’, which occurs e.g. over certain projective varieties $X_{22}\hookrightarrow \mathbb{C}P^{13}$ [@X22; @Mukai; @Fano; @3-folds] equipped with an *asymptotically rigid* bundle.'
author:
- 'Henrique N. Sá Earp'
title: '$G_2-$instantons over Kovalev manifolds II'
---
Introduction {#introduction .unnumbered}
============
The first paper [@G2I] was devoted to the Hermitian Yang-Mills (HYM) problem over A. Kovalev’s noncompact Calabi-Yau 3-folds $W$ with an exponentially asymptotically cylindrical (EAC) end. Its main theorem asserts the existence of such a metric on holomorphic bundles $\mathcal{E}\rightarrow W$ which are *asymptotically stable*, i.e., (slope-)stable over the divisor ‘at infinity’ $D\subset \bar{W}$. Moreover, solutions have ‘$C^\infty-$exponential decay’ along the tubular end to a reference metric, which extends the instanton metric on $\left.{\mathcal{E}}\right\vert_D$ in a prescribed way. On the other hand, it is known that a HYM connection $A$ on a holomorphic vector bundle over a Calabi-Yau $3-$fold $\left(W,\omega,\Omega\right)$ lifts to a $G_{2}-$instanton on the pull-back bundle $p_1^*\mathcal{E}\rightarrow M=W\times S^{1}$, hence the above result yields a non-trivial solution of the corresponding instanton equation: $$\label{sd part kahler -> G2}
F_A\wedge\ast\varphi
\doteq\tfrac{1}{2}F_{A}\wedge \left( \omega \wedge \omega -2{\textnormal {Re\,}} \Omega \wedge d\theta \right)=0$$ (where $d\theta$ is the coordinate 1-form on $S^1$ and $\varphi$ is the induced $G_2-$structure).
This paper will be concerned with the natural sequel, the gluing of two such solutions according to Kovalev’s twisted connected sum of the base manifolds. That construction joins two ‘truncated’ EAC products $W_{S}^{(i)}\times S^{1}$ in order to obtain a smooth *compact* $G_2-$manifold $$M_{S}=\left( W_{S}^{\prime }\times S^{1}\right) \cup _{F_{S}}\left(
W_{S}^{\prime \prime }\times S^{1}\right) \doteq W'\widetilde{\#}_{S}W'',$$where $S$ is the ‘neck-length’ parameter and $F_S$ is the product of a hyper-Kähler rotation on the divisors ‘near infinity’ and a nontrivial identification of the circle components of the boundary. In view of the exponential decay property of solutions over each end, one is led to expect that residual self-dual curvature, i.e., error terms from truncation in the corresponding $G_2-$instanton equation over $M_S$, can be dealt with by a perturbative, ‘stretch-the-neck’-type argument.
This paper’s main result \[*Theorem \[thm gluing\]*, *Subsection \[subsec Statement of gluing theorem\]*\] posits that this is indeed the case, albeit under a rigidity assumption on the ‘bundle at infinity’. In order to guarantee a right-inverse for the linearisation of the self-dual curvature operator $A\mapsto F_A\wedge\ast\varphi$ around a solution, one requires that the corresponding deformation complex over $\left.{\mathcal{E}}\right\vert_D$ be *acyclic*. In other words, the instanton ‘at infinity’ must be an isolated point in its moduli space. Examples satisfying this hypothesis are obtained from certain base manifolds $X_{22}\hookrightarrow\mathbb{C}P^{13}$ admissible by Kovalev’s construction and studied by Iskovskih [@X22] and Mukai [@Mukai; @Fano; @3-folds].
Readers familiar with the 4-dimensional model outlined by Donaldson in [@floer] will find that this paper mimics that source in all its essential aspects.
Acknowledgements {#acknowledgements .unnumbered}
================
This second paper was written at Unicamp, supported by post-doctoral grant 2009/10067-0 from Fapesp - São Paulo State Research Council. It contains some unpublished material from my thesis, which was funded by Capes Scholarship 2327-03-1, from the Brazilian Ministry of Education.
I would like to thank Simon Donaldson, for suggesting this outcome of the first article and guiding its initial steps, and Marcos Jardim, for many useful subsequent discussions.
$G_2-$instantons over EAC manifolds
===================================
Let us briefly recall the basic facts about the Calabi-Yau $3-$folds $W_{i} $ that one intends to glue together, via a nontrivial product with $S^1$. \[def base manifold (W,w)\] A *base manifold* for our purposes is a compact, simply-connected Kähler $3-$fold $\left( \bar{W},\bar{\omega}\right) $ carrying a $K3-$divisor $D \in \left\vert -K_{\bar{W}}\right\vert$ with holomorphically trivial normal bundle $\mathcal{N}_{D/\bar{W}}$ such that the complement $W\doteq\bar{W}\setminus D$ has $\pi _{1}\left( W\right) $ finite. Topologically $W$ looks like a compact manifold $W_{0}$ with boundary $D\times S^{1}$ and a cylindrical end attached:$$\label{eq cylindrical picture}
\begin{array}{c}
W=W_{0}\cup W_{\infty } \\
W_{\infty }\simeq D\times S_\alpha^{1}
\times \left(\mathbb{R}_{+}\right)_s .\end{array}$$
The $K3$ divisor $D$ is actually hyper-Kähler, with complex structure $I$ inherited from $\bar{W}$ and the additional structures $J$ and $K=IJ$ satisfying the quaternionic relations; denote by $\kappa_I,\kappa _{J}$ and $\kappa _{K}$ their associated Kähler forms. Then [@kovalevzinho Theorem 2.2] $W$ admits a complete Calabi-Yau structure $\omega $ and holomorphic volume form $\Omega $ *exponentially asymptotic* to the cylindrical model $$\begin{aligned}
\index{Kahler metric@K\"{a}hler metric!winf@$\omega _{\infty }$}
\begin{array}{r c l} \label{asymptotic forms}
\omega _{\infty } &=&\kappa _{I}+ds\wedge d\alpha \\
\Omega _{\infty } &=&\left( ds+\mathbf{i}d\alpha \right) \wedge \left(\kappa _{J}+\mathbf{i}\kappa _{K}\right) \text{,}
\end{array}
\end{aligned}$$ in the sense that $$\left.\omega \right\vert_{W_{\infty}}
=\omega _{\infty }+d\psi, \qquad
\left.\Omega \right \vert_{W_\infty}
=\Omega _{\infty }+d\Psi,$$where $\psi $ and $\Psi $ are smooth and decay exponentially in all derivatives along the tubular end.
As to the gauge-theoretic initial data [@G2I], let $z={\mathbf{e}^{-s+\bf{i}\alpha}} $ be the holomorphic coordinate along the tube and denote $D_z$ the corresponding $K3$ component of the boundary. \[def bundle E->W\] A bundle $\mathcal{E}\rightarrow W$ is called *asymptotically stable* (or *stable at infinity*) if it is the restriction of an indecomposable holomorphic vector bundle $\mathcal{E}\rightarrow \bar{W}$ such that $\left. \mathcal{E}\right\vert _{D}$ is stable (hence also $\left. \mathcal{E}\right\vert _{D_{z}}$ for $\left\vert z\right\vert <\delta $). Moreover, \[def reference metric H0\] a *reference metric* $H_{0}$ on such $\mathcal{E}\rightarrow W$ is (the restriction of) a smooth Hermitian metric on $\mathcal{E}\rightarrow \bar{W}$ such that $\left. H_{0}\right\vert _{D_{z}}$ are the corresponding HYM metrics on $\left. \mathcal{E}\right\vert _{D_z}$, $0\leq \left\vert z\right\vert <\delta$, and $H_{0}$ has finite energy: $\Vert\hat{F}_{H_{0}}\Vert _{L^{2}\left( W,\omega\right) }<\infty$.
Then, given an asymptotically stable bundle with reference metric $\left({\mathcal{E}},H_0 \right)$, a nontrivial smooth $G_2-$instanton on $p_1^*{\mathcal{E}}\rightarrow W\times S^1$ is obtained from every solution of the HYM problem over $W$. Moreover, such solutions have the property of exponential asymptotic decay in all derivatives to $H_0$ along the tubular end \[*ibid.*, Theorem 59\]: $$\index{metric!Hermitian Yang-Mills}
\label{eq HYM solution}
\fbox{$\begin{array}{c}
\hat{F}_{H}=0, \quad H {\overset{C^\infty}{\underset{S\rightarrow\infty}{\longrightarrow}}}H_0. \\
\end{array}$}$$ Here convergence takes place over cylindrical bands of fixed ‘size’:
\[Not finite cylinder\]Let $Q\rightarrow W$ be a bundle equipped with a fibrewise metric and denote $W_S$ the truncation of $W$ at ‘neck length’ $S$; given $S>r>0$, write $\Sigma _{r}\left( S\right) $ for the interior of the cylinder $\left( W_{S+r}\smallsetminus
W_{S-r}\right) $ of ‘length’ $2r$. We denote the *$C^k-$exponential tubular limit* of an element in $C^k\left(\Gamma(Q)\right)$ by: $$\begin{aligned}
\phi \overset{C^k}{\underset{S\rightarrow\infty}{\longrightarrow}}\phi_{0} & \dot{\Leftrightarrow}& \left\Vert \phi -\phi_{0} \right\Vert_{C^k\left( \Sigma_1(S),\omega \right)}=O\left( {\mathbf{e}^{-S}} \right).\end{aligned}$$
Finally, let us fix some vocabulary towards the statement of the main theorem. Denote $A_{0}$ the Chern connection of $H_0$; then by definition each $\left. A_{0}\right\vert _{D_{z}}
$ is ASD. In particular, $\left.
A_{0}\right\vert _{D}$ induces an elliptic deformation complex$$\label{eq ell complex over D}
\Omega ^{0}\left(\left.\mathfrak{g}\right\vert_D\right) \overset{d_{A_{0}}}{\rightarrow }\Omega ^{1}\left( \left.\mathfrak{g}\right\vert_D\right) \overset{d_{A_{0}}^{+}}{\rightarrow }\Omega _{+}^{2}\left( \left.\mathfrak{g}\right\vert_D\right)$$where $\left.\mathfrak{g}\right\vert_D={\textnormal {Lie\,}}\left( \left. \mathcal{G}\right\vert _{D}\right) $ generates the gauge group $\mathcal{G}$ $={\textnormal {End\,}}\mathcal{E}$ over $D$. Thus, the requirement that $\mathcal{E}$ be indecomposable restricts the associated cohomology: $$\mathbf{H}_{A_{0}\vert_D}^{0}=0.$$On the other hand, one might restrict attention to *acyclic* connections, i.e., whose gauge class $\left[ A_{0}\right] $ is *isolated* in ${\mathcal{M}}_D\doteq{\mathcal{M}}_{\left. \mathcal{E}\right\vert _{D}}$. The absence of infinitesimal deformations translates into the vanishing of the other cohomology group:$$\mathbf{H}_{A_{0}\vert_D}^{1}=0.$$
\[def asymp rigid\] A reference metric on $\mathcal{E}\rightarrow W$ is *asymptotically rigid* if the associated complex $(\ref{eq ell complex over D})$ over $D$ has trivial cohomology.
Suitable pairs and gluing
-------------------------
A $7-$dimensional product $W_{S}^{\prime }\times S^{1}$, where $W'$ is of the above form, has boundary $D^{\prime }\times S^{1}\times S^{1}$. Comparing the asymptotic model $\left( \ref{asymptotic forms}\right) $ with the standard form of the $G_{2}-$structure on a product $CY\times S^{1}$ we find that $W_{S}^{\prime }\times S^{1}$ carries a $G_{2}-$structure on a collar neighbourhood of the boundary that is asymptotic to:$$\index{G2@$G_{2}$!-structure@$-$structure}
\label{asymptotic G2-structure}
\varphi _{S}^{\prime }=\kappa _{I}^{\prime }\wedge d\alpha
+\kappa_{J}^{\prime }\wedge d\theta +\kappa _{K}^{\prime }
\wedge ds+d\alpha \wedge d\theta \wedge ds.$$Since the inclusion of the set of all $G_{2}-$structures $\mathcal{P}^{3}\left( W^{\prime
}\times S^{1}\right) \subset \Omega ^{3}\left( W^{\prime }\times
S^{1}\right) $ is open [@Joyce p. 243], $\varphi _{T}^{\prime }$ is itself a $G_{2}-$structure on $W^{\prime }\times S^{1}$ for large $S$.
Two manifolds $W^{\prime }$ and $W^{\prime \prime }$ as above will be suitable for the gluing procedure if there is a hyper-Kähler isometry $$f:D_{J}^{\prime }\rightarrow D^{\prime \prime }$$between $D^{\prime \prime }$ and the hyper-Kähler rotation of $D^{\prime
}$ with complex structure $J$. In this case the (pull-back) action on Kähler forms is$$\index{K3 surface!hyper-K\"{a}hler}f^{\ast }:\kappa _{I}^{\prime \prime }\mapsto \kappa _{J}^{\prime },\qquad
\kappa _{J}^{\prime \prime }\mapsto \kappa _{I}^{\prime },\qquad \kappa
_{K}^{\prime \prime }\mapsto -\kappa _{K}^{\prime }.
\label{matching of kahler forms}$$
Assuming this holds, define a map between collar neighbourhoods of the boundaries by $$\begin{aligned}
F_{S}:D^{\prime }\times S^{1}\times S^{1}\times \left[ S-1,S\right]
&\rightarrow&
D^{\prime \prime }\times S^{1}\times S^{1}\times \left[ S-1,S\right] \\
\left( y,\alpha ,\theta ,s\right)
&\mapsto&
\left( f\left( y\right) ,\theta,\alpha ,2S-1-s\right) .\end{aligned}$$ This identification gives a compact oriented $7-$manifold $$M_{S}=\left( W_{S}^{\prime }\times S^{1}\right) \cup _{F_{S}}\left(
W_{S}^{\prime \prime }\times S^{1}\right) \doteq W'\widetilde{\#}_{S}W''.$$
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The matching of Kähler forms $\left( \ref{matching of kahler forms}\right) $ guarantees that the respective $G_{2}-$structures $\left( \ref{asymptotic G2-structure}\right) $ on $W_{S}^{\prime }\times S^{1}$ and $W_{S}^{\prime \prime }\times S^{1}$ agree along the gluing region $\left[
S-1,S\right] $:$$\begin{aligned}
F_{S}^{\ast }\varphi _{S}^{\prime \prime } &=&F_{S}^{\ast }(\underset{\kappa
_{J}^{\prime }}{\underbrace{\kappa _{I}^{\prime \prime }}}\wedge \underset{d\theta ^{\prime }}{\underbrace{d\alpha ^{\prime \prime }}}+\underset{\kappa _{I}^{\prime }}{\underbrace{\kappa _{J}^{\prime \prime }}}\wedge \underset{d\alpha ^{\prime }}{\underbrace{d\theta ^{\prime \prime }}}+\underset{-\kappa _{K}^{\prime }}{\underbrace{\kappa _{K}^{\prime \prime }}}\wedge \underset{-ds^{\prime }}{\underbrace{ds^{\prime \prime }}}+\underset{d\theta ^{\prime }}{\underbrace{d\alpha ^{\prime \prime }}}\wedge \underset{d\alpha ^{\prime }}{\underbrace{d\theta ^{\prime \prime }}}\wedge \underset{-ds^{\prime }}{\underbrace{ds^{\prime \prime }}}) \\
&=&\kappa _{I}^{\prime }\wedge d\alpha ^{\prime }+\kappa _{J}^{\prime
}\wedge d\theta ^{\prime }+\kappa _{K}^{\prime }\wedge ds^{\prime }+d\alpha
^{\prime }\wedge d\theta ^{\prime }\wedge ds^{\prime } \\
&=&\varphi _{S}^{\prime }.\end{aligned}$$so we obtain a globally well-defined $G_{2}-$structure $\varphi _{S}$ on $M_{S}$. Thus, for large enough $S$, there is a $1-$parameter family $\left(
M_{S},\varphi _{S}\right) $ of compact oriented manifolds $M_{S}$ equipped with $G_{2}-$structures $\varphi _{S}$.
While it is possible to arrange $d\varphi _{S}=0$ for any $S$ [kovalevzao]{}, a pair $\left( M_{S},\varphi _{S}\right) $ is not in principle a $G_{2}-$manifold, as one has yet to satisfy the co-closedness condition:$$d\ast _{\varphi _{S}}\varphi _{S}=0.$$ In fact, although the cut-off functions involved in the asymptotic approximations leading to $\left( \ref{asymptotic G2-structure}\right) $ add error terms to $d\ast _{\varphi _{S}}\varphi _{S}$, these are controlled by the estimate [@kovalevzao *Lemma 4.25*] $$\left\Vert d\ast _{\varphi _{S}}\varphi _{S}\right\Vert _{L_{k}^{p}}
\leq C_{p,k}e^{-\lambda S},$$with $0<\lambda <1$. This exponential decay implies that, by ‘stretching the neck’ up to large enough $S_{0}$, one can make the error so small as to be compensated by a suitably small perturbation of $\varphi _{S}$ in $\mathcal{P}^{3}\left( M_{S}\right) $, $S>S_{0}$ [kovalevzinho]{}. Hence one achieves a $1-$parameter family of compact oriented $G_{2}-$manifolds:$$\index{G2@$G_{2}$!-manifold@$-$manifold}
\left( M_{S},\tilde \varphi _{S}\right) ,\qquad S>S_{0}.$$
Statement of the instanton gluing theorem {#subsec Statement of gluing theorem}
-----------------------------------------
Let $\left(M_S,\tilde\varphi_S\right)$ be a compact $G_2-$manifold with ${{\textnormal {Hol\,}}}(\tilde\varphi_S)= G_2$ as above, obtained from a Fano pair by Kovalev’s construction: $$M_S = W'{\tilde\#_S}W'' \doteq \left( W'_S \times S^1 \right)
\cup_{F_S} \left( W''_S \times S^1 \right).$$ Furthermore, let ${\mathcal{E}}^{(i)}\rightarrow W^{(i)}$ be asymptotically stable holomorphic bundles with same structure group $G={\textnormal {Aut\,}}({\mathcal{E}})$, such that there is a $G-$isomorphism $$g:\left.{\mathcal{E}}'\right\vert_{D_J'}\tilde\rightarrow \left.{\mathcal{E}}''\right\vert_{D''}.$$ One can define a holomorphic bundle over $M_S$ by an induced bundle gluing $${\mathcal{E}}^g_S\doteq {\mathcal{E}}'{\tilde\#_S}^g {\mathcal{E}}''\rightarrow M_S$$ of the following form. First, fix holomorphic trivialisations over neighbourhoods of infinity $U^{(i)}\subset W^{(i)}_\infty$ along the ends. Then, spreading $g$ along $U^{(i)}$ via pull-back by the fibration maps $\tau^{(i)}:W^{(i)}\rightarrow D^{(i)}$, we identify the fibres of $p_1^*{\mathcal{E}}'$ and $p^*_1{\mathcal{E}}''$ across the gluing zone of operation ${\tilde\#_S}$. Having said that, I will omit henceforth the superscript $g$ as well as any reference to the particular choice of trivialisations. This paper proves the following *Gluing theorem*:
\[thm gluing\] Let $\mathcal{E}^{(i)}\rightarrow W^{(i)}$, $i=1,2$, be asymptotically stable bundles of same semi-simple structure group $G$, with *asymptotically rigid* reference metrics $H_0^{(i)}$, admitting a $G-$isomorphism $g:\left.{\mathcal{E}}'\right\vert_{D_J'}\tilde\rightarrow \left.{\mathcal{E}}''\right\vert_{D''}$.
There exists $S_0>0$ such that the bundle $\mathcal{E}_S\rightarrow M_S\doteq W'{\tilde\#_S}W''$ admits a $G_2-$instanton, for every $S\geq S_0$.
Approximate instantons over $M_S$
=================================
In order to produce a $G_2-$instanton over the compact 7-manifold $M_S$, I use cut-offs to obtain a connection on ${\mathcal{E}}_S$ which is ‘approximately’ an instanton, then show that a certain surjectivity requirement is satisfied, in order to perturb it into a true solution.
Preliminary moduli theory {#subsec prelim moduli theory}
-------------------------
Following [@floer pp.83-87], given an asymptotically stable bundle ${{\mathcal{E}}}\rightarrow W$, we are interested in gauge classes of connections on $\tilde{{\mathcal{E}}}=p_1^*{\mathcal{E}}\rightarrow M=W\times S^1$ which are ‘asymptotically HYM’ \[cf. (\[eq ell complex over D\])\]. For every (smooth) reference metric $H_0$, we denote its Chern connection $A_0$ and pose \[cf. *Notation \[Not finite cylinder\]*\] $$\label{eq def A}
{\mathcal{A}}\doteq\left\{ p_1^*(A_0+a)
\left\vert
\begin{array}{ll}
\left. A_0\right\vert_D\in{\mathcal{M}}_D, & \\
\left\vert a\right\vert, \left\vert \nabla_{A_0} a \right\vert\in L^p_k, &
a{\overset{C^\infty}{\underset{S\rightarrow\infty}{\longrightarrow}}}0 \\
\end{array}
\right.\right\},$$ taking, for suitable integers $p,k$ \[cf. $(\ref{eq suitable p,k})$ below\], the $L_{k}^{p}-$norm induced by $\varphi $: $$\index{norm!of bundle-valued form}
\label{rem Sobolev norm of a}
\left\Vert f\right\Vert\doteq\left\Vert f\right\Vert _{L_{k}^{p}}
=\left(\int_{M}\sum_{l=0}^{k}\left\vert \nabla ^{i_{1}}...
\nabla^{i_{l}}f\right\vert ^{p}d{\textnormal {Vol\,}} \right) ^{\frac{1}{p}}.$$This has in view the use of Sobolev’s embedding (*Lemma \[Lemma Sobolev’s embedding\]*) in *Subsection \[subsect subbundle of kernels\]*. For notational clarity I will henceforth leave implicit the pull-back $p_1^*$.
Posing the gauge-equivalence condition $$A_1\sim A_2 \ \Leftrightarrow \ A_2=g(A_1)\overset{loc}{=}A_1-d_1 g.g^{-1}, \quad g\in L^p_{k+1,loc}({\textnormal {Aut\,}}{\mathcal{E}}),$$ and adopt accordingly the gauge group $${\mathcal{G}}\doteq \left\{ g\in{\textnormal {Aut\,}}\tilde{{\mathcal{E}}}
\left\vert \left\vert\nabla_0g.g^{-1} \right\vert\in L^p_k,
\quad g{\overset{C^\infty}{\underset{S\rightarrow\infty}{\longrightarrow}}}1\right.\right\},$$ whose (bundle of) Lie algebra(s) we denote $\mathfrak{g}$. Since every $\left.A_0\right\vert_D$ is assumed *irreducible* \[cf. p.\], ${\mathcal{G}}$ is in fact a Banach Lie group and the action ${\mathcal{G}}\times{\mathcal{A}}\rightarrow{\mathcal{A}}$ is smooth. Finally, the Coulomb gauge condition provides transversal slices for the action \[cf. (\[eq Hodge theory\]), below\]: $$\label{nbhds Te(A)}
U_{\varepsilon }\left( A\right)
=A+\left\{ a\in \Omega ^{1}\left( \mathfrak{g}\right)
\left\vert \ d_{A}^{\ast }a=0,
\quad\left\Vert a\right\Vert <\varepsilon,
\quad a{\overset{C^\infty}{\underset{S\rightarrow\infty}{\longrightarrow}}}0 \right.\right\},$$ so that the quotient ${\mathcal{B}}={\mathcal{A}}/{\mathcal{G}}$ is a Banach manifold [@4-manifolds Prop. 4.2.9, p.132]:
\[local model for B\] If $A$ is irreducible then, for small $\varepsilon $, the projection from $\mathcal{A}$ to $\mathcal{B}$ induces a homeomorphism from $T_{\varepsilon }\left( A\right)$ to a neighbourhood of $\left[ A\right] $ in $\mathcal{B}$.
\[def Moduli space of G2-instantons\] The *moduli space of (irreducible)* $G_{2}-$*instantons* on ${\mathcal{E}}$ is:$${\mathcal{M}}^+\doteq \left\{ \left[ A\right]\in \mathcal{B}
\mid p_{+}\left( F_{A} \right)\doteq F_A\wedge\ast\varphi=0\right\} .$$
NB.: In particular, the instantons obtained in the first paper [@G2I], as solutions of the HYM problem, decay exponentially in all derivatives to $H_0$ along the cylindrical end, hence ${\mathcal{M}}^+\neq \emptyset$ \[cf. (\[eq HYM solution\]) and (\[eq def A\])\].
For the local description of ${\mathcal{M}}^+$, define around a solution $A$ the map $$\begin{aligned}
\begin{array}{r c l}
\psi \; : \; U_{\varepsilon }\left( A\right) \subset \mathcal{A}
&\rightarrow&
\Omega^{6}\left( \mathfrak{g}_{E}\right) \label{psi} \\
a &\mapsto &\psi \left( a\right) \doteq p_{+} \left(F_{A+a}\right)=\left(d_{A}a + a\wedge a\right)\wedge\ast\varphi
\end{array}\end{aligned}$$and write $Z\left( \psi \right) \subset $ $T_{\varepsilon }\left( A\right) $ for its zero set. Slicing out by Coulomb gauge indeed makes $\psi$ a Fredholm map, and we have:
\[Prop local description of ME\] If $A\in \mathcal{A}$ is an irreducible $G_2-$instanton, then an $\varepsilon-$neighbourhood of $\left[ A\right] \in \mathcal{M}^+$ is modelled on $Z(\nu)$, where $\nu$ is the invertible map between finite-dimensional spaces defined by $$\begin{gathered}
\nu \; : \;
\underset{
\begin{tabular}{c}
$\cap $ \\
$\Omega ^{1}\left( \mathfrak{g}\right) $\end{tabular}}{\underbrace{\ker \left( d_{A}^{\ast }\oplus d_{A}^{+}\right) }}\;
\longrightarrow \;
\underset{\begin{tabular}{c}
$\cap $ \\
\;$ \Omega^{6}\left( \mathfrak{g}\right)$\end{tabular}}
{\underbrace{{\textnormal {coker\,}}d_{A}^{+} \;\cap\; \ker d_{A}}} \\
\quad
\nu \left( a\right) =\sigma \left( 0,a\right)\end{gathered}$$and $\sigma $ is the non-linear part of the local Fredholm decomposition of $\psi $.
The proof of *Proposition \[Prop local description of ME\]* is postponed to *Section \[sect local model moduli space\]*, as part of a more detailed discussion of the moduli theory.
Truncating instantons with decaying error term
----------------------------------------------
We may now start in earnest the proof of *Theorem \[thm gluing\]*. Let $A'$ be a $G_2-$instanton on $p^*_1{\mathcal{E}}'$ \[cf. (\[eq HYM solution\])\]. Along a neighbourhood of infinity down the tubular end of $W'$, we write $A'=A_{0}'+ a'$, where $A_0'$ is the lifted Chern connection associated to the reference metric $H_0'$ and $a'{\overset{C^\infty}{\underset{S\rightarrow\infty}{\longrightarrow}}}0$. Fix a smooth cut-off $$\begin{array}{c}
\chi:{\mathbb{R}}^+\rightarrow \left[ 0,1 \right] \\
\chi(s)=
\left\{
\begin{array}{c}
1, \quad s<0 \\
0, \quad s\geq 2 \\
\end{array} .
\right.\\
\end{array}$$ and truncate to $$A'_S\doteq A_{0}' +\chi\left( s-S+3 \right)a',$$ which agrees with $A_{0}'$ over the gluing region $[S-1,S]$ and has self-dual part $F_{A'_S}^+\simeq F_{A'_S}\wedge\ast\varphi$ supported in the (topological) cylinder segment $\Sigma_1(S-1)$. Clearly $$\left\Vert F_{A'_S}\wedge\ast\varphi\right\Vert_{L^p_k\left( M'\right)}
\leq C_{p,k}{\mathbf{e}^{-S}}$$ for $M'\doteq W'\times S^1$. Repeating the construction for $A''_S$ on $p_1^*{\mathcal{E}}''$, we may assume that the truncated connections $A^{(i)}_S$ match (via the bundle isomorphism $g$) over $[S-1,\infty[$, hence in particular over $[S-1,S]$, so they glue together to define a smooth connection $$A_S\doteq A'{\tilde\#_S}A'' \quad \text{on} \quad {\mathcal{E}}_S.$$ Since the hyper-Kähler rotation at infinity is assumed to be an isometry, we still have the asymptotic decay of the self-dual part $$\left\Vert F_{A_S}\wedge\ast\varphi \right\Vert_{L^p_k\left(M_S\right)}
\leq C_{p,k}{\mathbf{e}^{-S}}$$ from which we see that $A_S$ is *almost* an instanton.
Non-degeneracy under acyclic limits
-----------------------------------
Let $A=A^{(i)}$ as above, say, be a $G_2-$instanton on the pull-back bundle$$\tilde{{\mathcal{E}}}=p_1^*\mathcal{E}\rightarrow M
\doteq W\times S^1.$$ Then the hypothesis that the connection at infinity is *acyclic* implies that $A$ itself is acyclic, with respect to its own deformation complex: $$\label{extended complex 2}
\Omega ^{0}\left( \mathfrak{g}\right) \;
\overset{d_{A}}
{\underset{d_{A}^{\ast }}{\rightleftarrows }}\;
\Omega ^{1}\left( \mathfrak{g}\right) \;
\overset{d_{A}^{+}}
{\overbrace{\overset{d_{A}}{\longrightarrow }\;
\Omega^{2}\left( \mathfrak{g}\right) \;
\overset{\ast \varphi \wedge .}
{\longrightarrow }}}\;
\Omega ^{6}\left( \mathfrak{g}\right) \;
\overset{d_{A}}
{\underset{d_{A}^{\ast }}{\rightleftarrows }}\;
\Omega ^{7}\left( \mathfrak{g}\right).$$The goal of this *Subsection* is to prove that claim, in the following terms:
\[Prop asymp acyclic => acyclic\] When the reference metric $H_0$ is asymptotically rigid, then the induced $G_2-$instanton $A$ lifted from a HYM solution \[cf. $(\ref{sd part kahler -> G2})$ and $(\ref{eq HYM solution})$\] is acyclic, i.e., $\mathbf{}\mathbf{H}_A^0=0$ and $\mathbf{H}_A^1=0$ in the deformation complex $(\ref{extended complex 2})$.
Since our asymptotically stable bundle is, by definition, indecomposable, we have already $\mathbf{H}_A^0=0$, so one only needs to check the non-degeneracy condition $\mathbf{H}_A^1=0$. On the other hand, the complex is self-dual (under the Hodge star), so this is equivalent to showing $\mathbf{H}_A^2=0$, which means precisely that $A$ is an isolated point in its moduli space $\mathcal{M}^+$, in the light of the local model given by *Proposition $\ref{Prop local description of ME}$*. To check this fact, we resort to the Chern-Simons functional $$\rho(b)_A=\int_{W\times S^1}{{\textnormal {tr\,}}}F_A\wedge b_A\wedge \ast\varphi.$$
For any given direction $a\in T_{[A]}\mathcal{B}\;\widetilde{\subset}\;\Omega^1(\mathfrak{g}) $ with $\Vert a \Vert=1$ and possibly short length $\varepsilon>0$, we have $$\begin{aligned}
\left[A+ha\right]\in{\mathcal{M}}^+&\Leftrightarrow&
0\overset{\forall b}{=}\rho(b)_{A+\varepsilon a}=\underset{0}{\underbrace{\rho(b)_A}}
+\varepsilon.D\left[ \rho(b) \right]_A(a)+O(\varepsilon^2)\end{aligned}$$ for vector fields $b\in \Gamma\left( T\mathcal{A} \right)$. Explicitly, the first order variation is $$\begin{aligned}
D\left[ \rho(b) \right]_A(a)&=&\int_{W\times S^1}{{\textnormal {tr\,}}}\{d_Aa\wedge b_A
+ (Db)_A(a)\wedge
\underset{F_A^+=0}{\underbrace{F_A\}\wedge\ast\varphi}}\\
&=&\int_{W\times S^1}{{\textnormal {tr\,}}}a\wedge d_A b_A\wedge\ast\varphi
+\underset{0}{\underbrace{\lim_{S\rightarrow\infty}\int_{\partial W_S\times S^1}
{{\textnormal {tr\,}}}\left\{ a\wedge b_A \right\}\wedge\ast\varphi}}\\
&=&\int_{W\times S^1}{{\textnormal {tr\,}}}a\wedge d_A^+b_A\end{aligned}$$ since $H_0$ is asymptotically rigid and so $\left\vert a\right\vert\underset{S\rightarrow\infty}{\longrightarrow}0$, for small enough $\varepsilon$. Notice in passing that this is zero for any direction $a\in{{\textnormal {img\,}}}d_A\subset\Omega^1(\mathfrak{g})$ along gauge orbits, corresponding to the intuitive fact that the only ‘meaningful’ perturbations are those which descend nontrivially to ${\mathcal{B}}$. Choose now $0\neq\xi\in\Omega^6(\mathfrak{g})$ such that $$b_A\doteq\left(d_A^+\right)^*\xi\neq0
\text{ in }
\Omega^1(\mathfrak{g}).$$ By the orthogonal decomposition $(\ref{eq Hodge theory})$, this can be done in such a way that $d_A^+b_A\neq0$, so for any direction $a\in\Omega^1(\mathfrak{g})$ (transverse to gauge orbits) the number $$D\left[ \rho(b) \right]_A(a)\doteq N_A(a,\xi)\in{\mathbb{R}}$$ is not zero for a generic choice of $\xi$. Rescaling $\widetilde{b_A}\doteq \frac{1}{N_A(a,\xi)}b_A$, we find $$\rho(\tilde{b})_{A+ha}
=\varepsilon+O(\varepsilon^2)O(\Vert\tilde b_A \Vert)\neq0,
\quad \text{for } \;0\neq \varepsilon\ll 1.$$
Now, since $\mathcal{M}^+$ is finite-dimensional, there are tangent vectors ($1-$forms on the base) $u_1,\dots,u_n\in T_{[A]}\mathcal{B}$ such that any such perturbation is written as $$\varepsilon.a=\varepsilon.(a^1u_1+\dots+a^n u_n), \quad a^i\in {\mathbb{R}}.$$ But in the above way we can find, respectively for $u_1,\dots,u_n$, vector fields $\tilde b_1,\dots,\tilde b_n$ such that $\rho(\tilde{b_i})_{A+hu_i}= \varepsilon+O(\varepsilon^2)$. Consequently, for a generic linear combination $\mathbf{\tilde b}\doteq \beta^1 \tilde b_1+\dots+\beta^n \tilde b_n$, one has $$\rho(\mathbf{\tilde{b}})_{A+ha}=\varepsilon.\underset{\neq0}{\underbrace{\left( a^1\beta^1+...+a^n\beta^n \right)}}+O(\varepsilon^2).$$Hence there exists a (possibly small) value $\varepsilon_{0}>0$ such that $\rho_{A+\varepsilon a}\neq0$, as a $1-$form on $\mathcal{A}$, for any $\varepsilon.a\in U_{\varepsilon_0}(\left[A\right])$. In other words, there are no instantons in the open ball of radius $\varepsilon_{0}$ around $\left[A\right]$ in the moduli space (*q.e.d.*).
Perturbation theory over long tubular ends
==========================================
Following the standard approach, we may now look for a nearby exact solution $A=A_S+a$ to the $G_2-$instanton equation $$\label{eq nearby instanton}
d_{A_S}^+a+\left( a\wedge a \right)\wedge\ast\varphi
=-F_{A_S}\wedge\ast\varphi\doteq\epsilon(S).$$ We adopt all along the *acyclic assumption* that the operators $d_{A{^{(i)}}}^+$ have trivial cokernel, i.e., the (irreducible) connections $A^{(i)}$ are isolated points in the respective moduli spaces $\mathcal{M}^+_{\mathcal{E}^{(i)}}$ \[cf. *Proposition \[Prop asymp acyclic => acyclic\]*\].
Noncompact Sobolev estimates
----------------------------
For briefness, let us refer to ordered integers $k> l\geq0$ and $q\geq p$ as *suitable* if they satisfy the Sobolev condition: $$\label{eq suitable p,k}
\frac{1}{p}-\frac{1}{q}\leq \frac{k-l}{7}.$$ In particular, the pair $p,k$ will be *suitable* when $k=l+1$ and $2p=q$ are suitable.
\[Lemma noncompact Sobolev\] Let $ W$ be an asymptotically cylindrical $3-$fold; given suitable $k\geq l$ and $q\geq p$, there exists a constant $C\doteq C_{W,p,q,k,l}>0$ such that, for sections of any bundle over $W\times S^1$ (with metric and compatible connection), $$\left\Vert .\right\Vert_{L^q_l}\leq
C\left\Vert . \right\Vert_{L^p_k} .$$
Following [@floer pp.70-72], set $B_0\doteq W_0\times S^1$ and consider for $n\geq1$ the tubular segments of ‘length one’ $B_n=\left(W_n\setminus W_{n-1}\right)\times S^1$ along the tubular end. Then, using the usual Sobolev estimate for compact domains, $$\begin{aligned}
\left\Vert f \right\Vert_{L^q}^q &=&
\sum_{n\in \mathbb{N}} \int_{B_n} \left\vert f \right\vert^q\\
&\leq&\sum_{n\in \mathbb{N}} C_{n}\left( \int_{B_n}
\left\vert \nabla f \right\vert^p
+ \left\vert f \right\vert^p\right)^{q/p}\\
&\leq& \tilde C \left( \sum\int\left\vert \nabla f \right\vert^p
+ \left\vert f \right\vert^p \right)^{q/p}
= \tilde C \left\Vert f \right\Vert^q_{L^p_1}\end{aligned}$$ with $\tilde C=\lim\sup C_n<\infty$, since the segments are asymptotically cylindrical. This proves the statement, by induction on $l$.
Gluing right inverses
---------------------
We now investigate the behaviour of right-inverses under truncation and gluing:
For $S\gg0$, the operators $d_{A_S{^{(i)}}}$ admit bounded right inverses $Q_S{^{(i)}}$ satisfying $$\Vert Q_S{^{(i)}}\xi\Vert_{L^p_{k}}
\leq C{^{(i)}}_{p,k}\left\Vert \xi \right\Vert_{L^p_{k-1}}$$ for suitable $p,k\in\mathbb{N}$, where the bound $C{^{(i)}}_{p,k}$ depends only on $A{^{(i)}}$, not on $S$.
The operators $d_{A{^{(i)}}}$ correspond to the original instantons over each tubular component $W{^{(i)}}\times S^1$, hence by the acyclic assumption \[cf. *Proposition \[Prop asymp acyclic => acyclic\]*\] they admit bounded right inverses $Q{^{(i)}}$, independent of $S$.
The crucial fact is that $a^{i}_S\doteq A_S{^{(i)}}-A{^{(i)}}=O\left( {\mathbf{e}^{-S}} \right)$ so, for $S\gg0$, a right inverse for $d_{A{^{(i)}}}$ gives a right inverse $Q_S{^{(i)}}$ for $d_{A{^{(i)}}_S}$ with (approximately) the same uniform Lipschitz bound.
For $S\gg0$, there exist an ‘approximate’ right inverse $Q_{S}$ and a true right inverse $P_{S}$ for $d_{A_S}^+$: $$P_{S}=Q_{S}\left( d_{A_S}^+Q_{S} \right)^{-1}.$$ Moreover, for suitable $p,k\in\mathbb{N}$, there is a uniform bound $C_{p,k}$ on the operator norm of $P_{S}$: $$\left\Vert P_{S}(\xi)\right\Vert_{L^p_{k}}\leq C_{p,k}
\left\Vert \xi \right\Vert_{L^p_{k-1}}.$$
Following a standard argument [@floer §3.3 & §4.4], we may take truncation functions $\chi{^{(i)}}_S:W{^{(i)}}\rightarrow \left[ 0,1 \right]$ satisfying $$\left(\chi_{S}'\right)^2+\left( \chi_{S}'' \right)^2=1,
\quad {\textnormal {supp\,}}\chi_{S}{^{(i)}}\subset W{^{(i)}}_{\frac{3S}{2}},
\quad \Vert\nabla\chi_{S}{^{(i)}}\Vert_{L^\infty}=O({\mathbf{e}^{-S}}).$$ Then, denoting $r_{S}{^{(i)}}:M_S\rightarrow W{^{(i)}}\times S^1$ the maps given by restriction for $s\leq2S$ and extended by zero along the rest of the tubular end, we form $$Q_{S}\doteq\left(\chi'_{S}\right)^2 \left(Q'_S\circ r'_S\right)
+\left(\chi''_{S}\right)^2 \left(Q''_{S}\circ r''_S\right)$$ and we check that this is an approximate right inverse in the sense that $\Vert d_{A_S}^+Q_S-I\Vert=O({\mathbf{e}^{-S}})$; indeed we have $$d_{A_S}^+Q_S =\underset{I}{\underbrace{
\sum\nolimits_{i=1,2} \{\left(\chi{^{(i)}}_S\right)^2d_{A_S}^+ \left(Q{^{(i)}}_{S}\circ r{^{(i)}}_S \right)}
}+ 2\underset{O({\mathbf{e}^{-S}})}{\underbrace{ \left( \chi{^{(i)}}_S\nabla\chi{^{(i)}}_S\right)
\diamondsuit\left( Q{^{(i)}}_{S}\circ r{^{(i)}}_S \right)}}\}$$ where $\diamondsuit$ denotes an algebraic operation. It follows from the *Lemma* that the second summand is dominated by the decay of $\Vert\nabla\chi_{S}{^{(i)}}\Vert_{L^\infty}$. Then $$P_S\doteq Q_S \left( d_{A_S}^+Q_S \right)^{-1}$$ is a true right inverse for $d_{A_S}^+$, with uniformly bounded norm determined by $Q_S$ and $d_{A_S^+}$ itself.
Exact solution via the contraction principle
--------------------------------------------
For a solution of the form $a=P\xi$, equation $(\ref{eq nearby instanton})$ reads $$\label{eq contraction}
\left(I+G\right)(\xi)=\epsilon(S)$$ where $G(\xi)\doteq P\left( \xi\right)\wedge P\left( \xi\right)
\wedge\ast\varphi$ and $\left\Vert\epsilon(S)\right\Vert$ is small, as $S\gg0$. Thus if, given suitable $p,k$, the map \[cf. *Lemma \[Lemma noncompact Sobolev\]* for last inclusion\] $$G: L^{2p}_{k-1}\left(\Omega^6\right)
\hookrightarrow L^{p}_{k-1}\left(\Omega^6\right)
\longrightarrow L^{p}_{k}\left(\Omega^6\right)
\hookrightarrow L^{2p}_{k-1}\left(\Omega^6\right)$$ is Lipschitz with constant strictly smaller than $1$, then by the Banach contraction principle $I+G$ is continuously invertible between neighbourhoods of $\left(I+G\right)(0)=0$ and we obtain, for large $S$ say, a solution as $$a=P\left( I+G \right)^{-1}\epsilon(S).$$
In order to prove this, we need a uniform bound on exterior multiplication, which is an immediate consequence of Hölder’s inequality:
\[prop bounded multiplication\] For suitable $p,k\in\mathbb{N}$, there exists a uniform constant $M_{p,k}$ such that $$\left\Vert a\wedge b \right\Vert_{L^{p}_{k-1}}\leq M_{p,k}
\left\Vert a \right\Vert_{L^{2p}_{k-1}}
\left\Vert b \right\Vert_{L^{2p}_{k-1}}.$$
From this we infer that operator $G$ is bounded uniformly in $S$ for any suitable Sobolev norm, but so far we have no definite control over the actual bound. We can scale away this apparent difficulty using the fact that $G$ is homogeneous of degree 2; letting $\lambda=MC^{2}$, $\tilde\xi=\lambda\xi$ and $\tilde G=\frac{1}{\lambda}G$, equation $(\ref{eq contraction})$ becomes $$\left( I+\tilde G \right)(\tilde\xi)=\lambda\epsilon(S).$$ The point here is that now $\tilde G$ is a contraction over the ball $\tilde B_{1}\doteq\left\{ \Vert \tilde\xi\Vert\leq1 \right\}$: $$\Vert \tilde G (\tilde\xi) \Vert \leq
\frac{MC^{2}}{\lambda}\Vert \tilde\xi \Vert^2
\leq \Vert \tilde\xi\Vert.$$ Then indeed $I+\tilde G$ is a homeomorphism onto an interior domain $0\in U\subset \tilde B_1$, and one can choose $S\gg0$ so that $\lambda\epsilon(S)\in U$. One may check, by bootstrapping [@floer .96], that $A$ is in fact smooth. We have thus proved *Theorem \[thm gluing\]*.
In conclusion, it should be noticed that the acyclic hypothesis is a rather strong, non-generic requirement. The fact that it is not void should therefore be illustrated:
The prime Fano $3-$folds of type $X_{22}$ were discovered by Iskovskikh [@X22] and further studied by Mukai [@Mukai; @Fano; @3-folds §3]. On one hand, these appear in Kovalev’s list of suitable blocks for the gluing construction; in fact, they are extremal in the sense that a pair of base manifolds of type $X_{22}$ realises the lower bound on the third Betti number $b_3(M)=71$ [@kovalevzao pp.158-159].
On the other hand, crucially, these come equipped with an asymptotically stable bundle $E\rightarrow X_{22}$ which is *rigid* [@Mukai; @Moduli; @of; @bundles §3] over a divisor $D\in\left\vert -K_{X_{22}} \right\vert$. In other words, the holomorphic bundle $\left.E\right\vert_D$ corresponds to an isolated point in its moduli space, hence its associated HYM metric $H_0$ is indeed *acyclic*.
Local model for the moduli space {#sect local model moduli space}
================================
The moduli space $\mathcal{M}^+$ of $G_2-$instantons on ${\mathcal{E}}\rightarrow M$ is locally described as the zero set of a map $\psi $ \[cf. $\left( \ref{psi}\right) $\] between the Banach spaces $U_{\varepsilon }\left( A\right) \subset \mathcal{A}$ and $\Omega^6\left( \mathfrak{g}\right) $. Therefore, if our map $\psi $ is Fredholm on $Z\left( \psi \right) $, it is a matter of standard theory to model a neighbourhood of $\left[ A\right] $ in $\mathcal{M}^+$ on the finite-dimensional set $\nu ^{-1}\left( 0\right) $ \[cf. *Corollary \[cor zero set\]* in the *Appendix*\] . Foreseeing *Proposition \[prop projection between kernels\]*, we restrict attention to *irreducible* connections on an $SU\left(n\right) -$bundle ${\mathcal{E}}$.
Noncompact Fredholm theory {#Subsec Fredholm theory}
--------------------------
\[subsect extended elliptic complex\] \[subsect subbundle of kernels\]
As defined before, the map $\psi $ is just the self-dual part of the curvature, so $\psi \left( a\right) -\psi
\left( 0\right) =\left(p_+\circ d_{A}\right)a + O( \left\vert a\right\vert ^{2}) $ and$$\left( D\psi \right) _{0}=p_+\circ d_{A}.$$ Moreover, by the ‘slicing’ condition $\left( \ref{nbhds Te(A)}\right) $ across orbits, we consider in fact the restriction$$\label{da+ restr kerda*}
\index{gauge orbit}
\begin{array}{ccccc}
p_+\circ d_{A} & : & \ker d_{A}^{\ast } & \longrightarrow
& \Omega _{+}^{2}\left( \mathfrak{g}\right) . \\
& & \cap\ & & \\
& & \Omega ^{1}\left( \mathfrak{g}\right) & & \\
\end{array}$$
Since the map $L_{\ast \varphi }=`\ast \varphi \wedge.$’ now plays the role of ‘SD projection’, we denote henceforth $$\label{Da+}
d_{A}^{+}=L_{\ast \varphi }\circ d_{A}:\Omega ^{1}
\left( \mathfrak{g}\right) \rightarrow \Omega ^{6}
\left( \mathfrak{g}\right)$$ and consider the extended deformation complex$$\label{extended complex}
\Omega ^{0}\left( \mathfrak{g}\right) \;
\overset{d_{A}}
{\underset{d_{A}^{\ast }}{\rightleftarrows }}\;
\Omega ^{1}\left( \mathfrak{g}\right) \;
\overset{d_{A}^{+}}
{\overbrace{\overset{d_{A}}{\longrightarrow }\;
\Omega^{2}\left( \mathfrak{g}\right) \;
\overset{\ast \varphi \wedge .}
{\longrightarrow }}}\;
\Omega ^{6}\left( \mathfrak{g}\right) \;
\overset{d_{A}}
{\underset{d_{A}^{\ast }}{\rightleftarrows }}\;
\Omega ^{7}\left( \mathfrak{g}\right).$$ Using $d\ast \varphi =0$, we find $$\left[ L_{\ast \varphi },d_{A}\right] =0, \label{[Lfi,dA]=0}$$so, when $A$ is an instanton, $\left( \ref{extended complex}\right) $ is indeed a complex and the identification of the self-dual $2-$forms with the $6-$forms is consistent with the relevant differential operators (for more on elliptic complexes under the condition $d*\varphi=0$, see ). Moreover, this complex is elliptic:
\[lemma (Da+)\*=\*(Da+)\*\]The operator $d_{A}^{+}$ defined by $\left( \ref{Da+}\right) $ has formal adjoint $$\left( d_{A}^{+}\right) ^{\ast }
=\ast d_{A}^{+}\ast :\Omega ^{6}\left(\mathfrak{g}\right)
\rightarrow \Omega ^{1}\left( \mathfrak{g}\right) .$$
For $a\in \Omega ^{1}\left( \mathfrak{g}\right) $ and $\eta \in \Omega
^{6}\left( \mathfrak{g}\right) $, we have pointwise: $$\begin{aligned}
\left\langle d_{A}^{+}a,\eta \right\rangle \left( \ast 1\right) &=&\left(
\ast \varphi \wedge d_{A}a\right) \wedge \ast \eta =\left( d_{A}a\right)
\wedge \ast \left( \ast \left( \ast \varphi \wedge \ast \eta \right) \right)
\\
&=&\left\langle d_{A}a,\ast \left( L_{\ast \varphi }\ast \eta \right)
\right\rangle =\left\langle a,d_{A}^{\ast }\left( \ast L_{\ast \varphi }\ast
\eta \right) \right\rangle \left( \ast 1\right) \\
&=&\left\langle a,\ast \left( d_{A}L_{\ast \varphi }\right) \ast \eta
\right\rangle \left( \ast 1\right) \overset{\left( \ref{[Lfi,dA]=0}\right) }{=}\left\langle a,\ast \left( L_{\ast \varphi }d_{A}\right) \ast \eta
\right\rangle \left( \ast 1\right) \\
&=&\left\langle a,\left( \ast d_{A}^{+}\ast \right) \eta \right\rangle
\left( \ast 1\right) \end{aligned}$$
\[prop elliptic complex\] When $A$ is a $G_2-$instanton, the complex $\left(\ref{extended complex}\right)$ is elliptic.
First of all, since $\left( d_{A}^{+}\right) ^{\ast }=\ast d_{A}^{+}\ast $ \[*Lemma \[lemma (Da+)\*=\*(Da+)\*\]*\] and $d_{A}^{\ast }=\ast d_{A}\ast $, notice that our complex is self-dual with respect to the Hodge star:$$\begin{array}{ccccccc}
\Omega ^{0} & \overset{d_{A}}{\longrightarrow } & \Omega ^{1} &
\overset{d_{A}^{+}}{\longrightarrow } & \Omega ^{6} &
\overset{d_{A}}{\longrightarrow } & \Omega ^{7} \\
\shortparallel & & \shortparallel & & \shortparallel & & \shortparallel\\
\ast \Omega ^{7} & \underset{d_{A}^{\ast }}{\longleftarrow } &
\ast \Omega ^{6} & \underset{\left( d_{A}^{+}\right) ^{\ast }}
{\longleftarrow } & \ast\Omega ^{1} & \underset{d_{A}^{\ast }}
{\longleftarrow } & \ast \Omega ^{0}\end{array}.$$By *Corollary \[cor complex is elliptic iff dual is elliptic\]*, it suffices to show ellipticity at $\Omega ^{1}\left( \mathfrak{g}\right) $, as that is equivalent to the ellipticity of the dual $\ast \Omega ^{7}\overset{d_{A}^{\ast }}{\longleftarrow }\ast \Omega ^{6}\overset{\left(
d_{A}^{+}\right) ^{\ast }}{\longleftarrow }\ast \Omega ^{1}$, which is just $\Omega ^{1}\overset{d_{A}^{+}}{\longrightarrow }\Omega ^{6}\overset{d_{A}}{\longrightarrow }\Omega ^{7}$. Fixing a section $\xi $ of $T^{^{\prime }}M$ (the cotangent bundle minus its zero section), we have symbol maps$$0\rightarrow \pi ^{\ast }
\left( \Omega ^{0}\left( \mathfrak{g}\right)\right) _{\xi }
\overset{\xi .\left( .\right) }{\longrightarrow }\pi ^{\ast }
\left( \Omega ^{1}\left( \mathfrak{g}\right) \right) _{\xi }
\overset{\ast \varphi \wedge \xi \wedge \left( .\right) }
{\longrightarrow }\pi ^{\ast }\left( \Omega ^{6}
\left( \mathfrak{g}\right) \right) _{\xi}\longrightarrow \dots$$For $\alpha \in \Omega ^{1}\left( \mathfrak{g}\right) $ such that $\ast \varphi \wedge \xi \wedge \alpha =0$, exactness means $\alpha $ has to lie in $\xi .\Omega ^{0}\left( \mathfrak{g}\right) $. Since $G_{2}$ acts transitively on $S^{6}$, take $g\in G_{2}$ such that $g^{\ast }\xi =\left\Vert \xi \right\Vert .e^{1}$ and denote $\widetilde{\alpha }=g^{\ast }\alpha $, so that$$\ast \varphi \wedge e^{1}\wedge \widetilde{\alpha }=0.$$That is just the statement that $e^{1}\wedge \widetilde{\alpha }$ is anti-self-dual, but this cannot occur unless $e^{1}\wedge \widetilde{\alpha }=0$, as$$\left( e^{1}\wedge \widetilde{\alpha }\right) \wedge \varphi
=\widetilde{\alpha }\wedge \left( e^{1567}-e^{1345}-e^{1426}+e^{1237}\right)$$has non-vanishing components involving $e^{1}$ and $\ast \left( e^{1}\wedge
\widetilde{\alpha }\right) $ obviously has not. Therefore $\widetilde{\alpha
}=f.e^{1}$ for some $f\in \Omega ^{0}\left( \mathfrak{g}\right) $, and$$\alpha =\left( g^{\ast }\right) ^{-1}\left( f.e^{1}\right)
=\frac{f}{\left\Vert \xi \right\Vert }.\xi \in \xi .\Omega ^{0}
\left( \mathfrak{g}\right) .$$
In view of the isomorphism $\left. L_{\ast \varphi} \right\vert _{\Omega
_{+}^{2}}:\Omega_{+}^{2} \, \tilde{\rightarrow} \;\Omega^{6}$, taking the self-dual part of curvature via the $\mathcal{G}-$equivariant map $
L_{\ast \varphi}\circ
F^{+}:\mathcal{A}\rightarrow \Omega^{6}\left( \mathfrak{g}\right)$ defines a section $\Psi \left( \left[ A\right] \right) =F_{A} \wedge\ast\varphi $ of the Hilbert bundle $$\mathcal{A}\times _{\mathcal{G}}\Omega ^{6}
\left( \mathfrak{g}\right) \rightarrow \mathcal{B}.$$
The *intrinsic derivative*, i.e., the component of the total derivative tangent to the gauge-fixing slices in $\Omega ^{1}\left( \mathfrak{g}_{E}\right)$, of $\Psi $ at $\left[ A\right] $ is$$\begin{aligned}
\left( D\Psi \right) _{\left[ A\right] } :\ker d_{A}^{\ast }
&\rightarrow&\ker d_{A}\subset \Omega^{6}\left( \mathfrak{g}\right) \\
a&\mapsto& d_{A}^{+}a.\end{aligned}$$To see that $\left( D\Psi \right) _{\left[ A\right] }$ is Fredholm over $Z\left( \Psi \right) $, consider the extended operator $$\index{elliptic!operator}
\begin{array}{c c l}
{\mathbb{D}_{A}}: \;\Omega ^{1}\left( \mathfrak{g}\right) \oplus \Omega ^{7}\left( \mathfrak{g}\right)
&\rightarrow& \Omega ^{0}\left( \mathfrak{g}\right)
\oplus \Omega^{6}\left( \mathfrak{g}\right)\\
\qquad \left( a,f\right) &\mapsto&
\left( d_{A}^{\ast}a,\,d_{A}^{+}a+d_{A}^{\ast }f\right) .
\end{array}$$ If the base manifold was compact, then standard elliptic theory would imply ${\mathbb{D}_{A}}=d_{A}^{\ast }\oplus \left( d_{A}^{+}\oplus
d_{A}^{\ast }\right) $ is Fredholm, as the ‘Euler characteristic’ of an elliptic complex, by *Proposition \[prop elliptic complex\]*. However, on our manifolds with cylindrical ends $W\simeq W_0\cup W_\infty$, the parametrix patching method over the compact piece $W_0$ must be combined with tubular theory over $W_\infty$ under the acyclic assumption. This follows in all respects the proof of [@floer Prop. 3.6] and its preceding discussion, except that in this case we are in the much simpler situation where the exponential decay of $(a,f)\in\ker\mathbb{D}_{A}$ is guaranteed from the outset by our definitions \[cf. (\[eq def A\])\] and the fact that the bundle is indecomposable. Then we have:
If $\left[ A\right] \in Z\left( \Psi \right)$ is asymptotically rigid \[cf. *Definition \[def asymp rigid\]*\], ${\mathbb{D}_{A}}$ is a Fredholm operator.
In particular, ${\mathbb{D}_{A}}$ has closed range, so there is an orthogonal decomposition: $$\label{eq Hodge theory}
\Omega^6(\mathfrak{g})=\ker d_A\oplus {{\textnormal {img\,}}}d^*_A$$ and this implies that $\left( D\Psi \right) _{\left[ A\right]} $ is also Fredholm:
$$\begin{array}{r c c c c c c }
\ker\left(D\Psi\right)_{\left[ A\right]} &
\hookrightarrow & \ker{\mathbb{D}_{A}} & & & & \\
{\textnormal {coker\,}}\left( D\Psi \right) _{\left[ A\right] }
& = & {\textnormal {coker\,}}d_{A}^{+} & \cap &
{\textnormal {coker\,}}\left({ d_{A}^*}
\vert_{\Omega ^{7} }\right) &
\hookrightarrow & {\textnormal {coker\,}}{\mathbb{D}_{A.}} \\
& & \cap & & \| & & \\
& & \ker d_{A} & = & \ker d_{A} & & \\
\end{array}$$
Finally, the moduli space of $G_2-$instantons \[*Definition \[def Moduli space of G2-instantons\]*\] is cut out as its zero set $Z(\Psi)$. As an immediate consequence of $\left( \ref{[Lfi,dA]=0}\right) $ and the Bianchi identity, we have $\Psi \left( \left[ A\right] \right) \in \ker
d_{A}\subset \Omega ^{6}\left( \mathfrak{g}_{E}\right)$, so, intuitively, the image of $\Psi $ lies in the ‘subbundle of kernels of $d_{A}$’: $$\label{subbundle of kernels}
\begin{array}{ccccc}
\mathcal{A} & \times _{\mathcal{G}} & \ker d_{A} & \rightarrow & \mathcal{B} \\
& & \cap & & \\
& & \Omega ^{6}\left( \mathfrak{g}\right) & &
\end{array}$$When ${\mathcal{E}}$ is an $SU\left( n\right)
-$bundle, say, one can use the orthogonal projections $p_{a}:\ker
d_{A}\rightarrow \ker d_{A_{0}}$ to trivialise the fibres onto $\ker
d_{A_{0}}$ over a neighbourhood $U_{\varepsilon }\left( \left[ A_{0}\right] \right) \subset \mathcal{B}$, where $\varepsilon $ is a small global constant:
\[prop projection between kernels\] Let ${\mathcal{E}}$ be an $SU\left( n\right) -$bundle over a compact $G_{2}-$manifold $\left( M,\varphi
\right) $ and $A_{0}$ an irreducible connection; then there exists $\varepsilon >0$ such that the orthogonal projection$$p_{a}:\ker d_{A}\rightarrow \ker d_{A_{0}}$$in $\Omega ^{6}\left( \mathfrak{g}\right) $ is an isomorphism for all $A=A_{0}+a\in U_{\varepsilon }\left( A_{0}\right) $.
We consider, throughout, the operators \[cf. $\left( \ref{extended complex}\right) $ & $(\ref{eq Hodge theory})$\]: $$\Omega^{6}\left( \mathfrak{g}\right)
\overset{d_{A_{0}},d_{A}}{\underset{d_{A_{0}}^{\ast },
d_{A}^{\ast }}{\rightleftarrows }}\Omega^{7}
\left( \mathfrak{g}\right).$$Writing $\rho =d_{A_{0}}^{\ast }f$ for some $f\in \Omega ^{7}\left( \mathfrak{g}\right) $, we denote elements of $\Omega^6 \left( \mathfrak{g}\right) $ by $$\eta =(\eta _{0}\oplus \rho )\in
\left(\ker d_{A_{0}}\oplus {{\textnormal {img\,}}}d_{A_{0}}^{\ast }\right)
=\Omega^6\left( \mathfrak{g}\right).$$ ***Surjectivity***
Given $\eta _{0}\in \ker d_{A_{0}}$, write $g_{0}\doteq
-a\wedge \eta _{0}$; surjectivity of $p_{a}$ means finding $\rho \in
{{\textnormal {img\,}}}d_{A_{0}}^{\ast }\subset \Omega^6\left( \mathfrak{g}\right) $ such that $\eta =\eta _{0}\oplus \rho \in \ker d_{A}$, i.e., solving for $\rho $ the equation$$d_{A}\rho =g_{0}. \label{surjectivity}$$Since $A_{0}$ is irreducible, one has $\left( {{\textnormal {img\,}}}d_{A_{0}}\right) ^{\perp
}=\ker d_{A_{0}}^{\ast }=\left\{ 0\right\} $, therefore ${\textnormal {img\,}}d_{A_{0}}=\Omega ^{7}\left( \mathfrak{g}\right)$. Thus one may think of the restriction of $d_{A}$ to ${\textnormal {img\,}}d_{A_{0}}^{\ast }$ as $$\label{restriction of dA}
d_{A}:{{\textnormal {img\,}}}d_{A_{0}}^{\ast }\rightarrow {{\textnormal {img\,}}}d_{A_{0}}.$$Bijectivity of linear maps between Banach spaces is an open condition \[*Lemma \[Lemma Fine\]*\], so one can show that $\left( \ref{restriction of dA}\right) $ is invertible by checking that, for suitably small $a$, this map is arbitrarily close to the isomorphism $d_{A_{0}}:{{\textnormal {img\,}}}d_{A_{0}}^{\ast }\tilde{\rightarrow}\;{{\textnormal {img\,}}}d_{A_{0}}$. Indeed, writing $L_{a}:\eta \mapsto a\wedge \eta$, there exists a global constant such that$$\left\Vert d_{A}-d_{A_{0}}\right\Vert =\left\Vert L_{a}\right\Vert \leq
C\left\Vert a\right\Vert <C\varepsilon.$$Here we used *Lemma \[Lemma Sobolev’s embedding\]*, since our choice of $\left\Vert. \right\Vert^p_k$ suits Sobolev’s embedding theorem. So $\left( \ref{restriction of dA}\right) $ is also an isomorphism for $\varepsilon $ small enough, and we can find a unique $\rho \in \ker d_{A_{0}}^{\ast }$ solving $\left( \ref{surjectivity}\right) $.
***Injectivity***
Let $\eta \in \ker d_{A}\subset \Omega^6\left(
\mathfrak{g}\right) $; then $$\begin{aligned}
p_{a}\left( \eta \right) =0
&\Leftrightarrow &\rho =\eta \in \ker d_{A} \\
&\Leftrightarrow &d_{A}\rho =0 \\
&\Leftrightarrow &\rho =0\end{aligned}$$since $\rho \in {\textnormal {img\,}}d_{A_{0}}^{\ast }$ and we have just seen that $d_{A}:{{\textnormal {img\,}}}d_{A_{0}}^{\ast }\tilde{\rightarrow}\; {{\textnormal {img\,}}}d_{A_{0}}$ is an isomorphism (for suitably small $a$); so $\eta =\rho =0$.
Hence *Proposition $\ref{Prop local description of ME}$* is proved, in the terms of *Corollary \[cor zero set\]* \[cf. *Appendix*\].
Final comments: gluing families and transversality
--------------------------------------------------
We achieved in *Theorem \[thm gluing\]* this paper’s main goal of constructing a solution $A$ of the instanton equation over the compact $G_2-$manifold $M_S=W'{\tilde\#_S}W''$. To conclude, I will briefly outline two natural extensions of this theory.
First, one may consider instanton families, i.e., given (pre)compact sets $N{^{(i)}}\subset{\mathcal{M}}^+_{{\mathcal{E}}{^{(i)}}}$ of regular points on the moduli spaces over each end, define - for large neck length $S$ - an operation $$\tau_S:N'\times N''\rightarrow{\mathcal{M}}^+_{{\mathcal{E}}_S}.$$ This should be a diffeomorphism over its image, consisting itself of regular points. Moreover, given an adequate notion of ‘distance’ between a connection $A^{{\tilde\#_S}}$ on ${\mathcal{E}}_S$ and $A=A'{\tilde\#_S}A''$, any ‘nearby’ instanton is also obtained from such a sum. Namely, for a given integer $q$, one defines $$\ell^q_S\left(A^{{\tilde\#_S}} ;A',A''\right)\doteq
\inf_{g\in{\mathcal{G}}'}\Vert g.A^{{\tilde\#_S}}-A' \Vert_{L^q(W'_S)}+
\inf_{g\in{\mathcal{G}}''}\Vert g.A^{{\tilde\#_S}}-A'' \Vert_{L^q(W''_S)}$$ where $W{^{(i)}}_S$ are the truncations of $W{^{(i)}}$ at ‘length’ $S$. Then one should expect the following to hold:
Let $N{^{(i)}}\subset{\mathcal{M}}^+_{{\mathcal{E}}{^{(i)}}}$ be compact sets of regular points in the moduli spaces of $G_2-$instantons over $W{^{(i)}}\times S^1$; for sufficiently small $\delta>0$ and large $S>0$, there are neighbourhoods $V{^{(i)}}\supset N{^{(i)}}$ and a smooth map $$\tau_S:V'\times V''\rightarrow{\mathcal{M}}^+_{{\mathcal{E}}_S}$$ such that, for appropriate choices of $q>0$ (independent of $\delta$ and $S$),
1. $\tau_S$ is a diffeomorphism to its image, which consists of regular points;
2. $\ell^q_S\left(\tau_S(A',A'');A',A''\right)\leq\delta $, $\forall A{^{(i)}}\in V{^{(i)}}$;
3. any instanton $A\in{\mathcal{M}}^+_{{\mathcal{E}}_S}$ such that $\ell^q_S\left(\tau_S(A',A'');A',A''\right)\leq\delta $ for some $A{^{(i)}}\in V{^{(i)}}$ lies in the image of $\tau_S(V'\times V'')$.
Finally, under a generic Morse-Bott assumption, one may envisage relaxing the non-degeneracy requirement and deal with reducible connections via the ‘gluing parameter’ over the divisor at infinity. If the images of the obvious restriction maps as $r{^{(i)}}:{\mathcal{M}}^+_{{\mathcal{E}}{^{(i)}}}\rightarrow {\mathcal{M}}_D$ meet transversally, one would expect the moduli space ${\mathcal{M}}^+_{{\mathcal{E}}_S}$ over the compact base $M_S$ to be smoothly modelled on the product $${\mathcal{M}}^+_{{\mathcal{E}}'}\times_{{\mathcal{M}}_D}{\mathcal{M}}^+_{{\mathcal{E}}''}
=\left\{ (A',A'')\mid r'(A')=r''(A'') \right\}$$ for large values of the neck length $S$.
All of this follows strictly, of course, the general ‘programme’ of [@floer]. There are essentially two reasons to expect the analogy to carry through to our $G_2$ setting: the bounded geometry of Kovalev’s manofolds, which implies the uniform (Sobolev) bounds for the relevant operator norms \[cf. *Lemma \[Lemma noncompact Sobolev\]* and *Proposition \[prop bounded multiplication\]*\]; and the fact that our original solutions decay exponentially in all derivatives to a HYM connection over the divisor at infinity \[cf. (\[eq HYM solution\])\], so they are in $L^p_k(W)$ for any choice of $p,k$.
Chern-Simons formalism under holonomy $G_2$ {#Subsect Chern-Simmons}
===========================================
In (3+1)-dimensional gauge theory [floer]{}, the Chern-Simons functional is defined on $\mathcal{B=A}/\mathcal{G}$, with integer periods, its critical points being precisely the flat connections. A similar theory can be formulated in higher dimensions given a suitable closed $\left(
n-3\right) -$form [Donaldson-Thomas]{}[@Thomas]. Here, for suitable connections over a $G_2-$manifold $(M,\varphi)$, we use the Hodge-dual $\ast \varphi$.
Recall that the set of connections $\mathcal{A}$ is an affine space modelled on $\Omega ^{1}\left(
\mathfrak{g}_{P}\right) $ so, fixing a reference $A_{0}\in
\mathcal{A}$, we have ${\mathcal{A}}=A_{0}+\Omega ^{1}\left( \mathfrak{g}_{P}\right)$ and we can define $$\index{Chern-Simons!functional}
\vartheta \left( A\right) =\tfrac{1}{2}\int_{M}{\textnormal {tr\,}}\left( d_{A_{0}}a\wedge a +\frac{2}{3}a\wedge a\wedge a\right) \wedge \ast \varphi ,$$ fixing $\vartheta \left( A_{0}\right) =0$. Note in passing that, since only the condition $d\ast\varphi=0$ is required, the discussion extends to cases in which the $G_2-$structure $\varphi$ is not necessarily torsion-free. Moreover, the theory remains essentially unaltered if the compact base manifold is replaced by a manifold-with-boundary, say, under a setting of connections with suitable decay towards the boundary.
The above function is obtained by integration of the analogous $1-$form$$\label{ro ^ phi}
\index{Chern-Simons!1-form@$1-$form} \rho \left( a\right) _{A}=\int_{M} {\textnormal {tr\,}}\left( F_{A}\wedge a\right) \wedge \ast \varphi .$$We find $\vartheta $ explicitly by integrating $\rho $ over paths $A\left( t\right) =A_{0}+ta$:$$\begin{aligned}
\vartheta \left( A\right) -\vartheta \left( A_{0}\right)
&=&\int_{0}^{1}\rho _{A\left( t\right) }
\left( \dot{A}\left( t\right) \right) dt \\
&=&\int_{0}^{1}\int_{M}{\textnormal {tr\,}}\left( \left(F_{A_{0}}+td_{A_{0}}a
+t^{2}a\wedge a\right) \wedge a\right) \wedge \ast \varphi \\
&=&\tfrac{1}{2}\int_{M}{\textnormal {tr\,}}\left( d_{A_{0}}a\wedge a
+\frac{2}{3} a\wedge a\wedge a\right) \wedge \ast \varphi .\end{aligned}$$
It remains to check that $\left( \ref{ro ^ phi}\right) $ is closed, so that this doesn’t depend on the path $A\left( t\right) $. Using Stokes’ theorem and $d\ast \varphi =0$, the leading term of $\rho ,$ $$\rho \left( a\right) _{A+b}-\rho \left( a\right) _{A}=\int_{M}{\textnormal {tr\,}}\left( d_{A}b\wedge a\right) \wedge \ast \varphi +O\left( \left\vert
b\right\vert ^{2}\right) ,$$is indeed symmetric:$$\int_{M}{\textnormal {tr\,}}\left( d_{A}b\wedge a-b\wedge d_{A}a\right) \wedge \ast
\varphi =\int_{M}d\left( {\textnormal {tr\,}}\left( b\wedge a\right) \wedge \ast
\varphi \right) =0.$$Hence $$\rho \left( a\right) _{A+b}-\rho \left( a\right) _{A}=\rho \left( b\right)
_{A+a}-\rho \left( b\right) _{A}+O\left( \left\vert b\right\vert ^{2}\right),$$and we check that $\rho $ is closed comparing the reciprocal Lie derivatives on parallel vector fields $a,b$ around a point $A$:
$$\begin{aligned}
\label{closed 1-form}
d\rho \left( a,b\right) _{A}
&=&\left( \mathcal{L}_{b}\rho \left( a\right)\right) _{A}
-\left( \mathcal{L}_{a}\rho \left( b\right) \right) _{A} \\
&=&\lim\limits_{h\rightarrow 0}\frac{1}{h}\left\{ \left( \rho
\left(a\right) _{A+hb}-\rho \left( a\right) _{A}\right)
-\left( \rho \left(b\right) _{A+ha}
-\rho \left( b\right) _{A}\right) \right\} \\
&=&\lim\limits_{h\rightarrow 0}\frac{1}{h^{2}}\;
\underset{O\left( \left\vert h\right\vert ^{3}\right) }
{\underbrace{\left\{ \left( \rho \left( ha\right)_{A+hb}
-\rho \left( ha\right) _{A}\right) -\left( \rho \left( hb\right)_{A+ha}
-\rho \left( hb\right) _{A}\right) \right\} }} \\
&=&0.\end{aligned}$$
At least locally, then, the functional $\vartheta $ descends to the orbit space $\mathcal{B}$.
To obtain the periods of $\vartheta $ under gauge action, take $g\in \mathcal{G}$ and consider a path $\left\{ A\left( t\right) \right\} _{t\in \left[ 0,1\right] }\subset
\mathcal{A}$ connecting $A$ to $g.A$. The natural projection then induces a bundle$$\begin{array}{ccc}
\mathbf{E}_{g} & \overset{\widetilde{p_{1}}}{\longrightarrow }
& E \\
\downarrow & & \downarrow \\
M\times \left[ 0,1\right] & \overset{p_{1}}{\longrightarrow }
& M\end{array}$$and, using $g$ to identify the fibres $\left( \mathbf{E}_{g}\right) _{0}\overset{g}{\simeq }\left( \mathbf{E}_{g}\right) _{1}$, we think of $\mathbf{E}_{g}$ as a bundle over $M\times S^{1}$. Moreover, in some trivialisation, the path $A\left( t\right) =A_{i}\left( t\right) dx^{i}$ gives a connection $\mathbf{A}=\mathbf{A}_{0}dt+\mathbf{A}_{i}dx^{i}$ on $\mathbf{E}_{g}$: $$\begin{aligned}
\left( \mathbf{A}_{0}\right) _{\left( t,p\right) } &=&0 \\
\left( \mathbf{A}_{i}\right) _{\left( t,p\right) }
&=&A_{i}\left( t\right)_{p}.\end{aligned}$$The corresponding curvature $2-$form is $F_{\mathbf{A}}=\left( F_{\mathbf{A}}\right) _{0i}dt\wedge dx^{i}+\left( F_{\mathbf{A}}\right)
_{jk}dx^{j}\wedge dx^{k}$:$$\begin{aligned}
\left( F_{\mathbf{A}}\right) _{0i} &=&\dot{A}_{i}\left( t\right)\\
\left( F_{\mathbf{A}}\right) _{jk} &=&\left( F_{A}\right) _{jk}.\end{aligned}$$ The periods of $\vartheta$ are then of the form $$\begin{aligned}
\vartheta \left( g.A\right) -\vartheta \left( A\right) &=&
\int_{0}^{1}\rho_{A\left( t\right) }\left( \dot{A}\left( t\right) \right) dt \\
&=&\int_{M\times \left[ 0,1\right] }{\textnormal {tr\,}}(F_{A\left( t\right)}
\wedge\dot{A}_{i}\left( t\right) dx^{i})\wedge dt\wedge \ast \varphi\\
&=&\int_{M\times S^{1}}{\textnormal {tr\,}}F_{\mathbf{A}}\wedge F_{\mathbf{A}}
\wedge\ast \varphi \\
&=&\left\langle c_{2}\left( \mathbf{E}_{g}\right) \smallsmile
\left[ \ast\varphi \right] ,M\times S^{1}\right\rangle .\end{aligned}$$The Künneth formula for the cohomology of $M\times S^{1}$ gives$$H^{4}\left( M\times S^{1}\right)
=H^{4}\left( M\right) \oplus H^{3}\left(M\right)
\otimes \underset{\mathbb{Z}}{\underbrace{H^{1}\left( S^{1}\right)}}$$and obviously $H^{4}\left( M\right) \smallsmile \left[ \ast \varphi \right]
=0$ so, denoting $c_{2}^{\prime }\left( \mathbf{E}_{g}\right) $ the component lying in $H^{3}\left( M\right) $ and $S_{g}=\left[ c_{2}^{\prime
}\left( \mathbf{E}_{g}\right) \right] ^{PD}$ its Poincaré dual, we are left with$$\vartheta \left( g.A\right) -\vartheta \left( A\right) =\left\langle \left[
\ast \varphi \right] ,S_{g}\right\rangle .$$ Consequently, the periods of $\vartheta $ lie in the set$$\left\{ \left.\int_{S_{g}}\ast \varphi \;\right\vert S_{g}\in H_{4}\left( M,\mathbb{R}\right) \right\} .$$That may seem odd because in general this set is *dense* (there is no reason to expect $\ast\varphi$ to be an integral class). Nonetheless, as long as our interest remains in the study of the moduli space ${\mathcal{M}}^+=Z(\rho)$ of $G_{2}$-instantons, as the *critical set* of $\vartheta$, there is nothing to worry, for the gradient $\rho =d\vartheta $ is unambiguously defined on $\mathcal{B}$.
Theory of operators
===================
Here is a preliminary to the proofs of *Proposition \[prop elliptic complex\]* and *Proposition \[prop projection between kernels\]*:
For a sequence $A\overset{S}{\rightarrow }B\overset{T}{\rightarrow }C$ of linear operators (of dense domain) between Hilbert spaces, one has:$$\begin{aligned}
\ker T={{\textnormal {img\,}}}S
&\Leftrightarrow&
\ker S^{\ast }={{\textnormal {img\,}}}T^{\ast }.\end{aligned}$$
I claim $B={{\textnormal {img\,}}}S\oplus \ker S^{\ast }$: $$\begin{aligned}
b\in \left( {\textnormal {img\,}}S\right) ^{\perp }\subset B &\Leftrightarrow& \left\langle b,Sa\right\rangle =0,\quad \forall a\in A\\
&\Leftrightarrow& \left\langle S^{\ast }b,a\right\rangle =0,\quad \forall
a\in A\\
&\Leftrightarrow& b\in \ker S^{\ast }.\end{aligned}$$ Since $\overline{{\textnormal {Dom\,}}(S)}=A$, $S^{\ast }$ is closed [@Brezis II.16], so $\ker S^{\ast }\subset B$ is closed and $B=\left( \ker S^{\ast }\right) ^{\perp }\oplus
\ker S^{\ast }$. *Mutatis mutandis,* $B=\ker T\oplus {\textnormal {img\,}}T^{\ast }$, which yields the claim by uniqueness of the orthogonal complement.
\[cor complex is elliptic iff dual is elliptic\]Let $F\overset{L_{1}}{\rightarrow }G\overset{L_{2}}{\rightarrow }H$ be a complex of differential operators between vector bundles with fibrewise inner products; if the associated symbols satisfy $\sigma
\left( L_{i}^{\ast }\right) =\left( \sigma \left( L_{i}\right) \right)
^{\ast }$, then$$F\overset{L_{1}}{\rightarrow }G\overset{L_{2}}{\rightarrow }H\text{ is
elliptic }\Leftrightarrow H\overset{L_{2}^{\ast }}{\rightarrow }G\overset{L_{1}^{\ast }}{\rightarrow }F\text{ is elliptic.}$$
Ellip´tic complexes are closely related to Fredholm theory, which provides a local model for sets of moduli given as zeroes of sections, as discussed in *Subsection \[subsec prelim moduli theory\]* and the whole of *Section \[sect local model moduli space\]*. I adopt the notation for the Fredholm decomposition of a map [@4-manifolds 4.2.5]:
A Fredholm map $\Xi $ from a neighbourhood of $0$ is locally right-equivalent to a map of the form $$\begin{array}{rcl}
\widetilde{\Xi }:\quad U\times F & \rightarrow & V\times G \\
\widetilde{\Xi }\left( \xi ,\eta \right) & = &
\left( L\left( \xi \right),\sigma\left(\xi,\eta \right)\right)\\
\end{array}$$where $L=\left( D\Xi \right) _{0}:$ $U$ $ \tilde{\rightarrow }$ $V$ is a linear isomorphism, $F=\ker L$ and $G={\textnormal {coker\,}}L$ are finite-dimensional and $\left( D\sigma \right) _{0}=0$.
\[cor zero set\] A neighbourhood of $0$ in $Z\left( \Xi \right) $ is diffeomorphic to $Z\left( \nu \right) $, where $$\begin{aligned}
\begin{array}{rcl}
\nu \; : \; F &\rightarrow& G \\
\nu \left( \eta \right) &\doteq& \sigma \left( 0,\eta \right).
\end{array}\end{aligned}$$
Finally, a result borrowed from [@Joel] is essentially the generalisation to Banach spaces of the fact that the determinant of a linear map is continuous:
\[Lemma Fine\]Let $D:B_{1}\rightarrow B_{2}$ be a bounded invertible linear map of Banach spaces with bounded inverse $Q$. If $L:B_{1}\rightarrow
B_{2}$ is another linear map with$$\left\Vert L-D\right\Vert \leq \left( 2\left\Vert Q\right\Vert \right) ^{-1},$$then $L$ is also invertible with bounded right inverse $P$ satisfying $$\left\Vert P\right\Vert \leq 2\left\Vert Q\right\Vert .$$
For its application in the proof of *Proposition \[prop projection between kernels\]* we are also going to need the following *Lemma*, saying that the norm of the operator ‘multiplication by a function’ on $L^{p}$ is controlled by a suitable Sobolev norm of that function.
\[Lemma Sobolev’s embedding\] On a base manifold $M=W\times S^1$, fix $f\in L_{k}^{p}\left( M\right) $ with $k\geq \dfrac{7}{p}$; then there exists a constant $c=c\left( M\right) $ such that $$\left\Vert L_{f}\right\Vert \leq c\left\Vert f\right\Vert _{L_{k}^{p}},$$where$$\begin{array}{r c l}
L_{f}:L^{p}\left( M\right) &\rightarrow& L^{p}\left( M\right) \\
g&\mapsto& f.g\end{array}.$$
As a direct consequence of Sobolev’s embedding \[*Lemma \[Lemma noncompact Sobolev\]*\], one has $$\left\Vert f.g\right\Vert _{L^{p}}=\left( \int_{M}\left\vert f\right\vert
^{p}.\left\vert g\right\vert^{p} d{{\textnormal {Vol\,}}}\right) ^{\frac{1}{p}}\leq \left\Vert
f\right\Vert _{C^{0}}.\left\Vert g\right\Vert _{L^{p}}$$ and so$$\left\Vert L_{f}\right\Vert =\sup\limits_{\left\Vert g\right\Vert
_{L^{p}}=1}\left\Vert f.g\right\Vert _{L^{p}}\leq \left\Vert f\right\Vert
_{C^{0}}.$$
[99]{}
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Clifford H. Taubes, *Metrics, connections and gluing theorems*, AMS Reg. Conf. series in Math. **89** (1996).
Richard Thomas, D.Phil. Thesis, Univeristy of Oxford (1997).
|
---
abstract: |
It is argued that deep learning is efficient for data that is generated from hierarchal generative models. Examples of such generative models include wavelet scattering networks, functions of compositional structure, and deep rendering models. Unfortunately so far, for all such models, it is either not rigorously known that they can be learned efficiently, or it is not known that “deep algorithms" are required in order to learn them.
We propose a simple family of “generative hierarchal models" which can be efficiently learned and where “deep" algorithm are necessary for learning. Our definition of “deep" algorithms is based on the empirical observation that deep nets necessarily use correlations between features. More formally, we show that in a semi-supervised setting, given access to low-order moments of the labeled data and all of the unlabeled data, it is information theoretically impossible to perform classification while at the same time there is an efficient algorithm, that given all labelled and unlabeled data, perfectly labels all unlabelled data with high probability.
For the proof, we use and strengthen the fact that Belief Propagation does not admit a good approximation in terms of linear functions.
author:
- |
Elchanan Mossel[^1]\
MIT
title: Deep Learning and Hierarchical Generative Models
---
Introduction
============
We assume that the reader is familiar with the basic concepts and developments in deep learning. We do not attempt to summarize the big body of work studying neural networks and deep learning. We refer readers who are unfamiliar with the area to [@GoBeCo:16] and the references within.
We hypothesize that deep learning is efficient in learning data that is generated from generative hierarchical models. This hypothesis is in the same spirit of the work of Bruna, Mallat and others who suggested wavelet scattering networks [@bruna2013invariant] as the generative model, the work of Mhaskar, Liao and Poggio who suggested compositional functions as the generative model [@mhaskar2016learning] and work by Patel, Nguyen, and Baraniuk [@patel2015probabilistic] who suggested hierarchical rending models. Unfortunately so far, for all previous models, it is either not rigorously known that they can be learned efficiently, or it is not known that “deep algorithms" are required in order to learn them.
The approach presented in this paper is motivated by connections between deep learning and evolution. In particular, we focus on simple generative evolutionary models as our generative processes. While these models are [*not*]{} appropriate models for images or language inference, they provide several advantages:
- There are well established biological processes of evolution. Thus the generative model studied is not a human-made abstraction but the actual process that generated the data.
- The models are mathematically simple enough so we can provide very accurate answers as to the advantage of the depth in a semi-supervised setting.
We further note that some of the most successful applications of deep learning are in labeling objects that are generated in an evolutionary fashion such as the identification of animals and breeds from images, see e.g. [@LeBeHi:15] and the reference within. Let us consider the problem of identifying species from images. One form of this problem was tackled by Darwin. In his [*Evolution of Species*]{}, Darwin used phylogenetic trees to summarize the evolution of species [@Darwin:59]. The evolutionary tree, in turn, helps in identifying observed species. The problem of species identification took a new twist in the DNA age, where morphological characters of species were replaced by DNA sequences as the data for inference of the relationship between species [@Felsenstein:04; @SempleSteel:03].
Our hypothesis that deep learning is effective in recovering generative hierarchal models leads us to explore other generative models and algorithms to recover these models. While the models we will develop in the current work are restrictive, they represent an attempt to extend the phylogenetic theory from the problem of reconstructing trees based on DNA sequences to reconstructing relationships based on different types of representations with the ultimate goal of understanding “real" representations such as representations of natural languages and natural images. In what follows we introduce a family of models. We start with the phylogenetic model. The phylogenetic model we use, the symmetric Markov model, is a classical model. However, we study it from a new perspective:
- First - in addition to a DNA sequence, each node of the tree is associated with a label where different nodes might have the same label. For example, a node with a specific DNA sequence might have the label,”dog“ or ”mammal”.
- Second - we are interested in the semi-supervised learning problem, where the labels of a small subset of the data are known and the goal is to recover the labels of the remaining data.
We then define novel generative models which have additional features:
- Change of representation. In phylogenetic models, the representation is given by the DNA sequences (or RNA, proteins etc.), while it seems like in many deep learning situations, there isn’t necessarily a canonical representation. We model this by introducing a permutation on the alphabet between every node and each of its descendants.
- Interaction between features. In classical phylogenetic models each letter evolves independently, while in most deep learning scenarios, the interaction between features is key. We introduce a model that captures this property.
In order to establish the power of deep learning, we define two types of limited algorithms (which are not ”deep"):
- [*Local algorithms*]{}. Such algorithms have to determine the label of each data point based on the labeled data only. The notion of local algorithms is closely related to the notion of supervised learning. Note, however, that local algorithms do not output a classifier after observing the labeled data ; instead for each sample of unlabeled data we run the local algorithm.
- [*Shallow Algorithms*]{}. These algorithms only use summary statistics for the labeled data. In other words, such algorithms are not allowed to utilize high order correlations between different features of the labeled data (our results will apply to algorithms that can use bounded order correlations).
In our main results, we provide statistical lower bounds on the performance of local and shallow algorithms. We also provide efficient algorithms that are neither shallow nor local. Thus our results provide a formal interpretation of the power of deep learning. In the conclusion, we discuss a number of research directions and open problems.
Related Work
------------
Our work builds on work in theoretical phylogenetics, with the aim of providing a new theoretical perspective on deep learning. An important feature of our generative models is that they include both representation and labels. In contrast, most of the work in the deep learning literature focuses on the encoding within the deep network. Much of the recent work in deep learning deals with the encoding of data from one level to the next. In our model, we avoid this (important) aspect by considering only representations that are essentially 1 to 1 (if we exclude the effect of the noise or “nuisance variables"). Thus our main focus is in obtaining rigorous results relating multi-level hierarchical models and classes of semi-supervised learning algorithms whose goal is to label data generated from the models.
A main theme of research in the theory of deep networks is studying the expressive power of bounded width networks in terms of their depth, see e.g. [@CoShSa:16; @EldanShamir:16; @Telgarsky:16] and also [@mhaskar2016learning]. Our results show that learning of deep nets cannot be performed by simple methods that apply to shallow generative models. We also note that positive theoretical results of [@ABGM:14]. These, however, are not accompanied by lower bounds showing that deep algorithms are needed in their setup.
Hierarchical Generative Models
==============================
In this section, we will define the generative models that will be discussed in the paper.
The space of all objects - a tree
---------------------------------
All the models will be defined on a $d$-ary tree $T=(V,E)$ of $h$ levels, rooted at $v_0$.
The assumption that the tree is regular is made for simplicity. Most of the results can be extended to many other models of trees, including random trees.
Representations
---------------
The representation is a function $R : V \to [q]^k$. The representation of node $v \in V$ is given by $R(v)$.
In some examples, representations of nodes at different levels are of the same type. For example, if we think of $T$ as a phylogenetic tree, then $R(v)$ may represent the DNA sequence of $v$. In other examples, $R(v)$ has a different meaning for nodes at different levels. For example, if $T$ represents a corpus of images of animals, then $R(v)$ for a deep node $v$ may define the species and the type of background, while $R(v)$ at a lower level may represent the pixels of an image of an animal (this is an illustration - none of the models presented at the paper are appropriate for image classification).
Given a root to leaf path $v_0,\ldots,v_{\ell}$, we will consider $R(v_{\ell-1})$ as a higher level representation of $R(v_{\ell})$,. Similarly $R(v_{\ell-2})$ is a higher level representation of $R(v_{\ell-1})$ (and therefore of $R(v_{\ell})$. Thus, each higher level representation has many descendant lower level representations. In particular, all of the representations considered are derived from $R(v_0)$.
Labels
------
Each node $v \in V$ of the tree has a set of labels $L(v)$. $L(v)$ may be empty for some nodes $v$. We require that if $w$ is a descendant of $v$ then $L(v) \subseteq L(w)$ and that every two nodes $v_1,v_2$ that have the same label $\ell$ have a common ancestor $v_3$ with label $\ell$. In other words, the set of nodes labeled by a certain label is a node in the tree and all nodes below that node.
For example, a possible value for $L(v)$ is $\{ ``dog", ``german shepherd" \}$.
The inference problem
---------------------
Let $L_T$ denote the set of leaves of $T$. Let $S \subset L_T$. The input to the inference problem consists of the set $\{(R(v), L(v)) : v \in S\}$ which is the [*labeled data*]{} and the set $\{R(v) : v \in L_T \setminus S\}$ which is the [*unlabeled data*]{}.
The desired output is $L(v)$ for all $v \in L_T$, i.e., the labels of all the leaves of the tree.
Generative Models
-----------------
We consider a number of increasingly complex generative models. While all of the models are stylized, the more advanced ones capture more of the features of “deep learning" compared to the simpler models. All of the models will be Markov models on the tree $T$ (rooted at $v_0$). In other words, for each directed edge of the tree from a parent $v$ to child $w$, we have a transition matrix $M_{v,w}$ of size $[q]^k \times [q]^k$ that determines the transition probabilities from the representation $R(v)$ to the representation $R(w)$. We consider the following models:
### The i.i.d. Model (IIDM)
We first consider one of the simplest and most classical phylogenetic models given by i.i.d. symmetric Markov models. Special cases of this model, for $q=2$ or $q=4$ are some of the most basic phylogenetic evolutionary models. These models called the CFN and Jukes-Cantor model respectively [@JukesCantor:69; @Neyman:71; @Farris:73; @Cavender:78]. The model is defined as follows: If $w$ is the parent of $v$ then for each $1 \leq i \leq k$ independently, it holds that conditioned on $R(w)$ for all $a \in [q]$: $$P[R(v)_i = a] = \frac{1-\lambda}{q} + \lambda \delta(R(w)_i = a).$$ In words, for each letter of $R(w)$ independently, the letter given by the parent is copied with probability $\lambda$ and is otherwise chosen uniformly at random.
### The Varying Representation Model (VRM)
One of the reasons the model above is simpler than deep learning models is that the representation of nodes is canonical. For example, the model above is a classical model if $R(w)$ is the DNA sequence of node $w$ but is a poor model if we consider $R(w)$ to be the image of $w$, where we expect different levels of representation to have different “meanings". In order to model the non-canonical nature of neural networks we will modify the above representation as follows. For each edge $e = (w,v)$ directed from the parent $w$ to the child $v$, we associate a permutation $\sigma_e \in S_q$, which encodes the relative representation between $w$ and $v$. Now, we still let different letters evolve independently but with different encodings for different edges. So we let:
$$P[R(v)_i = a] = \frac{1-\lambda}{q} + \lambda \delta(R(w)_i = \sigma_e^{-1}(a)).$$ In words, each edge of the tree uses a different representation of the set $[q]$.
We say the the collection $\sigma = (\sigma_e : v \in E)$ is [*adversarial*]{} if $\sigma$ is chosen by an adversary. We say that it is [*random*]{} if $\sigma_e$ are chosen i.i.d. uniform. We say that $\sigma = (\sigma_e : v \in E)$ are [*shared parameters*]{} if $\sigma_{e}$ is just a function of the level of the edge $e$.
### The Feature Interaction Model (FIM)
The additional property we would like to introduce in our most complex model is that of an interaction between features. While the second model introduces some indirect interaction between features emanating from the shared representation, deep nets include stronger interaction. To model the interaction for each directed edge $e=(w,v)$, we let $\sigma_e \in S_{q^2}$. We can view $\sigma_e$ as a function from $[q]^2 \to [q]^2$ and it will be useful for us to represent it as a pair of functions $\sigma_e = (f_e,g_e)$ where $f_e : [q]^2 \to [q]$ and $g_e : [q]^2 \to [q]$. We also introduce permutations $\Sigma_1,\ldots,\Sigma_h \in S_k$ which correspond to rewiring between the different levels. We then let $$P[\widetilde{R(v)}_i = a] = \frac{1-\lambda}{q} + \lambda \delta(R(w)_i = a),$$ and $$R(v)_{2i} = f_{e}(\Sigma_{|v|}(\widetilde{R(w)}(2i)), \Sigma_{|v|}(\widetilde{R(w)}(2i+1))$$ $$R(v)_{2i+1} = g_{e}(\Sigma_{|v|}(\widetilde{R(w)}(2i)), \Sigma_{|v|}(\widetilde{R(w)}(2i+1)).$$
In words, two features at the parent mutate to generate two features at the child. The wiring between different features at different levels is given by some known permutation that is level dependent. This model resembles many of the convolutional network models and our model and results easily extend to other variants of interactions between features. For technical reasons we will require that for all $i,j$ it holds that $$\label{eq:rewire_cond}
\{ \Sigma_j(2i),\Sigma_j(2i+1) \} \neq \{2i, 2i + 1 \}.$$ In other words, the permutations actually permute the letters. It is easy to extend the model and our results to models that have more than two features interact.
The parameter sharing setup
---------------------------
While the traditional view of deep learning is in understanding one object - which is the deep net, our perspective is different as we consider the space of all objects that can be encoded by the network and the relations between them. The two-point of view are consistent in some cases though. We say that the VRM model is [*fixed parametrization*]{} if the permutation $\sigma_e$ are the same for all the edges at the same level. Similarly the FIM is [*parameter shared*]{} if the functions $(f_e,g_e)$ depend on the level of the edge $e$ only. For an FIM model with a fixed parametrization, the deep network that is associated with the model is just given by the permutation $\Sigma_i \in S_k$ and the permutations $(f^1,g^1),\ldots,(f^h,g^h)$ only. While our results and models are stated more generally, the shared parametrization setup deserves special attention:
- Algorithmically: The shared parametrization problem is obviously easier - in particular, one expects, that as in practice, after the parameters $\sigma$ and $\Sigma$ are learned, classification tasks per object should be performed very efficiently.
- Lower bounds: our lower bounds hold also for the shared parametrization setup. However as stated in conjecture \[conj:stronger\], we expect much stronger lower bounds for the FIM model. We expect that such lower bound hold even in the shared parametrization setup.
Shallow, Local and Deep Learning
================================
We define “deep learning" indirectly by giving two definitions of “shallow" learning and of “local" learning. Deep learning will be defined implicitly as learning that is neither local nor shallow.
A key feature that is observed in deep learning is the use of the correlation between features. We will call algorithms that do not use this correlation or use the correlation in a limited fashion shallow algorithms.
Recall that the input to the inference problem $D$ is the union of the labeled and unlabeled data $$D := \{(R(v), L(v)) : v \in S\} \cup \{R(v) : v \in L_T \setminus S\}.$$
Let $A = (A_1,\ldots,A_j)$ where $A_i \subset [k]$ for $1 \leq i \leq j$. The compression of the data according to $A$, denoted $C_A(D)$, is $$C_A(D) := \{R(v) : v \in L_T \setminus S\}
\cup \left( n_D(A_i,x,\ell) : 1 \leq i \leq j, x \in [q]^{|A_i|}, \ell \mbox{ is a label } \right),$$ where for every possible label $\ell$, $1 \leq i \leq j$ and $x \in [q]^{|A_i|}$, we define $$n_D(A_i,x,\ell) := \# \{ v \in S : L(v) = \ell, R(v)_{A_i} = x\}.$$ The canonical compression of the data, $C_{\ast}(D)$, is given by $C_{A_1,\ldots,A_k}(D)$, where $A_i = \{ i \}$ for all $i$.
In words, the canonical compression gives for every label $\ell$ and every $1 \leq i \leq k$, the histogram of the $i$’th letter (or column) of the representation among all labeled data with label $\ell$. Note that the unlabeled data is still given uncompressed.
The more general definition of compression allows for histograms of joint distributions of multiple letters (columns). Note in particular that if $A = (A_1)$ and $A_1 = [k]$, then we may identify $C_A(D)$ with $D$ as no compression is taking place.
We say that an inference algorithm is $s$-shallow if the output of the algorithm as a function of the data depends only on $C_A(D)$ where $A = (A_1,\ldots,A_j)$ and each $A_i$ is of size at most $s$. We say that an inference algorithm is shallow if the output of the algorithm as a function of the data depends only on $C_{\ast}(D)$.
In both cases, we allow the algorithm to be randomized, i.e., the output may depend on a source of randomness that is independent of the data.
We next define what local learning means.
Given the data $D$, we say that an algorithm is local if for each $R(w)$ for $w \in L_T \setminus S$, the label of $v$ is determined only by the representation of $w$, $R(w)$, and all labeled data $\{(R(v), L(v)) : v \in S\}$.
Compared to the definition of shallow learning, here we do not compress the labeled data. However, the algorithm has to identify the label of each unlabeled data point without access to the rest of the unlabeled data.
Main Results
============
Our main results include positive statements establishing that deep learning labels correctly in some regimes along with negative statements that establish that shallow or local learning do not. Combining both the positive and negative results establishes a large domain of the parameter space where deep learning is effective while shallow or local learning isn’t. We conjecture that the lower bounds in our paper can be further improved to yield a much stronger separation (see conjecture \[conj:stronger\]).
Main parameters
---------------
The results are stated are in terms of
- the branching rate of the tree $d$,
- the noise level $1-\lambda$,
- the alphabet size $[q]$ and
- The geometry of the set $S$ of labeled data.
Another crucial parameter is $k$, the length of representation. We will consider $k$ to be between logarithmic and polynomial in $n = d^h$.
We will also require the following definition.
Let $\ell$ be a label. We say that $\ell$ is [*well represented*]{} in $S$ if the following holds. Let $v \in V$ be the vertex closest to the root $v_0$ that is labeled by $\ell$ (i.e., the set of labels of $v_0$ contains $\ell$). Then there are two edge disjoint path from $v$ to $v_1 \in S$ and to $v_2 \in S$.
The following is immediate
If the tree $T$ is known and if $\ell$ is well represented in $S$ then all leaves whose label is $\ell$ can be identified.
In general the set of leaves $L_{\ell}$, labeled by $\ell$ contains the leaves $L'$ of the subtree rooted at the most common ancestor of all the elements of $S$ labeled by $\ell$. If $\ell$ is well represented, then $L_{\ell} = L'$.
The power of deep learning
--------------------------
\[thm:deep\] Assume that $d {\lambda}^2 > 1$ or $d {\lambda}> 1 + {\varepsilon}$ and $q \geq q({\varepsilon})$ is sufficiently large. Assume further that $k \geq C \log n$. Then the following holds for all three models (IIDM, VRM, FIM) with high probability:
- The tree $T$ can be reconstructed. In other words, there exists an efficient (deep learning) algorithm that for any two representation $R(u)$ and $R(v)$, where $u$ and $v$ are leaves of the tree, computes their graph distance.
- For all labels $\ell$ that are well represented in $S$, all leaves labeled by $\ell$ can be identified.
The weakness of shallow and local learning
------------------------------------------
We consider two families of lower bounds - for local algorithms and for shallow algorithms.
In both cases, we prove information theory lower bounds by defining distributions on instances and showing that shallow/local algorithms do not perform well against these distributions.
### The Instances and lower bounds
Let $h_0 < h_1 < h$. The instance is defined as follows. Let $\operatorname{dist}$ denote the graph distance
\[def:instance\] The distribution over instances $I(h_0,h_1)$ is defined as follows. Initialize $S = \emptyset$.
- All nodes $v_1$ with $\operatorname{dist}(v_1,v_0) < h_0$ are not labeled.
- The nodes with $\operatorname{dist}(v_1,v_0) = h_0$ are labeled by a random permutation of $''1",\ldots,''d^{h_0}!"$.
- For each node $v_1$ with $\operatorname{dist}(v_1,v_0) = h_0$, pick two random descendants $v_2, v_2'$ with $\operatorname{dist}(v_2',v_0) = \operatorname{dist}(v_2,v_0) = h_1$, such that the most common ancestor of $v_2, v_2'$ is $v_1$. Add to $S$ all leaves in $L_T$ that are descendants of $v_2$ and $v_2'$.
\[thm:local\] Given an instance drawn from \[def:instance\] and data generated from IIDM, VRM or FIM, the probability that a local algorithm labels a random leaf in $L_T \setminus S$ correctly is bounded by $$d^{-h_0}(1+ O(k \lambda^{h-h_1} q)).$$
Note that given the distribution specified in the theorem, it is trivial to label a leaf correctly with probability $d^{-h_0}$ by assigning it any fixed label. As expected our bound is weaker for longer representations. A good choice for $h_1$ is $h_0 + 1$ (or $h_0+2$ if $d=2$), while a good choice of $h_0$ is $1$, where we get the bound $$d^{-1}(1+O(k \lambda^h) q),$$ compared to $d^{-1}$ which can be achieved trivially.
\[thm:shallow\] Consider a compression of the data $C_A(D)$, where $A = (A_1,\ldots,A_m)$ and let $s = max_{j \leq m} |A_{j}|$. If $d {\lambda}^2 < 1$ and given an instance drawn from \[def:instance\] and data generated from IIDM, VRM or FIM, the probability that a shallow algorithm labels a random leaf in $L_T \setminus S$ correctly is at most $$d^{-h_0} + C m d^{h_0} \exp(-c (h-h_1)),$$ where $c$ and $C$ are positive constants which depend on $\lambda$ and $s$.
Again, it is trivial to label nodes correctly with probability $d^{-h_0}$. For example if $h_0 = 1, h_1 = 2$ (or $=3$ to allow for $d=2$) and we look at the canonical compression $C_*(D)$, we obtain the bound $d^{-1} + O(k \exp(-c h)) = d^{-1} + O(k n^{-\alpha})$ for some $\alpha > 0$. Thus when $k$ is logarithmic of polylogarithmic in $n$, it is information theoretically impossible to label better than random.
Proof Ideas
===========
A key idea in the proof is the fact that Belief Propagation cannot be approximated well by linear functions. Consider the IIDM model with $k=1$ and a known tree. If we wish to estimate the root representation, given the leaf representations, we can easily compute the posterior using Belief Propagation. Belief Propagation is a recursive, thus deep algorithm. Can it be performed by a shallow net?
While we do not answer this question directly, our results crucially rely on the fact that Belief Propagation is not well approximated by one layer nets. Our proofs build on and strengthen results in the reconstruction on trees community [@MosselPeres:03; @JansonMossel:04] by showing that there is a regime of parameters where Belief Propagation has a very good probability of estimating the roof from the leaves. Yet, $1$ layer nets have an exponentially small correlation in their estimate.
The connection between reconstructing the roof value for a given tree and the structural question of reconstructing the root has been studied extensively in the phylogenetic literature since the work of [@Mossel:04a] and the reminder of our proof builds on this connection to establish the main results.
Discussion
==========
Theorem \[thm:local\] establishes that local algorithms are inferior to deep algorithms if most of the data is unlabeled. Theorem \[thm:shallow\] shows that in the regime where $\lambda^{-1} \in (\sqrt{d},d)$ and for large enough $q$, shallow algorithms are inferior to deep algorithms. We conjecture that stronger lower bound and therefore stronger separation can be obtained for the VRM and FIM models. In particular:
\[conj:stronger\] In the setup of Theorem \[thm:shallow\] and the VRM model, the results of the theorem extend to the regime $d {\lambda}^4 < 1$. In the case of the FIM model it extends to a regime where ${\lambda}< 1-\phi(d,h)$, where $\phi$ decays exponentially in $h$.
Random Trees
------------
The assumption that the generative trees are regular was made for the ease of expositions and proofs. A natural follow up step is to extend our results to the much more realistic setup of randomly generated trees.
Better models
-------------
Other than random trees, better models should include the following:
- The VRM and FIM both allow for the change of representation and for feature interaction to vary arbitrarily between different edges. More realistic models should penalize variation in these parameters. This should make the learning task easier.
- The FIM model allows interaction only between fixed nodes in one level to the next. This is similar to convolutional networks. However, for many other applications, it makes sense to allow interaction with a small but varying number of nodes with a preference towards certain localities. It is interesting to extend the models and results in such fashion.
- It is interesting to consider non-tree generating networks. In many applications involving vision and language, it makes sense to allow to “concatenate" two or more representations. We leave such models for future work.
- As mentioned earlier, our work circumvents autoencoders and issues of overfitting by using compact, almost 1-1 dense representations. It is interesting to combine our “global’ framework with “local" autoencoders.
More Robust Algorithms
----------------------
The combinatorial algorithms presented in the paper assume that the data is generated accurately from the model. It is interesting to develop a robust algorithm that is effective for data that is approximately generated from the model. In particular, it is very interesting to study if the standard optimization algorithms that are used in deep learning are as efficient in recovering the models presented here. We note that for the phylogenetic reconstruction problem, even showing that the Maximum Likelihood tree is the correct one is a highly non-trivial task and we still do not have a proof that standard algorithms for finding the tree, actually find one, see [@RochSly:15].
Depth Lower bounds for Belief Propagation
-----------------------------------------
Our result suggest the following natural open problem:
Consider the broadcasting process with $k=1$ and large $q$ in the regime $\lambda \in (1/d, 1/\sqrt{d})$. Is it true that the BP function is uncorrelated with any network of size polynomial in $d^h$ and depth $o(h)$?
The power of deep learning: proofs
==================================
The proof of all positive results is based on the following strategy:
- Using the representations $\{ R(v) : v \in L_T\}$ reconstruct the tree $T$.
- For each label $\ell$, find the most common $w$ ancestor of $\{v : v \in L_T, R(v) \in S, L(v) = \ell\}$ and label all nodes in the subtree root at $w$ by $\ell$.
For labels that are well represented, it follows that if the tree constructed at the first step is the indeed the generative tree, then the identification procedure at the second step indeed identifies all labels accurately.
The reconstruction of the tree $T$ is based on the following simple iterative “deep" algorithm in which we iterate the following. Set $h' = h$.
1. This step computes the Local Structure of the tree: For each $w_1,w_2$ with $\operatorname{dist}(v_0,w_1) = \operatorname{dist}(v_0,w_2) = h'$, compute $\min(\operatorname{dist}(w_1,w_2),2r +2)$. This identifies the structure of the tree in levels $\min(h'-r,\ldots,h')$.
2. If $h'-r \leq 0$ then $EXIT$, otherwise, set $h' := h'-r$.
3. Ancestral Reconstruction. For each node $w$ with $\operatorname{dist}(v_0,w) = h'$, estimate the representation $R(w)$ from all its descendants at level $h'+r$.
This meta algorithm follows the main phylogenetic algorithm in [@Mossel:04a]. We give more details on the implementation of the algorithm in the $3$ setups.
IIDM
----
We begin with the easiest setup and explain the necessary modification for the more complicated ones later.
The analysis will use the following result from the theory of reconstruction on trees.
\[prop:ancestral\] Assume that $d {\lambda}^2 > 1$ or $d {\lambda}> 1 + {\varepsilon}$ and $q \geq q({\varepsilon})$ is sufficiently large. Then there exists ${\lambda}_1 > 0$ and $r$ such that the following holds. Consider a variant of the IIDM model with $r$ levels and $k=1$. For each leaf $v$, let $R'(v) \sim {\lambda}_2 \delta_R(v) + (1-{\lambda}_2) U$, where ${\lambda}_2 > {\lambda}_1$ and $U$ is a uniform label. Then there exists an algorithm that given the tree $T$, and $(R'(v) : v \in L_T)$ returns $R'(v_0)$ such that $R'(v) \sim {\lambda}_3 \delta_R(v) + (1-{\lambda}_3) U$ where ${\lambda}_3 > {\lambda}_1$.
For the case of $d {\lambda}^2 > 1$ this follows from[@KestenStigum:66; @MosselPeres:03]. In the other case, this follows from [@Mossel:01].
Let ${\lambda}(h')$ denote the quality of the reconstructed representations at level $h'$. We will show by induction that ${\lambda}(h') > {\lambda}_1$ and that the distances between nodes are estimated accurately. The base case is easy as we can accurately estimate the distance of each node to itself and further take ${\lambda}(h) = 1$.
To estimate the distance between $w_1$ and $w_2$ we note that the expected normalized hamming distance $d_H(R(w_1),R(w_2))$ between $R(w_1)$ and $R(w_2)$ is: $$\frac{q-1}{q} (1-{\lambda}(h')^2 {\lambda}^{\operatorname{dist}(w_1,w_2)})$$ and moreover the Hamming distance is concentrated around the mean. Thus if $k \geq C({\lambda}',q,r) \log n$ then all distances up to $2 r$ will be estimated accurately and moreover all other distances will be classified correctly as being larger than $2r + 2$. This establishes that the step LS is accurate with high probability. We then apply Proposition \[prop:ancestral\] to recover $\hat{R}(v)$ for nodes at level $h'$. We conclude that indeed ${\lambda}(h') > {\lambda}_1$ for the new value of $h'$.
VRM
---
The basic algorithm for the VRM model is similar with the following two modifications:
- When estimating graph distance instead of the Hamming distance $d_H(R(w_1),R(w_2))$, we compute $$\label{eq:rel_h_dist}
d_H'(R(w_1),R(w_2)) = \min_{\sigma \in S_q} d_H \left(\sigma \left( R(w_1) \right),R(w_2) \right),$$ i.e., the minimal Hamming distance over all relative representations of $w_1$ and $w_2$. Again, using standard concentration results, we see that if $k \geq C({\lambda}',q,r) \log n$ then all distances up to $2 r$ will be estimated accurately. Moreover, for any two nodes $w_1, w_2$ of distance at most $2 r$, the minimizer $\sigma$ in (\[eq:rel\_h\_dist\]) is unique and equal to the relative permutation with high probability. We write $\sigma(w_1,w_2)$ for the permutation where the minimum is attained.
- To perform ancestral reconstruction, we apply the same algorithm as before with the following modification: Given a node $v$ at level $h'$ and all of its descendants at level $r+h'$, $w_1,\ldots,w_{h^d}$. We apply the reconstruction algorithm in Proposition \[prop:ancestral\] to the sequences $$R(w_1), \sigma(w_1,w_2)(R(w_2)),\ldots, \sigma(w_1,w_{h^d})(R(w_{h^d})),$$ where recall that $\sigma(w_1,w_j)$ is the permutation that minimizes the Hamming distance between $R(w_1)$ and $R(w_j)$. This will insure that the sequence $\hat{R}(v)$ has the right statistical properties. Note that additionally to the noise in the reconstruction process, it is also permuted by $\sigma(w_1,v)$.
FIM
---
The analysis of FIM is similar to VRM. The main difference is that while in VRM model, we reconstructed each sequence up to a permutation $\sigma \in S_q$, in the FIM model there are permutations over $S_{q^2}$ and different permutations do not compose as they apply to different pairs of positions. In order to overcome this problem, as we recover the tree structure, we also recover the permutation $(f_e,g_e)$ up to a permutation $\sigma \in S_q$ that is applied to each letter individually.
For simplicity of the arguments, we assume that $d \geq 3$. Let $w_1,w_2, w_3$ be three vertices that are identified as siblings in the tree and let $w$ be their parent. We know that $R(w_1), R(w_2)$ and $R(w_3)$ are noisy versions of $R(w)$, composed with permutations $\tau_1,\tau_2, \tau_3$ on $S_{q^2}$. We next apply concentration arguments to learn more about $\tau_1,\tau_2,\tau_3$. To do so, recall that $k \geq C({\lambda}) \log n$. Fix $x = (x_1,x_2) \in [q]^2$, and consider all occurrences of $\tau_1(x)$ in $R(w_1)$. Such occurrences are correlated with $\tau_2(x)$ in $R(w_2)$ and $\tau_3(x)$ in $R(w_3)$. We now consider occurrences of $\tau_2(x)$ in $R(w_2)$ and $\tau_3(x)$ in $R(w_3)$. Again the most common co-occurrence in $w_1$ is $\tau_1(x)$. The following most likely occurrences values will be the $2q-1$ values $y$ obtained as $\tau_1(x_1,z_2))$, or $\tau_1((z_1,x_2))$ where $z_1 \neq x_1, z_2 \neq x_2$.
In other words, for each value $\tau_1(x)$ we recover the set $$A(x) = B(x) \cup C(x),$$ where $$B(x) = \{ \tau_1(x_1,z_2) : z_2 \neq x_2 \}, \quad
C(x) = \{ \tau_1(z_1,x_2) : z_1 \neq x_1 \}.$$ Note that if $y^1, y^2 \in B(x)$ then $y^2 \in A(y_1)$ but this is not true if $y^1 \in B(x)$ and $y^2 \in C(x)$. We can thus recover for every value $\tau_1(x)$ not only the set $A(x)$ but also its partition into $B(x)$ and $C(x)$ (without knowing which one is which).
Our next goal is to recover $$\{ (x,B(x)) : x \in [q]^2 \}, \quad \{ (x,C(x)) : x \in [q]^2 \}$$ up to a possible [*global*]{} flip of $B$ and $C$. In order to do so, note that $$\{ x \} \cup B(x) \cup_{y \in C(x)} B(y) = [q]^2,$$ and if any of the $B(y)$ is replaced by a $C(y)$, this is no longer true. Thus once we have identified $B(y)$ for one $y \in C(x)$, we can identify $B(y)$ for all $y \in C(x)$. Repeating this for different values of $x$, recovers the desired $B$ and $C$.
We next want to refine this information even further. WLOG let $x = \tau_1(0,0)$ and let $y = \tau_1(a,0) \in C(x)$. and $z = (0,b) \in B(x)$. And note that $C(y) \cap B(z)$ contains a single element, i.e., $\tau_1(a,b)$. We have thus recovered $\tau_1$ up to a permutation of $S_{[q]}$ as needed.
After recovering $\tau_1,\tau_2,\tau_3$ etc. we may recover the ancestral state at their parent $w$ up to the following degrees of freedom (and noise)
- A permutation of $S_q$ applied to each letter individually.
- A global flip of the sets $B$ and $C$.
In general the second degree of freedom cannot be recovered. However if $v_2$ is a sister of $v_1$ (with the same degrees of freedom), then only the correct choice of the $B/C$ flips will minimize the distance defined by taking a minimum over permutations in $S_{q^2}$. Thus by a standard concentration argument we may again recover the global $B/C$ flip and continue recursively. Note that this argument is using condition (\[eq:rewire\_cond\]).
The limited power of limited algorithms
=======================================
The Limited Power of Local Algorithms
-------------------------------------
To prove lower bounds it suffices to prove them for the IIDM model as is a special case of the more general models. We first prove Theorem \[thm:local\].
Let $R(w)$ be an unlabeled leaf representation. Let $M = d^{h_0}$. Let $u_1,\ldots,u_M$ denote the nodes level $h_0$ and denote their labels by $\ell_1,\ldots,\ell_M$. Let $v_i, v_i'$ denote the nodes below $u_i$ at level $h_1$ with the property that the leaves of the tree rooted at $v_i$ are the elements of $S$ with label $\ell_i$.
Let $u_i$ be the root of the tree $w$ belongs to and let $x_i$ be the lowest intersection between the path from $w$ to $u_i$ and the path between $v_i$ and $v_i'$. We write $h'$ for $\operatorname{dist}(w,x_i)$. Note that $h' \geq h-h_1$. For $j \neq i$ let $x_j$ be the node on the path between $v_j$ and $v_j'$ such that $\operatorname{dist}(v_j,x_j) = \operatorname{dist}(v_i,x_i)$. We assume that in addition to the labeled data we are also given $h'$ and $$D' = (\ell_1,R(x_1)),\ldots,(\ell_M,R(x_m)).$$ Note that we are not given the index $i$.
Of course having more information reduces the probability of error in labeling $R(w)$. However, note that $R(w)$ is independent of $\{(R(v), L(v)) : v \in S\}$ conditioned on $D'$ and $h'$. It thus suffices to upper bound the probability of labeling $R(w)$ correctly given $D'$. By Bayes: $$P[L(w) = \ell_i | D', h']
= \frac{P[R(w) | D', L(w) = \ell_i, h']}{\sum_{j=1}^M P[R(w) | D', L(w) = \ell_j, h']}
= \frac{P[R(w) | R(x_i), h']}{\sum_{j=1}^M P[R(w) | R(x_j), h']}$$ We note that $$\left( \frac{(1-\lambda^{h'})/q}{(\lambda^{h'} + (1-\lambda^{h'})/q)} \right)^k \leq
\frac{P[R(w) | R(x_i), h']}{P[R(w) | R(x_j), h']}
\leq \left( \frac{(\lambda^{h'} + (1-\lambda^{h'})/q)}{(1-\lambda^{h'})/q} \right)^k$$ So the ratio is $$1 + O(k \lambda^{h'} q) = 1 + O(k \lambda^{h-h_1} q)$$ and therefore the probability of correct labeling is bounded by $$\frac{1}{M} (1+ O(k \lambda^{h-h_1} q))$$ as needed.
On count reconstruction {#subsec:counts}
-----------------------
We require the following preliminary result in order to bound the power of local algorithms.
\[lem:counts\] Consider the IIDM with $d \lambda^2 < 1$ and assume that all the data is labeled and that is compressed as $C_A(D)$, where $A = (A_1)$ and $A_1 = [k]$, i.e, we are given the counts of the data. Let $P_{x,h}$ denote the distribution of $C_A(D)$ conditional on $R(v_0) = x$. There there exists a distribution $Q = Q^h$ such that for all $x$, it holds and $$\label{eq:count_coupling}
P_{x,h}= (1-\eta) Q + \eta P'_{x,h} \quad \eta \leq C \exp(-c h),$$ where $Q$ is independent of $x$ and $c,C$ are two positive constant which depend on $\lambda$ and $k$ (but not on $h$).
Our proof builds on the a special case of the result for $k=1$, where [@MosselPeres:03] show that the “count reconstruction problem is not solvable" which implies the existence of $\eta(h)$ which satisfies $\eta(h) \to 0$ as $h \to \infty$. The statement above generalizes the result to all $k$. Moreover, we obtain an exponential bound on $\eta$ in terms of $h$.
Assume first that $k=1$. The proof that the threshold for “count reconstruction is determined by the second eigenvalue" [@MosselPeres:03] implies the statement of the lemma with a value $\eta = \eta(h)$ which decays to $0$ as $h \to \infty$. Our goal in the lemma above is to obtain a more explicit bound showing an exponential decay in $h$. Such exponential decay follows from [@JansonMossel:04] for a different problem of [*robust reconstruction*]{}. Robust reconstruction is a variation of the reconstruction problem, where the tree structure is known but the value of each leaf is observed with probability $\delta > 0$, independently for each leaf. [@JansonMossel:04] proved that if $d \lambda^2 < 1$ and if $\delta(d,\lambda) > 0$ is small enough then the distributions $S_x$ of the the partially observed leaves given $R(v_0) = x$ satisfy $$\label{eq:count_coupling2}
S_x = (1-\eta) S + \eta S'_X, \quad \eta \leq C \exp(-c h).$$
From the fact that the census reconstruction problem is not solvable [@MosselPeres:03], it follows that there exists a fixed $h' = h'(\delta)$ such that for the reconstruction problem with $h'$ levels, the distribution of the counts at the leaves can be coupled for all root values except with probability $\delta$. We can now generate $P_{x,h+h'}$ as follows: we first generate $Q_{x,h}$ which is the representations at level $h$. Then each node at level $h$ is marked as [*coupled*]{} with probability $1-\delta$ and [*uncoupled*]{} with probability $\delta$ independently. To generate the census $P_{x,h+h'}$ from $Q_{x,h}$, as follows: for each coupled node at level $h$, we generate the census of the leaves below it at level $h + h'$ conditioned on the coupling being successful (note that this census is independent of the representation of the node at level $h$). For uncoupled nodes, we generate the census, conditioned on the coupling being unsuccessful. From the description it is clear that $P_{x,h+h'}$ can be generated from $Q_{x,h}$. Since $Q_{x,h}$ has the representation (\[eq:count\_coupling2\]), it now follows that $P_{x,h+h'}$ has the desired representation (\[eq:count\_coupling\]) as needed.
The case of larger $k$ is identical since the chain on $k$ sequences has the same value of $\lambda$. Therefore (\[eq:count\_coupling\]) follows from (\[eq:count\_coupling2\]).
The limited power of shallow algorithms
---------------------------------------
We now prove Theorem \[thm:shallow\].
The idea of the proof is to utilize Lemma \[lem:counts\] to show that the compressed labeled data is essentially independent of the unlabeled data. Write $M = d^{h_0}$. For each permutation $\sigma$ of the $M$ labels $1,\ldots,M$, we write $P_\sigma$ for the induced distribution on the compressed labeled data $C_A(D)$. Our goal is to show we can write $$\label{eq:couple_perm}
P_{\sigma} = (1-\eta) P + \eta P'_{\sigma}$$ where $\eta$ is small. Note that (\[eq:couple\_perm\]) implies that the probability of labeling a unlabeled leaf accurately is at most $M^{-1} + \eta$. Indeed we may consider a problem where in addition to the sample from $P_{\sigma}$ we are also told if it is coming from $P$ or from $P'_{\sigma}$. If it is coming from $P$, we know it is generated independently of the labels and therefore we cannot predict better than random (i.e. $M^{-1}$).
Let $R(v_1(1)),R(v_2(1)),\ldots,R(v_1(M)),R(v_2(M))$ denote the representations at the roots of the subtrees of the labeled data. Let $I$ denote all of the representations and let $P_I$ denote the distribution of $C_A(D)$ conditioned on these representations. By convexity, to prove the desired coupling it suffices to prove $$P_I = (1-\eta) P + \eta P'_{I}$$ By applying Lemma \[lem:counts\] to each of the trees rooted at $v_1(1),v_2(1),\ldots,v_1(M),v_2(M)$ and to each of the sets $A_i$, we obtain the desired results.
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[^1]: Supported by ONR grant N00014-16-1-2227 and NSF CCF-1665252 and DMS-1737944 [[email protected]]{}
|
---
abstract: |
A recently proposed method to preserve the electron beam polarization at the VEPP-4M collider during acceleration with crossing the integer spin resonance energy E=1763 MeV has been successfully applied. It is based on full decompensation of $ 0.6\times3.3$ Tesla$\times$meter integral of the KEDR detector longitudinal magnetic field due to s ’switched-off’ state of the anti-solenoids.
Usage
:
PACS numbers
: [29.27.Hj, 29.27.Bd, 29.27.Eg ]{}
Structure
:
author:
- 'A.K. Barladyan'
- 'A.Yu. Barnyakov'
- 'S.A. Glukhov'
- 'Yu.M. Glukhovchenko'
- 'S.E. Karnaev'
- 'E.B. Levichev'
- 'S.A. Nikitin'
- 'I.B. Nikolaev'
- 'I.N. Okunev'
- 'P.A. Piminov'
- 'A.G. Shamov'
- 'A.N.Zhuravlev'
bibliography:
- 'paper.bib'
title: 'Integer spin resonance crossing at VEPP-4M with conservation of beam polarization '
---
,
,
\[sec:level1\]Motivation
========================
A set of the beam energy values in the Hadron-Muon Branching Ratio measurement with the KEDR detector [@KEDR] at the electron-positron VEPP-M collider [@VEPP-4M] in the region between $J/\psi$ and $\psi(2S)$ resonances includes several critical points. In particular, $E=1764$ MeV and $1814$ MeV. Since the beam energy calibration in this experiment is performed with the Resonant Depolarization technique (RD) so polarized beams are required. Polarization is obtained due to the natural radiative mechanism at the VEPP-3 booster storage ring. Both mentioned energy values are in the so-called ’Polarization Downfall’ $-$ the VEPP-3 energy range of approximately 160 MeV width where obtaining of the polarization of a fairly high degree is significantly hampered because of strong depolarization effect of the guide field imperfections. The ’Polarization Downfall’ range was found in the 2003 year experiment with the polarimeter based on the internal polarized target [@DYUG].
The center of that critical range is the energy value $E_4=1763$ MeV which corresponds to the integer spin resonance $\nu=\nu_k=4$ (in the conventional storage rings $\nu=\gamma a$ is the spin tune parameter equal to a number of the spin vector precessions about the vertical guide field axis per a turn subtracting one; $\gamma$ is the Lorentz factor; $a=(g-2)/2$ ). Nevertheless, one can obtain polarized beams at VEPP-4M with the energies from the ’Polarization Downfall’ excepting a small island in the vicinity of $E_4$ if using the method of ’auxiliary energy point’. In the given experiment the magnetization cycle of the collider is of the ’upper’ type. It means that the ’auxiliary energy points’ should be below the energies of experiment as well as below 1660 MeV taking into account the ’Polarization Downfall’ region lower boundary. In the cases when the method is valid the beam polarization in VEPP-3 is achieved at the ’auxiliary’ energy. Then the beam is injected into the collider ring. After that its energy is raised to the energy of experiment. Radiative spin relaxation time in the collider ring is two orders larger than that in the booster ($\tau_p\approx 80$ h at $E=1.8$ GeV at VEPP-4M). This allows us to use the beam polarization in the RD energy calibration procedures even at a rather small detuning from the dangerous spin resonances. For instance, the RD calibrations of the beam energy in the tau-lepton mass measurement experiment [@TAU] at energies close to the tau production threshold ($E=1777$ MeV) were carried out with a delay of about half an hour after injection of the beam at the detuning $\delta \nu\approx 0.03$ from the resonance $\nu_k=4$ ($\Delta E\approx 13$ MeV in the energy scale) and less. In the case under consideration one can apply the described method at the energy points below 1763 MeV at the detuning of $\Delta E\approx 9 \div 13$ MeV as the tau mass measurement experiment. But there are the special additional measures required for the point $E=1814$ MeV and, apparently, for the points somewhat below the $\psi(2S)$ peak energy (for example. 1839 MeV). The reason is the necessity to cross the integer spin resonance at 1763 MeV during acceleration starting from the ’auxiliary’ energy.
\[sec:level1\] Inability of fast or slow crossing of the spin resonance
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Fast change of the beam energy in a storage ring (during acceleration or deceleration) enables the polarization of particles to be preserved when crossing any spin resonance $\nu_0=\nu_k$ (in a general case the spin tune $\nu_0$ is not coincident with the definition given above for the storage rings with an unidirectional guide field) if the following condition is fulfilled [@DERB]: $$\frac{d \varepsilon}{dt}=\dot \varepsilon \gg |w_k|^2\omega_0,
\label{eq:eq1}$$ $\varepsilon(t)=|\nu_0(t)-\nu_k|$ is a time-dependent resonant detuning; $w_k$ is a resonant harmonic amplitude of the field perturbations; $\omega_0$ is an angular frequency of particle revolution. For instance, in the tau-mass measurement experiment the rate of change of the detuning during the beam energy lowering down to the tau-production threshold was $dE/dt\approx 1$ Mev$/$s or $\dot\varepsilon \approx 2.3\times 10^{-3}$ s$^{-1}$. Since the intersection of the combination spin resonances $\nu+m\nu_x+n\nu_y=k$, $m,n=\pm
1,\pm 2 ...$, with consideration of the betatron tunes, was successful (as proven by the fact of the RD energy calibrations in the final state), then the likely power of each of these minor resonances was significantly less than the amount $|w_k|^2\sim 2\times 10^{-5}$ ($\omega_0=2\pi f_0$, $f_0=819$ kHz). The data on measurement of the polarization lifetime $\tau_d$ due to radiative depolarization at $E=1777$ MeV, the tau-lepton production threshold, gives the information about the natural power of the spin resonance $\nu_k=\nu=4$ related to the sources of the vertical closed orbit perturbations. The polarization lifetime was adjusted to the level $\tau_d\ge 1$ hour [@DYUG]. One can associate this quantity with the formal estimate of the resonant spin harmonic amplitude using the known equation [@DERB2; @Derbenev:1973ia]: $$\tau_d\approx\frac{\tau_p}{1+\frac{11}{18}\frac{|w_k|^2\nu^2}{\varepsilon^4}}
\label{eq:eq2}$$ where $\varepsilon=\nu_0-4\approx 0.03 \ll 1$ , $\tau_p$ is the Sokolov-Ternov polarization time [@SokolovTernov]. For VEPP-4M at $E=1777$ MeV $\tau_p=87$ hours, $\tau_d=1$ hour, $\varepsilon\approx 0.03$, so the estimate $|w_k|\sim 2.8\times 10^{-3}$ is obtained. Therefore, the maximal necessary rate of the resonance crossing is $\dot\varepsilon \gg20$ s$^{-1}$, or, $dE/dt \gg 10^{4}$ MeV/s. In practice, the maximal rate of the VEPP-4M energy change without any notable losses of particles usually does not exceed 5 MeV/s. So, a fast crossing of the integer spin resonance $E=1763$ MeV is impossible.
Otherwise, if [@DERB; @DERB2] $$\dot\varepsilon \ll |w_k|^2\omega_0,
\label{eq:eq3}$$ then a spin resonance occurs adiabatically (slowly). Basing on the estimates made above, one can conclude that the rate of energy change of $1\div10$ MeV/s, in principle, may be appropriate. In the theory of adiabatic crossing of the spin resonances the polarization has a chance to survive when (3) is fulfilled and it reverses a sign as a result of the crossing. Despite the feasibility of the condition of slow crossing, it is necessary to bear in mind that there is a lower limit on the rate of the crossing due to the depolarizing effect of radiative diffusion and friction. Radiative depolarization time related to the vertical closed orbit distortions declines very quickly when decreasing the detuning from the integer spin resonance: $\tau_d\propto
\varepsilon^{-4}$. For example, this time decreases as 16 times at $ E=1770$ MeV as compared with the value $\tau_d=1$ h at $E=1777$ MeV. In the case of high power resonance the spin diffusion depends also on a decrement $\Lambda$, the parameter of radiative friction. In the resonant area where $|\varepsilon|\sim |w_k|$ and in the case $\omega_0|w_k| \gg \Lambda$ the depolarization time owing to radiative diffusion and friction achieves a minimal value $\tau_d\sim \Lambda^{-1}\sim 100$ msec. Given estimates [@Nikitin-BINP-2015-1] allow us to infer that an adiabatic crossing the integer spin resonance at $E=1763$ MeVas well as the fast crossing discussed above will result in a loss of the beam polarization.
\[sec:level1\]Detuning from integer spin resonance due to the KEDR field decompensation
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The simple method to preserve the VEPP-4M beam polarization in the conditions under consideration has been proposed and numerically substantiated in [@Nikitin-BINP-2015-1] basing on some features of the KEDR detector magnetic system. Two cases of the polarization kinematics in the storage rings at the energy of an integer spin resonance $\nu=4$ are clarified in Fig. \[fig:Vect\]. If one switches off the current in the anti-solenoid coils, the 0.6 Tesla KEDR detector longitudinal magnetic field integral becomes un-compensated that results in the spin and velocity rotation angle of about 0.34 rad at $E=1.75$ GeV. This causes a shift of the spin tune with regard to an unperturbed value $\nu$. The latter is proportional to the beam energy and describes a spin precession in the storage ring with a flat closed orbit. Thus, a non-integer part of the perturbed spin frequency $\nu_0$ does not accept a null value at the critical point near $E = 1763$ MeV. This fact is used as a basis for the proposal to preserve the beam polarization during acceleration.
![a) There is no the preferred direction of spin polarization in an ideal storage ring with a flat closed orbit if the spin precession parameter $\nu$ takes an integer value (in our case $\nu=4$)). b) If a solenoid with an arbitrary spin rotation angle $\varphi$ is inserted and $\nu$ is an integer, then there exists an equilibrium axis of polarization, the vector $\vec{n}$. This vector rotates in a median plane periodically with an azimuth $\vartheta$ and is always directed along velocity at the location of solenoid. []{data-label="fig:Vect"}](fig1.png){width="85mm"}
The equilibrium polarization axis as a function of an azimuth in a storage ring containing the insertion with longitudinal magnetic field is calculated using the known formulae [@Derbenev:1978hv; @NikSaldin]: $$\begin{split}
n_x(\vartheta)&=\pm\frac{\sin{\nu(\vartheta-\pi)}}{\sin{}\xi}\cdot\sin{\frac{\varphi}{2}}, \\
n_y(\vartheta)&=\mp\frac{\cos{\nu(\vartheta-\pi)}}{\sin{}\xi}\cdot\sin{\frac{\varphi}{2}}, \\
n_z(\vartheta)&=\mp\frac{\sin{\pi\nu}}{\sin{}\xi}\cdot\cos{\frac{\varphi}{2}} ,\\
\sin\xi&=\sqrt{1-\cos^2{\pi\nu}\cos^2{\frac{\varphi}{2}}}.
\end{split}$$ Here $$\varphi\approx \frac{\pi}{4.6\nu}\cdot\int H_{||}ds$$ is an angle of electron spin rotation in the longitudinal magnetic field with the integral of $\int H_{||}ds$ in Tesla$\cdot$meter. The symbols $x,y,z$ denote the horizontal, longitudinal and vertical orts of the movable coordinate basis, respectively. We use an approximation of the isomagnetic storage ring in which the azimuth $\vartheta$ can be considered as equal to the angle of a particle velocity rotation ($\vartheta=0$ at the solenoid location). The effective spin precession tune is determined from an equation $$\cos{\pi\nu_0}=\cos{\pi\nu}\cos{\frac{\varphi}{2}}.$$ In our case $\int H_{||}ds=H_{KEDR}\cdot L_{eff}$, where $H_{KEDR}=0.6$ Tesla, the KEDR detector field; an effective KEDR solenoid length $L_{eff}=3.3$ m if the anti-solenoids are switched off (and $L_{eff}=2.5$ m in the case of full compensation of the detector field integral). Fig. \[fig:Tune\] shows that the calculated spin tune shift $|\nu_0-\nu|\propto \Delta
E$ makes about $\Delta E\approx 18 $ MeV in the vicinity of the critical energy 1763 MeV at full decompensation that is two times larger than the minimal detuning (9 MeV), which took a place in the RD calibrations of beam energy in the tau-mass measurement experiment.
![Spin tune shift in the energy units as function of the beam energy in the cases of no and full decompensation of the KEDR detector field integral of $0.6\times3.3$ Tesla$\times$meter. []{data-label="fig:Tune"}](fig2.png){width="80mm"}
\[sec:level1\]Radiative depolarization rate during acceleration
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Characteristic time of the radiative depolarization $\tau_d$ due to quantum fluctations in the presence of a strong perturbation in the form of the KEDR detector longitudinal field can be found from the generalized equation [@Derbenev:1973ia]: $$\tau_d\approx\frac{\tau_p}{\left< 1-\frac{2}{9}(\vec n \vec \beta)^2+\frac{11}{18}\vec d^2 \right>}$$ Here $\tau_p$ is the Sokolov-Ternov polarization time (it is proportional to $E^{-5}$ and makes up 72 hours at the VEPP-4M energy of 1.85 GeV ); $\vec d^2$ is a square of the spin-orbit coupling vector function perodically depending on the azimuth. In our case the spin-orbit coupling is excited by an uncompensated part of the KEDR field integral. It can be defined as a derivative of the polarization axis vector $\vec n$ (4) with respect to the particle Lorentz-factor $\gamma$ [@NikSaldin]: $$\begin{split}
d_x&=\gamma\frac{\partial n_x}{\partial \gamma}=\pm\bigg\{F\sin{\nu(\vartheta-\pi)}\cdot\sin{\frac{\varphi}{2}}+\\
& \quad \frac{1}{\sin{\xi}}\left[\nu(\vartheta-\pi)\cdot\cos{\nu(\vartheta-\pi)}\cdot\sin{\frac{\varphi}{2}}-\right.\\
& \quad \left.\frac{\varphi}{2}\cdot\sin{\nu(\vartheta-\pi)}\cdot\cos{\frac{\varphi}{2}}\right] \bigg\},\\
d_y&=\gamma\frac{\partial n_y}{\partial \gamma}=\mp\bigg\{F\cos{\nu(\vartheta-\pi)}\cdot\sin{\frac{\varphi}{2}}-\\
& \quad \frac{1}{\sin{\xi}}\left[\nu(\vartheta-\pi)\cdot\sin{\nu(\vartheta-\pi)}\cdot\sin{\frac{\varphi}{2}}+\right.\\
& \quad \left.\frac{\varphi}{2}\cdot\cos{\nu(\vartheta-\pi)}\cdot\cos{\frac{\varphi}{2}}\right] \bigg\},\\
d_z & =\gamma\frac{\partial n_z}{\partial \gamma}=\mp\bigg\{F\sin{\pi\nu}\cdot\cos{\frac{\varphi}{2}}+\\
& \quad \frac{1}{\sin{\xi}}\left[\pi\nu\cdot\cos{\pi\nu}\cdot\cos{\frac{\varphi}{2}}+\right.\\
& \quad \left.\frac{\varphi}{2}\cdot\sin{\pi\nu}\cdot\sin{\frac{\varphi}{2}}\right] \bigg\},\\
F & =-\frac{1}{2\sin^3{\xi}}\Big(\pi\nu\sin{2\pi\nu}\cdot\cos^2{\frac{\varphi}{2}}- \\
& \quad - \frac{\varphi}{2}\sin{\varphi}\cdot\cos^2{\pi\nu}\Big).
\end{split}$$ Another contribution to a spin-orbit coupling is given by betatron oscillations excited by SR fluctuations. Besides an integer spin resonance this coupling relates to combination spin resonances with betatron tunes $\nu_{x,z}$, in particular, of the type $\nu\pm\nu_{x,z}=k_{x,z}$ in the linear approximation ($k_{x,z}$ are integer). When considering the case of a significant integral of longitudinal magnetic field the betatron contribution to the depolarization rate near an integer spin resonance turns to be small as compared with a similar effect of the polarization axis “chromatism” [@NikSaldin; @NikCtau; @NikRupac]. By this reason we neglect the betatron oscillation effect in our estimate of the spin radiative kinetics.
The depolarization time is calculated using the formula (6) and is plotted in Fig. \[fig:Time\] versus the beam energy at $100\%$ and $50\%$ extent of decompensation of the KEDR field integral. Minimal depolarization time $\tau_d
= 10$ seconds. A width of the energy area where $10$ s $< \tau_d < 100$ s is about 30 MeV. It takes time of about 30 seconds to cross this area at a nominal rate of energy change $dE/dt = 1$ MeV/s.
The result shows that it is almost impossible to have time to check the presence of polarization, using RD or monitoring by the Touschek polarimeter, when the energy value E = 1764 MeV because of the expected fast depolarization in a time of about 80 seconds. At the energy of 1810 MeV the time $\tau_d$ becomes 2000 seconds. This gives a chance to measure the energy by a spin frequency. Theoretical behavior of the polarization degree during the process of acceleration beginning from the injection (’auxiliary’) energy E = 1650 MeV with relatively safe crossing of the integer spin resonance energy at the expense of the KEDR field decompensation is shown in Fig. \[fig:Degree\] for two values of the acceleration rate. The current value of the degree in units of the initial one ($P_0$) is calculated from the equation $$\frac{P}{P_0}\approx \exp{\left[ -\int_{E_1}^{E_2}\frac{dE}{(dE/dt)\cdot\tau_d}\right]}.$$
It is seen from the calculations that it is advantageous to apply the full decompensation of the KEDR field and accelerate with a rate not below $2$ MeV/sec. At best, it can provide about $80\%$ of the initial polarization degree in the final state. RD calibration of the beam energy should be performed only after resetting the anti-solenoid field that leads to cancelation of the spin tune shift and, with this, a systematic error in the energy value. Moreover, the polarization lifetime increases manyfold if the KEDR field is compensated.
![The radiative depolarization time vs. the beam energy under the influence of 0.6 T KEDR field decompensation.[]{data-label="fig:Time"}](fig3.png){width="80mm"}
![Calculated change of the polarization degree relative to the initial one in a process of beam acceleration at a rate characterized by the parameter $dE/dt$ in the case of full decompensation of the KEDR field. []{data-label="fig:Degree"}](fig4.png){width="80mm"}
\[sec:level1\]Two skew quad compensation of betatron coupling from the KEDR field
=================================================================================
If the anti-solenoids are switched off the special measures are needed to provide the alternative relevant operation modes of VEPP-4M at the beam injection and process of acceleration. In this case, it is convenient to use the scheme of betatron coupling localization proposed by K.Steffen [@Steffen1982] and based on application of two skew quadrupole lenses (rotated by an angle of $45^\circ$) - see Fig.5. Previously, this scheme has already been successfully tested at VEPP-4M [@NikProt1999; @Nikitin:2001ej].
Transport matrix for the vector of betatron variables $(x,x',z,z')$ at the section from the skew lens SQ$_+$ to SQ$_-$ including KEDR can be approximately written as $$M=Q_-\cdot M_-\cdot M_s\cdot L^2\cdot M_s\cdot M_+\cdot Q_+.$$ Here $Q_\pm$ is the ’thin’ skew quad matrices; $L$ is the empty section matrix for the length $l=L_s/4$; $L_s=3.3$ m, the effective length of the KEDR main solenoid ($L_s=2.5$ m in the case when the anti-solenoids are switched on); $M_s$ is the half-solenoid matrix in the ’thin magnet’ approximation ($\chi=\varphi/2$): $$M_s=\left(\begin {array}{cccc}
1 & 0 & - \chi & 0 \\
0 & 1 & 0 & -\chi \\
\chi & 0 & 1 & 0 \\
0 & \chi & 0 & 1
\end{array}\right);$$ $M_\pm$ are the matrices for transformation from the solenoid edge to the corresponding skew quad. The skew quads are placed symmetrically relative to the solenoid at the ’magic’ azimuths for which some elements of the matrix $M$ strictly or approximately satisfy a certain equation. Strengths of the $SQ_\pm$ lenses are found from another equation to be proportional to $\chi$, similar in value and opposite in sign. If you set these found skew quad strengths, then the matrix $M$ does not contain the off-diagonal (coupling) 2x2 blocks or becomes close to such kind. The simplicity of the scheme is based on the mirror symmetry of the magnetic structure at the section with the solenoid. Betatron coupling is localized at this section. Vertical and horizontal oscillations excited beyond the section are mutually independent with the accuracy the compensation scheme is designed and made. The scheme provides a minimal split of the normal betatron mode frequencies of the order of $10^{-3}$ (in units of the revolution frequency). If no compensation is applied, this split achieves 0.1, and this does not allow a sustainable maintenance of the beam during acceleration.
![Scheme of compensation of the betatron coupling caused by the KEDR main solenoid field with the help of two skew quadrupoles (SQ+ and SQ-) located near the VEPP-4M Final Focus lenses. Anti-solenoids (AS) are switched off.[]{data-label="fig:Scheme"}](fig5.png){width="80mm"}
\[sec:level1\]Touschek polarimeter
==================================
Relative polarization degree is measured by the Touschek polarimeter [@TSH-POL-NIM-2002; @EMS-REVIEW-NIM-2009], which consists of eight plastic scintillator counters located inside the accelerator vacuum chamber. Counting rate of the Intra-Beam Scattering (IBS) depends on the beam polarization [@bayer1969; @BKS; @STRAKH; @TSH-CALC-JETP-2012]. The relaxation time of polarization (’polarization lifetime’) is measured by a time evolution of the scattered (Touschek) particle counting rate. Correct determination of the polarization lifetime requires taking into account the Touschek beam lifetime, the spin dependence of IBS and scattering on residual gas by solving an equation for the beam particle population $N$: $$- \frac{dN}{dt} = \frac{1}{\tau_{tsh}} \frac{N^2(t)}{N(0)} \frac{V(0)}{V(t)} \bigl(1-\delta(t)\bigr)+ \frac{N(t)}{\tau_{bg}}.
\label{eq:eq4}$$ Here, the first term corresponds to IBS and $\tau_{tsh}$ is the characteristic Touschek beam lifetime; the second term describes background scattering on residual gas with $\tau_{bg}$, the characteristic background lifetime; $\delta(t)$ is the polarization contribution to IBS proportional polarization degree squared; $V(t)$ is the beam volume.
We use a compensation technique by normalizing the counting rate from a polarized beam by the counting rate from an unpolarized one. This technique allows us to suppress some count rate fluctuations related to the beam orbit or beam size instabilities.
During the experiment the beam volume $V$ (more precisely, the transverse beam sizes because it is assumed that the longitudinal size varies slightly), the beam current $I_{1, 2}$ and the count rate $f_{1, 2}$ for the first (polarized) and the second (unpolarized) bunches are measured. The relative count rate difference $\Delta = f_1/f_2-1$ is calculated. These experimental data are fitted using the following formulae: $$\begin{split}
f_{i}(t) & = \frac{p_{tsh}}{\tau_{tsh}} \frac{I_{i}^2(t)}{I_{i}(0)I_e}
\frac{1+\alpha_V I_{i}(0)}{ 1 + \alpha_V I_{i}(t)}
\bigl(1-\delta_{i}(t)\bigr)
+ \frac{p_{bg}}{\tau_{bg}} \frac{I_{i}(t)}{I_e} \\
I_{i}(t) & = \frac{I_{i}(0) e^{-t/\tau_{bg}}}{1 + (1 - e^{-t/\tau_{bg}})\frac{\tau_{bg}}{\tau_{tsh}} - \int_0^t e^{-t/\tau_{bg}}(\delta_{i}(t) + \delta V_i(t)) \frac{dt}{\tau_{tsh}}} \\
V(t) & = V_0 \biggl( 1 + \alpha_V \frac{I_{1} (t) + I_{2}(t) }{2} \biggr) \hspace{0.1\textwidth} \\
\Delta(t) & \approx - \epsilon_1(t) e^{-t/\tau_{bg}}\bigl(\delta_1(t) - \delta_2(t)\bigr) + \\
& \quad + \frac{\epsilon_1(t)}{\epsilon_2(t)} \frac{\delta N + \int_0^t e^{-t/\tau_{bg}} (\delta_1(t)-\delta_2(t)) \frac{dt}{\tau_{tsh}}}{1+(1 - e^{-t/\tau_{bg}})\frac{\tau_{bg}}{\tau_{tsh}}}.
\end{split}$$ Here $i=1, 2$ denotes first and the second bunch; $\epsilon_x(t)$ is the factor taking into account the registration efficiency: $$\epsilon_{x=1,2}(t) = \biggl [
\frac{p_{bg}}{p_{tsh}} \left(
1 + \frac{\tau_{tsh}}{\tau_{bg}}
\right) +
\left(x - \frac{p_{bg}}{p_{tsh}}\right)
e^{-\frac{t}{\tau_{bg}}} \biggr ]^{-1};$$ $I_e = e f_0$ is the single electron current, $f_0$ is the revolution frequency; $\delta_{1, 2}(t)$ are the polarization contribution to the Touschek intra-beam scattering for the first and the second bunches respectively: $$\begin{split}
\delta_1(t) & = \delta_0 \left [ P_1 e^{-t/\tau_d} - P_2 (1-e^{-t/\tau_d})\right ]^2 \\
\delta_2(t) & = \delta_0 \left [P_2 (1-e^{-t/\tau_d})\right ]^2.
\end{split}$$ Initial polarization of the first bunch $P_1=0.8$ corresponding to the assumed beam state after acceleration (see Fig.4) is fixed to the calculated value due to strong correlation with the parameter of the Touschek polarization effect $\delta_0$; $P_2 = 8\sqrt{3}/15\, \tau_d/\tau_p$, the equilibrium polarization degree where $\tau_p$ is the Sokolov-Ternov radiative polarization time ($87$ hours at $1777$ MeV). $\delta V_i(t)$ is the relative beam volume as a function of time: $$\begin{aligned}
\delta V_i(t) & = & \alpha_V ( I_i(t) - I_i(0)) \approx \nonumber
\\ & \approx & - \alpha_V I_i(0) \frac{(1 - e^{-t/\tau_{bg}})(\tau_{tsh} + \tau_{bg})}{ \tau_{tsh} + \tau_{bg}(1 - e^{-t/\tau_{bg}})}.\end{aligned}$$
The following free parameters are used for the fitting: $\delta_0 \approx 1\div2\%$ is the polarization Touschek effect; $\delta N = N_1/N_2 - 1 $ is the relative number of particles difference in the bunches; $\tau_d $ is the polarization life time which is anobject of interest; $\tau_{tsh} \approx 5\, 000 \div 20\, 000$ s is the Touschek life time; $\tau_{bg} \approx 10\, 000 \div 20\, 000$ s is the background life time; $p_{tsh} \approx 0.2$ is a relative probability to register Touschek particles; $p_{bg} \approx 0.05$ is a relative probability to register residual gas scattered particles; $I_{1, 2}(0) \approx 2$ mA are initial current of the first and the second bunches, respectively; $V_0$ is the initial beam volume; $\alpha_V\approx 0.1\% \mbox{mA}^{-1}$ is coefficient of dependence of the beam volume on the beam current.
An evolution in time of the measured quantity $\Delta (t)$ in the conditions when the acceleration up to the target energy is just done but the compensating solenoids stay switched off can be described as follows. At the ’auxiliary energy’ the ratio of the bunch currents is adjusted to a level of $(I_1/I_2-1)\times 100\approx -(1\div2)\%<0$. This is due to the necessity to minimize a slope of the dependence $\Delta (t)$ as a whole and an associated systematic error. For this purpose we kick out portion by portion the redundant bunch particles using the VEPP-4M inflector. If the bunch current ratio mentioned above is provided then $\Delta(t)>0$ during all time of observation. Depolarization process is enhanced during crossing of the critical energy area and goes on after completion of the acceleration and setting of the target energy. By this reason the quantity $\Delta (t)$ grows in a positive direction by the law close to the exponential one. The characteristic time of this growth for the given target energy is calculated (see Fig.3). Polarization in the beam drops to zero and then another process becomes dominating - a relaxation due to a difference of the bunch currents. Because of a difference in the IBS beam lifetime the quantity $\Delta (t)$ begins to change in the negative direction. Asymptotically, it goes to zero.
\[sec:level1\]Experimental results
==================================
The results of the RD beam energy calibration in the ’auxiliary energy’ mode are presented in Fig.6. Basing on these data, one can get an idea of the magnitude of the polarization effect measured by the Touschek polarimeter, as well as its repeatability . The time allowed for the radiative polarization at the VEPP-3 booster ring at energy $E =1.65$ GeV was 5000-6000 seconds at the estimated characteristic time of polarization $\tau_d\approx4000$ seconds.
![Depolarization jumps during the resonant depolarization scans in two typical runs at the ’auxiliary’ energy (1655 MeV). The energy value at the bottom of each plot is measured with an accuaracy better than 10 keV. []{data-label="fig:JmpAux"}](fig6a.pdf "fig:"){width="85mm"} ![Depolarization jumps during the resonant depolarization scans in two typical runs at the ’auxiliary’ energy (1655 MeV). The energy value at the bottom of each plot is measured with an accuaracy better than 10 keV. []{data-label="fig:JmpAux"}](fig6b.pdf "fig:"){width="85mm"}
First of all, an efficiency of the proposed method for acceleration with crossing of the critical energy $E=1763$ MeV has been tested by an observation of the beam polarization relaxation (depolarization) in the final energy mode at $E\approx 1.81$ GeV. Evolution in time of the normalized Touschek electron counting rate after completion of acceleration in the case when the anti-solenoids remained switched off for all the time is shown in Fig. \[fig:Rlx\]. Qualitatively, the relaxation process proceeds as described in the previous section. Relaxation (depolarization) time of $\tau=1399\pm92$ determined from the data in Fig. \[fig:Rlx\]$a$ is in good agreement with the estimated time of about 1400 s ( Fig. \[fig:Time\]). Compliance with the calculation in the data in Fig. \[fig:Rlx\]$b$ looks a little worse and can be explained by the influence of beam instability on a systematic error.
![The relaxation process of beam polarization following the acceleration from the auxiliary energy up to 1806 MeV with the rate of 5 MeV/s ($a$) and up to 1808 MeV with the rate of 2 MeV/s ($b$). Compensatory solenoids shut down before acceleration remain in the same state.[]{data-label="fig:Rlx"}](fig7a.pdf "fig:"){width="85mm"}\
*(a)*\
![The relaxation process of beam polarization following the acceleration from the auxiliary energy up to 1806 MeV with the rate of 5 MeV/s ($a$) and up to 1808 MeV with the rate of 2 MeV/s ($b$). Compensatory solenoids shut down before acceleration remain in the same state.[]{data-label="fig:Rlx"}](fig7b.pdf "fig:"){width="80mm"}\
*(b)*\
The fact of beam polarization preservation has been fully confirmed in the runs on the beam energy measurement by the resonant depolarization technique performed after the acceleration (Fig. \[fig:Jmp1810\]).
![Resonant depolarization beam energy calibrations at the target energy of 1.81 GeV with the compensatory solenoid field restoration after completion of acceleration.[]{data-label="fig:Jmp1810"}](fig8a.pdf "fig:"){width="80mm"} ![Resonant depolarization beam energy calibrations at the target energy of 1.81 GeV with the compensatory solenoid field restoration after completion of acceleration.[]{data-label="fig:Jmp1810"}](fig8b.pdf "fig:"){width="80mm"} ![Resonant depolarization beam energy calibrations at the target energy of 1.81 GeV with the compensatory solenoid field restoration after completion of acceleration.[]{data-label="fig:Jmp1810"}](fig8c.pdf "fig:"){width="80mm"}
In contrast to the ’spin relaxation’ runs, the storage ring mode with anti-solenoid field switched on was restored before every start of the RD procedure. This measure stops the non-resonant (radiative) depolarization process related to the contribution of strong longitudinal magnetic fields to spin-orbit coupling. An additional time of 385 seconds was required to restore the anti-solenoid field and, concurrently, to set necessary corrections in the collider magnetic structure. If the acceleration stops at the level of 1810 MeV with $P/P_0=0.81$ then the relative degree falls down to 0.74 in 385 seconds.
According to the calculation, the degradation of the depolarization jump amplitude during acceleration makes about $(P/P_ 0)^2\approx0.5$. The measured values of three jumps in Fig. \[fig:Jmp1810\] are in the range $(0.85\div0.36)\%$ while the jumps measured at the ’auxiliary’ energy and presented in Fig. \[fig:JmpAux\] make up $0.9\%$ and $1.2\%$. The accumulated experience in the calibration of energy by resonant depolarization at VEPP-4M says that the stability of a beam polarization degree obtained at VEPP-3 under the same controlled conditions, is at the level of $10-20\%$. Therefore, comparing the specified data, we can conclude that the experiment and the calculation are in satisfactory quantitative agreement.
\[sec:level1\]Discussion
========================
The method to preserve an electron beam polarization at crossing of the integer spin resonance energy successfully realized and practically applied in the HEP experiment is based on the use of a longitudinal magnetic field of the detector incorporated into the storage ring structure. It differs from the well-known Siberian Snake technique which requires much greater integral of magnetic field ($4.6\nu$ Tesla$\times$meter) to rotate a spin of electron through an angle of $180^\circ$ around the velocity vector. The corresponding spin tune is $1/2$ or about 220 MeV in the energy units and so a detuning from integer spin resonances is maximal at any energy. A single Siberian snake solenoid ($\varphi=\pi$) increases a radiative depolarization rate by a factor of $\tau_p/\tau_d\approx (11/54)\pi^2\nu^2$. With approaching to an integer spin resonance a weak solenoid ($\varphi << 1$) brings a more stronger spin-orbit coupling than the Siberian Snake. In an accordance with (7) and (8), the ratio of the respective depolarization times is ($\nu=k$) $$\frac{\tau_d(\varphi=\pi)}{\tau_d (\varphi<<1)}\approx \frac{12}{\varphi ^2}.$$ In our case, a comparison of the necessary field integral in the moderate approach and the Siberian snake variant looks like $0.6\times3.3\approx 2$ Tesla$\times$meter against $18.4$ Tesla$\times$meter at $ E=1763$ MeV. The ratio of times is about $102$, $\tau_d(\varphi=\pi)=2.7$hours, $\tau_d(\varphi=0.34\,rad)=97$ seconds.
To date, an application of the Siberian Snakes is limited, mainly, to acceleration of the polarized proton and heavy ion beams in synchrotrons. In such machines the role of radiative processes in the spin kinetics is reduced to zero. At the electron-positron storage ring-collider VEPP-4M the radiative depolarization can limit the effectiveness of the developed method especially with growth of a target beam energy. Estimates show that using this method at full decompensation of the KEDR field integral one can cross the resonance $E=1322$ MeV ($\nu=3$) practically without polarization loss during beam deacceleration with a rate of 2 MeV/s starting from 1550 MeV. But acceleration in the same manner from 1.85 GeV up to 2.4 GeV with crossing of the resonance $E=2203$ MeV $(\nu=5$) leads to a three-fold decrease of the polarization degree.
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---
abstract: 'A measurement technique for the spin Seebeck effect is presented, wherein the normal metal layer used for its detection is exploited simultaneously as a resistive heater and thermometer. We show how the various contributions to the measured total signal can be disentangled, allowing to extract the voltage signal solely caused by the spin Seebeck effect. To this end we performed measurements as a function of the external magnetic field strength and its orientation. We find that the effect scales linearly with the induced rise in temperature, as expected for the spin Seebeck effect.'
author:
- Michael Schreier
- Niklas Roschewsky
- Erich Dobler
- Sibylle Meyer
- Hans Huebl
- Rudolf Gross
- 'Sebastian T. B. Goennenwein'
bibliography:
- 'Current\_heating\_induced\_spin\_Seebeck\_effect.bib'
title: Current heating induced spin Seebeck effect
---
The spin Seebeck effect [@Uchida2008; @Uchida2010; @Uchida2010a; @Uchida2010b] (SSE) is one of the hot topics in spin caloritronics. [@Bauer2012] In analogy to the charge Seebeck effect, where a charge current is driven by an applied temperature gradient, in the spin Seebeck effect a spin current is driven by a temperature gradient. [@Xiao2010] Since there is no direct meter for spin currents in present experiments usually a ferromagnet/normal metal (F/N) bilayer structure is used to convert the spin current into an electric signal: A temperature gradient applied perpendicular to the F/N bilayer drives a spin current across the F/N interface. This spin current is then converted into a charge current in N by virtue of the inverse spin Hall effect. Since most spin Seebeck effect measurements are performed using open circuit boundary conditions, the experimental signature of the spin Seebeck effect is a spin Hall electric field - viz. the corresponding spin Seebeck voltage - which is oriented perpendicular to both the applied temperature gradient and the magnetization in F.\
Nowadays most spin Seebeck experiments are performed in the so-called longitudinal geometry, in which the temperature gradient and the spin current are parallel, and oriented perpendicular to the F/N interface. To rule out anomalous Nernst effect voltages in F, [@Huang2011; @Huang2012] this geometry however requires that the (ferro- or ferri-) magnetic constituent is insulating. In most longitudinal spin Seebeck experiments to date, the magnetic insulator yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$, YIG) is used for F, and the high Z metals Pt or Au are used for N. [@Uchida2008; @Uchida2010; @Uchida2010a; @Uchida2010b; @Weiler2012; @Qu2013]\
In experiments, the controlled generation and quantification of temperature gradients represents a challenge. The temperature gradients are most often established by clamping the F/N sample between two heat reservoirs, acting as heat source and sink. [@Uchida2008; @Uchida2010; @Uchida2010a; @Uchida2010b; @Jaworski2010; @Qu2013; @Meier2013; @Kikkawa2013] An important issue in this type of spin Seebeck effect setup is good thermal coupling between the heat reservoirs and the sample. Laser heating [@Weiler2012] is an alternative technique, which enables scannable, local temperature gradient generation. The temperature gradients thus generated, however, can be quantified only from numerical temperature profile calculations. [@Schreier2013]\
![Sketch of the setup used for the current heating induced spin Seebeck experiments. The samples consists of magnetic insulator (YIG) thin films on single crystalline GGG or YAG substrates covered by a thin normal metal (Pt) film. The YIG/Pt bilayer is patterned into a Hall bar mesa structure. A dc-current source is used to drive a large current $I_\mathrm{d}$ through the Hall bar while the voltage drop $V_\mathrm{t}$ transverse to the current direction is measured with a nanovoltmeter. An external, in-plane, magnetic field is applied at an angle $\alpha$ to the current direction. Due to the resistive (Joule) heating by $I_\mathrm{d}$ a temperature gradient across the F/N interface emerges, giving rise to the spin Seebeck effect.[]{data-label="fig:setup"}](figure1.pdf){width="\columnwidth"}
In this paper we present a third, very simple technique to generate large thermal gradients across the F/N interface. The main idea is to use the sample’s normal metal layer itself as a resistive heater. In other words, we drive a large dc-current $I_\mathrm{d}$ through N, and simultaneously record the thermal (spin Seebeck) voltage in the direction transverse to the driving current (Fig. \[fig:setup\]).\
Since the current heating induced spin Seebeck voltage $V_\mathrm{iSSE}$ originates form the inverse spin Hall effect, we expect $V_\mathrm{iSSE}\propto \pmb j_\mathrm{s}\times\hat{\pmb s}$, where $\pmb j_\mathrm{s}$ is the direction of the spin current and $\hat{\pmb s}$ is its polarization vector. This can be used to discriminate $V_\mathrm{iSSE}$ from other possible signal contributions. For $H_\mathrm{ext}$ in the sample plane along $I_\mathrm{d}$ (along $x$, $\alpha=0^\circ$), a spin Seebeck voltage will arise along $y$. The large voltage drop $V_\mathrm{d}$ arising along the direction of current flow thus will not influence the spin Seebeck measurement.\
The spin Seebeck effect is, in fact, driven by the temperature difference ${\Delta T_\mathrm{me}}$ between magnons in F and electrons in N rather than the thermal gradient across the layers. [@Xiao2010] While the precise determination of ${\Delta T_\mathrm{me}}$ is very challenging [@Schreier2013] and depends crucially on e.g. sample dimension and the material parameters at the chosen sample temperature, ${\Delta T_\mathrm{me}}$ will, in first order approximation, be directly proportional to the temperature increase (decrease) of N with respect to the heat sink in the experiment (i.e. it is proportional to the the thermal gradient). [@Xiao2010; @Schreier2013] This temperature increase is in turn directly proportional to the dissipated electrical power (the Joule heating) $P_\mathrm{Joule}=V_\mathrm{d}I_\mathrm{d}=I_\mathrm{d}^2R$ where $R$ is the sample resistance. Using an insulating ferromagnet (YIG) greatly simplifies the interpretation of the experimental results since the current will only flow in the normal metal (platinum). [The heat in our experiment is generated uniformly within the entire N layer as compared to an injection through the top interface only for the clamping technique or the nonuniform heating for the laser method. Nevertheless, for a fixed amount of heat, in steady state and since the spin Seebeck effect is generated at the F/N interface rather than within N, the thermal gradient at the F/N interface should be very similar among the techniques.]{} In summary, we thus expect $$V_\mathrm{iSSE}\propto I_\mathrm{d}^2\cos\alpha,
\label{eq:Vsseprop}$$ However, the voltage [$V_\mathrm{t}=E_\mathrm{t}\times w$ ($E_\mathrm{t}$ and $w$ being the transverse electric field under open circuit conditions and the width of the Hall bar, respectively)]{} transverse to $I_\mathrm{d}$ will have contributions from the spin Seebeck effect and from magnetoresistive effects, such as the newly discovered spin Hall magnetoresistance. [@Nakayama2013; @Althammer2013a; @Chen2013a] Typically, these magnetoresistive transverse voltages will be much larger than the $V_\mathrm{iSSE}$ of interest. Additionally, the longitudinal resistance can contribute to $V_\mathrm{t}$ due to a slight misalignment of the transverse contacts. Since these effects are linear, or odd in $I_\mathrm{d}$, they can be distinguished from thermal effects, proportional to $P_\mathrm{Joule}$ or $I_\mathrm{d}^2$, by comparing two measurements with reversed driving current direction. The resistive contributions and the cross-coupling obey $V_\mathrm{res}(+I_\mathrm{d})=-V_\mathrm{res}(-I_\mathrm{d})$ while the spin Seebeck voltage obeys $V_\mathrm{iSSE}(+I_\mathrm{d})=+V_\mathrm{iSSE}(-I_\mathrm{d})$. $V_\mathrm{iSSE}$ can thus be obtained by adding [$V_\mathrm{t}(+I_\mathrm{d})$ to $V_\mathrm{t}(-I_\mathrm{d})$]{} such that $$\begin{aligned}
V_\mathrm{t}(+I_\mathrm{d}) + V_\mathrm{t}(-I_\mathrm{d}) &=& V_\mathrm{res}(+I_\mathrm{d}) + V_\mathrm{res}(-I_\mathrm{d}) +\notag\\
&& V_\mathrm{iSSE}(+I_\mathrm{d}) + V_\mathrm{iSSE}(-I_\mathrm{d})\notag\\
&=& V_\mathrm{res}(+I_\mathrm{d}) - V_\mathrm{res}(+I_\mathrm{d}) +\notag\\
&& V_\mathrm{iSSE}(+I_\mathrm{d}) + V_\mathrm{iSSE}(+I_\mathrm{d})\notag\\
&=& 2V_\mathrm{iSSE}(+I_\mathrm{d})
\label{eq:VpmI}.\end{aligned}$$ It is fair to argue that with increasing $I_\mathrm{d}$, the sample’s resistance $R=R(T)=R(I_\mathrm{d}^2)$ will increase due to the induced temperature changes. This also influences the resistive contributions by introducing higher order terms which, in good approximation, should be odd powers of $I_\mathrm{d}$ since $$V_\mathrm{res}\propto I_\mathrm{d}{\times} R\propto I_\mathrm{d}{\times} I_\mathrm{d}^2.
\label{eq:Vresodd}$$ Thus they should cancel out in the aforementioned procedure.\
[We would also like to add that it is possible to extract $V_\mathrm{iSSE}$ from the longitudinal voltage as well, albeit generally with a worse signal to noise ratio due to the large background signal ($V_\mathrm{d}$).]{}\
The samples in our experiment consists of YIG thin films grown by pulsed laser deposition on $\SI{500}{\micro m}$ thick gadolinium gallium garnet (Gd$_3$Ga$_5$O$_{12}$, GGG) or yttrium aluminium garnet (Y$_3$Al$_5$O$_{12}$, YAG) substrates. On top of the YIG layer few $\SI{}{\nano m}$ thick Pt films were then deposited in situ, without breaking the vacuum, using electron beam evaporation (more details on the sample growth can be found in Refs. ). One sample was fabricated with an additional gold spacer layer between YIG and Pt. After removing the samples from the growth chamber, the Pt (Au) and the YIG were patterned into Hall bar mesa structures (length $\SI{950}{\micro m}$, width $\SI{80}{\micro m}$) using optical lithography and argon ion beam milling. Afterwards the samples are mounted onto copper heat sinks.\
The measurements in this paper have all been performed in vacuum ($p\lesssim\SI{1}{\milli\bar}$) in a cryostat with variable temperature insert, with the sample stabilized at a base temperature of $\SI{250}{K}$. Note, however, that measurements under ambient conditions in an electromagnet at room temperature (not shown) gave very similar results. We furthermore measured the temperature dependence of $\rho_\mathrm{Pt}$ by systematically changing the cryostat base temperature. In this way, the Pt resistance can be used for on-chip thermometry [@Ri1991; @Ri1993] in the subsequent experiments.\
![Recorded transverse voltage on the GGG/YIG($\SI{61}{\nano m}$)/Pt($\SI{11}{\nano m}$) sample as a function of the external magnetic field strength for $\alpha=45^\circ$. [The arrows indicate the sweep direction of the external magnetic field in the experiment.]{} **(a)** For $I_\mathrm{d}<0$ a positive offset voltage signal is recorded that exhibits the typical features of the spin Hall magnetoresistance. **(b)** Reversing the direction of $I_\mathrm{d}$ also inverts the observed voltage signal. **(c)** Adding $V_\mathrm{t}(+I_\mathrm{d})$ and $V_\mathrm{t}(-I_\mathrm{d})$ reveals the much smaller, thermal (spin Seebeck) component. The large spikes close to the YIG’s coercive fields are attributed to either domain reconfiguration or spin torque effects. The inset shows $V_\mathrm{iSSE}$ at large fields for $\alpha=0^\circ$. Here $V_\mathrm{iSSE}$ stays constant for fields of up to $\SI{7}{T}$. []{data-label="fig:2"}](figure2.pdf){width="\columnwidth"}
The $V_\mathrm{iSSE}$ extraction procedure is visualized in Fig. \[fig:2\] for a fixed angle $\alpha=45^\circ$ between the Hall bar and the external magnetic field (*cf.* Fig. \[fig:setup\]) on a GGG/YIG($\SI{61}{\nano m}$)/Pt($\SI{11}{\nano m}$) sample. Here the transverse voltage is recorded as a function of the external magnetic field magnitude, which is varied from $+\SI{0.4}{T}$ to $-\SI{0.4}{T}$ and back to $+\SI{0.4}{T}$. For a pure spin Seebeck signal one would expect the observed signal’s shape to closely mimic that of the magnetic hysteresis loop of YIG, but apparently this is not the case. Clearly the signal shown in panel **(a)** is dominated by the transverse component of the in-plane spin Hall magnetoresistance, [@Althammer2013a] which changes sign upon changing the current direction \[panel **(b)**\]. Upon adding the two curves, however, the hysteresis loop becomes visible \[panel **(c)**\]. [For $H_\mathrm{ext}>0$ we observe a positive $V_\mathrm{iSSE}$ as observed in earlier experiments. [@Uchida2010]]{} The large additional peaks close to the coercive fields may stem from torque induced magnetization dynamics [@Fan2013] in the YIG, which affect the spin current flow in the Pt [or may be an artifact due to stray Oersted fields]{}. Dedicated experiments will be required to pinpoint the exact origin of these signal contributions. We can, however, exclude a proximity effect [@Huang2011] induced origin since these peaks appear in *all* samples, including the one with the additional gold layer between the YIG and the Pt. Here, we are interested only in the spin-Seebeck-like signal at high fields, which stays constant up to $\SI{7}{T}$ (inset Fig. \[fig:2\]).\
![Thermal voltage as a function of the external magnetic field direction $\alpha$ on the GGG/YIG($\SI{61}{\nano m}$)/Pt($\SI{11}{\nano m}$) sample. The magnitude of the external magnetic field remains fixed at $\SI{1}{T}$ throughout the entire measurement. **(a)** For $I_\mathrm{d}<0$ a positive offset voltage is recorded with the visible $\sin^2\alpha$ variation stemming from the spin Hall magnetoresistance. **(b)** Inverting the current ($I_\mathrm{d}>0$) also reverses the voltage signal, but upon adding up $V_\mathrm{t}(+I_\mathrm{d})$ and $V_\mathrm{t}(-I_\mathrm{d})$ and dividing the result by two \[**(c)**\] a $\cos\alpha$ component remains, consistent with the spin Seebeck effect \[Eq. \].[]{data-label="fig:V(alpha)"}](figure3.pdf){width="\columnwidth"}
To investigate the expected $\cos\alpha$ dependence of $V_\mathrm{iSSE}$ we keep the applied magnetic field at a fixed value of $\SI{1}{T}$ and record the transverse voltage $V_\mathrm{t}$ while varying the field orientation with respect to the Hall bar. The field value is chosen large enough to rule out any remanent magnetic features of the YIG and ensure that its magnetization is truly parallel to the external magnetic field. Figure \[fig:V(alpha)\] shows the result of this measurement on the GGG/YIG($\SI{61}{\nano m}$)/Pt($\SI{11}{\nano m}$) sample. As in Fig. \[fig:2\] the measured signal is dominated by the spin Hall magnetoresistance, featuring its characteristic $\sin^2\alpha$ dependence, which reverses sign as the current direction is inverted. Once again, by adding the signals obtained for opposite current direction the resistive effects cancel out and the spin Seebeck component remains. As predicted by Eq. the signal follows a $\cos\alpha$ dependence.\
![**(a)** Thermal component ($V_\mathrm{iSSE}$) of the recorded voltage (full blue symbols) and Pt temperature (open circles) as a function of the square of applied current on the GGG/YIG($\SI{61}{\nano m}$)/Pt($\SI{11}{\nano m}$) sample. To obtain the individual $V_\mathrm{iSSE}(I_\mathrm{d})$ data points the external magnetic field was rotated at a fixed field strength of $\SI{1}{T}$, from which the spin Seebeck voltage is extracted \[$V_\mathrm{iSSE}\equiv V_\mathrm{iSSE}(\alpha=0^\circ)$\] for each value of the driving current $\pm I_\mathrm{d}$. The observed $V_\mathrm{iSSE}$ scales quadratically with $I_\mathrm{d}$ as does $T_\mathrm{Pt}$. [The indicated error for $T_\mathrm{Pt}$ is an upper estimate for the uncertainty in the fitting algorithm and accounts for the fact that the two rightmost data points were extrapolated. The inset shows the resistivity of the Pt film as a function of temperature.]{} **(b)** Generated spin current per applied heating power for the samples investigated in this letter. The numbers in parentheses give the layer thicknesses in $\SI{}{\nano m}$. Taking the smaller spin mixing conductance of the YIG/Au interface into account, all samples give very similar spin current generation efficiencies.[]{data-label="fig:V(I)"}](figure4.pdf){width="\columnwidth"}
To confirm that the recorded signals indeed stem from a thermal effect this procedure is repeated as a function of the applied current. Panel **(a)** in Fig. \[fig:V(I)\] shows $V_\mathrm{iSSE}=\frac12[V_\mathrm{t}(+I_\mathrm{d})+V_\mathrm{t}(-I_\mathrm{d})]$ as a function of $I_\mathrm{d}^2$ on the GGG/YIG($\SI{61}{\nano m}$)/Pt($\SI{11}{\nano m}$) sample. $V_\mathrm{iSSE}$ clearly shows a quadratic dependence on the applied current, supporting the notion of Eq. that the measured spin Seebeck effect should scale quadratically with $I_\mathrm{d}$. Moreover the effect quickly drops below the noise floor for small currents for which the spin Hall magnetoresistance is still clearly visible in $V_\mathrm{t}$. [@Althammer2013a] Furthermore, by simultaneously measuring the resistance of the Pt Hall bar along the current direction we are also able to determine its temperature by comparing the measured resistance value to a $R(T)$ calibration curve recorded separately at a small current value [(inset Fig.\[fig:V(I)\])]{}. As expected, the temperature increase of the Pt film is directly proportional to $I_\mathrm{d}^2$ as well. In other words, $V_\mathrm{iSSE}$ is directly proportional to the temperature increase of the Pt film as suggested above. Panel **(b)** in Fig. \[fig:V(I)\] compares the results of the different samples investigated in this letter. Here the recorded spin Seebeck voltage is normalized as to extract the spin current density per applied heating power $J_\mathrm{s}/P_\mathrm{Joule}$. This is achieved by dividing $V_\mathrm{iSSE}$ by the sample resistance, Joule heating power and the correction factor for spin diffusion in the normal metal. [@Schreier2013] The latter is calculated using a spin diffusion length of $\SI{1.5}{\nano m}$ for Pt [@Althammer2013a] and assuming that the gold layer does not affect the spin current. The value of $J_\mathrm{s}/P_\mathrm{Joule}$ is very similar for the investigated YIG/Pt samples, while the GGG/YIG($\SI{15}{\nano m}$)/Au($\SI{8}{\nano m}$)/Pt($\SI{7}{\nano m}$) sample shows about half the value of the YIG/Pt samples, fully consistent with the smaller spin mixing conductance of the YIG/Au interface. [@Althammer2013a; @Weiler2013; @Burrowes2012] [Generally samples with thinner YIG films give smaller voltage signals, the number of samples investigated here is, however, too small and the individual samples too different to confidently read any trend [@Kehlberger2013] from this observation.]{}\
The arguments brought forward here would also apply to a potential contamination via the anomalous Nernst effect. However, recent experiments [@Geprags2012; @Kikkawa2013; @Geprags2013] show that pure Pt does not get proximity polarized by the YIG layer and hence no anomalous Nernst can occur. This is also confirmed by the measurement on the GGG/YIG($\SI{61}{\nano m}$)/Au($\SI{8}{\nano m}$)/Pt($\SI{7}{\nano m}$) sample which gives voltage signals very similar to those on the samples without the gold spacer layer \[Fig. \[fig:V(I)\]**(c)**\].\
In summary, we have demonstrated that the longitudinal spin Seebeck effect can be measured by simply using the normal metal (Pt) layer as a Joule heater to create the required thermal gradient at the ferromagnet/normal metal interface. Measurements as a function of the magnitude and orientation of the applied magnetic field show the characteristic dependencies of the spin Seebeck effect and scale quadratically with the applied current, as expected for a thermal effect. We thus conclude that the simple Joule heating technique indeed enables the detection of the spin Seebeck effect in yttrium iron garnet/platinum thin film hybrid structures.\
\
We thank Timo Kuschel for valuable discussions and gratefully acknowledge financial support from the DFG via SPP 1538 “Spin Caloric Transport” (project GO 944/4-1) and the German Excellence Initiative via the Nanosystems Initiative Munich (NIM).
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---
abstract: 'In this article, we give a geometric description for any invertible operator on a finite dimensional inner–product space. With the aid of such a description, we are able to decompose any given conformal transformation as a product of planar rotations, a planar rotation or reflection and a scalar transformation. Also, we are able to conclude that an orthogonal transformation is a product of planar rotations and a planar rotation or a reflection.'
author:
- 'Srikanth K.V. and Raj Bhawan Yadav [^1]'
title: Decomposition Of Invertible And Conformal Transformations
---
Introduction
============
A classical theorem on orthogonal operators on a finite dimensional inner–product space decomposes the given operator into planar rotations and possibly a planar reflection[@Decm05; @Kost.; @and; @Manin]. The standard proof of this theorem relies on the fundamental theorem of algebra applied to the characteristic polynomial of the complexified operator. Such a proof lacks geometric intuition regarding these planar rotations or reflections which arise in the decomposition of an orthogonal operator. One of the aims of this article is to provide a more geometrically inclined proof of this theorem. In Section \[sec:prelim\] we fix notation and recall a few elementary definitions. Section \[sec:axis\] introduces elementary but novel concepts such as an *axial–vector* and the *axis* of a basis of a finite dimensional real inner–product space. In Section \[sec:invertible\] we identify a class termed *axonal* operators. Theorem \[thm:decomposelinear\] gives a geometric description of invertible operators using axonal operators. Finally, in Section \[sec:decompose\], Theorem \[thm:decomposeconformal\] gives a decomposition of conformal operators into planar rotations followed possibly by a reflection and a scalar transformation. A similar theorem for orthogonal operators is noted in Theorem \[thm:decomposeorthogonal\].
Preliminaries {#sec:prelim}
=============
Throughout this article, $V$ denotes a finite dimensional inner– product space of dimension $n\in\mathbb{N}$ over the field of real numbers $\mathbb{R}$. The set of all invertible operators on $V$ shall be denoted by GL($V$).
Let $W$ be a two dimensional subspace of $V$. Suppose that $\{u, v \}\subset V$ is an orthonormal basis of $W$. For any fixed real number $\theta$, the assignments\
$u\mapsto \left(\cos\theta \right)u+\left(\sin\theta\right) v$ and $v\mapsto \left(-\sin\theta\right) u+\left(\cos\theta\right) v$\
determine a unique linear operator on $W$ termed as a **rotation** of $W$.
Similarly the assignments\
$u\mapsto u$ and $v\mapsto -v$\
determine a unique linear operator on $W$ termed as a **reflection** of $W$.
Let $T$ be a linear operator on $V$ of the form $\rho\oplus id$ where $\rho$ is an operator on a two dimensional subspace $W$ of $V$ and $id$ is the identity operator on $W^\perp$. We say that $T$ is a **planar rotation** or a **planar reflection** accordingly as $\rho$ is a rotation or a reflection of $W$. A planar rotation and a planar reflection are also termed as **rotational** and **reflectional** operators respectively.
When $V$ is uni–dimensional, we allow the identity operator on $V$ to be termed a rotational operator.
Axis Of A Basis {#sec:axis}
===============
A basis $\{u_i\}_{i=1}^n$ of the inner–product space V is **equimodular** if there exists a real $\delta > 0$ such that $\|u_i\|=\delta$, for each $i$ in $\{1,2,\ldots , n\}$. When such a $\delta=1$, we say that the basis is **unimodular**.
Let $\alpha$ be a non–zero vector in an inner– product space $V$ and $\{u_i\}_{i=1}^n$ be a basis of $V$. If $$\langle u_i,\alpha\rangle \langle u_j, u_j \rangle^\frac{1}{2} = \langle u_j,\alpha\rangle \langle u_i, u_i \rangle^\frac{1}{2}\;\mathrm{ for\; all }\; i,j \in \{1,2,\ldots , n\},$$ we say that $\alpha$ is an **axial–vector** of the given basis $\{u_i\}_{i=1}^n$.
\[lem:existaxial\] Let $\alpha$ be an axial–vector of a basis $\{u_i\}_{i=1}^n$ of an inner–product space $V$.
1. Clearly, the ratio $\frac{\left< u_i, \alpha \right>}{\|u_i\| \|\alpha\|}$ is independent of the choice of $i\in \{1,2,\ldots , n\}$. Thus, the axial–vector $\alpha$ makes the same angle with each of the basis vectors $u_i$ for $i \in \{1,2,\ldots , n\}$.
2. When $n=1$, $\frac{\langle u_1,\alpha \rangle}{\|u_1\|\|\alpha\|}$ is either $-1$ or $1$.
3. However, when $n\ge 2$, if $\frac{\langle u_i,\alpha \rangle}{\|u_i\|\|\alpha\|}$ is either $-1$, $0$ or $+1$, it forces $\{u_i\}_{i=1}^n$ to be linearly dependent. In other words, the common angle between an axial vector and the basis vectors can not be $0$, $\frac{\pi}{2 }$ or $\pi$.
Every equimodular basis of a finite dimensional inner–product space has an axial–vector. Further, any two axial–vectors of a given equimodular basis are linearly dependent.
Suppose $\{u_i\}_{i=1}^n$ is an equimodular basis of $V$. Given a non–zero real $\omega$, we prove the existence of a non–zero $\alpha \in V$ such that $\langle u_i,
\alpha\rangle = \omega$ for all $i \in \{1,2,\ldots , n\}$. Such an $\alpha$ would be an axial–vector of the given equimodular basis.
Let $A$ be the $n\times n$ matrix with $A_{ij}=\langle u_i,u_j\rangle $ for $i, j \in \{1,2,\ldots , n\}$. Let $\alpha =\sum_{i=1}^nx_i u_i$ for undetermined real numbers $\{x_i\}_{i=1}^n$. Further, let $X$ denote the column vector $(x_1,x_2,...,x_n)^\mathrm{T}$. Then for $i \in \{1,2,\ldots , n\}$, the collection of $n$ equations $\langle u_i, \alpha \rangle=\langle u_i, \sum_{j=1}^nx_ju_j\rangle=\omega$ is the system $AX = \Omega$, where $\Omega$ is the column vector $(\omega, \omega, \ldots ,\omega)^T$. Existence of a solution $X$ to the latter system suffices to prove the existence of $\alpha$. In fact, we show that for each $\Omega$ there is a unique solution $X$ by proving that $A$ is invertible.
Suppose to the contrary that $A$ is not invertible. Then, there exists a non–zero column vector $Y=(y_1, y_2, \ldots, y_n)^\mathrm{T}$ such that $AY=0$. Set $s=\sum_{i=1}^n y_iu_i$. Then, $s\neq 0$ and consequently $\|s\|\neq 0$. However, $\|s\|^2=\langle \sum_{i=1}^n y_iu_i,
\sum_{j=1}^n y_ju_j\rangle
=Y^TAY =0$, a contradiction.
If $\omega$ is non–zero, it is evident from $X = A^{-1}\Omega$ that $X\neq 0$ and hence $\alpha$ is non–zero. This proves the existence of an axial vector for the given basis.
Suppose that $\alpha$ and $\tilde{\alpha}$ are two axial–vectors of a given equimodular basis $\{u_i\}_{i=1}^n$ of $V$. Let $\alpha= \sum_{i=1}^n x_i u_i$ and $\tilde{\alpha}= \sum_{i=1}^n \tilde{x}_i u_i$ for some real numbers $\{x_i\}_{i=1}^n$ and $\{\tilde{x}_i\}_{i=1}^n$. Set $X=(x_1,x_2,\ldots, x_n)^T$ and $\tilde{X}=(\tilde{x}_1,\tilde{x}_2,\ldots, \tilde{x}_n)^T$. Corresponding to the axial vectors $\alpha$ and $\tilde{\alpha}$, there are two non–zero real numbers $\omega$ and $\tilde{\omega}$ such that $\langle u_i, \alpha\rangle = \omega$ and $\langle u_i, \tilde{\alpha}\rangle = \tilde{\omega}$ for all $i\in\{1,2,\ldots , n\}$. These two sets of $n$ equations are the two systems $AX=\Omega$ and $A\tilde{X}=\tilde{\Omega}$ where $\Omega=(\omega,\omega,\ldots,\omega)^T$ and $\tilde{\Omega}=(\tilde{\omega},\tilde{\omega},\ldots,\tilde{\omega})^T$. Clearly the column vectors $\Omega$ and $ \tilde{\Omega}$ are linearly dependent. Hence the corresponding solutions $X=A^{-1}\Omega$ and $ \tilde{X}=A^{-1}\tilde{\Omega}$ are linearly dependent. We can now conclude that the axial–vectors $\alpha$ and $\tilde{\alpha}$ are linearly dependent.
\[thm:existaxial\] Every basis of a finite dimensional inner–product space has an axial–vector. Any two axial–vectors of a given basis are linearly dependent.
If $\{v_i\}_{i=1}^n$ is a basis of $V$, then the collection $u_i :=
\frac{v_i}{\|v_i\|}$ is an equimodular basis. The latter has an axial–vector $\alpha$ by Lemma \[lem:existaxial\]. Consequently $ \langle u_i, \alpha \rangle =\langle u_j, \alpha \rangle$ for all $i, j\in \{1,2,\ldots , n\}$. Substituting for $u_i$, we get that $ \langle v_i, \alpha \rangle \langle v_j, v_j \rangle =\langle v_j, \alpha \rangle \langle v_i, v_i \rangle $ for all $i, j\in \{1,2,\ldots , n\}$. Thus, $\alpha$ is an axial–vector of the given basis $\{v_i\}_{i=1}^n$.
Further, if $\alpha$ and $\tilde{\alpha}$ are two axial–vectors of the given basis $\{v_i\}_{i=1}^n$, then $\alpha$ and $\tilde{\alpha}$ are also axial–vectors of the equimodular $\{u_i\}_{i=1}^n$ and again by Lemma \[lem:existaxial\] are linearly dependent.
Theorem \[thm:existaxial\] ensures the existence and uniqueness of the $\textbf{axis}$ of a basis defined below.
Let $\alpha$ be an axial–vector of a basis $\mathcal{B}$ of a finite dimensional inner–product space. The $\textbf{axis}$ of $\mathcal{B}$ is defined to be the span of $\{\alpha\}$.
Given a non–zero $\alpha\in V$ and a real number $\theta \in [0,\pi]$, we define the **cone** around $\alpha$ of **vertex angle** $\theta$ by $$\Lambda_\alpha^\theta = \left\{x\in V|\langle x,\alpha\rangle = \|x\|\|\alpha \|\cos \theta \right\}.$$ The span of $\{\alpha \}$ shall be termed as the **axis** of the cone $\Lambda_\alpha^\theta$.
1. When $V$ is one dimensional, the cone around any non–zero vector is empty if the angle $\theta \neq 0,\pi$.
2. When the dimension of $V$ is at least two, every such cone $\Lambda_\alpha^\theta$ is non–empty.
3. From Theorem \[thm:existaxial\], if $\mathcal{B}$ is any basis of $V$, there exists a cone in $V$ whose vertex is the zero vector, whose axis is the axis of the basis and whose vertex angle is the common angle between each of the basis elements and an axial–vector of $\mathcal{B}$. Such a cone is termed as the **associated cone** of basis $\mathcal{B}$.
Geometric Description Of Invertible Operators {#sec:invertible}
=============================================
An invertible linear operator $T:V\rightarrow V$ is called **axonal** if it maps an equimodular basis $\mathcal{B}$ to an equimodular basis $\mathcal{B}^{'}$ such that $\mathcal{B}$ and $\mathcal{B}^{'}$ share a common axis.
We denote the set of all axonal operators on $V$ by AX($V$). The latter is a subset of GL($V$).
1. It would be of interest to know examples of axonal operators on $V$. Indeed, given any two equimodular bases sharing a common axis, we have an example of an axonal operator which would map one of these bases to the other.
2. Axonal linear operators which map an equimodular basis $\mathcal{B}$ to $\mathcal{B'}$ may be classified into two kinds: those for which the associated cones of $\mathcal{B}$ and $\mathcal{B'}$ are same and those for which the associated cones are distinct. Proposition \[prop:axonalexample\] below provides examples of axonal operators of the latter kind.
\[prop:axonalexample\] Suppose that $V$ is an inner–product space of dimension at least two. Let $\{u_i\}_{i=1}^n$ be a basis of $V$ with $\alpha$ as its axial–vector. Assume that for each $i \in \{1,2,\ldots , n\}$, $u_i$ is rotated in the space spanned by $u_i$ and $\alpha$ to get $v_i$ such that each $v_i$ makes the same angle $\phi$ with $\alpha$. If $\phi \not\in \{0,\frac{1}{2}\pi, \pi \}$, then $\{v_i\}_{i=1}^n$ is a basis of $V$.
Without any loss of generality assume that $\{u_i\}_{i=1}^n$ is an equimodular basis. Let $S$ be the span of $\{\alpha\}$. Since each of the $u_i's$ have the same norm and make the same angle with $\alpha$, they have the same component, say $s$, in $S$. For each $i\in\{1,2,\ldots , n\}$, orthogonally decompose the two collections of vectors to get $$\begin{aligned}
u_i&=s+w_i, \text{where }s\in S \text{ and } w_i\in S^\perp \\
v_i&=ks+hw_i, \text{ for some real numbers } h,k \neq 0.\end{aligned}$$ We note that if either $h$ or $k$ equals 0, angle $\phi\in\{0,\frac{1}{2}\pi, \pi\}$, contradicting hypothesis.
Next, assume a linear relation of the form $\sum_{i=1}^n \lambda_i v_i = 0$ for some real numbers $\{\lambda_i\}_{i=1}^n$. Using Eq. (2) from above, we get $$\begin{aligned}
k\left(\sum_{i=1}^n\lambda_i\right)s+h\left(\sum_{i=1}^n\lambda_i w_i\right)=0.\end{aligned}$$ The summands are in orthogonal complements $S$ and $S^\perp$ while $h, k \neq 0$. Hence we conclude $$\begin{aligned}
\left(\sum_{i=1}^n\lambda_i\right)s=0\quad \text{and}\quad \sum_{i=1}^n\lambda_i w_i=0.\end{aligned}$$ Adding the two equalities, we get $\sum_{i=1}^n\lambda_i \left(s+w_i\right)=0$ and hence $\sum_{i=1}^n\lambda_i u_i=0$. From the linear independence of $\{u_i\}_{i=1}^n$, we conclude $\lambda_i=0$ for all $i\in\{1,2,\ldots , n\}$. This proves that $\{v_i\}_{i=1}^n$ are linearly independent and hence form a basis of $V$.
Let $k$ be a positive integer with $k\le n$. A linear transformation $S:V\rightarrow V$ is called a **$k$–shear** if there exists a $k$–dimensional subspace $W$ of $V$ such that the restriction $S_{|W^\perp}$ is the identity while $S_{|W}$ is an axonal transformation which maps an equimodular basis $\{u_i\}_{i=1}^k$ to an equimodular basis $\{v_i\}_{i=1}^k$ of $W$ with the same axis so that each $v_i$ is obtained by rotating $u_i$ by the same angle $\theta$ in the two dimensional subspace containing $u_i$ and the axis of $\{u_i\}_{i=1}^k$ .
\[rem:axonalshear\] Any axonal transformation $A:V\rightarrow V$ can be written as $A=A'\circ S$, where $S$ is a $n$-shear and $A'$ is an axonal transformation which maps an equimodular basis to another equimodular basis such that the two bases have the same associated cones.
\[thm:decomposelinear\] Let $T:V\rightarrow V$ be any invertible linear operator on $V$. Then, there exist a diagonal operator $D$, an axonal operator $A$ and a rotational operator $R$ on $V$ such that $T=D\circ A\circ R$.
Suppose $\{u_i\}_{i=1}^n$ is any equimodular basis of $V$. Let $v_i = T(u_i)$ and $w_i=\frac{v_i}{\|v_i\|}$ for all $i\in\{1,2,\ldots , n\}$. Since $T$ is invertible, $\{w_i\}_{i=1}^n$ is a unimodular basis of $V$. Let $L_1$ be the axis of the basis $\{u_i\}_{i=1}^n$ and $L_2$ be the common axis of the bases $\{v_i\}_{i=1}^n$ and $\{w_i\}_{i=1}^n$. There exists a rotational operator $R$ of $V$ which maps $L_1$ to $L_2$. Clearly, every rotational operator is invertible and norm–preserving and hence $\{R(u_i)\}_{i=1}^n$ is a unimodular basis whose axis is $L_2$. Further the linear operator $A$ which maps $R(u_i)$ to $w_i$ for all $i \in \{1,2,\ldots , n\}$ is axonal. Let $D$ be the diagonal operator on $V$ which maps $w_i$ to $v_i$. Since both $T$ and $D\circ A\circ R$ map the basis $\{u_i\}_{i=1}^n$ to the basis $\{v_i\}_{i=1}^n$ we can conclude that $T=D\circ A\circ R$.
From Remark \[rem:axonalshear\], a further characterization of axonal transformations which maps an equimodular basis to another equimodular basis sharing the same associated cone would provide a finer description of invertible transformations on the lines of Theorem \[thm:decomposelinear\].
Decomposition Of Conformal And Orthogonal Operators {#sec:decompose}
===================================================
Recall that a linear $f:V\rightarrow V$ is said to be conformal if there exists a real $\lambda >0$ such that $\langle f(u), f(v) \rangle = \lambda \langle u, v\rangle$ for all $u, v \in V$.
A linear function is conformal if and only if it is angle preserving.
\[thm:decomposeconformal\]
Given any conformal $T \in GL(V)$, we can write $$T=D\circ \mathcal{R}\circ R_{n-2}\circ R_{n-3}\circ\dots \circ R_{2}\circ R_1,$$ where $R_k$ is a rotational operator on $V$ for each $k\in\{1,2,\ldots, n-2\}$, $\mathcal{R}$ is either a rotational or a reflectional operator on $V$ and $D$ is a scalar operator on $V$.
We induct on $n$ – the dimension of $V$. For unidimensional $V$, every $T$ is a scalar operator. When dimension of $V$ is two, it is easily verified that every conformal $T$ is of the form $D\circ \mathcal{R}$ where $D$ is a scalar operator on $V$ and $\mathcal{R}$ is either a rotation or a reflection of $V$.
Assume next that $V$ has dimension at least three. By Theorem \[thm:decomposelinear\], we have $T=D\circ A_1 \circ R_1$ for linear operators $D, A_1$ and $R_1$ on $V$ such that $D$ is diagonal, $A_1$ is axonal and $R_1$ is rotational. Suppose the axonal $A_1$ maps an equimodular basis $\{u_i\}_{i=1}^n$ to an equimodular basis $\{v_i\}_{i=1}^n$ which share a common axis. By rescaling $\{u_i\}_{i=1}^n$, we can assume that the basis $\{u_i\}_{i=1}^n$ is unimodular. By rescaling $D$, if necessary, we can assume that $\{v_i\}_{i=1}^n$ is unimodular. Assume that for some real numbers $\lambda_i$ the diagonal operator $D$ maps $v_i$ to $\lambda_iv_i$ for each $i \in \{1,2,\ldots, n\}$. The scalars $\{\lambda_i\}_{i=1}^n$ are non–zero as $T$ is invertible.
Write $A_1=D^{-1}\circ T\circ R_1^{-1}$ and in the latter composition $R_1^{-1}$ and $T$ are angle preserving. Now $T\circ R_1^{-1}$ maps $u_i$ to $\lambda_i v_i$ and hence the angle between $u_i$ and $u_j$ equals the angle between $\lambda_i v_i$ and $\lambda_j v_j$ for $i,j\in\{1,2,\ldots, n\}$. Note that $D^{-1}$ being diagonal along $\{v_i\}_{i=1}^n$ keeps the angle between $\lambda_i v_i$ and $\lambda_j v_j$ equal to the angle between $v_i$ and $v_j$ for $i,j\in\{1,2,\ldots, n\}$. Consequently, if we take $\{u_i\}_{i=1}^n$ to be orthonormal, then $\{v_i\}_{i=1}^n$ is an orthonormal basis. We conclude that $A_1$ is orthogonal.
Now that $A_1$ and $R_1$ are conformal and $T$ is given to be conformal, we conclude that $D=T\circ R_1^{-1}\circ A_1^{-1}$ is conformal. Hence, $D$ is a scalar operator.
Suppose $\alpha$ is an axial–vector of the basis $\{u_i\}_{i=1}^n$. If the common angle between $\alpha$ and each of these basis vectors is $\theta$, then $A_1(\alpha)$ makes the same angle $\theta$ with each of the vectors from the basis $\{v_i\}_{i=1}^n$ as $A_1$ is orthogonal. Hence $A_1(\alpha)$ is an axial–vector for the basis $\{v_i\}_{i=1}^n$. However, these two bases share a common axis, say, $W$. Since $W=\mathrm{Span}\{\alpha\}=\mathrm{Span}\{A_1(\alpha)\}$ and $A_1$ is orthogonal, $A_1(\alpha)=\pm \alpha$. By replacing $D$ by $-D$ if necessary, we assume $A_1(\alpha)=\alpha$ and hence $A_1$ is the identity on $W$. Since $A_1$ is orthogonal, $W^\perp$ is an invariant subspace of $A_1$. Thus we may write $A_1=id\oplus T_2$ where $id$ is the identity operator on $W$ and $T_2$ is an orthogonal operator on $W^\perp$ of dimension $n-1$. We apply induction hypothesis to the orthogonal $T_2$ to realize this as a composition $$T_2= \tilde{D}_2\circ\tilde{\mathcal{R}}\circ \tilde{R}_{n-2}\circ \tilde{R}_{n-3}\circ\cdots \circ\tilde{R}_2,$$ where $\tilde{D}_2$ is scalar while $\tilde{\mathcal{R}}$ is either reflectional or rotational and $\tilde{R}_2, \tilde{R}_3, \ldots , \tilde{R}_{n-2}$ are rotational operators on $W^\perp$. Since $T_2$ is orthogonal, $\tilde{D}_2$ is the identity on $W^\perp$. We extend $\tilde{\mathcal{R}}, \tilde{R}_2, \tilde{R}_3, \ldots \tilde{R}_{n-2}$ respectively to $\mathcal{R}, R_2, R_3, \ldots R_{n-2}$ by declaring the latter operators to be identity on $W$. We now have\
$T=D\circ \mathcal{R}\circ R_{n-2}\circ R_{n-3}\circ\dots \circ R_{2}\circ R_1$.
\[thm:decomposeorthogonal\]
Given any orthogonal $T \in GL(V)$, we can write $$T= \mathcal{R}\circ R_{n-2}\circ R_{n-3}\circ\dots \circ R_{2}\circ R_1,$$ where $R_k$ is a rotational operator on $V$ for each $k\in\{1,2,\ldots , n-2\}$ and $\mathcal{R}$ is either a rotational or a reflectional operator on $V$.
$T$ being orthogonal is conformal. From Theorem \[thm:decomposeconformal\], we may write $$T=D\circ \mathcal{R}\circ R_{n-2}\circ R_{n-3}\circ\dots \circ R_{2}\circ R_1,$$ where $R_k$ is a rotational operator on $V$ for each $k\in\{1,2,\ldots , n-2\}$, $\mathcal{R}$ is either a rotational or a reflectional operator on $V$ and $D$ is a scalar operator on $V$. Since $T$ is orthogonal, the scalar operator $D$ has to be the identity and hence\
$T= \mathcal{R}\circ R_{n-2}\circ R_{n-3}\circ\dots \circ R_{2}\circ R_1.$\
[12]{}
Decomposition of Orthogonal operators as Rotations and Reflections. 2005 \[cited 2013 Aug 27\]. Available from: http://planetmath.org/decompositionoforthogonaloperatorsasrotationsandreflections. Alexei I Kostrikin, Yu I Manin. Linear Algebra and Geometry. New York:Gordon and Breach Science Publishers;1989.
[^1]: Dept. of Mathematics, IIT Guwahati, Assam 781039, INDIA
|
---
abstract: |
Under certain conditions on an integrable function $ P
$ having a real-valued Fourier transform $ \hat{ P }$ and such that $ P(0)
= 0 $, we obtain an estimate which describes the oscillation of $ \hat{P}
$ in the interval $ [ -C \| P' \|_{ \infty} / \| P \|_{\infty} , C \| P'
\|_{ \infty} / \| P \|_{\infty}] $, where $C$ is an absolute constant, independent of $P$. Given $ \lambda > 0 $ and an integrable function $
\phi $ with a non-negative Fourier transform, this estimate allows us to construct a finite linear combination $ P_{ \lambda} $ of the translates $
\phi ( \cdot + k \lambda ) , \ k \in {\bf Z}$ such that $ \| P_{ \lambda}
'\|_{ \infty } > c \| P_{ \lambda} \|_{ \infty } / \lambda $ with another absolute constant $c>0$. In particular, our construction proves sharpness of an inequality of Mhaskar for Gaussian networks.
address:
- 'Szilárd Gy. Révész A. Rényi Institute of Mathematics Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary'
- 'Noli N. Reyes Department of Mathematics College of Science University of the Philippines Quezon City, 1101, Philippines'
author:
- 'Szilárd Gy. Révész, Noli N. Reyes and Gino Angelo M. Velasco'
title: 'Oscillation of Fourier Transforms and Markov-Bernstein Inequalities'
---
[^1]
[^2]
Introduction
============
The original A.A. Markov inequality states that $ || P'||_{ L^{ \infty }(I) } \leq n^{2} || P ||_{ L^{ \infty }(I) } $ for any algebraic polynomial $P$ of degree $n$. Here, $ I= [-1,1] $. This inequality becomes an equality if $P$ is the Chebyshev polynomial $ P(x) = \cos nt $ where $ x = \cos t $. The reader may find the details of this in page 40 of [@lorentz].
Upper estimates of the derivative norm by that of the function itself are usually termed Markov-Bernstein inequalities. There is an extensive literature on such inequalities, which play an important role in inverse theorems, where smoothness of a function is deduced from rates of convergence of polynomial approximations. For an excellent survey on Markov-Bernstein and related inequalities, the reader may consult the book [@borwein] of P. Borwein and T. Erdélyi.
By imposing additional assumptions on the zeros of the polynomials, one can obtain estimates which give lower estimates for the norm of a derivative in terms of the norm of the function. These results are usually termed inverse Markov-Bernstein inequalities or Turán type inequalities. For instance, Turán [@turan] proved that $$|| P'||_{ L^{ \infty }(I) } \geq \frac{ \sqrt{n}}{ 6 } || P ||_{ L^{ \infty }(I) }$$ for any polynomial $P$ of degree $n$, provided that all of its zeros lie in the interval $I = [-1,1]$. We also refer the reader to a valuable paper of Eröd [@erod].
There is an upsurge of interest in such estimates, with a number of recent results dealing with the topic ([@erdelyi2], [@levenberg], [@szilard], [@zhou]). For instance, in [@zhou], Zhou showed that if $ 0 <r \leq q \leq \infty $ and $ 1 \geq 1/r - 1/q$, then $$|| P'||_{ L^{ r }(I) } \geq C n^{\alpha} || P ||_{ L^{ q }(I) }$$ for every polynomial $P$ whose zeros lie in the interval $I$. Here, $ \alpha = \frac{1 }{2 } - \frac{ 1}{2r } + \frac{1 }{ 2q} $.
More related to our work are results of Erdélyi and Nevai [@erdelyi] where they obtained $$\lim_{ n \rightarrow \infty} \frac{ ||p_{n}'||_{X} }{ ||p_{n} ||_{Y} } = \infty$$ for sequences of polynomials $ p_{n}$ whose zeros satisfy cetain conditions .
Markov-Bernstein inequalities have also been obtained for other classes of functions such as Gaussian networks. For instance, in [@mhaskar], Mhaskar showed that for some constant $c$, $ ||g' ||_{p} \leq c m || g ||_{p} $ for any function $g$ defined on the real line of the form $$g(x) \ = \ \sum_{k=1}^{N} a_{k} \exp ( - ( x-x_{k} )^{2} ),$$ where $ | x_{j} - x_{k} | \geq 1/m $ for $ j \neq k $, and $ \log N =
{\mathcal O} (m^{2}) $.
One of our goals in this note is to show that under certain conditions on an integrable function $ P :{\bf R} \longrightarrow {\bf R} $ having a real-valued Fourier transform $ \hat{ P }$ with $ P(0) = 0 $, $$r \geq C \frac{\| P'\|_{ \infty } }{\| P\|_{ \infty }} \ \
\Longrightarrow \ \ \int_{ -r}^{r } ( \hat{P })_{ \pm } \ \geq
\frac{\sqrt{2\pi}}{4}\ || P ||_{ \infty } .
\label{eqn:oscillation}$$ Here, we can take $C= 8^{3} / \pi $. This estimate not only tells us that $ \hat{P} $ will have a zero in the interval $ [-r,r] $, but also provides an effective estimate on how it oscillates in the interval.
For a fixed function $ \phi $, let $$\label{eqn:Endef}
E_n(\lambda):=\left\{ \sum_{k= -n}^{n } b_{k} \phi (x + \lambda k)
~:~ b_k\in {\bf R},~k=-n,\dots,-1,0,1,\dots,n \right\}.$$ The estimate in (\[eqn:oscillation\]) allows us to construct $ P_{ \lambda } \in E_n( \lambda ) $ for each $ \lambda > 0$ and for sufficiently large positive integers $n$ (depending on $ \lambda$) such that $ \| P_{ \lambda} '\|_{ \infty } > c
\| P_{ \lambda} \|_{ \infty } / \lambda $ with some absolute constant $c>0$. In particular, our construction proves sharpness of the above-mentioned inequality of Mhaskar [@mhaskar] for Gaussian networks.
Notations and preliminaries
===========================
For any integrable function $f$ on the real line, we write for its Fourier transform $$\hat{ f }( \omega ) \ = \ \frac{1}{ \sqrt{ 2 \pi}}
\int_{ {\bf R}} f(x) e^{ -i \omega x} \ dx \ .$$ Given a real number $x$, its positive and negative parts are $x_{+} = \max \{ x,0 \} $ and $ x_{-} = \max \{-x,0 \} $ respectively.
We will write $h$ for the Fejér kernel, that is $$h( x ) \ := \ \frac{1}{\sqrt{ 2 \pi } }
\left( \frac{ \sin x/2 }{ x/2 } \right)^{2} \ .$$ Its Fourier transform is given by $$\hat{ h } ( \omega ) \ = \
\max \{ 1 - | \omega |, \ 0 \} \ .$$
For the rest of the paper we fix an auxiliary function $H$. We could use any even, $ 2 \pi $-periodic, and e.g. twice continuously differentiable function\
$ H: {\bf R} \longrightarrow {\bf R} $, not identically one, such that $ H(x) = 1 $ if $ |x| \leq \pi/2$. The special constants and values in the following choice are not relevant, only some order is essential. Nevertheless, for definiteness and more explicit calculation we take e.g. $$H(x)= \left\{
\begin{array}{ll}
1, \qquad\qquad & \mbox{if} \ |x| \leq \pi /2; \\
\sin^2x, & \mbox{if } \pi/2 < |x| \leq \pi .
\end{array}
\right. \label{eqn:Hdef}$$ Then $H$ has the Fourier cosine series development $$H(x) \ = \ \sum_{k=0}^{\infty } a_{k} \cos kx$$ where $a_k$ are the Fourier cosine coefficients of $H$. Although precise values are not needed here, a calculation leads to $a_0=3/4$, $a_1=4/(3\pi)$, $a_2=\frac{-1}{4}$ and $$a_k= \frac{-4\sin\frac{k\pi}{2}}{\pi k(k^2-4)} =
\left\{
\begin{array}{ll}
0, & k \mbox{ even;} \\
\frac{-4}{\pi k(k^2-4)}, & k \equiv 1 \mbox{ mod } 4; \\
\frac{4}{\pi k(k^2-4)}, & k\equiv 3 \mbox{ mod } 4.
\end{array}
\right. ~~~~ \hbox{for}~k \geq 3, ~ k\in {\bf N}.
\label{eqn:akdef}$$ It is immediate that $|a_k| \leq k^{-2} $ for all $k \in {\bf N}$; moreover, a direct calculation yields $$\sum_{k=1}^{\infty} |a_k| = 1 +\frac{5}{3\pi} = 1.530516...<1.6
\qquad \textrm{and}\qquad \sum_{k=1}^{\infty} a_k^2 = \frac{9}{8}.
\label{eqn:akabssum}$$
Oscillation of Fourier transforms
=================================
\[l:oscillation\] Let $ P: {\bf R} \longrightarrow {\bf R} $ be bounded, differentiable, and integrable such that $ \hat{ P } $ is real-valued. Suppose $ P(0) = 0$ and let $$r\ > \ \frac{8^{3} \ ||P'||_{\infty }}{\pi\ || P||_{\infty}} \ ,
\label{eqn:lambda}$$ then $$\frac{4}{\sqrt{ 2 \pi}} \int_{ -r}^{r } ( \hat{P })_{\pm} \ \geq
\ || P ||_{ \infty }$$
[**Proof of Lemma \[l:oscillation\]**]{}: There is nothing to prove if $ || P' ||_{ \infty} = \infty $. Hence, we assume $ || P'
||_{ \infty} < \infty $. Fix $ r $ satisfying (\[eqn:lambda\]) and define $$f(x) \ = \ P \star h_{ r } (x) \ = \ \frac{1}{ \sqrt{ 2 \pi }}
\int_{ {\bf R}} P( x-t ) h_{ r } (t) dt$$ where $ h_{ r } (t) = r h ( r t) $. Since $ ( 2 \pi )^{ -1/2} \int_{ {\bf R}} h_{ r } =1 $, for any real number $x$, $$f(x) - P(x) = S(x) + L(x)
\label{eqn:split}$$ where $$S(x) = \frac{1}{ \sqrt{ 2 \pi }} \int_{ |t| < \delta } ( P(x-t ) - P(x) )
h_{ r } (t) \ dt \ ,$$ $$L(x) = \frac{1}{ \sqrt{ 2 \pi }} \int_{ |t| \geq \delta } ( P(x-t ) - P(x) )
h_{ r } (t) \ dt \$$ and $ \delta > 0 $ is chosen such that $ 8 \delta q = 1 $ with $ q = || P'||_{ \infty } / || P ||_{ \infty } $. Combining the inequalities $$| S(x)| \ \leq \ \delta \ || P' ||_{ \infty } \ = \
\frac{ || P ||_{ \infty }}{ 8}
\ \ \mbox{ and } \ \ | L(x) | \ \leq \
\frac{ 8 \ || P||_{ \infty } }{ \pi r \delta } \ < \
\frac{|| P||_{ \infty }}{ 8 }$$ with (\[eqn:split\]), we obtain for any real number $x$, $$|f(x) - P(x) | < || P||_{ \infty } / 4 \ .
\label{eqn:main}$$
Since $f$ and $ \hat{f} $ are both integrable, the inversion formula for the Fourier transform shows that $$\sqrt{ 2 \pi } || f ||_{ \infty } \ \leq \ \int_{ {\bf R}} | \hat{ f } |
\ = \ \int_{ {\bf R}} \left( \hat{ f } + 2 ( \hat{ f } )_{-} \right)
\ = \ \sqrt{ 2 \pi } f(0) + 2 \int_{ {\bf R}}( \hat{ f } )_{-} .$$ Applying (\[eqn:main\]) with $ x= 0$ and noting that $ P(0) = 0
$, we conclude that $$|| f ||_{ \infty } \ \leq \ \frac{1}{4}|| P ||_{ \infty } +
\frac{2}{ \sqrt{2 \pi } } \int_{ {\bf R}}( \hat{ f } )_{-} \ .$$ Making use once more of (\[eqn:main\]) and the last inequality gives $$|| P ||_{ \infty } \ \leq \ || f ||_{ \infty } + \frac{1}{4}|| P
||_{ \infty } \ \leq \ \frac{1}{2}|| P ||_{ \infty } + \frac{2}{
\sqrt{ 2 \pi } } \int_{ {\bf R}}( \hat{ f } )_{-}$$ and therefore $$|| P ||_{ \infty } \ \leq \
\frac{4}{\sqrt{ 2 \pi}} \int_{ {\bf R}}( \hat{ f } )_{-} \ .$$
Finally, we observe that $ \hat{f} ( \omega ) = \hat{P} ( \omega ) \hat{h} ( r^{-1} \omega )$, $ 0 \leq \hat{h} \leq 1 $ and $ \hat{h} = 0 $ outside $ [-1,1]$. These imply that $ ( \hat{ f } )_{-} = 0 $ outside $ [-r,r] $ and $ ( \hat{ f } )_{-} \leq ( \hat{ P})_{- } \ $. Therefore $$|| P ||_{ \infty } \ \leq \
\frac{4}{\sqrt{ 2 \pi}} \int_{ -r}^{r } ( \hat{ P } )_{-} \ .$$ A similar argument leads to the same inequality for $ ( \hat{ P } )_{+} \ $. $ \Box $
Construction of sums of translates with\
large oscillation
========================================
Let $ \phi : {\bf R} \longrightarrow {\bf R}$ be an even, continuous, integrable function such that $ \phi (0) =1 $. In addition, suppose that its Fourier transform $ \hat{ \phi } $ is nonnegative, integrable and analytic on $ {\bf R} $. Given $ \lambda > 0 $, then there exist a positive integer $n$ and $P\in E_n(\lambda)$, with $E_n(\lambda)$ defined in (\[eqn:Endef\]), such that $$\frac{|| P '||_{\infty }}{|| P ||_{\infty }} \ \geq
\frac{ C}{ \lambda} .$$ Here, we could take $ C = \pi^{2}/2^{10} $.
[**Proof:**]{} For each positive integer $n$ and for each real number $x$, we define $$\label{eqn:pndef}
P_{n}(x) = 2 A_{n} \phi (x) \ + \ \sum_{ k=1}^{n} a_{k} ( \phi ( x
+ \lambda k) + \phi ( x - \lambda k) )$$ and also $$P_{ \infty } (x) \ := \ \lim_{ n \rightarrow \infty} P_{n} (x) =2
A_{\infty}(\lambda)\phi (x) \ + \ \sum_{ k=1}^{\infty} a_{k} (
\phi ( x + \lambda k) + \phi ( x - \lambda k) ),
\label{eqn:pinfty}$$ where the coefficients $a_k$ are the Fourier cosine coefficients of $H$ in (\[eqn:akdef\]), and $$\label{eqn:Andef}
A_{n} := A_n(\lambda):=- \sum_{k=1}^{n} a_{k} \phi ( \lambda k),
\qquad A_{\infty} := A_\infty(\lambda):=- \sum_{k=1}^{\infty}
a_{k} \phi (\lambda k).$$
We start with showing that $ P_{ \infty } $ is not identically zero.
\[l:Pnotvanish\] Under the assumptions of Theorem 1, we have $ || P_{ \infty } ||_{
\infty } > 0 $.
[**Proof of lemma \[l:Pnotvanish\]:**]{} For each $ \omega \in
{\bf R} $ and $n\in{\bf N}$ we define $$T_{n} ( \omega ) \ := \ \sum_{k=1}^{n} a_{k} ( \cos ( k \lambda
\omega ) - \phi ( \lambda k ) ) \label{eqn:trig}$$ and also $$T_{ \infty }( \omega ) \ := \ \lim_{n \rightarrow \infty} T_{n} (
\omega ) = \sum_{k=1}^{\infty} a_{k} ( \cos ( k \lambda \omega ) -
\phi ( \lambda k ) ). \label{eqn:triginfty}$$ Thus, $ \hat{ P}_{ \infty } ( \omega ) \ = \hat{\phi}(\omega) 2 T_{
\infty }( \omega ) = 2 \hat{ \phi } ( \omega ) ( H( \lambda \omega ) - F(
\lambda ) ) $, where $$\label{eqn:Fdef}
F( \lambda) := \sum_{ k=0}^{\infty} a_{k} \phi(\lambda k ) =
a_0-A_{\infty}(\lambda)~$$ is a uniformly convergent sum of bounded functions of $\lambda$. By the Fourier inversion formula, $ P_{ \infty } \equiv 0 $ if and only if $ \hat{P}_{ \infty} \equiv 0 $. Thus, it suffices to show that for any given $ \lambda
> 0$, $ \hat{ \phi } ( \omega ) ( H( \lambda \omega ) - F( \lambda
) ) $ does not vanish identically.
Note that for any $ \lambda > 0$, $ H ( \lambda \omega) \neq 1 $ for $ \ \omega \in {\mathcal I} = ( \frac{ \pi }{ 2 \lambda } ,
\frac{ 3\pi }{ 2 \lambda } )
+ (2 \pi /\lambda) {\bf Z} $ while $ H ( \lambda \omega) = 1 $ for $ \ \omega \in {\mathcal J} = [ - \frac{ \pi }{ 2 \lambda } ,
\frac{ \pi }{ 2 \lambda } ]
+ (2 \pi /\lambda) {\bf Z} $. Therefore, if $ F( \lambda) =1 $, then $ F( \lambda ) \neq H(
\lambda \omega) $ for $ \omega \in {\mathcal I} $, while if $ F(
\lambda) \neq 1 $, then $ F( \lambda ) \neq H( \lambda \omega) $ for $ \omega \in {\mathcal J} $. In any case, $ H( \lambda \omega
) - F( \lambda ) \neq 0 $ for $ \omega $ in a union of non-empty open intervals. If $ \hat{P}_{ \infty} \equiv 0$, then $ \hat{
\phi } $ would have to be zero on these intervals, which is impossible since $ \hat{ \phi } $ is assumed to be analytic on $
{\bf R}$. This completes the proof of lemma 2. $ \Box $
To finish the proof of the theorem it suffices to show the next assertion.
\[claim:punchline\] If a positive integer $n$ is chosen such that $ 20 \sum_{ k > n } | a_{k}| < || P_{ \infty} ||_{ \infty} $, then $$\frac{ || P_{n} '||_{ \infty }}{ || P_{n} ||_{ \infty }} \ \geq
\ \frac{ \pi^{2} }{ 2^{10} \lambda }.$$
[**Proof of Lemma \[claim:punchline\]:**]{} Recall $\hat{P}_{n} (
\omega ) = 2 \hat{ \phi} ( \omega ) T_{n} ( \omega ) $ with $T_{n} $ in (\[eqn:trig\]). We also define $ \Delta_{n} ( \omega
) = T_{n} ( \omega ) - H ( \lambda \omega ) + F( \lambda) $ for $
\omega\in {\bf R}$, with $F(\lambda)$ in (\[eqn:Fdef\]).
Meanwhile, in view of the assumptions that $ \hat{ \phi } \geq 0 $ and $ \hat{ \phi } \in L^{1} $, the inversion formula for the Fourier transform shows that $ ||\phi||_{\infty} = \phi(0) = 1$. With this in mind, we obtain for every positive integer $n$ $$|| \Delta_{n} ||_{ \infty } \leq \ 2 \sum_{ k > n } | a_{k}|~,
\qquad \mbox{and} \qquad
||P_{\infty} - P_{n} ||_{ \infty } \leq \ 4 \sum_{k > n } |a_{k}|.
\label{eqn:error}$$
Suppose $ 0 < r \leq \pi / ( 2 \lambda ) $. Then $ H( \lambda \omega ) =1 $ for $ | \omega | \leq r$. Therefore, if $ F( \lambda ) \geq 1 $, then $$\int_{ -r}^{r } ( \hat{ P}_{n} )_{ +} \ = \ 2 \int_{ -r}^{r } \hat{ \phi
}~
( 1 - F( \lambda )+ \Delta_n )_{ +}
\ \leq \ 4 \sqrt{ 2 \pi } \sum_{ k > n } | a_{k}| .$$ Here, we’ve again made use of the conditions $ \hat{ \phi } \geq 0$ and $ \phi (0 ) =1 $. Similarly, if $ F( \lambda ) < 1 $, we also obtain $$\int_{ -r}^{r } ( \hat{ P}_{n} )_{- } \ \leq \ 4
\sqrt{ 2 \pi } \sum_{ k > n } | a_{k}| .$$ Thus, we’ve shown that if $ 0 < r \leq \pi / ( 2 \lambda ) $, then for each positive integer $n$, $$\min \left( \int_{ -r}^{r } ( \hat{ P}_{n} )_{- }, \int_{ -r}^{r }
( \hat{ P}_{n} )_{+ } \right)
\ \leq \ 4 \sqrt{ 2 \pi } \sum_{ k > n } | a_{k}|.
\label{eqn:summary}$$ On the other hand, lemma \[l:oscillation\] together with the second inequality in (\[eqn:error\]) asserts that if $r > (8^{3}
/ \pi) || P_{n}' ||_{ \infty } / || P_{n}||_{ \infty }$, then $$\label{eqn:lower-bound}
\frac{ 4}{ \sqrt{ 2 \pi} } \min \left( \int_{ -r}^{r } ( \hat{
P}_{n} )_{- }~, \int_{ -r}^{r } ( \hat{ P}_{n} )_{+ } \right)
\geq || P_{n}||_{ \infty} \geq || P_{\infty}||_{ \infty} - 4
\sum_{ k
> n } | a_{k}|.$$ Combining (\[eqn:summary\]) and (\[eqn:lower-bound\]) we conclude that if $ (8^{3} / \pi) || P_{n}'||_{ \infty} / || P_{n}
||_{ \infty} < \pi /(2 \lambda)$, then $ || P_{ \infty } ||_{
\infty }- 4\sum\limits_{ k > n } | a_{k}| \leq 16 \sum\limits_{ k
> n } | a_{k}|$ and therefore $ || P_{ \infty } ||_{ \infty } \leq 20 \sum\limits_{ k > n } | a_{k}| $. This proves the lemma which gives the conclusion of the theorem. $\Box$
Application to Gaussian networks
================================
Our goal in this section is to prove sharpness of an inequality of Mhaskar (mentioned in the introduction of this paper) for Gaussian networks. We shall apply Theorem 1 (in particular, lemma 1 in the proof) with $ \phi (x ) = \exp ( -x^{2} ) $. In this section $E_n(\lambda)$ is defined according to (\[eqn:Endef\]) with our above given Gaussian $\phi$.
The following theorem is the main result of this section.
Let $ n \in {\bf N} $ and $ \lambda \in (0,1) $ satisfy $$\label{eqn:Nzerocond}
n > N_0:= C_0 \lambda \exp\left( \frac{\pi^2}{2\lambda^2}
\right)\qquad \qquad \left( C_0:=\frac{1280}{3\pi}\right).$$ Then there exists $P\in E_n(\lambda)$, such that $$\frac{|| P '||_{ \infty }}{|| P ||_{ \infty }}
\geq \frac{ \pi^{2} }{ 2^{10} \lambda }.$$
[**Remark.**]{} Note $\log N_0 = O (1/\lambda^2)$, in complete agreement with the above mentioned result of H. N. Mhaskar. Thus the result proves sharpness of the result in [@mhaskar] for an arithmetic progression of shifts $x_k:=\lambda k$ with separation $1/m=\lambda$. $$$$ We retain the function $H$ from (\[eqn:Hdef\]) and its Fourier coefficients $a_k$ in (\[eqn:akdef\]) also in this section. With these Fourier coefficients $a_k$ and for each $ \lambda > 0 $ and $x \in {\bf R}$, $P_{\infty}(\lambda,x)$ will again be as in (\[eqn:pinfty\]) with $
A_{\infty} ( \lambda ) $ defined in (\[eqn:Andef\]). However, in contrast to the proof of Theorem 1, $ \lambda $ is no longer fixed.
As we are dealing with the Gaussian function $\phi(x):=\exp(-x^2)$, a number of properties are immediate.
First of all, the fact that $ \phi $ is even and decreasing on $[0, \infty )$ implies that for each $ \lambda > 0 $ and for any real number $x$, $$\ \sum_{ k \in {\bf Z}} \phi ( k \lambda - x )
\ \leq \ 1 + \frac{1}{\lambda } \int_{ {\bf R}} \phi = 1 +
\frac{\sqrt{\pi}}{\lambda}. \label{eqn:riemann}$$ Indeed, all values of $\phi ( k \lambda - x )$ can be replaced by the $\int$ over the interval of length $\lambda$ from $k \lambda - x$ towards $0$, except perhaps the function value at the (single, if $x\ne
\pm\lambda/2$) point which is closest to $0$ (and thus is estimated by 1).
Also, the Fourier transform of $\phi$ is given by $ \hat{ \phi} ( \omega )
= (1/\sqrt{ 2 }) \exp ( - \omega ^{2} / 4) $. Keeping only the term with maximal absolute value, we easily obtain $$\label{eqn:phihatest}
\sum_{l=-\infty}^{\infty} \left|\hat{\phi} \left( \frac{\omega +
2\pi l} {\lambda} \right) \right|^{2} \geq \hat{\phi}^{2}
\left(\frac{\pi}{\lambda}\right) \qquad\qquad \left( \forall
~\omega \in {\bf R}\right).$$
For the function (\[eqn:pinfty\]) we have $$|P_{\infty}(
\lambda, x)| \leq \frac{24}{1+x^2} \qquad \qquad \left( x\in {\bf
R}\right), \label{eqn:decay-of-p}$$ uniformly for all $\lambda \in (0,1)$. \[lemma:unidecay\]
[**Proof of Lemma \[lemma:unidecay\]:**]{} Using $\phi(\lambda k)\leq 1$ and (\[eqn:akabssum\]) we obtain $$|A_\infty(\lambda)| \leq \sum_{k=1}^\infty |a_k| \leq 1.6~.$$ As $\max\limits_{\bf R}(1+x^2)\phi(x)= \max\limits_{[0,\infty)}
(1+t)e^{-t}=1$, we get $$\label{Aestim}
|2A_\infty(\lambda)\phi(x)| \leq \frac{3.2}{1+x^2}.$$ It follows that we indeed have $$|P_{\infty}( \lambda, x)| \ \leq \frac{3.2}{1+x^2} +
\ \sum_{k \in {\bf Z}\setminus 0} |a_k| \phi ( x - \lambda k),
\label{eqn:decroiss}$$ where $ a_{k} = a_{-k} $ if $ k <0$. As $\|\phi\|_{\infty} = \phi(0)= 1$, in case $|x|\le 2$ this immediately leads to $|P_{\infty}( \lambda, x)|
\leq {3.2}/(1+x^2) + 3.2 < 20/(1+x^2)$, hence (\[eqn:decay-of-p\]).
Because the right hand side of (\[eqn:decroiss\]) is even, it remains to take $x>2$.
Now let $\mathcal A$ be the set of all nonzero integers $k$ such that $|x-\lambda k | < x /2 $. Observe that for $k\in\mathcal A~$, $\lambda |k|
\geq x/2 $ and thus $|k| \geq x/(2\lambda)$, which gives by $|a_k|\leq
1/k^2$, also $|a_k| \leq 4\lambda^2/x^2 \leq 5 \lambda^{2}/(1+x^2)$ for $x>2$. Therefore, taking into account also (\[eqn:riemann\]) and $x>2$, we are led to $$\sum_{ k \in {\mathcal A}} |a_k| \phi ( x - \lambda k) \leq \frac{5
\lambda^2}{1+x^2} \left( 1 + \frac{\sqrt{\pi}}{\lambda} \right) \leq
\frac{5\lambda^2+5 {\sqrt{\pi}}{\lambda}}{1+x^2} . \label{eqn:nearx}$$ On the other hand, in view of (\[eqn:akabssum\]) and $$\max\limits_{[2,\infty)} ( 1+x^{2} ) \phi(x/2)=
\max\limits_{[4,\infty)}(1+t)e^{-t/4}=5/e ,$$ we have $$\sum_{ k \not\in {\mathcal A} } |a_k|\phi ( x - \lambda k) \leq \
\phi\left(\frac{x}{2}\right) 2 \sum_{k=1}^{\infty} |a_k| \leq
\frac{10}{e(1+x^{2})} \sum_{k=1}^{\infty} |a_k| <
\frac{6}{1+x^2}. \label{eqn:farx}$$ Recalling $0<\lambda<1$ a combination of (\[eqn:decroiss\]), (\[eqn:nearx\]) and (\[eqn:farx\]) gives the result of the lemma. $ \Box$ $$$$
We shall also make use of an explicit lower bound for the $L^{2}$-norm of $ P_{ \infty } ( \lambda , \cdot ) $ in terms of the $ l^{2}$-norm of its coefficients. Actually, a more general phenomenon can be observed here.
Let $ \lambda > 0 $ be fixed and $c_k\in{\bf C}$ $(k\in{\bf Z})$ be arbitrary coefficients satisfying $\sum_{k\in{\bf Z}}
|c_k|^{2} < \infty$, i.e., $(c_k)\in\ell_2({\bf Z})$. Consider the function $f(\lambda,x):=\sum_{k=-\infty}^{\infty} c_k \phi(x-\lambda k)$. We then have $$\label{eqn:independence}
|| f (\lambda, \cdot ) ||^{2}_{2}
\ \geq \ \mu ( \lambda ) \sum_{ k =-\infty}^{ \infty} |c_{k}|^{2}$$ where $$\mu ( \lambda ) : = \ \frac{ 2 \pi }{ \lambda } \inf_{ \omega \in {\bf R}}
\sum_{ l \in {\bf Z}} \left| \hat{\phi} \left( \frac{\omega + 2 \pi
l}{\lambda } \right) \right|^{2}. \label{eqn:riesz-bound}$$ \[lemma:independence\]
[**Proof of Lemma \[lemma:independence\]:**]{} First of all, for a fixed $ \lambda > 0$, the series defining $ f:=f (\lambda , \cdot ) $ converges in $ L^{2} ( {\bf
R}) $. To see this, we consider its sequence $ f_{n} ( \lambda , x ) = \sum_{|k| \leq n} c_k \phi(x-\lambda k)$ of partial sums. The Fourier transform of $ f_{n}:=f_n (\lambda ,
\cdot ) $ is given by $ \hat{f}_{n} ( \lambda , t) = \hat{ \phi
}(t) \sum_{|k| \leq n } c_{k} e^{ - ik \lambda t}$. Applying Plancherel’s theorem to $ || f_{n}(\lambda , \cdot ) - f_{m}(\lambda , \cdot )||^{2}_{2} $ and writing the resulting integral as a sum of integrals over the intervals $ [ 2 \pi \lambda^{-1}l, 2 \pi \lambda^{-1}(l+1) ] , \
l \in {\bf Z} $, we obtain $$|| f_{n}(\lambda , \cdot ) - f_{m}(\lambda , \cdot ) ||^{2}_{2} \ = \
\frac{1}{ \lambda } \sum_{ l = - \infty}^{ \infty }
\int_{ 0}^{ 2 \pi } \left| \hat{ \phi }
\left( \frac{ \omega + 2 \pi l}{ \lambda } \right)
\sum_{ m < |k| \leq n } c_k e^{ ik \omega} \right|^{2}
d \omega$$ for $m < n $. The rapid decay of $ \hat{ \phi }$ assures the finiteness of $$M(\lambda) \ := \ \frac{2 \pi}{ \lambda } \
\sup\limits_{ \omega \in {\bf R}} \
\sum_{ l = - \infty }^{ \infty} \left| \hat{ \phi }
\left( \frac{ \omega + 2 \pi l}{ \lambda } \right)
\right|^{2}$$ and therefore by Parseval’s theorem, $$|| f_{n}(\lambda , \cdot ) - f_{m}(\lambda , \cdot ) ||^{2}_{2}
\ \leq \ M_{ \lambda } \sum\limits_{m< |k| \leq n }
| c_{k}|^{2} \ \longrightarrow 0$$ as $ n > m \longrightarrow \infty $. This proves convergence in $ L^{2} $ of the series defining $ f (\lambda , \cdot ) $.
A similar argument furnishes the conclusion of the lemma except that we take the infimum $\mu(\lambda)$ (as defined in (\[eqn:riesz-bound\])), instead of the supremum $M(\lambda)$ above. $ \Box$
[**Proof of Theorem 2:**]{} First of all, we estimate $ ||
P_{\infty}( \lambda, \cdot ) ||_{ \infty } $ from below by $ ||
P_{\infty}( \lambda, \cdot ) ||_{ 2 } $. In view of lemma \[lemma:unidecay\] we have $$|P_{\infty}( \lambda, x )| \leq \frac{C}{|x|} \qquad
(\textrm{with}~~C=12)\label{eqn:decay}$$ for all real numbers $x\ne 0$ and for each $\lambda >0$.
Now let the parameter $\sigma$ be chosen so that $$\sigma := \frac{4C^{2}}{|| P_{\infty}(\lambda, \cdot ) ||_{ 2 }^2} .$$ Note that $P_{\infty}$ does not vanish identically, hence $ \sigma >0$. We write $ || P_{\infty}( \lambda, \cdot ) ||_{ 2 }^{2} $ as a sum of integrals over $ [- \sigma , \sigma ] $ and over $ {\bf R} \setminus
[- \sigma , \sigma ]$. Estimating trivially in $[- \sigma, \sigma ]$ and applying (\[eqn:decay\]) to the second integral yields $$|| P_{\infty}( \lambda, \cdot ) ||_{ 2 }^{2} \leq
2 \sigma || P_{\infty}( \lambda, \cdot ) ||_{ \infty }^{2} +
2 C^{2} \sigma^{-1} .$$ Thus, a short calculation with the chosen value of $ \sigma$ leads for each $ \lambda> 0$ $$|| P_{\infty}( \lambda, \cdot ) ||_{ 2 }^{2} \ \leq \ 4C
|| P_{\infty}( \lambda, \cdot ) ||_{ \infty }.
\label{eqn:infty-two}$$
To evaluate $\|P_{\infty}(\lambda,\cdot)\|_2$ we note $P_{\infty}(\lambda,x)= \sum_{ k \in {\bf Z} } \alpha_{k} \phi(x-
k \lambda)$, where $\alpha_{k} = a_{|k|}$ if $ k \neq 0$, and $
\alpha_{0} = 2 A_{\infty}(\lambda)$, with $A_{\infty}(\lambda)$ defined in (\[eqn:Andef\]). For this function we clearly have $\sum_{k\in{\bf Z}} |\alpha_k|^2 \geq 2 \sum_{k=1}^{\infty}
|a_k|^2=9/4$ in view of (\[eqn:akabssum\]).
Meanwhile, we consider the function $ \mu(\lambda) $ defined in (\[eqn:riesz-bound\]). Recalling (\[eqn:phihatest\]) and the explicit form of $\widehat{\phi}$ provides for each $ \lambda > 0$ the estimate $$\mu( \lambda ) \geq \frac{\pi}{ \lambda } \exp \left( -
\frac{\pi^2}{2\lambda^2} \right).$$ Combining this with lemma \[lemma:independence\] and (\[eqn:infty-two\]) we obtain $$|| P_{\infty}( \lambda, \cdot ) ||_{ \infty } \geq \frac{\pi}{4C\lambda}
\exp \left( - \frac{\pi^2}{2\lambda^2} \right) \sum_{k=-\infty}^{\infty}
|\alpha_k|^2 = \frac{9\pi}{16C\lambda} \exp \left( -
\frac{\pi^2}{2\lambda^2} \right)~. \label{eqn:l2}$$
Now recalling $|a_k|\leq 1/k^2$ we obtain $ \sum_{ k>n} |a_{k}| < 1/n$ for each positive integer $n$. Recalling also $C=12$, this and (\[eqn:l2\]) yields that whenever (\[eqn:Nzerocond\]) holds, then $$20 \sum_{k>n} |a_{k}| < 20/n < 20/N_0 = \frac{3\pi }{64 \lambda}
\exp\left(-\frac{\pi^2}{2\lambda^2}\right) < || P_{\infty}(
\lambda, \cdot ) ||_{ \infty }.$$ Therefore, an application of lemma \[claim:punchline\] concludes the proof of Theorem 2. $ \Box $
[99]{} P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, NY, 1995. T. Erdélyi and P. Nevai, Lower bounds for derivatives of polynomials and Remez type inequalities, Trans.Amer.Math.Soc. [**349**]{} (1997), 4953-4972. T. Erdélyi, Bernstein type inequalities for linear combinations of shifted Gaussians, manuscript, www.math.tamu.edu/tamas.erdelyi/papers-online/list.html. T. Erdélyi, Inequalities for exponential sums via interpolation and Turán type reverse Markov inequalities, manuscript, www.math.tamu.edu/tamas.erdelyi/papers-online/list.html. J. Erőd, Bizonyos polinomok maximumának alsó korlátjáról, Mat. Fiz. Lapok, [**46**]{} (1939), 58-82 (in Hungarian). N. Levenberg and E. Poletsky, Reverse Markov inequalities, Ann. Acad. Fenn., [**27**]{} (2002), 173-182. G.G. Lorentz, “Approximation of Functions," Chelsea Publishing Company, New York, 1986. H.N. Mhaskar, A Markov-Bernstein inequality for Gaussian networks in “Trends and Applications in Constructive Approximation", ed. M.G. de Bruin, D.H. Mache, and J. Szabados, International Series of Numerical Mathematics Vol 1, Birkhauser Verlag Basel, 2005. Sz. Gy. Révész, Right order Turán-type converse Markov inequalities for convex domains on the plane, manuscript, http://arxiv. org/abs/math.CA/0504416. P. Turán, Über die Ableitung von Polynomen, Compositio Math. [**7**]{} (1939), 89-95. S.P. Zhou, Some remarks on Turán’s inequality III: The completion, Anal. Math. [**21**]{} (1995), 313-318.
[^1]: The second author acknowledges the support of the Natural Sciences Research Institute of the University of the Philippines.
[^2]: The third author was supported in part through the Hungarian-French Scientific and Technological Governmental Cooperation, Project \# F-10/04, the Hungarian-Spanish Scientific and Technological Governmental Cooperation, Project \# E-38/04 and by the Hungarian National Foundation for Scientific Research, Project \#s T-049301, T-049693 and K-61908.
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---
abstract: 'We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different types. We also study properties of the augmentation representation.'
author:
- Honghao Gao
title: Augmentations and link group representations
---
Introduction
============
A link is a disjoint union of simple closed curves. A link invariant is an algebraic construction associated to links which is well-defined within each isotopy class. Research on link invariants serves the goal of not only distinguishing links, but also understanding fundamental properties of links and related subjects.
Link invariants appear in various guises. Considering the complement space of a link and taking its fundamental group, we get a group as a link invariant which is known as the link group. In a more complicated form, another algebraic construction we study in this paper is the framed cord algebra, which is a non-commutative algebra generated by paths beginning and ending on the framing longitudes of the link.
They are powerful invariants. In the case of knots, both invariants in their further enhanced forms become complete invariants [@W; @ENS], meaning that distinct isotopy classes result in non-isomorphic invariants. However, robust invariants can be impractical to distinguish knots or links, because sometimes it is difficult to show that two groups or two non-commutative algebras are not isomorphic.
A possible trade-off is to study more computable invariants, such as representations of the group or the algebra. An augmentation is a rank one representation of the framed cord algebra. We want to understand augmentations in terms of representations of the link group, since a group appears simpler than a non-commutative algebra. We approach this goal by constructing the *augmentation representation*.
Let $K$ be an oriented link with its Seifert framing. Let $\textrm{Cord}(K)$ be the framed cord algebra and $\pi_K$ be the link group.
(Theorem-Definition \[MainConstruction\]) Let $\epsilon: \textrm{Cord}(K)\rightarrow k$ be an augmentation of the framed cord algebra. By writing $K$ as a braid closure, we construct a representation of the link group $$\rho_\epsilon: \pi_K\rightarrow GL(V_\epsilon).$$
(Theorem \[MainThm\]) Up to isomorphism, $(\rho_\epsilon,V_\epsilon)$ is well-defined for the augmentation $\epsilon$. In particular, it does not depend on the choice of the braid in the construction.
The slogan of the construction is “action by interpolation”. Placing the link as the closure of a braid, we can select a set of standard cords and arrange their augmented values into a square matrix. The column vectors of the matrix span a vector space underlining the augmentation representation. A based loop acts on a matrix entry by interpolating the standard cord with the based loop, see Figure \[Fig:Interpolation\].
In the next part of the paper, we look into properties of the augmentation representation.
Let $K$ be an oriented link with its Seifert framing, and $\epsilon: \textrm{Cord}(K)\rightarrow k$ an augmentation of the framed cord algebra. The augmentation representation $(\rho_\epsilon,V_\epsilon)$ satisfy the following properties.
- (Proposition \[propSimpleness\]) Microlocal simpleness.
For any meridian $m$, there is a subspace $W\subset V_\epsilon$ such that $\rho_\epsilon(m)|_W = \textrm{id}_{W}$.
- (Proposition \[propVanishing\]) Vanishing.
Suppose $K'\subset K$ is a sublink. If $\epsilon(\gamma) = 0$ for either (1) every framed cord $\gamma$ starting from $K'$, or (2) every framed cord $\gamma$ ending on $K'$, then $\rho(m') = \textrm{id}_{V_\epsilon}$ for any meridian $m'$ of $K'$.
- (Proposition \[separability\]) Separability.
Suppose $K = K_1\sqcup K_2$ is the union of two sublinks. If $\epsilon(\gamma) = 0$ for all mixed cords between $K_1$ and $K_2$, then $(\rho_\epsilon, V_\epsilon)$ is a direct sum of two representation $(\rho_1, V_1)$ and $(\rho_2, V_2)$, where each $(\rho_i,V_i)$ is a representation of $\pi_{K_i}$.
These properties can be reinterpreted in terms of microlocal sheaf theory. In microlocal theory, one considers the “micro-support” in the cotangent bundle, generalizing the usual notion of the support of a sheaf on the base manifold. The augmentation representation is defined over the link group, which is equivalent to a locally constant sheaf on the link complement. Taking the underived push-forward, we obtain a sheaf ${{\mathcal{F}}}$ microsupported within the conormal bundle of the link.
The three properties of $(\rho_\epsilon,V_\epsilon)$ can be rephrased in terms of ${{\mathcal{F}}}$. The first property, $(\rho_\epsilon,V_\epsilon)$ being microlocally simple, is equivalent to say ${{\mathcal{F}}}$ is a simple sheaf along its micro-support (in the convention of [@KS]), or microlocal rank $1$ along its micro-support (in the convention of [@STZ]). The vanishing property states a sufficient condition when ${{\mathcal{F}}}$ is microsupported outside the conormal of the sublink $K'$. The separability states a sufficient condition when ${{\mathcal{F}}}$ splits into two subsheaves, each microsupported along a sublink.
The results of augmentation representations outreach to several directions. First off, the construction explicitly connects rank $1$ representations of the framed cord algebra to higher rank representations of the link group. The framed cord algebra can be thought of as being generated by a subcategory of the fundamental groupoid of the link complement. One may guess from functoriality that representations should be pulled back in some sense. Our construction is a concrete way to realizes this idea.
A character variety is a moduli space of representations of a finitely generated group. The character variety of the link group is the space of flat bundles on the link complement, which plays an important role in knot theory and three manifolds. The $SL(2)$-character variety has been intensively studied. We mention [@CS; @KM] and the $A$-polynomial discussed in the next paragraph. The $SL(n)$-character variety in higher ranks becomes increasingly complicated [@AH; @GTZ; @GW; @HMP; @MP]. Our results show that the “full augmentation variety”, which is the moduli space of augmentations, cuts off a closed subvariety in the link group character variety. It suggests a direction of future research to understand the subvariety characterized by microlocally simple representations.
Introduced by Copper et al. [@CCGLS], the $A$-polynomial of a knot defines a complex plane curves (which we call the “$A$-variety” for now) which parametrizes the $SL(2)$-character variety projected to a torus determined by the peripheral subgroup. It can be used to detect knots [@DG; @BZ; @NiZh]. The $A$-polynomial is closely related to the augmentation polynomial, whose vanishing locus is the moduli space of augmentations projected to the same torus, or the “augmentation variety”. When $K$ is a knot, both polynomials have two variables, denoted as $A_{K}(\lambda,\mu)$ and $Aug_K(\lambda,\mu)$. Ng proved that $A_{K}(\lambda,\mu)$ divides $Aug_K(\lambda,\mu^2)$ [@Ng3]. It is conjectured in [@AENV] that the augmentation polynomial as a generalization of the $A$-polynomial produces a new notion of mirror symmetry. To see the relation between two polynomials using our result, augmentation representations are microlocally simple, and a generic rank $2$ microlocally simple representation is in one-to-one correspondence with $SL(2)$-representations. When $K$ is a link, we no long have polynomials since the ideals of vanishing function are not principal, but a similar result holds for the same reason — the $A$-variety is a closed subvariety of the augmentation variety.
Finally the augmentation representation builds up to the correspondence between augmentations and sheaves for links. The correspondence was proven for Legendrian links [@NRSSZ] and connected Legendrain surfaces defined from knot conormals [@Ga2] or cubic graphs (by Sackel in appendix of [@CM]). For a brief motivation, augmentations of a framed cord algebra correspond to augmentations of the conormal tori dga [@Ng3], whose geometric counterparts are Lagrangian filling in the sense of SFT [@El; @EGH]. Fillings are sheaves through microlocalization [@Na; @NZ], and further determine link group representations via the Radon transform [@Ga1]. For knots, the correspondence in both ways are constructed, see [@Ga2]. A technical reason why the construction is more complicated in the case of links than knots is that the framed cord algebra of a knot is generated by elements in the knot group, but the similar statement does not hold for links. In this paper, we focus on the geometric origin of the theory and construct the augmentation representation. As explain earlier in the slogan and Figure \[Fig:Interpolation\], the action of the link group underlines a geometric meaning, namely based loops acts by interpolating the framed cords. This construction makes a direct connection from augmentations of a framed cord algebra to simple sheaves, without the lengthy detour in the motivation story.
This paper is organized in a simple fashion. We construct the augmentation representation is Section \[Sec:AugRep\] and study its properties in Section \[Sec:Properties\].
**Acknowledgements.** We thank Lenhard Ng, Stéphane Guillermou and Eric Zaslow for helpful conversations. This work is supported by ANR-15-CE40-0007 “MICROLOCAL”.
Augmentation representation {#Sec:AugRep}
===========================
The goal of this section is to construct the augmentation representation. It is a link group representation associated to an augmentation. We will first introduce some preliminary concepts, including the framed cord algebra and the augmentation associated to the framed cord algebra, then we construct the augmentation representation.
We fix some conventions. Let $k$ be a commutative field. It is the ground field where we will define augmentations and group representations. Throughout the paper, we set $X= {{\mathbb{R}}}^3$ or $S^3$. Let $K$ be an oriented $r$-component link in $X$. We label the components as $$K = K_1\sqcup K_2\sqcup \dotsb \sqcup K_r.$$
Suppose $p:[0,1]\rightarrow X$ is a path or a loop, we write $p^{-1}$ for the reversed path: $$p^{-1}(t) = p(1-t).$$ We denote by $p_1\cdot p_2$ the concatenation of two composable paths. Namely if $p_1(1) = p_2(0)$, we define $$p_1\cdot p_2(t) =
\begin{cases}
p_1(2t), & 0\leq t\leq 1/2, \\
p_2(2t-1), & 1/2\leq t\leq 1.
\end{cases}$$ We work with paths up to homotopy, hence the concatenation induces an associative product.
Braids {#Sec:Braid}
------
To construct an augmentation representation, we need to represent the link as the closure of a braid. A braid can be expressed as an element in Artin’s braid group of $n$ strands: $$Br_n = {{\langle}}\, \sigma_1^{\pm 1},\dotsb, \sigma_{n-1}^{\pm 1} \,|\, \sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}, \textrm{ and } \sigma_j\sigma_k = \sigma_k\sigma_j \textrm{ for } |j-k|\geq 2\,{{\rangle}}.$$
Geometrically, a braid is a collection of strands in a solid cylinder, with endpoints fixed on boundary disks. Braids can be realized via the mapping class group of a configuration space. Let $D$ be an oriented disk with sufficiently large radius, equipped with two-tuple coordinates $(-,-)$. Let $y_1 = (1,0),\dotsb, y_n = (n,0)$ be $n$ marked points, and $D^\circ \subset D$ be the $n$-punctured disk with marked points removed. The braid group $Br_n$ is isomorphic to the mapping class group $\textrm{MCG}(D^\circ) := \pi_0(\textrm{Diff}\,^+(D^\circ))$, where $\textrm{Diff}\,^+(D^\circ)$ is the topological group of orientation preserving diffeomorphisms of $D^\circ$. For each $[h]\in \textrm{MCG}(D^\circ)$, we can extend $h$ to a homeomorphism $\tilde h: D\rightarrow D$. Let $H: D\times [0,1]\rightarrow D$ be a $C^0$ isotopy between $\textrm{id}_D$ and $h$, namely $H(-,0) = \textrm{id}_D$, $H(-,1) = h$, and $H(-,t): D\rightarrow D$ is a homeomorphism for each $t\in [0,1]$. Then $H^{-1}(\{y_1,\dotsb,y_n\})\subset D\times [0,1]$ is the braid associated to $[h]\in \textrm{MCG}(D^\circ) \cong Br_n$.
\[Whiteheadbraid\] In Figure \[Fig:Whiteheadbraid\], we plot a $3$-strand braid whose braid word is $$\sigma_1^2\sigma_2^2\sigma_1^{-1}\sigma_2^{-2}.$$ The closure of the braid is a two-component link, known as the Whitehead link.
The *configuration disk* $D$ with $n$ marked points will be extensively used in the upcoming constructions. In practice, one can plot the figure in a more symmetric fashion. We keep the coordinates for easier description.
Given a braid $B \in D\times [0,1]$, one can close the solid torus by identifying $D\times \{0\}$ and $D\times \{1\}$. Since the marked points are fixed on the boundaries, the braid can be closed to a one dimensional compact submanifold ${{\langle}}B {{\rangle}}$ in the solid torus. If we take a sectional disk, in particular which we glued, we obtain the configuration disk where the marked points equal to $D\cap {{\langle}}B {{\rangle}}$. If the solid torus is further embedded in $X = {{\mathbb{R}}}^3$ or $S^3$ as the tubular neighborhood of a unknot, we obtain a link $K = {{\langle}}B {{\rangle}}\subset X$. A braid carries a natural orientation (induced from the orientation of $[0,1]$), and hence the braid closure is an orientated link. Alexander’s theorem asserts that every oriented link can be expressed (not in a unique way) as a braid closure. See [@Al] for the original construction, or [@Ya; @Vo] for the improved Yamada-Vogel algorithm.
Suppose an $r$-component link $K = K_1\sqcup \dotsb \sqcup K_r$ is the closure of an $n$-strand braid $B$. The strands in $B$ can be indexed by the first coordinate in the transverse disk $D$. We define the *component function* to be a map $$\label{IndexFun}
\{ - \}: \{1,\dotsb, n\}\rightarrow \{1,\dotsb, r\},$$ such that the strand $t$ of the braid belongs to the component $K_{\{t\}}$.
The strand index admits a natural linear ordering by the $x$-coordinate of the marked points in the configuration disk. A partition of a linearly ordered set is *ordered* if for any two parts $P_1,P_2$ of the partition, one has either $$a< b \textrm{ for all } a\in P_1, b\in P_2, \quad \textrm{or} \quad a> b \textrm{ for all } a\in P_1, b\in P_2.$$
\[sortingLem\] If $K$ is the closure of an $n$-strand braid, then it can be represented by an $n$-strand braid which admits an ordered partition where strands lie in the same part if and only if they belong to the same component of $K$.
Suppose $K = {{\langle}}B{{\rangle}}$ for some $n$-strand braid $B$. For any half twist $\sigma_i$, there is ${{\langle}}B {{\rangle}}= {{\langle}}\sigma_i \sigma_i^{-1} B {{\rangle}}= {{\langle}}\sigma_i^{-1} B \sigma_i{{\rangle}}$. The conjugated braid $\sigma_i^{-1} B \sigma_i$ is also an $n$-strand braid, with $i$-th and $(i+1)$-th strands switched. We can conjugate the braid finitely many times and obtain the desired braid.
Suppose an $m$-component link $K = K_1\sqcup \dotsb \sqcup K_m$ is the closure of an $n$ strand braid $B$. By Lemma \[sortingLem\], we can assume there exist integers $$0 = n_0 < n_1 < \dots < n_{m-1}< n_m = n,$$ such that the closure of the strands $\{n_{i-1}+1, n_{i-1}+2, \dotsb, n_i \}$ is the component $K_i$ of $K$. In other words, we can assume the component function is non-decreasing.
Link group
----------
Let $X = {{\mathbb{R}}}^3$ or $S^3$. Let $K \subset X$ be an oriented $r$-component link, $K = K_1\sqcup \dotsb \sqcup K_r$. The link group $\pi_K$ is the fundamental group of the link complement, i.e. $\pi_K = \pi_1(X\setminus K)$. The link group has the following properties:
- It is finitely generated by meridians, and finitely presented.
- There are $r$ conjugacy classes, labelled by components of the link.
These properties follow easily from the Wirtinger presentation of the link group, (for example, see [@Ro]). We recall the construction here. Thinking of an oriented link by it two-dimensional diagram with under-crossings, in a generic position, the diagram has finite arcs and finite under-crossings. We take the base point of the fundamental group far away from the plane. Each arc corresponding to a loop in the fundamental group, which travels from the base point to the plane, wraps around the arc and then travels back. After fixing a convention, the orientation of the loop is determined by the orientation of the link. Note this loop is a meridian, which is by definition the boundary of a disk intersecting the link transversely at a point. Finally, each under-crossing imposes a conjugation relation among the meridian generators, giving the Wirtinger presentation.
When the link $K$ is the the closure of an $n$-strand braid, the link group $\pi_K$ can be necessarily generated by $n$ meridians, (though $n$ may not be the minimum number of meridian generators). To see this, we scan the braid diagram from left to right. The braid is given by a word of half twists. At the beginning, each of the $n$ strands determines a meridian, which we denote $m_t$ for $1\leq t\leq n$. Each half twist introduces an under-crossing, and therefore a new meridian in the Wirtinger presentation. The new meridian can be expressed as a word of previous meridians. Iterating the procedure, we see that the set of meridians $\{m_t\}_{1\leq t\leq n}$ generates the whole link group. In the rest of the paper, we will call $$\{m_t\}_{1\leq t\leq n}$$ the *generating set of meridians* of an $n$-strand braid closure.
Recall the configuration disk $D$ with $n$ marked points. If the base point of the link group is $(0, -\delta)\in D\subset X$, where $\delta$ is a small positive real number, then the meridian generators $\{m_t\}_{1\leq t\leq n}$ can be plotted on the configuration disk. Namely, $m_t$ is the loop in $D$ wrapping around the marked point $y_t = (t,0)$.
To be compatible with the convention in [@CELN], we fix the orientation of the knot in the configuration disk pointing inward to the paper, and the meridian generators wrap clockwisely around marked points. See Figure \[Fig:ConfigurationDisk\] for an example.
The planar diagram of the Whitehead link is plotted in Figure \[Fig:Whiteheadlink\].
There are five strands, giving five meridian generators in the Wirtinger presentation of the link group. Each crossing in the diagram gives a conjugation relation. Therefore the link group is isomorphic to $$\pi_K = {{\langle}}m_1, m_2,m_3, m_4,m _5 {{\rangle}}/ \sim,$$ modulo relations $$\begin{aligned}
m_1\cdot m_5 &= m_4\cdot m_1,\quad m_3\cdot m_5 = m_5\cdot m_2,\quad m_1\cdot m_3 = m_3\cdot m_2, \\
&\quad m_3\cdot m_4 = m_4\cdot m_1,\quad m_4\cdot m_2 = m_2\cdot m_5.\end{aligned}$$
We have seen that the Whitehead link can be expressed as the closure of a $3$-strand braid, as in Example \[Whiteheadbraid\]. Therefore the link group can be generated by three meridians instead of five. As for the Wirtinger presentation we just computed, we can write $m_3,m_5$ in terms of $m_1,m_2, m_4$, reducing the number of meridian generators to three.
Framed cord algebra
-------------------
The cord algebra first appeared in [@Ng1; @Ng2]. The framed version was introduced in [@Ng3], which models the degree zero knot contact homology [@EENS].
Suppose $K\subset X$ is an oriented knot, and $n(K)$ is a small tubular neighborhood of $K$. A *framing* of $K$ is a push off the knot to the boundary of the tubular neighborhood. In other words, a curve $\ell\subset {{\partial}}(n(K))$ whose homology class in $H_1(n(K))$ agrees with $[K]\in H_1(n(K))$. A Seifert surface is an oriented surface $S$ with ${{\partial}}S = K$. The *Seifert framing* is $\ell = S\cap n(K)$, which is up to homotopy independent from the choice of the Seifert surface. The Seifert framing has zero linking number with $K$.
A framing of a link $K = K_1\sqcup \dotsb \sqcup K_r$ is a choice of a framing $\ell_i$ for each component $K_i$. We decorate each $\ell_i$ with a marked point $\ast_i\in \ell_i$. We write $\ast := \{\ast_1,\dotsb, \ast_r\}$, and $$L = \ell_1\sqcup \dotsb \sqcup \ell_r.$$ We write $(K,L)$ for a framed link, namely an oriented link with a choice of a framing.
Upcoming, we define the framed cord algebra of a framed link, which is a mild generalization of the definition for knots as in [[@CELN Definition 2.5]]{}.
Suppose $(K, L)\subset X$ is a framed link.
A framed cord of $K$ is a continuous map $c: [0,1]\rightarrow X\setminus K$ such that $c(0), c(1) \in L \setminus \ast$. Two framed cords are homotopic if they are homotopic through framed cords. We write $[c]$ the homotopy class of cord $c$.
We now construct a noncommutative unital ring ${{\mathcal{A}}}$ as follows: as a ring, ${{\mathcal{A}}}$ is freely generated by homotopy classes of framed cords and extra generators $\lambda_i^{\pm 1}, \mu_i^{\pm 1}$, $1\leq i\leq r$, modulo the relations $$\lambda_i \cdot \lambda_i^{-1} = \lambda_i^{-1}\cdot \lambda_i = \mu_i\cdot \mu_i^{-1} = \mu_i^{-1}\cdot \mu_i = 1,\quad \textrm{ for } 1\leq i\leq r,$$ and $$\begin{cases}
\lambda_i \cdot \mu_j = \mu_j\cdot \lambda_i \\
\lambda_i \cdot \lambda_j = \lambda_j\cdot \lambda_i \quad \textrm{ for } 1\leq i,j\leq r.\\
\mu_i \cdot \mu_j = \mu_j\cdot \mu_i
\end{cases}$$ Thus ${{\mathcal{A}}}$ is generated as a ${{\mathbb{Z}}}$-module by noncommutative words in homotopy classes of cords and powers of $\lambda_i$ and $\mu_j$. The powers of the $\lambda_i$ and $\mu_i$ commute with each other, but do not with any cords.
The framed cord algebra is the quotient ring $$\textrm{Cord}(K, L) = {{\mathcal{A}}}/{{\mathcal{I}}},$$ where ${{\mathcal{I}}}$ is the two-sided ideal of ${{\mathcal{A}}}$ generated by the following relations:
- (Normalization)
\[Fig:Normalization\]
- (Meridian)
\[Fig:Meridian\]
- (Longitude)
\[Fig:Longitude\]
- (Skein relations)
\[Fig:Skeinrelation\]
The definition of a framed cord algebra, at first glance, depends on the choices of decorations $\ast_i$ on framings. Different choices of decorations give isomorphic algebras. Consider the following argument. Since the generators are framed cords up to cord homotopy, we can choose a base point on each framing and homotope the endpoints of each cord to the base point. For example, we can choose base points to be a positive push off of marked points $\ast_i$ along each component of the framing. For two different sets of marked points, one set of marked points can be homotoped to the other set along the framing. Therefore we can use the same homotopy to move base points along the framing, underlining an isomorphism of the framed cord algebra.
\[rmkframing\] For different choices of framings $L_1,L_2$, the resulting framed cord algebras are isomorphic as ${{\mathbb{Z}}}$-algebras. We can construct $$\textrm{Cord}(K,L_1) \cong \textrm{Cord}(K,L_2),$$ by keeping $\mu_i$ and framed cords, and sending $\lambda_i$ to $\lambda_i\cdot \mu_i^{\textrm{lk}_i(K,L_2)-\textrm{lk}_i(K,L_1)}$. Here $\textrm{lk}_i(K,L)$ is the linking number between $K_i$ and its framing.
If $L$ is the Seifert framing, we simply write the framed cord algebra as $\textrm{Cord}(K)$.
If we take the quotient of $\textrm{Cord}(K,L)$ where $\lambda_i,\mu_i$ commute with everything, the the result $\textrm{Cord}^c(K,L)$ is a ${{\mathbb{Z}}}[\lambda^{\pm1}_1,\mu^{\pm1}_1,\dotsb, \lambda^{\pm1}_r, \mu^{\pm1}_r]$-algebra. Following [@Ng3], $\textrm{Cord}^c(K,L)$ is isomorphic to the degree $0$ homology of a differential ${{\mathbb{Z}}}[\lambda^{\pm1}_1,\mu^{\pm1}_1,\dotsb, \lambda^{\pm1}_r, \mu^{\pm1}_r]$-algebra of the conormal tori of the link.
For different framings $L_1,L_2$, there is also the isomorphism $\textrm{Cord}^c(K,L_1) \cong \textrm{Cord}^c(K,L_2)$. Note it is an isomorphism of ${{\mathbb{Z}}}$-algebras, but not of ${{\mathbb{Z}}}[\lambda^{\pm1}_1,\mu^{\pm1}_1,\dotsb, \lambda^{\pm1}_r, \mu^{\pm1}_r]$-algebras.
\[3fcas\] There are various versions of the (framed) cord algebra in literature ${{\mathcal{C}}}_K$ [@Ng3], $\textrm{Cord}^c(K)$ [@CELN], and ${{\mathcal{P}}}_K$ [@Ng4]. Each cord algebra is a quotient ring of the free non-commutative algebra generated by some version of cords over the same coefficient ring ${{\mathbb{Z}}}[\lambda^{\pm1}_1,\mu^{\pm1}_1,\dotsb, \lambda^{\pm1}_r, \mu^{\pm1}_r]$. In each version, the free algebra quotients out four relations: normalizations, meridian relations, longitude relations, and skein relations. We shall compare their definitions with a focus on skein relations.
1. [@Ng3] ${{\mathcal{C}}}_K$ is defined for a link $K$. Choose a marked point $\ast_i\in K_i$ for each $1\leq i\leq r$. A cord is a continue map $c: [0,1]\rightarrow X\setminus K$, with $c(0), c(1)\in K\setminus \{\ast_1\dotsb, \ast_r\}$. The skein relations are
\[Fig:SkeinrelationCKrmk\]
2. [@CELN] $\textrm{Cord}^c(K,L)$ is defined for a framed link $(K,L)$. Choose a maked point $\ast_i\in \ell_i$ for each $1\leq i\leq r$. A cord is a continue map $c: [0,1]\rightarrow X\setminus K$, with $c(0), c(1)\in L\setminus \{\ast_1\dotsb, \ast_r\}$. The skein relations are
\[Fig:Skeinrelationrmk\]
3. [@Ng4] ${{\mathcal{P}}}_K$ is defined for a knot $K$, namely $r =1$. A cord $[\gamma]\in {{\mathcal{P}}}_K$ is represented by a based loop $\gamma\in \pi_K$. In particular, we forget about the group structure in $\pi_K$ and consider it as a set, then take the set elements as free generators. Let $m$ be the meridian in the peripheral subgroup of $\pi_K$. The skein relations are given by $$[\gamma_1\cdot \gamma_2] = [\gamma_1\cdot m \cdot \gamma_2] + [\gamma_1][\gamma_2].$$ We remind the reader again that the ${{\mathcal{P}}}_K$ version of the framed cord algebra is only defined for knots, but not links.
If $L$ is the Seifert framing, then $(1) = (2)$. If in addition that $K$ is a single component knot, then $(1) = (2) = (3)$. Namely, there are ${{\mathbb{Z}}}[\lambda^{\pm1}_1,\mu^{\pm1}_1,\dotsb, \lambda^{\pm1}_r, \mu^{\pm1}_r]$-algebra isomorphisms $${{\mathcal{C}}}_K \cong \textrm{Cord}^c(K) \cong {{\mathcal{P}}}_K.$$ From ${{\mathcal{C}}}_K$ to $\textrm{Cord}^c(K)$, one can push the end points of a cord off the link to the framing. There are different choices of the push off which leads to ambiguity. One can fix this issue by multiply copies of $\mu_{i}$ depending on the linking number. From $\textrm{Cord}^c(K)$ to ${{\mathcal{P}}}_K$, one can choose a base point on each framing, then move the points of each framed cord to the base point along the oriented of the framing.
Cords in a framed cord algebra can be classified into pure and mixed cords depending on the points where the cord starts and ends.
Let $K$ be a link.
Suppose $K'\subset K$ is a sublink. A cord is a *pure cord* of $K'$ if it starts and ends on the framing of $K'$.
Suppose $K_1,K_2\subset K$ are two disjoint sublinks. A cord is a *mixed cord* between $K_1$ and $K_2$ if either (1) it starts on the framing of $K_1$ and ends on the framing of $K_2$, or (2) it starts on the framing of $K_2$ and ends on the framing of $K_1$.
If we simply say a pure cord or a mixed cord without decorations, it means the underlying sublink is a knot.
Recall the configuration disk $D$ with $n$-marked points $\ast$. It can be regarded as a locally closed submanifold in $X$ with $\ast = D\cap K$. We can perturb the framing such that $D\cap L = \{x_1, x_2,\dotsb, x_n \} = \{(1, -\delta), (2, -\delta),\dotsb, (n, -\delta)\}$, where $\delta$ is a small positive number. Recall we had a marked point $\ast_i$ on each $\ell_i$, where $1\leq i\leq r$. We assume that $\{\ast_i\}_{1\leq i\leq r}$ and $\{x_j\}_{1\leq j\leq n}$ are all distinct.
The orientation of the link $K$ induces an orientation of the framing $L$. After a suitable perturbation, the framing $\ell_{\{i\}}$ can be viewed as a (framed) cord, with the choice of end points $x_i$, denoted as $[\ell_{\{i\}}^{(i)}]$. We call it a *framing cord*.
Recall that we set the based point of $\pi_K$ at $x_0 = (0, -\delta)$. Define the *capping path* $p_i$ to be the linear map from $x_0$ to $x_i$, for $i=1,\dotsb, n$.
If we move the based point of $\pi_K$ to one of $x_i$, a meridian generator $m_t$ can also be reviewed as a framed cord, which we term as a *meridian cord*. A meridian based at different longitudes are different framed cords in $\textrm{Cord}(K)$. Hence it is necessary to remember the base point. Let $m_t^{(i)}$ be the meridian cord based at $x_i$ wrapping around $y_t$. Base point change subjects to the following relation: $$m_t = p_i \cdot m_t^{(i)}\cdot p_i^{-1}.$$
Define the *standard cord* to be $\gamma_{ij} := p_i^{-1} \cdot p_j$, for $i,j \in\{1, \dotsb, n\}$.
Standard cords satisfy the following relations:
- $\gamma_{ij}\cdot \gamma_{ji} = \gamma_{ii} = e_{\{i\}}$, where $e_{\{i\}}$ is the trivial cord on $\ell_{\{i\}}$,
- $\gamma_{ij}\cdot \gamma_{jk} = \gamma_{ik}$.
In terms of standard cords, conjugations of meridian cords satisfy $$m_t^{(i)} = \gamma_{ij}\cdot m_t^{(j)} \cdot \gamma_{ji}.$$
The relations of framed cords on the configuration disk can be written in the following way. Let $c_{ij}$ denote a framed cord from $x_i$ to $x_j$.
- Normalization $$[\gamma_{ii}] = 1 -\mu_{\{i\}},$$
- Meridian $$[m_i^{(i)}\cdot c_{ij}] = \mu_{\{i\}}[c_{ij}],\quad [c_{ij} \cdot m_j^{(j)}] = [c_{ij}]\mu_{\{j\}},$$
- Longitude $$[\ell^i_{\{i\}}\cdot c_{ij}] = \lambda_{\{i\}} [c_{ij}], \quad [c_{ij}\cdot \ell^j_{\{j\}}] = [c_{ij}]\lambda_{\{j\}},$$
- Skein relations $$\label{ConfDsr}
[c_{it} \cdot c_{tj}] = [c_{it} \cdot m_t^{(t)}\cdot c_{tj}] + [c_{it}][c_{tj}].$$
The longitude relation is not really visible in the configuration disk, but we write it down anyway. Note a longitude is a word of meridian generators in the configuration disk. For most of calculations over the disk, it suffices to use the other three relations.
\[StdCordsGen\] Standard cords generate the framed cord algebra.
By generation, we mean that every element in $\textrm{Cord}(K)$ can be written as a sum of words consisting of standard cords and $\lambda_i^{\pm1}, \mu_i^{\pm1}$. We start by proving two claims.
\(1) Every framed cord is cord homotopic to one from $x_i$ to $x_j$ for some $i,j \in\{1,\dotsb, n\}$.
Suppose a cord $[c]$ starts at $x\in\ell_s\setminus \{\ast_s\}$. There exists $x_i$ such that $\{i\}= s$. Take a path $p\subset \ell_s$ connecting $x_i$ and $x$, without passing through point $\ast_s$. The concatenation $[p\cdot c]$ is isomorphic to $c$ in the framed cord algebra, and the staring point of $p\cdot c$ is $x_i$. Similarly one can similarly homotope the end point to some $x_j$.
\(2) Every framed cord from $x_i$ to $x_j$ is cord homotopic to the concatenation of a loop $\gamma^{(i)} \in \pi_1(X\setminus K, x_i)$ and the standard cord $\gamma_{ij}$.
Suppose $[c_{ij}]$ is a framed cord from $x_i$ to $x_j$. We set $\gamma^{(i)} = c_{ij}\cdot \gamma_{ji}$, then $[c_{ij}] = [c_{ij}\cdot \gamma_{ji} \cdot \gamma_{ij}] = [\gamma^{(i)} \cdot \gamma_{ij}]$, proving the statement.
Now we prove the lemma. By statements (1) and (2), it sufficed to prove the assertion for a framed cord in the form of $[c] = [\gamma^{(i)} \cdot \gamma_{ij}]$. Since $\gamma^{(i)} \in \pi_1(X\setminus K, x_i)$, it is generated by meridians. We prove by induction on the word length of $\gamma^{(i)}$. The initial step is trivial, since $[c] = [\gamma_{ij}]$ is already a standard cord.
Suppose the induction hypothesis that $[\gamma^{(i)}\cdot \gamma_{ij}]$ is generated by standard cords holds for any loop $\gamma^{(i)}\in \pi_K$ with word length less or equal to $s \in {{\mathbb{N}}}$, we need to show that the induction hypothesis also holds for word length $s+1$. It is equivalent to prove that $[(m_t^{(i)})^{\pm 1}\cdot \gamma^{(i)}\cdot \gamma_{ij}]$ can also be generated by standard cords. There is $$\begin{aligned}
[m_t^{(i)}\cdot \gamma^{(i)} \cdot \gamma_{ij}]
&= [\gamma_{it} \cdot m_t^{(t)} \cdot \gamma_{ti} \cdot \gamma^{(i)} \cdot \gamma_{ij}] \\
&= [\gamma_{it} \cdot \gamma_{ti} \cdot \gamma^{(i)} \cdot \gamma_{ij}] - [\gamma_{it}] [ \gamma_{ti} \cdot \gamma^{(i)} \cdot \gamma_{ij}] \\
&= [\gamma^{(i)} \cdot \gamma_{ij}] - [\gamma_{it}] [ \gamma_{ti} \cdot \gamma^{(i)} \cdot \gamma_{it}\cdot \gamma_{ti}\cdot \gamma_{ij}] \\
&= [\gamma^{(i)} \cdot \gamma_{ij}] - [\gamma_{it}] [ \gamma^{(t)} \cdot \gamma_{tj}].\end{aligned}$$ The second equality uses the skein relation, and the other equalities are cord identities. Similarly we have $$\begin{aligned}
[(m_t^{(i)})^{-1}\cdot \gamma^{(i)} \cdot \gamma_{ij}]
&= [\gamma_{it} \cdot (m_t^{(t)})^{-1} \cdot \gamma_{ti} \cdot \gamma^{(i)} \cdot \gamma_{ij}] \\
&= [\gamma^{(i)} \cdot \gamma_{ij}] + [\gamma_{it}] [(m_t^{(t)})^{-1} \cdot \gamma_{ti} \cdot \gamma^{(i)} \cdot \gamma_{ij}] \\
&= [\gamma^{(i)} \cdot \gamma_{ij}] + \mu_{\{t\}}^{-1}[\gamma_{it}] [ \gamma^{(t)} \cdot \gamma_{tj}].\end{aligned}$$ The last equation uses the meridian relation. We complete the induction, as well as the proof of the lemma.
It follows the lemma that meridian cords are generated by standard cords. Suppose $[m_t^{(b)}]$ is a meridian cord wrapping around strand $t$, and based at $x_b$. That is to say, the underlining loop is $m_t \in \pi_1(X\setminus K, x_b)$. Then $$\begin{aligned}
[m_t^{(b)}] = [\gamma_{bb}] - [\gamma_{bt}][\gamma_{tb}], \qquad
[(m_t^{(b)})^{-1}] = [\gamma_{bb}] + \mu_{\{t\}}^{-1} [\gamma_{bt}][\gamma_{tb}]. \end{aligned}$$ Here $\{t\}$ is the component function defined in (\[IndexFun\]). It is evident from the equations that a meridian cord depends on the base point as well as the homotopy class of the underlining loop.
Augmentations
-------------
An augmentation $\epsilon$ of the framed cord algebra $\textrm{Cord}(K)$ of an oriented link $K$ is a unit preserving algebra morphism $$\epsilon: \textrm{Cord}(K)\rightarrow k,$$ where $k$ is any commutative field.
Recall we have a variant framed cord algebra $\textrm{Cord}^c(K)$ where $\lambda_i^{\pm1},\mu_i^{\pm1}$ commutes with cords. We can similarly define its augmentation as a unit preserving algebra morphism $$\epsilon: \textrm{Cord}^c(K)\rightarrow k.$$ Since $k$ is commutative, the image of any non-commutative generators in either $\textrm{Cord}(K)$ or $\textrm{Cord}^c(K)$ becomes commutative as elements in $k$. Therefore, the set of augmentations for $\textrm{Cord}(K)$ and $\textrm{Cord}^c(K)$ are canonically isomorphic.
In Remark \[rmkframing\] we explained that different framings give isomorphic ${{\mathbb{Z}}}$-algebras but not ${{\mathbb{Z}}}[\lambda^{\pm1}_1,\mu^{\pm1}_1,\dotsb, \lambda^{\pm1}_r, \mu^{\pm1}_r]$-algebras. Hence there is a bijection between augmentations in one framing and augmentations in the other, but if we restricts to augmentations sending $\lambda_i,\mu_i$ to particular values in $k^*$, then there is no bijection.
Given a framed cord $[c]$, its augmented value $\epsilon([c])$ will be abbreviated as $\epsilon(c)$. This abbreviation should cause no confusions. For example, an augmentation applied to the skein relation within a configuration disk (\[ConfDsr\]) can be expressed as $$\epsilon(c_{it} \cdot c_{tj}) = \epsilon(c_{it} \cdot m_t^{(t)}\cdot c_{tj}) + \epsilon(c_{it})\epsilon(c_{tj}).$$
Suppose $K$ is the closure of an $n$-strand braid and $\epsilon: \textrm{Cord}(K)\rightarrow k$ is an augmentation. We can organize augmented values of standard cords into an $n\times n$ square matrix $R$, by setting $R_{ij} = \epsilon(\gamma_{ij})$. Namely, $$R =
\begin{pmatrix}
\epsilon(\gamma_{11}) & \dotsb & \epsilon(\gamma_{1n}) \\
\vdots & \ddots &\vdots \\
\epsilon(\gamma_{n1}) & \dotsb & \epsilon(\gamma_{nn})
\end{pmatrix}.$$
Since Lemma \[StdCordsGen\] asserts that standard cords generate $\textrm{Cord}(K)$, it is natural to ask whether the matrix $R$ in turn determines the augmentation $\epsilon$. The answer is sometimes. For example when $K$ is a knot as in [@Ga2 Theorem 4.16], in the first two cases one can reconstruct the augmentation $\epsilon$ from the associated matrix $R$, while in the third case one needs to specify in addition the data of $\epsilon(\lambda)$ to recover $\epsilon$ from $R$.
Augmentation representation {#augmentation-representation}
---------------------------
We defined a matrix $R$ out of an augmentation $\epsilon: \textrm{Cord}(K)\rightarrow k$. Let $R_j$ be the column vectors of $R$. Define the $k$-vector space $$V_\epsilon := \textrm{Span}_k\{R_j\}_{1\leq j\leq n}.$$
We adopt the following convention to represent a column vector of size $n$. For any path $c_j$ from $x_0$ to $x_j$, define $$\epsilon({p}_\alpha^{-1} \cdot c_j) = \big(\epsilon({p}_1^{-1} \cdot c_j),\dotsb, \epsilon(p_n^{-1}\cdot c_j)\big)$$
\[MainConstruction\] Suppose $K$ is an oriented link equipped with its Seifert framing. Let $\epsilon: Cord(K)\rightarrow k$ be an augmentation of its framed cord algebra. The following map defines a link group representation $\rho_\epsilon: \pi_K\rightarrow GL(V)$, $$\label{DefAugRep}
\rho_\epsilon(\gamma) R_j: = \epsilon({p}_\alpha^{-1} \cdot \gamma \cdot p_j), \textrm{ where } \gamma\in \pi_1(X\setminus K,x_0).$$ We call $(\rho_\epsilon,V_\epsilon)$ the *augmentation representation* associated to $\epsilon$.
In this to be verified representation, it may be difficult to compute the action for a general $\gamma$, but the action of meridians on standard cords takes a much simpler form. Check out the following lemma.
\[meridanRj\] For $1\leq t, i, j\leq n$, there are $$\label{MeridianAction}
\rho_\epsilon(m_t) R_j = R_j - \epsilon(\gamma_{tj})R_t, \qquad \rho_\epsilon(m_t^{-1}) R_j = R_j + \mu_{\{t\}}^{-1}\epsilon(\gamma_{tj})R_t.$$
We recall some notations and identities. For $1\leq t,i,j \leq n$, $p_i$ is a capping path from $x_0$ to $x_i$, $\gamma_{ij} = p_i^{-1} \cdot p_j$ is a standard cord from $x_i$ to $x_j$, $m_t$ is a meridian based at $x_0$, and $m_t^{(t)} = p_t^{-1}\cdot m_t \cdot p_t$ is a meridian cord based at $x_t$.
To verify the first identity, by definition and interpolating $(p_t\cdot p_t^{-1})$, there is $$\rho_\epsilon(m_t) R_j := \epsilon(p_\alpha^{-1}\cdot m_t \cdot p_j)
= \epsilon(p_\alpha^{-1}\cdot (p_t\cdot p_t^{-1}) \cdot m_t \cdot (p_t\cdot p_t^{-1}) \cdot p_j)
= \epsilon(\gamma_{\alpha t} \cdot m_t^{(t)}\cdot \gamma_{tj}).$$ By the skein relation, there is $$\epsilon(\gamma_{\alpha t} \cdot m_t^{(t)}\cdot \gamma_{tj})
= \epsilon(\gamma_{\alpha t}\cdot \gamma_{tj}) - \epsilon(\gamma_{\alpha t})\epsilon(\gamma_{tj})
= R_j - \epsilon(\gamma_{tj})R_t.$$ Combining these equations, we conclude that $$\rho_\epsilon(m_t) R_j = R_j - \epsilon(\gamma_{tj})R_t.$$
In a similar fashion, we can derive for the second identity in the assertion: $$\begin{aligned}
\rho_\epsilon(m_t^{-1}) R_j
&= \epsilon(p_\alpha^{-1}\cdot m_t^{-1} \cdot p_j) = \epsilon(\gamma_{\alpha t} \cdot (m_t^{(t)})^{-1}\cdot \gamma_{tj}) \\
&= \epsilon(\gamma_{\alpha t}\cdot (m_t^{(t)})\cdot(m_t^{(t)})^{-1}\cdot \gamma_{tj}) + \epsilon(\gamma_{\alpha t})\epsilon((m_t^{(t)})^{-1}\cdot \gamma_{tj}) \\
& = R_j + \epsilon((m_t^{(t)})^{-1}\cdot \gamma_{tj})R_t.\end{aligned}$$ From the meridian relation, we know that $\epsilon((m_t^{(t)})^{-1}\cdot \gamma_{tj}) = \mu_{\{t\}}^{-1}\epsilon(\gamma_{tj})$. We conclude that $$\rho_\epsilon(m_t^{-1}) R_j = R_j + \mu_{\{t\}}^{-1}\epsilon(\gamma_{tj})R_t.$$
Now we are ready to prove that the construction is indeed a representation.
We write $(\rho,V)$ instead of $(\rho_\epsilon, V_\epsilon)$ in this proof.
\(1) To see that $\rho(\gamma)$ is a closed linear map, it suffices to prove for meridian generators. For $\gamma = m_t^{\pm 1}$ and any vector $R_j$, there is $\rho(m_t)R_j\in V$. By Lemma \[meridanRj\], we have $$\label{MeridianAction1}
\rho(m_t) R_j = R_j- \epsilon({p}_t^{-1}\cdot p_j) R_t, \quad \rho(m_t^{-1}) R_j = R_j + \epsilon(p_t^{-1}\cdot m_t^{-1}\cdot p_j)R_t.$$ For fixed $t$ and $j$, $\epsilon({p}_t^{-1}\cdot p_j)$ and $\epsilon(p_t^{-1}\cdot m_t^{-1}\cdot p_j)$ are constants in the field $k$. Therefore $\rho(m_t^{\pm 1})$ is closed.
\(2) The identity is straightforward to check. Let $e$ be the identity loop based at $x_0$, then $\rho(e)R_j = R_j$ for all $j$ by definition.
\(3) To verify the composition in the group action, one needs $\rho(\gamma_1)\rho(\gamma_2) = \rho(\gamma_1\cdot \gamma_2)$. Since each loop in $\pi_K$ can be expressed as a word of meridian generators, we can prove by induction on the word length of $\gamma_2$.
For the initial step, we take $\gamma_1 \in \pi_K$, and $\gamma_2 = m_t^{\pm 1}$. It suffices to prove that for any $R_j$, $\rho(\gamma_1)\rho(m_t^{\pm 1})R_j = \rho(\gamma_1\cdot m_t^{\pm 1})R_j$. For $m_t$, the left hand side is $$\begin{aligned}
\rho(\gamma_1)\rho(m_t)R_j
&= \rho(\gamma_1) \big( R_j- \epsilon({p}_t^{-1}\cdot p_j) R_t \big) \\
&= \epsilon(p_\alpha^{-1}\cdot \gamma_1 \cdot p_j) -\epsilon({p}_t^{-1}\cdot p_j)\epsilon(p_{\alpha}^{-1}\cdot \gamma\cdot p_t). \end{aligned}$$ The first equation follows (\[MeridianAction\]) and the second equation is by definition (\[DefAugRep\]). Continuing with the right hand side, there is $$\begin{aligned}
\rho(\gamma_1\cdot m_t)R_j
&= \epsilon(p_\alpha^{-1}\cdot \gamma_1\cdot m_t \cdot p_j) \\
&= \epsilon(p_\alpha^{-1}\cdot \gamma_1\cdot p_j) - \epsilon(p_\alpha^{-1}\cdot \gamma_1 \cdot p_t) \epsilon(p_t^{-1}\cdot p_j).\end{aligned}$$ The first equation is by definition (\[DefAugRep\]) and the second equation is by skein relations. Comparing the results, we see that $\rho(\gamma_1)\rho(m_t)R_j = \rho(\gamma_1\cdot m_t)R_j$ for any $j= 1,\dotsb n$.
The proof for the case $\gamma_2 = m_t^{-1}$ is similar, except the skein relation is applied in the other way. In brief, we get $$\rho(\gamma_1)\rho(m_t^{-1})R_j = \epsilon(p_{\alpha}^{-1}\cdot \gamma_1\cdot p_j) + \epsilon( p_t^{-1}\cdot m_t^{-1}\cdot p_j)\epsilon (p_{\alpha}^{-1}\cdot \gamma_1 \cdot p_t) = \rho(\gamma_1\cdot m_t^{-1})R_j,$$ completing the initial step.
Proceeding to the induction step, we assume that $\rho(\gamma_1)\rho(\gamma_2') = \rho(\gamma_1\cdot \gamma_2')$ for any $\gamma_1\in \pi_K$, and any $\gamma_2'\in \pi_K$ that can be written as a word of meridian generators with length less or equal to $s$. Assume $\gamma_2 = \gamma_2' m_t^{\pm 1}$ has word length less or equal to $s+1$, then $$\rho(\gamma_1)\rho(\gamma_2' m_t^{\pm 1})=\rho(\gamma_1)\rho(\gamma_2')\rho(m_t^{\pm 1}) = \rho(\gamma_1\gamma_2')\rho(m_t^{\pm 1}) = \rho(\gamma_1\gamma_2'm_t^{\pm 1}).$$ Each of the three equalities follows from the induction hypothesis. Therefore we complete the induction and show that $\rho(\gamma_1)\rho(\gamma_2) = \rho(\gamma_1\cdot \gamma_2)$.
\(4) We prove that $\rho(\gamma)$ is a well-defined linear map, namely if $I \subset \{1,\dotsb, n\}$ is a subset and $\sum_{i\in I} a_i R_i=0$ for some constants $a_i$, then $\sum_{i\in I} a_i \rho(\gamma)R_i=0$ as well. If $\gamma = m_t$, by Lemma \[meridanRj\], there is $$\sum_{i\in I} a_i \rho(\gamma)R_i = \sum_{i\in I} a_i R_i - \sum_{i\in I} a_i \epsilon(\gamma_{ti}) R_t.$$ The first summand is zero by hypothesis. In the second summand, $\sum_{i\in I} a_i \epsilon(\gamma_{ti})$ equals to the $t$-th row of $\sum_{i\in I} a_i R_i=0$, which is also zero. The argument for $\gamma =m_t^{-1}$ is similar. Continuing by induction on word length of $\gamma$ in terms of meridian generators, we prove the well-definedness.
Since a framed link isotopy can be extended to an ambient isotopy and every framed cord finds its counterpart by the isotopy, it is clear that $\textrm{Cord}(K)$ is a framed link invariant. An augmentation $\epsilon:\textrm{Cord}(K)\rightarrow k $, algebraically defined as a algebra morphism, is also well-defined for isotopy classes of framed links.
However, to construct an associated augmentation representation $(\rho_\epsilon, V_\epsilon)$, one needs to choose a braid, which is additional data. It is not immediately clear about the dependent of this choice from the definition. In the following theorem, we address this question.
\[MainThm\] Suppose $K$ is an oriented link equipped with its Seifert framing. Let $\epsilon: {\textrm{Cord}}(K)\rightarrow k$ be an augmentation of its framed cord algebra. Up to isomorphism, the augmentation representation $(\rho_\epsilon,V_\epsilon)$ is well-defined for the augmentation $\epsilon$. In particular, it does not depend on the choice of the braid in the construction.
Suppose $B$ is a braid whose closure is $K$. Let $(\rho_\epsilon,V_\epsilon)$ be the augmentation representation of $\epsilon$ constructed for $B$ as in Theorem-Definition \[MainConstruction\]. Suppose $\tilde{B}$ is another braid whose closure is also $K$, and let $(\tilde{\rho}_\epsilon,\tilde{V}_\epsilon)$ be the representation constructed with respect to $\tilde{B}$. We shall prove that $(\rho_\epsilon,V_\epsilon)$ and $(\tilde{\rho}_\epsilon,\tilde{V}_\epsilon)$ are isomorphic as $\pi_K$-representations.
In the rest of the proof, we drop the subscript and write $(\rho,V), (\tilde{\rho},\tilde{V})$ for $(\rho_\epsilon,V_\epsilon), (\tilde{\rho}_\epsilon,\tilde{V}_\epsilon)$
To prove the isomorphism, we will construct a linear map $T: V\rightarrow \tilde{V}$ which intertwines with link group actions. Namely, for any $\gamma\in \pi_K$, there should be a commutative diagram,
(A)[$V$]{}; (B)\[right of=A, node distance=2cm\][$\tilde{V}$]{}; (C)\[below of=A, node distance=2cm\][$V$]{}; (D)\[right of=C, node distance=2cm\][$\tilde{V}$]{}; (A) to node \[\] [$T$]{} (B); (A) to node \[swap\] [$\rho(\gamma)$]{} (C); (B) to node \[\][$\tilde\rho(\gamma)$]{}(D); (C) to node \[\][$T$]{}(D);
which can be restated as $\tilde{\rho}(\gamma)\circ T = T\circ \rho(\gamma)$. Since the link group is generated by a set of meridian generators, it suffices to check the commutativity for $\gamma$ being meridian generators.
By Markov’s theorem, two braids have isotopic closures if and only if they are related by a sequence of equivalence relations: (1) they are equivalent braids, (2) they are conjugate braids, (3) one braid is a positive/negative stabilization of the other braid. It is clear that the construction of the augmentation representation is well-defined within an equivalence class of braids. We will verify the well-definedness in other cases.
Suppose $B$ is an $n$-strand braid. Let $\sigma_s$, $1\leq s\leq n-1$ be the positive half twists in the braid group $Br_n$. Let $\tilde B = \sigma_s B\sigma_s^{-1}$ be a conjugation.
We write $\gamma_{ij}$ for the standard cords in the configuration disk $D$ that is used to define $(\rho,V)$, and $\tilde{\gamma}_{ij}$ (abbreviated from a more rigorous notation $\tilde{\gamma}_{\tilde{i}\tilde{j}}$) for the standard cords in the configuration disk $\tilde{D}$ that is used to define $(\tilde{\rho},\tilde{V})$. These standard cords are related as elements in $\textrm{Cord}(K)$, expressed as follows: for $i,j \neq s,s+1$, $$\begin{aligned}
\label{conjugationcords}
\tilde{\gamma}_{ij} & = \gamma_{ij}, \nonumber\\
\tilde\gamma_{i,s+1} &= \gamma_{is}, & \tilde{\gamma}_{is} &= \gamma_{is} \cdot m_s^{(s)} \cdot \gamma_{s,s+1}, \nonumber\\
\tilde\gamma_{s+1,j} &= \gamma_{sj}, & \tilde{\gamma}_{sj} &= \gamma_{s+1,s} \cdot (m_s^{(s)})^{-1} \cdot \gamma_{sj},\\
\tilde{\gamma}_{s,s} &= \gamma_{s+1,s+1}, & \tilde{\gamma}_{s,s+1} &= \gamma_{s+1,s}\cdot (m_s^{(s)})^{-1}, \nonumber\\
\tilde{\gamma}_{s+1,s+1} &= \gamma_{s,s}, & \tilde{\gamma}_{s+1,s} &= m_{s}^{(s)}\cdot \gamma_{s,s+1}.\nonumber\end{aligned}$$ We related the two sets of meridian generators $\{m_t\}_{1\leq t\leq n}$, $\{\tilde{m}_t\}_{1\leq t\leq n}$ in a similar way: $$\begin{aligned}
&\tilde{m}_{s+1} = m_s,\quad \tilde{m}_s = m_s\cdot m_{s+1}\cdot m_s^{-1},\\
&\tilde{m}_t = m_t \quad \textrm{ for } \; t\neq s,s+1.\end{aligned}$$ Again, $\tilde{m}_{t}$ is an abbreviation of $\tilde{m}_{\tilde{t}}$.
We define matrices $R,\tilde{R}$ by $R_{ij} = \epsilon(\gamma_{ij}), \tilde{R}_{ij} = \epsilon(\tilde\gamma_{ij})$. Following the construction of the augmentation representation, we define $$V := \textrm{Span}_k\{R_j\}_{1\leq j\leq n}, \quad \tilde{V} := \textrm{Span}_k\{\tilde{R}_j\}_{1\leq j\leq n}.$$ We first verify that $V$ and $\tilde{V}$ have the same rank, which is not obvious from the definition. We will argue that $\tilde{R}$ equals to $R$ after a sequence of row operations and column operations, namely there exist $n\times n$ matrices $M_L$ and $M_R$ such that $\tilde{R} = M_L R M_R$. The isomorphism follows from the fact that these operations do not change the rank of the space spanned by column vectors. We claim $$M_L =
\begin{pmatrix}
I_{s-1} & & \\
& M_L' & \\
& & I_{n-s-1}
\end{pmatrix},
\quad
M_R =
\begin{pmatrix}
I_{s-1} & & \\
& M_R' & \\
& & I_{n-s-1}
\end{pmatrix},$$ where $M_L'$ and $M_R'$ are $2\times 2$ matrices $$M_L' =
\begin{pmatrix}
\epsilon(\gamma_{s+1,s}\cdot(m_s^{(s)})^{-1}) & 1 \\
1 & 0
\end{pmatrix},
\quad
M_R' =
\begin{pmatrix}
-\epsilon(\gamma_{s,s+1}) & 1 \\
1 & 0
\end{pmatrix}.$$ We verify that $M_L R M_R = \tilde R$ at each entry. Observe that $M_L,M_R$ only affect rows and columns indexed by $s,s+1$. If neither the row index nor column index is in $\{s,s+1\}$, it is clear that $$(M_L R M_R)_{ij} =R_{ij} = \epsilon(\gamma_{ij}) = \epsilon(\tilde{\gamma}_{ij}) = \tilde{R}_{ij}.$$ If only one of the row or column index is in $\{s,s+1\}$, then $$\begin{aligned}
(M_L R M_R)_{sj} = (M_LR)_{sj} &= \epsilon(\gamma_{s+1, s}\cdot (m_s^{(s)})^{-1}) \epsilon(\gamma_{sj}) + \epsilon(\gamma_{s+1,j}) \\
&= \epsilon (\gamma_{s+1,s} \cdot (m_s^{(s)})^{-1} \cdot \gamma_{sj}) = \epsilon(\tilde{\gamma}_{sj}) = \tilde{R}_{sj},\\
(M_L R M_R)_{s+1, j} = (M_LR)_{s+1,j} &= \epsilon(\gamma_{sj}) = \epsilon(\tilde \gamma _{s+1,j}) = \tilde{R}_{s+1,j},\\
(M_L R M_R)_{is} = (RM_R)_{is} &= -\epsilon(\gamma_{is})\epsilon(\gamma_{s, s+1}) + \epsilon(\gamma_{i,s+1}) \\
&= \epsilon (\gamma_{is} \cdot m_s^{(s)} \cdot \gamma_{s,s+1}) = \epsilon(\tilde{\gamma}_{is}) = \tilde{R}_{is},\\
(M_L R M_R)_{i, s+1} = (RM_R)_{i, s+1} &= \epsilon(\gamma_{is}) = \epsilon(\tilde \gamma_{i,s+1}) = \tilde{R}_{i,s+1}.\end{aligned}$$ Finally we verify for the $2\times 2$ sub-matricies whose indices are contained in $[s,s+1]^2$. We write $A_{[s,s+1]^2}$ for a $2\times 2$ submatrix of $A$ consisting entries with both row and column indices in $\{s,s+1\}$. Let $\epsilon(\gamma_{s+1,s}) = a,\epsilon(\gamma_{s,s+1}) = b$. $$\begin{aligned}
(M_L R M_R)_{[s,s+1]^2} &= M_L' R _{[s,s+1]^2} M_R' \\
&=
\begin{pmatrix}
a\mu_{\{s\}}^{-1} & 1 \\
1 &
\end{pmatrix}
\begin{pmatrix}
1-\mu_{\{s\}} & b \\
a & 1- \mu_{\{s+1\}}
\end{pmatrix}
\begin{pmatrix}
-b & 1 \\
1 &
\end{pmatrix} \\
&=
\begin{pmatrix}
1- \mu_{\{s+1\}} & a\mu_{\{s\}}^{-1} \\
b\mu_{\{s\}}& 1-\mu_{\{s\}}
\end{pmatrix}= \tilde{R}_{[s,s+1]^2},\end{aligned}$$ where the last equality is because of $\epsilon(\tilde{\gamma}_{s,s+1}) = \epsilon(\gamma_{s+1,s}\cdot (m_s^{(s)})^{-1})= \epsilon(\gamma_{s+1,s})\mu_{\{s\}}^{-1} = a\mu_{\{s\}}^{-1}$, and similarly $\epsilon(\tilde{\gamma}_{s,s+1}) = b\mu_{\{s\}}$. Therefore $\tilde{R} = M_L R M_R$.
We have verified that $V$ and $\tilde{V}$ are isomorphic as vector spaces. We define a linear map $T: V\rightarrow \tilde{V}$. We define $T$ over the spanning vectors: $$\begin{aligned}
T(R_s) &= \tilde{R}_{s+1}, \quad T(R_{s+1}) = \tilde{R}_s + \epsilon(\gamma_{s,s+1})\tilde{R}_{s+1},\\
T(R_j) &= \tilde{R}_j \;\; \textrm{for} \;\; j\neq s,s+1.\end{aligned}$$ The inverse map $T^{-1}$ is thusly determined: $$\begin{aligned}
T^{-1}(\tilde{R}_s) &= {R}_{s+1} - \epsilon(\gamma_{s,s+1})R_s, \quad T^{-1}(\tilde{R}_{s+1}) = {R}_s,\\
T^{-1}(\tilde{R}_j) &= {R}_j \;\; \textrm{for} \;\; j\neq s,s+1.\end{aligned}$$ In terms of the spanning vectors, $T$ and $T^{-1}$ can be expressed as $n\times n$ matrices: $$\label{CongT}
T =
\begin{pmatrix}
I_{s-1} &&&\\
& 0 & 1 & \\
& 1 & \epsilon(\gamma_{s,s+1}) & \\
& & & I_{n-s-1}
\end{pmatrix},
\quad
T^{-1}
=
\begin{pmatrix}
I_{s-1} &&&\\
& -\epsilon(\gamma_{s,s+1}) & 1 & \\
& 1 & 0 & \\
& & & I_{n-s-1}
\end{pmatrix}.$$ Recall that $V$ and $\tilde{V}$ are related by row and column operations. Since row operations do not change the linear dependence relations among the column vectors and the matrix $M_R$ for column operations is consistent with $T^{-1}$, we conclude that the map $T$ descents to a linear transform from $V$ to $\tilde {V}$.
Finally we show that $\tilde{\rho}(\gamma)\circ T = T\circ \rho(\gamma)$ for $\gamma = m_t$, $t = 1, \dotsb, n$. We introduce some notations to prepare the calculation. Following the calculations in Lemma \[meridanRj\], $\rho(m_t)$ and $\tilde{\rho}(\tilde{m}_t)$ can be written as $n\times n$ matrices respecting formal linear combinations of their spanning vectors: $$\label{CongM}
M_t: =\rho(m_t)= I_n - \sum_{j=1}^n\epsilon(\gamma_{tj})E_{tj}, \quad \tilde{M}_t :=\tilde\rho(\tilde m_t)= I_n - \sum_{j=1}^n\epsilon(\tilde{\gamma}_{tj})E_{tj},$$ where $E_{ij}$ is the matrix having $1$ at the $(i,j)$-th entry and $0$ elsewhere.
We use Kroneker delta $\delta^i_j$ to denote a number which equals to $1$ when $i=j$ and $0$ otherwise. We do not use Einstein’s convention for contractions. All summations will be indicated explicitly.
Introducing new matrices $N_t := I_n - M_t$ and $\tilde{N}_t := I_n - \tilde{M}_t$, then $$\begin{aligned}
\label{SubstitutionMN}
\begin{split}
(M_t)_{ij} &= \delta^i_j - \epsilon(\gamma_{tj})\delta^t_i,\quad (N_t)_{ij} = \epsilon(\gamma_{tj})\delta^t_i, \\
(\tilde{M}_t)_{ij} &= \delta^i_j - \epsilon(\tilde\gamma_{tj})\delta^t_i, \quad (\tilde{N}_t)_{ij} = \epsilon(\tilde\gamma_{tj})\delta^t_i.
\end{split}\end{aligned}$$
We are ready to proceed into details. Recall our goal is to verify $$\label{MainThmCongRepCom}
\tilde{\rho}(m_t) \circ T = T \circ \rho(m_t).$$ We remind the reader that on the left hand side, it is $\tilde{\rho}(m_t)$ instead of $\tilde{\rho}(\tilde{m}_t)$. We will consider the following three cases based on the value of $t$: (A) $t\neq s,s+1$, (B) $t=s$ and (C) $t=s+1$.
$\bullet$ Case (A).
If $t\neq s,s+1$, then $m_t = \tilde{m}_t$. Equation (\[MainThmCongRepCom\]) becomes $$\tilde{M}_t T = T M_t.$$ Invoking the relations $N_t = I_n - M_t$, $\tilde{N}_t = I_n - \tilde{M}_t$ and then simplify, we only need to check $$\tilde{N}_t T = T{N}_t.$$ Observing matrix $T$ from (\[CongT\]), we compare the $(i,j)$-entries based on the following cases.
If $i\neq t$, then $(\tilde{N}_t T)_{tj} = 0 = (T{N}_t)_{tj}$.
If $i =t$, and $j \neq s,s+1$, then $(\tilde{N}_t T)_{tj} = \epsilon(\gamma_{tj})= (T{N}_t)_{tj}$.
Otherwise, it remains to check entries $(t,s)$ and $(t,s+1)$, namely verifying $$\begin{pmatrix}
(\epsilon(\tilde{\gamma}_{ts}), \epsilon(\tilde{\gamma}_{t,s+1})
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & \epsilon(\gamma_{s,s+1})
\end{pmatrix}
=
\begin{pmatrix}
(\epsilon({\gamma}_{ts}), \epsilon({\gamma}_{t,s+1}).
\end{pmatrix}$$ The $(t,s)$-entry is easy. By (\[conjugationcords\]), we have $\epsilon(\tilde{\gamma}_{t,s+1}) = \epsilon({\gamma}_{ts})$, proving $$(\tilde{N}_t T)_{ts} = (T{N}_t)_{ts}$$ For the $(t,s+1)$-entry, we will need to use the skein relation (second equality below) $$\begin{aligned}
(\tilde{N}_t T)_{t, s+1}
&=\epsilon(\tilde{\gamma}_{ts}) + \epsilon(\tilde{\gamma}_{t,s+1}) \epsilon(\gamma_{s,s+1}) \\
&= \epsilon(\gamma_{ts} \cdot m_s^{(s)} \cdot \gamma_{s,s+1}) + \epsilon({\gamma}_{ts})\epsilon(\gamma_{s,s+1}) \\
&= \epsilon(\gamma_{t,s+1}) = (T{N}_t)_{t,s+1}.\end{aligned}$$
$\bullet$ Case (B).
If $t = s$, then $m_s = \tilde{m}_{s+1}$ and equation (\[MainThmCongRepCom\]) becomes $\tilde{M}_{s+1} T = T{M}_s$. Again substituting $M$ by $N$, it suffices to verify $$\tilde{N}_{s+1} T = T{N}_s.$$
If $i \neq s,s+1$, then $(\tilde{N}_{s+1} T)_{ij} = 0 = (T{N}_s)_{ij}$.
If $i=s,s+1$ and $j\neq s,s+1$, then $$\begin{aligned}
(\tilde{N}_{s+1} T)_{sj} &= 0 = (T{N}_s)_{sj}, \\
(\tilde{N}_{s+1}T)_{s+1, j} &= \epsilon(\tilde{\gamma}_{s+1,j}) = \epsilon({\gamma}_{sj}) = (T{N}_s)_{s+1, j}.\end{aligned}$$
If $i = s, s+1$, $j = s, s+1$, then there is $$\begin{aligned}
(\tilde{N}_{s+1}T)_{[s,s+1]^2} &= \begin{pmatrix}
0 & 0 \\
\epsilon(\tilde\gamma_{s+1,s}) & \epsilon(\tilde{\gamma}_{s+1,s+1})
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & \epsilon(\gamma_{s,s+1})
\end{pmatrix}
\\
&=
\begin{pmatrix}
0 & 0 \\
\epsilon(\tilde{\gamma}_{s+1,s+1}) & \epsilon(\tilde\gamma_{s+1,s}) + \epsilon(\tilde{\gamma}_{s+1,s+1})\epsilon(\gamma_{s,s+1})
\end{pmatrix},\end{aligned}$$ and $$\begin{aligned}
(T{N}_s)_{[s,s+1]^2} &= \begin{pmatrix}
0 & 1 \\
1 & \epsilon(\gamma_{s,s+1})
\end{pmatrix}
\begin{pmatrix}
\epsilon(\gamma_{s,s}) & \epsilon(\gamma_{s,s+1}) \\
0 & 0
\end{pmatrix} \\
&=
\begin{pmatrix}
0 & 0 \\
\epsilon(\gamma_{s,s}) & \epsilon(\gamma_{s,s+1})
\end{pmatrix} .$$ Top row entries are obviously equal. For bottom left, it follows from (\[conjugationcords\]) that $\epsilon(\tilde{\gamma}_{s+1,s+1}) = \epsilon(\gamma_{s,s})$. For bottom right, we first compute that $$\epsilon(\tilde{\gamma}_{s+1,s+1}) = \epsilon(\gamma_{s,s}) = 1- \mu_{\{s\}},$$ and using the normalization and the meridian relation, that $$\epsilon(\tilde{\gamma}_{s+1,s}) = \epsilon(m_{s}^{(s)}\cdot \gamma_{s,s+1})= \mu_{\{s\}}\epsilon(\gamma_{s,s+1}).$$ Then the bottom right entries are equal because $$\epsilon(\tilde\gamma_{s,s+1}) + \epsilon(\tilde{\gamma}_{s+1,s+1})\epsilon(\gamma_{s,s+1}) = \mu_{\{s\}}\epsilon(\gamma_{s,s+1}) + (1- \mu_{\{s\}}) \epsilon(\gamma_{s,s+1}) = \epsilon (\gamma_{s,s+1}).$$
$\bullet$ Case (C).
If $t= s+1$, then $m_{s+1} = \tilde{m}_{s+1}^{-1}\cdot \tilde{m}_s\cdot \tilde{m}_{s+1}$. This time equation (\[MainThmCongRepCom\]) is more complicated, becoming $$\tilde{M}_{s+1}^{-1}\tilde{M}_s \tilde{M}_{s+1} T = TM_{s+1},$$ which further simplifies to $$\label{congcase3eq}
\tilde{M}_{s+1}^{-1} \tilde{N}_s \tilde{M}_{s+1} T = TN_{s+1}.$$
We first compute the right hand side of (\[congcase3eq\]). At the $(i,j)$-entry, there is $$(TN_{s+1})_{ij} = \sum_{d}T_{id}(N_{s+1})_{dj} = \sum_{d}T_{id}\,\epsilon(\gamma_{s+1,j})\delta^d_{s+1} = T_{i,s+1}\epsilon(\gamma_{s+1,j}).$$ Observe that row $s+1$ of $T$ is mostly zero except $i = s,s+1$. Therefore the right hand side of (\[congcase3eq\]) reads $$\label{congCaseCRHS}
\begin{cases}
(TN_{s+1})_{ij} =0, \qquad i \neq s,s+1, \\
(TN_{s+1})_{sj} = \epsilon(\gamma_{s+1,j}), \\
(TN_{s+1})_{s+1, j} = \epsilon({\gamma}_{s,s+1})
\epsilon(\gamma_{s+1,j}).
\end{cases}$$
Next we compute the left hand side of (\[congcase3eq\]). We make a claim that the the $(i,j)$ entry admits the following expansion
$$\label{CongCaseCLHSexpansion}
(\tilde{M}_{s+1}^{-1}\tilde{M}_s \tilde{M}_{s+1} T)_{ij} = (\tilde{M}_{s+1}^{-1})_{is} \cdot \sum_f \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,f})T_{fj}.$$
Proof of claim (\[CongCaseCLHSexpansion\]): Using (\[SubstitutionMN\]), we find that $$\begin{aligned}
(\tilde{M}_{s+1}^{-1}\tilde{M}_s \tilde{M}_{s+1} T)_{ij}
&= \sum_{d,e,f} (\tilde{M}_{s+1}^{-1})_{id}(\tilde{N}_s)_{de}(\tilde{M}_{s+1})_{ef}T_{fj} \\
&= \sum_{d,e,f} (\tilde{M}_{s+1}^{-1})_{id}\,\epsilon(\tilde{\gamma}_{se})\delta^s_d(\tilde{M}_{s+1})_{ef}T_{fj} \\
&=(\tilde{M}_{s+1}^{-1})_{is} \cdot \sum_{e,f} \epsilon(\tilde{\gamma}_{se})(\tilde{M}_{s+1})_{ef}T_{fj} \end{aligned}$$ Continuing with the term $\epsilon(\tilde{\gamma}_{se})(\tilde{M}_{s+1})_{ef}$ in the summand, we have $$\begin{aligned}
\epsilon(\tilde{\gamma}_{se})(\tilde{M}_{s+1})_{ef} &= \epsilon(\tilde{\gamma}_{se}) (\delta^e_f - \epsilon(\tilde\gamma_{s+1,f})\delta^{s+1}_e) \\
&= \epsilon(\tilde\gamma_{sf}) - \epsilon(\tilde\gamma_{s+1,f})\epsilon(\tilde\gamma_{s,s+1}) \\
&= \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,f}).\end{aligned}$$ Combining these terms and we obtain (\[CongCaseCLHSexpansion\]).
We have verified equation (\[CongCaseCLHSexpansion\]). Observing the result is the product of two term, one depends only on $i$ and the other depends only on $j$, we calculate them separately.
Consider the term with $i$, $(\tilde{M}_{s+1}^{-1})_{is}$. Given the expression of $\tilde{M}_t$, it is straightforward to compute its inverse $$\tilde{M}_t^{-1} = I_n + \sum_{j=1}^{n} \tilde{\mu}_{\{t\}}^{-1}\epsilon(\tilde{\gamma}_{tj})E_{tj},
\qquad (\tilde{M}_t^{-1})_{ij} = \delta^i_j + \tilde{\mu}_{\{t\}}^{-1} \epsilon(\tilde{\gamma}_{tj}) \delta^t_i.$$ We set $t= s+1$ and get $$(\tilde{M}_{s+1}^{-1})_{is} = \delta^i_s + \tilde{\mu}_{\{s+1\}}^{-1} \epsilon(\tilde{\gamma}_{s+1, s}) \delta^{s+1}_i.$$
- If $i\neq s,s+1$, $(\tilde{M}_{s+1}^{-1})_{is} = 0$.
- If $i =s$, then $(\tilde{M}_{s+1}^{-1})_{is} = 1$.
- If $i=s+1$, then $(\tilde{M}_{s+1}^{-1})_{is} = \tilde{\mu}_{\{s+1\}}^{-1}\epsilon(\tilde{\gamma}_{s+1, s})=\epsilon(\gamma_{s,s+1})$, because $$\tilde{\mu}_{\{s+1\}}^{-1}\epsilon(\tilde{\gamma}_{s+1, s})= \epsilon((\tilde{m}_{s+1}^{(s+1)})^{-1}\cdot \tilde{\gamma}_{s+1,s}) = \epsilon((m_s^{s})^{-1}\cdot m_s^{s}\cdot \gamma_{s,s+1}) = \epsilon(\gamma_{s,s+1}).$$
Consider the term involving $j$. We first present the answer, which is $$\label{congCaseCtermj}
\sum_f \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,f})T_{fj} = \epsilon(\gamma_{s+1, j}).$$ Multiplying it with the term involving $i$, we immediately see that the result matches (\[congCaseCRHS\]), proving (\[congcase3eq\]).
We verify (\[congCaseCtermj\]) in the following cases.
- If $j\neq s,s+1$, then $T_{fj} = \delta^f_j$, and $$\begin{aligned}
\sum_f \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,f})T_{fj}
&= \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,j}) \\
&= \epsilon(\gamma_{s+1,s}\cdot (m_s^{(s)})^{-1}\cdot m_s^{(s)}\cdot \gamma_{sj}) = \epsilon(\gamma_{s+1,j})\end{aligned}$$
- If $j=s$, then $T_{fs} = \delta^{f}_{s+1}$, and $$\begin{aligned}
\sum_f \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,f})T_{fs} &= \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,s+1}) \\
&= \epsilon(\gamma_{s+1,s}\cdot (m_s^{(s)})^{-1}\cdot m_s^{(s)}\cdot \gamma_{ss}) = \epsilon(\gamma_{s+1,s}) \end{aligned}$$
- If $j=s+1$, then $T_{f,s+1} = \delta^{f}_{s} + \epsilon(\gamma_{s,s+1})\delta^{f}_{s+1}$, and $$\begin{aligned}
&\sum_f \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,f})T_{f,s+1} \\
=& \epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,s}) + \epsilon({\gamma}_{s,s+1})\epsilon(\tilde{\gamma}_{s,s+1} \cdot \tilde{m}_{s+1}^{(s+1)} \cdot \tilde{\gamma}_{s+1,s+1}) \\
=& \epsilon(\gamma_{s+1,s}\cdot (m_s^{(s)})^{-1}\cdot m_s^{(s)}\cdot m_s^{(s)} \cdot \gamma_{s,s+1}) + \epsilon({\gamma}_{s,s+1}) \epsilon(\gamma_{s+1,s}) \\
=& \epsilon(\gamma_{s+1,s}\cdot m_s^{(s)} \cdot \gamma_{s,s+1}) + \epsilon(\gamma_{s+1,s})\epsilon({\gamma}_{s,s+1}) \\
=& \epsilon(\gamma_{s+1,s+1}).\end{aligned}$$ Here the second equality follows from (\[conjugationcords\]), and the last equality follows from the skein relations.
In the past two pages, we have verified Case (C) of (\[MainThmCongRepCom\]). Together with the other two cases, we have proven (\[MainThmCongRepCom\]), which says that the vector space isomorphism $T$ respects group actions. To summarize, the augmentation representations of a braid before and after a conjugation are isomorphic as representations.
A stabilization of a braid closure is given by adding one strand and perform a positive/negative a half twist with the outmost strand. The braid closure of a stabilized braids is equivalence to the closure of the original braid by a Reidemeister I move. Therefore stabilizations do not change the isotopy class of a link.
We fix conventions. Let $\iota: Br_n\rightarrow Br_{n+1}$ be the natural inclusion by adding a trivial strand labelled $n+1$. Suppose $B$ is an $n$-strand braid, its positive stabilization is $\sigma_{n}\iota(B)$ and its its negative stabilization is $\sigma_{n}^{-1}\iota(B)$.
Let $B$ be an $n$-strand braid whose closure is $K$, let $\tilde{B} = \sigma_{n}\iota(B)$ be its positive stabilization. We adopt similar notations as before. Coefficients involving $\mu$, are related by $$\begin{aligned}
\begin{split}
&\tilde\mu_{\{n\}} = \tilde{\mu}_{\{n+1\}} = \mu_{\{n\}},\\
&\tilde\mu_{\{t\}} = \mu_{\{t\}} \quad \textrm{ for } \; t\neq n, n+1.
\end{split}\end{aligned}$$
Let $\gamma_{ij}, 1\leq i,j\leq n$ and $\tilde{\gamma}_{ij}, 1\leq i,j\leq n+1$ be standard cords for $B$ and $\tilde{B}$. Different from the case of conjugations, this time $B$ and $\tilde{B}$ no longer have the same number of strands. As a consequence, the indices for the two sets of framed cords are different. For $i,j\neq n,n+1$, there are $$\begin{aligned}
\tilde\gamma_{ij} &= \gamma_{ij}, &\nonumber\\
\tilde\gamma_{i,n+1} &= \gamma_{in}, & \tilde\gamma_{i,n} &= \gamma_{i,n}\cdot m_n^{(n)},\nonumber\\
\tilde\gamma_{n+1, j} &= \gamma_{nj}, & \tilde\gamma_{n,j} &= (m_n^{(n)})^{-1} \cdot \gamma_{nj}, \\
\tilde{\gamma}_{nn} &= \gamma_{nn}, & \tilde\gamma_{n,n+1} &= (m_n^{(n)})^{-1}, \nonumber\\
\tilde\gamma_{n+1,n} &= m_n^{(n)}, & \tilde{\gamma}_{n+1,n+1} &= \gamma_{nn}. \nonumber\end{aligned}$$
The meridian generators are related by: $$\begin{aligned}
\label{meridiansPoSt}
\begin{split}
&\tilde{m}_{n+1} = \tilde{m}_n = m_n, \\
&\tilde{m}_t = m_t \quad \textrm{ for } \; t\neq n,n+1.
\end{split}\end{aligned}$$
We define matrices $R,\tilde{R}$ by $R_{ij} = \epsilon(\gamma_{ij}), \tilde{R}_{ij} = \epsilon(\tilde\gamma_{ij})$. Following the construction of the augmentation representation, we define $$V := \textrm{Span}_k\{R_j\}_{1\leq j\leq n}, \quad \tilde{V} := \textrm{Span}_k\{\tilde{R}_j\}_{1\leq j\leq n+1}.$$
Different from the previous case of conjugations, this time the two matrices have different sizes. Out first goal is to show that $V$ and $\tilde{V}$ have the same rank. Recall the identities $\epsilon(m_{n}^{(n)}) = \mu_{\{n\}}\epsilon(\gamma_{nn})$ and $\epsilon((m_{n}^{(n)})^{-1}) = \mu_{\{n\}}^{-1}\epsilon(\gamma_{nn})$. We see that $$\tilde{R} =
\begin{pmatrix}
\epsilon(\gamma_{11}) & \dotsb & \epsilon(\gamma_{1,n-1}) & \mu_{\{n\}}\epsilon(\gamma_{1n}) & \epsilon(\gamma_{1n}) \\
\vdots & \ddots & \vdots & \vdots &\vdots \\
\epsilon(\gamma_{n-1,1}) & \dotsb &\epsilon(\gamma_{n-1,n-1}) & \mu_{\{n\}}\epsilon(\gamma_{n-1,n}) & \epsilon(\gamma_{n-1,n})\\
\mu^{-1}_{\{n\}}\epsilon(\gamma_{n1}) & \dotsb & \mu^{-1}_{\{n\}}\epsilon(\gamma_{n, n-1}) & \epsilon(\gamma_{nn}) & \mu^{-1}_{\{n\}}\epsilon(\gamma_{nn})\\
\epsilon(\gamma_{n1}) & \dotsb & \epsilon(\gamma_{n, n-1}) & \mu_{\{n\}}\epsilon(\gamma_{nn}) & \epsilon(\gamma_{nn})
\end{pmatrix}.$$ Set $M_L = I_{n+1} - \mu_{\{n\}}E_{n+1,n}$ and $M_R = I_{n+1} - \mu_{\{n\}}^{-1}E_{n,n+1}$, then $$M_L \tilde{R} M_R =
\begin{pmatrix}
R & 0 \\
0 & 0
\end{pmatrix}.$$ Now it is clear that $V$ and $\tilde{V}$ have the same rank.
Define a linear transform $T: V\rightarrow \tilde{V}$ over spanning vectors: $$\begin{aligned}
&T(R_n) = \tilde{R}_{n+1},\\
&T(R_j) = \tilde{R}_j, \quad \textrm{ for } \; j\neq n.\end{aligned}$$ Because of the linear relation $\mu_{\{n\}}^{-1}\tilde{R}_{n} + \tilde{R}_{n+1} = 0$ in $\tilde{V}$, the linear map $T$ is a surjection, and further an isomorphism of vector spaces. In terms of spanning vectors, $T$ can be written as a $(n+1)\times n$ matrix: $$T =
\begin{pmatrix}
I_{n-1} & 0 \\
0 & 0\\
0 & 1
\end{pmatrix}.$$
Finally we prove that $T$ is an isomorphism of representations, namely for $1\leq t\leq n$, $$\label{PNsisorep}
\tilde{\rho}(m_t)\circ T = T\circ \rho(m_t)$$ Let $M_t =\rho(m_t)$ and $\tilde{M}_t = \tilde\rho(m_t)$, there is $$M_t = I_n - \sum_{j=1}^n\epsilon(\gamma_{tj})E_{tj}, \quad \tilde{M}_t = I_n - \sum_{j=1}^n\epsilon(\tilde{\gamma}_{tj})E_{tj}.$$
If $t\neq n$, then $m_t= \tilde{m_t}$, $\gamma_{tj} = \tilde{\gamma}_{tj}$. It is equivalent to check $(I_{n+1} - \tilde{M}_t)\circ T = T\circ (I_n - M_t)$. By a straightforward calculation, both hand sides equal to a $(n+1)\times n$ matrix where the $(t,j)$ entry equals to $\epsilon(\gamma_{tj})$ and zero otherwise.
If $t= n$, then we take $m_t = \tilde{m}_{n+1}$. It remains to check $(I_{n+1} - \tilde{M}_{n+1})\circ T = T\circ (I_n - M_n)$. Both hand sides equal to a $(n+1)\times n$ matrix where the $(n+1,j)$ entry equals to $\epsilon(\gamma_{nj})$ and zero otherwise.
We have proven that the group actions of the meridian generators are compatible on both $V$ and $\tilde{V}$, therefore so is the entire link group $\pi_K$. We conclude that the construction of the augmentation representations before and after a positive stabilization are isomorphic.
The proof for a negative stabilization is similar, which we do not repeat here. For reference, we record some identities and the matrix $\tilde{R}$ after a negative stabilization. For $1\leq i, j \leq n-1$, the framed cords are identified by the following relations $$\begin{aligned}
\tilde\gamma_{ij} &= \gamma_{ij}, \nonumber\\
\tilde{\gamma}_{i,n} &= \tilde\gamma_{i,n+1} = \gamma_{in} \cdot (m_{n}^{(n)})^{-1},\nonumber\\
\tilde\gamma_{n,j} &= \tilde\gamma_{n+1, j} = (m_n^{(n)}) \cdot \gamma_{nj}, \nonumber \\
\tilde{\gamma}_{nn} &= \tilde\gamma_{n,n+1} = \tilde\gamma_{n+1,n} = \tilde{\gamma}_{n+1,n+1} =\gamma_{nn}, \nonumber\end{aligned}$$ the meridian generators are identified in the same way as in (\[meridiansPoSt\]), and the matrix $\tilde{R}$ is
$$\tilde{R} =
\begin{pmatrix}
\epsilon(\gamma_{11}) & \dotsb & \epsilon(\gamma_{1,n-1}) & \mu_{\{n\}}^{-1}\epsilon(\gamma_{1n}) & \mu_{\{n\}}^{-1}\epsilon(\gamma_{1n}) \\
\vdots & \ddots & \vdots & \vdots &\vdots \\
\epsilon(\gamma_{n-1,1}) & \dotsb &\epsilon(\gamma_{n-1,n-1}) & \mu_{\{n\}}^{-1}\epsilon(\gamma_{n-1,n}) & \mu_{\{n\}}^{-1}\epsilon(\gamma_{n-1,n})\\
\mu_{\{n\}}\epsilon(\gamma_{n1}) & \dotsb & \mu_{\{n\}}\epsilon(\gamma_{n, n-1}) & \epsilon(\gamma_{nn}) &\epsilon(\gamma_{nn})\\
\mu_{\{n\}}\epsilon(\gamma_{n1}) & \dotsb & \mu_{\{n\}}\epsilon(\gamma_{n, n-1}) & \epsilon(\gamma_{nn}) & \epsilon(\gamma_{nn})
\end{pmatrix}.$$
So far we have proven that the augmentation representation constructed from a braid is invariant under either a positive or a negative stabilization.
**Summary.** In previous arguments, we discussed in different cases that the augmentation representation $\rho_\epsilon: \pi_K \rightarrow GL(V_\epsilon)$ constructed from $\epsilon: \textrm{Cord}(K) \rightarrow k$, where the link $K$ is represented by a braid closure, is invariant under a conjugation or a positive/negative stabilization of the braid. Together with the relatively obvious case of braid equivalences, we can apply Markov’s theorem and prove that the augmentation representation $(\rho_\epsilon,V_\epsilon)$ is well-defined up to isomorphism, namely it is independent from the choice of the braid representative of the link.
In this remark, we compare the construction in this paper and that in [@Ga2].
In the case when $K$ is a knot, we can either write it as a braid closure and apply Theorem-Definition \[MainConstruction\] to construct the augmentation representation for $\epsilon: \textrm{Cord}(K)\rightarrow k$, or we can choose a set of meridian generators and apply [@Ga2 Propsition 4.11] to construct a representation for $\epsilon: {{\mathcal{P}}}_K\rightarrow k$.
Both constructions require additional data, and in both cases, the representation does not depend on additional data. The arguments for independence are different. In this paper, we use Marko’s theorem to prove that Markov moves give isomorphic representations. In [@Ga2], we use the correspondence between augmentations and sheaves (which is the main theorem of *loc.cit.*) to argue that the construction does not depend on the choice of the generating set of meridians.
Now we explain that the current paper manifests the underlining geometry of the construction in [@Ga2]. It is not an immediate identification of the two constructions using the ${{\mathbb{Z}}}$-algebra isomorphism $\textrm{Cord}^c(K) \cong {{\mathcal{P}}}_K$ in Remark \[3fcas\]. Thanks to the fact that in neither constructions the representation depends on the additional data, we can make a preferred choice for the position of the knot as well as the set of meridian generators.
Suppose knot $K$ is the braid closure of an $n$-strand braid $B$ and let $\{m_t\}_{1\leq t\leq n}$ be the set of meridian generators in the configuration disk associated to $B$. We choose $x_1$ in the configuration disk to be the base point of the knot group $\pi_K$, then our preferred set of meridian generators is $$\{m_t^{(1)}\}_{1\leq t\leq n}.$$ Since any two meridians are conjugate to each other in knot group, there exists $g_t\in \pi_K$ for each $t$, such that $m_t^{(1)} = g_t^{-1}\cdot m_1^{(1)}\cdot g_t$.
Both constructions starts with a square matrix $R$. Given our set up, both $R$ matrices are of size $n$. For $\textrm{Cord}(K)$, the $R$ matrix is given by augmented values of standard cords, $$(R_{\textrm{Cord}(K)})_{ij} = \epsilon_{\textrm{Cord}(K)}(\gamma_{ij}).$$ For ${{\mathcal{P}}}_K$, the $R$ matrix is given by $$(R_{{{\mathcal{P}}}_K})_{ij} = \epsilon_{{{\mathcal{P}}}_K}(g_i\cdot g_j^{-1}).$$
We present a entry-wise identification of the two $R$ matrices. The orientation of knot $K$ induces an orientation of its Seifert framing $\ell$. Recall that $\ast \in \ell$ is the marked point used in the longitude relation in the cord algebra. For each $t\in \{1,\dotsb, n\}$, there is a unique path $d_{t1}$ from $x_t$ to $x_1$ that is contained in $L\setminus \{\ast\}$ with compatible orientation. There are $$m_t^{(1)} = \gamma_{1t} \cdot m_t^{(t)}\cdot \gamma_{t1},\quad m_t^{(t)} = d_{t1} \cdot m_1^{(1)}\cdot d_{t1}^{-1}.$$ Combining these two equations with $m_t^{(1)} = g_t^{-1}\cdot m_1^{(1)}\cdot g_t$, we can solve for $$\label{gtchoice}
g_t = (\gamma_{1t}\cdot d_{t1})^{-1}.$$ Note the $g_t$ here is really a choice we made to fit into the equation $m_t^{(1)} = g_t^{-1}\cdot m_1^{(1)}\cdot g_t$. This choice is not unique in general. For example, the longitude commutes with the meridian in the peripheral subgroup. The concatenation of a choice of $g_t$ with the longitude gives another choice. Fix the chosen $g_t$ in (\[gtchoice\]), we have $$g_i\cdot g_j^{-1} = (\gamma_{1i}\cdot d_{i1})^{-1}\cdot (\gamma_{1j}\cdot d_{j1}) = d_{i1}^{-1}\cdot \gamma_{ij}\cdot d_{ji}.$$ Note that if $c_{ij}$ is a framed cord, then $d_{i1}^{-1}\cdot \gamma_{ij}\cdot d_{ji}$ is a based loop in $\pi_K$. Moreover, $[c_{ij}]\mapsto [d_{i1}^{-1}\cdot \gamma_{ij}\cdot d_{ji}]$ defines an isomorphism $\textrm{Cord}(K) \xrightarrow {\sim} {{\mathcal{P}}}_K$. Therefore $$\epsilon_{{{\mathcal{P}}}_K}(g_i\cdot g_j^{-1}) = \epsilon_{\textrm{Cord}(K)}(\gamma_{ij}),$$ proving $R_{\textrm{Cord}(K)} = R_{{{\mathcal{P}}}_K}$. It is similar to check that the group actions also match.
As a conclusion, the construction in this paper explains the underlining geometry of the correspondence between augmentations and sheaves for knots.
A microlocal digression
=======================
Given an augmentation of the framed cord algebra of a link, we defined the associated augmentation representation in the previous section. It is a representation of the fundamental group of the link complement, which is equivalent to a locally system on the link complement. We will study properties of the augmentation representation in the next section. It is easier to understand these properties from the perspective of microlocal sheaf theory. Indeed, a local system can be viewed as a locally constant sheaf, and we can proceed to study it using microlocal methods. In this section, we give a quick introduction to microlocal sheaf theory.
A key concept is the micro-support of a sheaf. We introduce the definition in a general setting. Let $k$ be a commutative field. Let $Y$ be a smooth manifold. Let ${Mod}(Y)$ be the abelian category of sheaves of $k$-modules on $Y$, and $Sh(Y)$ be the bounded dg derived category. For any object ${{\mathcal{F}}}\in Sh(Y)$, its microsupport $SS({{\mathcal{F}}})\subset T^*Y$ is a closed conic subset, point-wisely defined as follows (also see [@KS Definition 5.1.1]).
Let ${{\mathcal{F}}}\in Sh(Y)$ and let $p = (y_0,\xi_0)\in T^*Y$. We say that $ p \notin {SS}({{\mathcal{F}}})$ if there exists an open neighborhood $U$ of $p$ such that for any $y\in Y$ and any real $C^1$ function $\phi$ on $Y$ satisfying $d\phi(y_0) \in U$ and $\phi(y_0)=0$, we have $$\label{microsupport}
R\Gamma_{\{\phi(y)\geq 0\}}({{\mathcal{F}}})_{y_0}\cong 0.$$
Kashiwara-Schapira defined the microlocal hom bifunctor [@KS Definition 4.1.1] $$\label{muhom}
\mu hom: Sh(Y)^{op}\otimes Sh(Y)\rightarrow Sh(T^*Y).$$ giving a quantitative description of the micro-support. Suppose ${{\mathcal{F}}},{{\mathcal{G}}}\in Sh(Y)$, then $$\label{muhomSSbound}
\textrm{supp } \mu hom ({{\mathcal{F}}},{{\mathcal{G}}}) \subset {SS}({{\mathcal{F}}})\cap {SS}({{\mathcal{G}}}).$$
Let $T^\infty Y$ be the unit cosphere bundle of $Y$. It admits a natural a contact induced form from the canonical form of the cotangent bundle. Let $\Lambda\subset T^\infty Y$ be a (not necessarily connected) smooth Legendrian submanifold. Define a dg subcategory $$Sh_\Lambda(Y) := \{{{\mathcal{F}}}\in Sh(Y) \,|\, SS({{\mathcal{F}}})\cap T^\infty Y \subset \Lambda\}.$$ Following a result of Guillermou-Kashiwara-Schapira [@GKS], $Sh_\Lambda(Y)$ is a Legendrian isotopy invariant. The abelian category $Mod(Y)$ can be viewed as a dg subcategory of $Sh(Y)$ consisting of objects concentrated in homological degree zero. Then $Mod_\Lambda(Y) := Sh_\Lambda(Y)\cap Mod(Y)$ is also a Legendrian isotopy invariant of $\Lambda$.
A sheaf ${{\mathcal{F}}}\in Sh_\Lambda(Y)$ is simple along $\Lambda$ if one of the following equivalent conditions holds:
- For any $p\in \Lambda$, the microlocal Morse cone defined as in (\[microsupport\]) has rank $1$, namely $$R\Gamma_{\{\phi(y)\geq 0\}}({{\mathcal{F}}})_{y_0} \cong k[d],\quad \textrm{for some }\ d\in {{\mathbb{Z}}}.$$
- The self microlocal hom restricts to a constant sheaf supported on $\Lambda$, namely $$\mu hom({{\mathcal{F}}},{{\mathcal{F}}})|_{T^\infty Y} = k_{\Lambda}.$$
We write $Sh^s_\Lambda(Y)\subset Sh_\Lambda(Y)$ for the subcategory of simple sheaves along $\Lambda$. The triangulated structure is lost when we pass to this subcategory.
There is a distinction between a sheaf ${{\mathcal{F}}}\in Sh_\Lambda(Y)$ being “simple along $\Lambda$” or “simple along its micro-support”. The two notions are equivalent if $SS({{\mathcal{F}}})\cap T^\infty Y = \Lambda$. However in the definition $Sh_\Lambda(Y)$, an object ${{\mathcal{F}}}$ is only required to have its micro-support intersecting $T^\infty Y$ in a subset of $\Lambda$. When $\Lambda$ has multiple connected components, such as in our case the conormal tori of links, the first notion is strictly stronger than the second (defining fewer objects).
In the case that we consider in this paper when $Y = X = {{\mathbb{R}}}^3$ or $S^3$, and $\Lambda = \Lambda_K$ the Legendrian conormal tori of a link $K$, simple sheaves microsupported along $\Lambda$ admit an easier description. The derivation is parallel to the case studied in [@Ga2] when $K$ is a knot. It is because both the micro-support and the simpleness are local properties, and the local pictures for knots and links are no different.
We outline the argument with references in [@Ga2]. For simplicity, we consider a sheaf ${{\mathcal{F}}}\in Mod_\Lambda(X)$ concentrated at homological degree $0$. The micro-support constraints force that ${{\mathcal{F}}}$ restricted to each component of the link, or the link complement is a local system [@Ga2 Lemma 3.1]. In terms of group representations, these local systems are equivalent to $$\begin{aligned}
&\rho: \pi_K \rightarrow GL(V), \\
&\rho_i: {{\mathbb{Z}}}_{K_i} \rightarrow GL(W_i), \;\;\textrm{for}\;\; 1\leq i\leq r.\end{aligned}$$ Here $\pi_K$ is the link group and ${{\mathbb{Z}}}_{K_i} := \pi_1(K_i)$. Conversely, we can reconstruct the sheaf from these local systems by gluing. The gluing data is an element in an extension class, and according to [@Ga2 Lemma 3.2, Lemma 3.3], it is equivalent to a collection of linear maps $$T_i: W_i\rightarrow V, \;\;\textrm{for}\;\; 1\leq i\leq r$$ which satisfy the following compatibility conditions. For each $1\leq i\leq r$, let $m_i, \ell_i$ be the meridian and longitude in the peripheral subgroup, then $T_i$ satisfies (a) $\rho(\ell_i)\circ T_i = T_i\circ \rho_i(K_i)$, and (b) $m_i$ acts on the image of $T_i$ as identity. We conclude that the sheaf ${{\mathcal{F}}}$ is equivalent to the collection of data $(\rho, V, \rho_i, W_i, T_i)$.
The sheaf ${{\mathcal{F}}}$ is simple along $\Lambda$ if and only if $cone(T_i)$ has rank $1$ for each $1\leq i\leq r$. It is simple along its singular support if and only if $cone(T_i)$ has rank at most $1$ for each $1\leq i\leq r$. These statements follow from the first of the equivalent definitions of simpleness.
In this paper, we focus on the representation of the link group, namely $(\rho,V)$. The simpleness imposes strong restrictions to this representation. If $cone(T_i)$ has rank $1$, then $T_i$ is either injection with a rank $1$ cokernel, or $T_i$ is surjective with a rank $1$ kernel. Consider the first case when $T_i$ is injective, then $W_i$ can be regarded as a subspace of $V$ of codimension $1$. The condition (b) of $T_i$ requires $m_i$ acting on a codimension one space as identity. In the other cases, including when rank$(cone(T_i)) =1$ with surjective $T_i$, or when rank$(cone(T_i)) = 0$, the action on $m_i$ is entirely trivial. Combining these cases, we see that for ${{\mathcal{F}}}$ to be simple, it is necessary to require $(\rho,V)$ satisfying that the action of each meridian $m_i$ fixes a subspace of codimension at most $1$.
Properties of the augmentation representation {#Sec:Properties}
=============================================
Given the framed cord algebra of a framed link, we have constructed a link group representation for each augmentation of the algebra. In this section, we study the properties of these representations.
Microlocal simpleness
---------------------
Let $Mod_{\Lambda_K}(X)$ be the abelian category of sheaves on $X$ microsupported along the link conormal $\Lambda_K$. Let $Mod^s_{\Lambda_K}(X) \subset Mod_{\Lambda_K}(X)$ be the full subcategory of simple sheaves. Suppose ${{\mathcal{F}}}\in Mod^s_{\Lambda_K}(X)$, then $j^{-1}{{\mathcal{F}}}$ is a local system on $X\setminus K$. The local system is equivalent to a representation $\rho: \pi_K\rightarrow GL(V)$. The simpleness of ${{\mathcal{F}}}$ requires that any meridian acts on $V$ as identity on a subspace of codimension $1$ or $0$.
Based on this observation, we make the following notion of simpleness.
Let $V$ be a vector space. A linear automorphism $T \in GL(V)$ is *almost identity* if there is a subspace $W\subset V$ of codimension $1$ such that $T|_W = \textrm{id}_W$.
Equivalently, $T$ is almost identity if and only if the rank of $(\textrm{id}_V - T)$ is at most $1$.
Suppose $K\subset X$ is a link. A link group representation $$\rho: \pi_K \rightarrow GL(V)$$ is *microlocally simple* if $\rho(m)$ is almost identity for every meridian $m\in \pi_K$.
\[propSimpleness\] Augmentation representations are microlocally simple.
Let $(\rho_\epsilon, V_\epsilon)$ be the augmentation representation associated to an augmentation $\epsilon: \textrm{Cord}(K)\rightarrow k$. Recall that $R$ is the $n\times n$ matrix determined by $\epsilon$. As a vector space, $V_\epsilon$ is spanned by the column vectors $R_j$, $1\leq j \leq n$. To verify that a linear automorphism $T \in GL(V_\epsilon)$ is almost identity, it is sufficient to show that $(\textrm{id}_V - T)R_j$ is contained in a rank $1$ subspace of $V$ for all $1\leq j \leq n$.
To prove $(\rho_\epsilon, V_\epsilon)$ is microlocally simple, we need to verify that every $T = \rho_\epsilon(m)$ with $m$ being a meridian is almost identity. Since any two meridians with coherent orientations belonging to the same component $K_i\subset K$ are conjugate, it suffices to select a meridian $m_i$ for each component $K_i$ and prove that $\rho_\epsilon(m_i)^{\pm 1}$ are both almost identity. It is easy to see that if $\rho_\epsilon(m_i)$ is almost identity, so is $\rho_\epsilon(m_i)^{-1}$. Finally recall that $\{m_t\}_{1\leq t\leq n}$ are meridian generators of the link group $\pi_K$. We observe that the generating set contains at least one meridian for each component.
Following these arguments, the assertion in the proposition reduces to show that for any generating meridian $m_t$, there exists a rank $1$ subspace $U_t\subset V_\epsilon$ such that $$(\textrm{id}_V-\rho_\epsilon(m_t))R_j \subset U_t, \quad\textrm{for } 1\leq j \leq n.$$ Recall the formula in (\[MeridianAction\]), $\rho_\epsilon(m_t) R_j = R_i - \epsilon({p}_t^{-1}\cdot p_j) R_t$. We take $U_t = \textrm{Span}_k\{R_t\}$, then $$(\textrm{id}_V -\rho_\epsilon(m_t))R_j = R_j - (R_i - \epsilon({p}_t^{-1}\cdot p_j) R_t) = \epsilon({p}_t^{-1}\cdot p_j) R_t \subset U_t.$$
We prove the desired result.
Microlocally simple knot group representations are called “KCH representations” or “unipotent KCH representations” in earlier papers [@Ng4; @Co; @Ga2]. We stop using these names for two reasons. First, the abbreviation “KCH” as an prefix of the representation emphasizes more on contact topology and its relation to sheaf theory, instead of the property of the representation itself. It is more revealing to borrow the notion of simpleness from microlocal sheaf theory. For the second reason, it is a locally property whether a meridian action is diagonalizable or unipotent. When we work with links, the previous naming system fails to generalize in a concise way when meridians of different components act differently. On the other hand, the word “microlocal” has a good implication that one should consider the local behaviors separately for each component of the link.
Vanishing
---------
Suppose $K'\subset K$ is a sublink. The sublink group is $\pi_{K'} = \pi_1(X\setminus K')$. Note this fundamental group is taken over the complement of $K'$ in $X$, forgetting that $K'$ is a sublink of $K$. The natural open inclusion $j: X\setminus K\rightarrow X\setminus K'$ induces a map on fundamental groups: $$\pi_K\rightarrow \pi_{K'}.$$ The composition defines a functor of abelian categories $Rep(\pi_{K'})\rightarrow Rep(\pi_K)$. If we identify representations of a fundamental group as local systems, the functor coincides the pull back functor $j^{-1}: loc(X\setminus K')\rightarrow loc(X\setminus K)$.
In the previous subsection, we proved that augmentations are microlocally simple. It follows from the definition that the group action of a meridian has two possibilities. It either fixes the entire vector space, or defines an invariant subspace of codimension one. In this subsection, we focus on the first case, and give a sufficient condition on augmentations so that the associated augmentation representations fit into this situation.
Let $K\subset X$ be a link and $K'\subset K$ be a sublink. A simple link representation $\rho: \pi_K\rightarrow GL(V)$ vanishes on $K'$ if $\rho(m' ) = \textrm{id}_V$ for any meridian $m'$ of $K'$.
The definition comes from an observation in the sheaf theory. Let ${{\mathcal{E}}}\in loc(X\setminus K)$ be the local system determined by a link group representation $(\rho,V)$. Let $j: X\setminus K\rightarrow X$ be the open embedding. We consider the underived push forward ${{\mathcal{F}}}= j_*{{\mathcal{E}}}$. If $(\rho,V)$ is vanishes on $K'$, then ${{\mathcal{F}}}$ is microsupported on the components other than $K'$, i.e. $$SS({{\mathcal{F}}}) \cap T^\infty X \subset \Lambda_{K\setminus K'}.$$
\[propVanishing\] Let $K\subset X$ be a link and $K'\subset K$ sublink. Suppose an augmentation $\epsilon: \textrm{Cord}(K)\rightarrow k$ satisfies either one the following conditions:
- for any framed cord $\gamma$ starting from $K'$, $\epsilon(\gamma) = 0$; or
- for any framed cord $\gamma$ ending on $K'$, $\epsilon(\gamma) = 0$,
then the associated augmentation representation $(\rho_\epsilon, V_\epsilon)$ vanishes on $K'$.
By Lemma \[sortingLem\], we can assume there is an integer $s$ with $1\leq s\leq n$, such that the closure of strands $\{1,\dotsb, s\}$ is precisely the sublink $K'$.
By definition, the associated augmentation representation vanishes on $K'$ if for any meridian $m_t$, $1\leq t\leq s$, the action $\rho_\epsilon(m_t)$ is identity. Recall from (\[MeridianAction\]) that $$\rho_\epsilon(m_t) R_j = R_j- \epsilon({p}_t^{-1}\cdot p_j) R_t.$$ Therefore it suffices to show that either $\epsilon({p}_t^{-1}\cdot p_j) = 0$ for any $j \in \{1,\dotsb, n\}$, or $R_t$ is the zero vector.
Suppose $\epsilon$ maps all cords starting on $K'$ to zero. For any $t\in \{1,\dotsb, s\}$ and any $j \in \{1,\dotsb, n\}$, $p_t^{-1}\cdot p_j$ is a cord starting on $K'$. Hence $\epsilon({p}_t^{-1}\cdot p_j) = 0$ and the assertion follows.
Suppose $\epsilon$ maps all cords ending on $K'$ to zero. For any $t\in \{1,\dotsb, s\}$, $R_t$ consists of augmentations of cords ending on $K'$. Therefore $R_t$ is a zero vector as expected.
In each of the cases, we have proven the assertion.
Separability
------------
In this section, we discuss a sufficient condition for which the augmentation representation is separable. We begin with an example to motivate the notion of separability.
We consider the two-component unlink and Hopf link (Figure \[Fig:UnlinkandHopf\]). Each of them has two components, and every component is an unknot. A link is *split* if it is the union of two sublinks that lie in two disjoint solid balls. In our example, the unlink is split while Hopf link is non-split.
In either case, the framed cord algebra is generated over ${{\mathbb{Z}}}[\mu_1^{\pm 1}, \lambda_2^{\pm 1}, \mu_2^{\pm 1},\lambda_2^{\pm 1}]$ by four cords, $a_{11}, a_{12}, a_{21}, a_{22}$. The subscripts label the link components to which end points of a cord belong. (They are not standard cords, which is why we did not use $\gamma_{ij}$.) In $\textrm{Cord}^c(K_{\textrm{Hopf}})$, generators subject to the following relations: $$\begin{aligned}
& (\lambda_1\mu_1\lambda_2^{-1}\mu_2^{-1}-1)a_{12}=0,\\
& a_{21}(1-\lambda_1^{-1}\mu_1^{-1}\lambda_2\mu_2)=0,\\
& 1-\lambda_1-\mu_1 +\lambda_1\mu_1+\lambda_1\mu_1\mu_2^{-1}a_{12}a_{21}=0,\\
& 1 -\lambda_2-\mu_2 +\lambda_2\mu_2 + \lambda_2a_{21}a_{12}=0, \\
& a_{12} +\lambda_1\mu_2^{-1}a_{12}(\mu_1\mu_2 - \mu_1 - \mu_2 + \mu_1a_{12}a_{21})=0, \\
& \lambda_2a_{21}-\mu_1a_{21}=0.\end{aligned}$$ Pure cords do not appear in these equations, because in these two cases, a pure cord is cord homotopic to a constant cord, and can be replaced by $1-\mu_{\{i\}}$ using meridian relations. An augmentation $\epsilon: {\textrm{Cord}^c(K_{\textrm{Hopf}})}\rightarrow k$ is in one the following two non-exclusive cases (we abbreviate $\epsilon(\mu_i),\epsilon(\lambda_i)$ as $\mu_i,\lambda_i$):
1. $\mu_1 = \lambda_2$, $\mu_2 = \lambda_1$, and $\epsilon(a_{12})\epsilon(a_{21}) = (1-\mu_1^{-1})(1-\mu_2)$; or
2. $\epsilon(a_{12}) = \epsilon(a_{21}) = 0$, and $1 - \lambda_1 - \mu_1 + \lambda_1\mu_1 = 1 - \lambda_2 - \mu_2 +\lambda_2\mu_2 = 0$.
Recall the augmentation variety in [@Ng4], $$V_K =\big\{\big(\epsilon(\mu_1), \epsilon(\lambda_1),\dotsb, \epsilon(\mu_r), \epsilon(\lambda_r)\big)\,|\, \epsilon: {\textrm{Cord}^c(K_{\textrm{Hopf}})}\rightarrow k\big\}\subset (k^*)^{2r}.$$ Therefore he augmentation variety of Hopf link is $$V_{\textrm{Hopf}}
=\left \{
\begin{matrix}
\mu_1 = \lambda_2 \\
\mu_2 = \lambda_1
\end{matrix}
\right \}
\cup
\left \{
\begin{matrix}
1 - \lambda_1 - \mu_1 + \lambda_1\mu_1 =0\\
1 - \lambda_2 - \mu_2 +\lambda_2\mu_2 =0
\end{matrix}
\right\} = V_A \cup V_B.$$ We can similarly compute the augmentation variety of the two-component unlink $$V_{\textrm{unlink}}
=
\left \{
\begin{matrix}
1 - \lambda_1 - \mu_1 + \lambda_1\mu_1 =0\\
1 - \lambda_2 - \mu_2 +\lambda_2\mu_2 =0
\end{matrix}
\right\} = V_B.$$
We observe that the augmentation variety of the unlink is contained in that of Hopf link, corresponding to case (B) when $\epsilon(a_{12}) = \epsilon(a_{21}) =0$. In this case, all mixed cords are augmented to zero. Though Hopf link is non-split, some augmentations behave as if the framed cord algebra came from a split link. We remark that the idea of sending mixed cords to $0$ has been considered in contact geometry such as: [@Mi; @AENV].
We propose a counterpart of this phenomenon on the sheaf side. The following lemma will show that the direct sum of two simple sheaves which are microsupported along disjoint Legendrians is again simple.
\[DirectSum\] Let $Y$ be a manifold, and $\Lambda_1,\Lambda_2\subset T^\infty Y$ two disjoint Legendrian submanifolds. If ${{\mathcal{F}}}_1 \in D^{b,s}_{\Lambda_1}(Y), {{\mathcal{F}}}_2 \in D^{b,s}_{\Lambda_2}(Y)$, then ${{\mathcal{F}}}_1\oplus {{\mathcal{F}}}_2 \in D^{b,s}_{\Lambda_1\sqcup\Lambda_2}(Y)$.
Because $\Lambda_1,\Lambda_2$ are disjoint, $\mu hom({{\mathcal{F}}}_1,{{\mathcal{F}}}_2)|_{T^\infty Y} = \mu hom({{\mathcal{F}}}_2,{{\mathcal{F}}}_1)|_{T^\infty Y} = 0$. The simpleness of ${{\mathcal{F}}}_i$ yields $\mu hom({{\mathcal{F}}}_i,{{\mathcal{F}}}_i)|_{T^{\infty}Y} = k_{\Lambda_i}$. Finally we have $$\begin{aligned}
\mu hom({{\mathcal{F}}}_1\oplus {{\mathcal{F}}}_2, {{\mathcal{F}}}_1\oplus {{\mathcal{F}}}_2)|_{T^\infty Y}
&= \mu hom({{\mathcal{F}}}_1, {{\mathcal{F}}}_1)|_{T^\infty Y} \oplus \mu hom({{\mathcal{F}}}_2, {{\mathcal{F}}}_2)|_{T^\infty Y} \\
&= k_{\Lambda_1}\oplus k_{\Lambda_2} = k_{\Lambda_1\sqcup \Lambda_2}.\end{aligned}$$
Now we present a sufficient condition on augmentations so that the associated augmentation representations split into direct summands.
Suppose $K = K_{1} \sqcup K_{2}$ is the union of two sublinks $K_1, K_2$. A link group representation $\rho: \pi_K\rightarrow GL(V)$ is separable with respect to the partition if there exists link group representations $\rho_i: \pi_{K_i} \rightarrow GL(V_i)$ for each $i= 1,2$, such that $$(\rho, V) = (\rho_1,V_1)\oplus (\rho_2, V_2),$$ where $V_i$ is considered as a $\pi_K$-representation through the composition $$\pi_K\rightarrow \pi_{K_i}\rightarrow GL(V_i).$$
\[separability\] Suppose $K = K_{1} \sqcup K_{2}$ is the union of two sublinks $K_1, K_2$. If an augmentation $\epsilon: \textrm{Cord}(K)\rightarrow k$ maps all mixed cord between $K_1$ and $K_2$ to zero, the induced augmentation representation is separable with respect to the partition.
To simplify the presentation of the proof, we apply Lemma \[sortingLem\] and assume there is an integer $s$ with $1\leq s\leq n-1$, such that the closure of strands $P_1 := \{1,\dotsb, s\}$ is $K_1$, and the closure of strands $P_2 :=\{s+1,\dotsb, n\}$ is $K_2$.
We define the sublink group representation $\rho_1: \pi_{K_1}\rightarrow GL(V_1)$ for $K_1$. The construction for $K_2$ is similar. Consider pure cords of $K_1$, whose augmented values form an $s\times s$ matrix: $$(\tilde{R}_1)_{ij} = \epsilon(\gamma_{ij}), \quad 1\leq i,j\leq s.$$ Let $\tilde{R}_j$, $1\leq j\leq s$ be the column vectors of $\tilde{R}_1$. We define $$V_1:= \textrm{Span}_k\{\tilde{R}_j\}_{1\leq j\leq s}.$$
The sublink group $\pi_{K_1}$ acts on $V_1$ in the following way. For $\gamma\in \pi_{K_1}$, we define $${\rho_1}(\gamma) \tilde{R}_j = \epsilon(p_\alpha^{-1} \cdot \gamma \cdot p_j),\quad \textrm{for } 1\leq j\leq s.$$
The procedure to check that $(\rho_1, V_1)$ is a representation of $\pi_{K_1}$ is similar to that in Theorem \[MainConstruction\]. We remark two subtleties in the construction. First, $\pi_{K_1}$ is generated by meridians $m_t$, $1\leq t\leq s$, which is a subset of meridian generators of $\pi_K$, but the quotient relations are different from those in $\pi_K$. Second, we take skein relations in $\textrm{Cord}(K_1)$ instead of those in $\textrm{Cord}(K)$.
From now on, we abuse notations and regard $(\rho_i,V_i), i=1,2$ as a representation of $\pi_K$. Taking the direct sum, we get a representation $$\rho_1 \oplus \rho_2: \pi_K \rightarrow GL(V_1 \oplus V_2).$$
We prove that $(\rho_1 \oplus \rho_2, V_1\oplus V_2)$ is isomorphic to the augmentation representation $(\rho_\epsilon,V_\epsilon)$. Recall that $V_1$ is spanned by $\{\tilde{R}_{j}\}_{1\leq j\leq s}$, $V_2$ is spanned by $\{\tilde{R}_{j}\}_{s+1\leq j\leq n}$, and $V_\epsilon$ is spanned by $\{R_j\}_{1\leq j\leq n}$. We define a morphism between vector spaces: $$\varphi: V_1\oplus V_2 \rightarrow V_\epsilon, \qquad \varphi(\tilde{R}_j) = R_j.$$ It is an isomorphism of vector spaces. By hypothesis, $R_{ij} =0 $ for both $1\leq i \leq s< j \leq n$ and $1\leq j \leq s< i \leq n$. Therefore the intersection of $\textrm{Span}\{R_j\}_{1\leq j\leq s}$ and $\textrm{Span}\{R_j\}_{s+1\leq j\leq n}$ is trivial. Further by definition, linear relations in $V_1$ are preserved in $\textrm{Span}\{R_j\}_{1\leq j\leq s}$, and similar for $V_2$. We conclude the vector space isomorphism.
It remains to check that $\varphi$ preserves the group action. Namely, for any $v \in V_1\oplus V_2$ and any $\gamma\in \pi_K$, we want $$\label{GrpActionForSep}
\rho_\epsilon(\gamma)\cdot \varphi(v) = \varphi ((\rho_1 \oplus \rho_2)(\gamma)\cdot v).$$
It suffices to verify the equation for any spanning vector $\tilde{R}_j$ and any generating meridian $m_t$. The left hand side of (\[GrpActionForSep\]) becomes, $$\rho_\epsilon(m_t) \cdot \varphi (\tilde{R}_j) = \rho_\epsilon(m_t) {R}_j = \rho_\epsilon(m_t) R_j = R_j- \epsilon({p}_t^{-1}\cdot p_j) R_t.$$ Recall $\{-\}$ is the component function. If $K_{\{t\}}$ and $K_{\{j\}}$ belong to different sublink, then $\epsilon({p}_t^{-1}\cdot p_j) = 0$ by hypothesis. The equation can be further simplified to $$\rho_\epsilon(m_t) \cdot \varphi (\tilde{R}_j) = R_j.$$ For the right hand side of (\[GrpActionForSep\]), we compute in two cases.
- If $K_{\{t\}}$ and $K_{\{j\}}$ belong to the same sublink, then $(\rho_1 \oplus \rho_2)(m_t)\cdot \tilde{R}_j = \tilde{R}_j- \epsilon({p}_t^{-1}\cdot p_j) \tilde{R}_t$, and $$\varphi ((\rho_1 \oplus \rho_2)(m_t)\cdot \tilde{R}_j) = {R}_j- \epsilon({p}_t^{-1}\cdot p_j) {R}_t.$$
- If $K_{\{t\}}$ and $K_{\{j\}}$ belong to different sublinks, then $(\rho_1 \oplus \rho_2)(m_t)\cdot \tilde{R}_j = \tilde{R}_j$, and $$\varphi ((\rho_1 \oplus \rho_2)(m_t)\cdot \tilde{R}_j) = R_j.$$
Comparing the two sides of (\[GrpActionForSep\]), we have $\rho_\epsilon(\gamma)\cdot \varphi(v) = \varphi ((\rho_1 \oplus \rho_2)(\gamma)\cdot v)$ as desired.
We complete the proof.
In the proof, we define an induced representation for a sublink. It is natural to ask whether this construction is functorial. In general it is not possible, because for a sublink $K_1\subset K$, one does not have a natural embedding $\textrm{Cord}(K_1)\rightarrow \textrm{Cord}(K)$.
The natural morphism between these two framed cord algebras is in the opposite direction: $$\textrm{Cord}(K)\rightarrow \textrm{Cord}(K_1).$$ Under the hypothesis of Proposition \[separability\], an augmentation $\epsilon: \textrm{Cord}(K)\rightarrow k$ factors through this morphism, giving an induced augmentation $\epsilon_1$. The augmentation representation of $\epsilon_1: \textrm{Cord}(K_1)\rightarrow k$ is isomorphic to $\rho_1: \pi_{K_1}\rightarrow GL(V_1)$ constructed in the proof. It is similar for $\epsilon_2: \textrm{Cord}(K_2)\rightarrow k$.
A natural question is whether the augmentation representation is irreducible. The answer is no. If $K$ is a link, then any example in Proposition \[separability\] such that $V_1,V_2$ are non-trivial is reducible.
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|
---
abstract: 'Schemes for topological quantum computation with Majorana bound states rely heavily on the ability to measure products of Majorana operators projectively. Here, we employ Markovian quantum measurement theory, including the readout device, to analyze such measurements. Specifically, we focus on the readout of Majorana qubits via continuous charge sensing of a tunnel-coupled quantum dot by a quantum point contact. We show that projective measurements of Majorana products $\prod_i\hat{\gamma}_i$ can be implemented by continuous charge sensing under quite general circumstances. Essential requirements are that a combined local parity $\hat{\pi}$, involving the quantum dot charge along with the Majorana product of interest, be conserved, and that the two eigenspaces of the combined parity $\hat{\pi}$ generate distinguishable measurement signals. We find that qubit readout may have to rely on measuring noise correlations of the quantum-point-contact current. The average current encodes the qubit readout only transiently for fine-tuned parameters or in the presence of relaxation processes. We also discuss the corresponding measurement and decoherence times and consider processes such as residual Majorana hybridizations which are detrimental to the measurement protocol. Finally, we emphasize that the underlying mechanism – which we term symmetry-protected readout – is quite general and has further implications for both, Majorana and non-Majorana systems.'
author:
- 'Jacob F. Steiner and Felix von Oppen'
bibliography:
- 'refs.bib'
title: Readout of Majorana Qubits
---
Introduction
============
Qubits based on Majorana bound states promise key advantages for quantum computing, including long intrinsic lifetimes deriving from the nonlocal encoding of quantum information [@kitaev_fault-tolerant_2003; @nayak_non-abelian_2008; @Oreg2020] and topologically protected single-qubit gates based on braiding or, alternatively, on exploiting measurements in all Pauli bases. A popular Majorana qubit – known as Majorana box qubit [@plugge_majorana_2017] or tetron and hexon [@karzig_scalable_2017] – is based on semiconductor quantum wires proximity coupled to a superconductor [@lutchyn_majorana_2010; @oreg_helical_2010; @alicea_non-abelian_2011]. These qubits are believed to be within experimental reach [@lutchyn_majorana_2018] and quantum computing architectures have been developed on their basis [@vijay_majorana_2015; @plugge_roadmap_2016; @karzig_scalable_2017; @litinski_quantum_2018]. Quantum computation with Majorana qubits is expected to rely heavily on projective qubit measurements, with all Clifford gates implemented using single and two-qubit measurements [@Zilberberg2008; @karzig_scalable_2017; @litinski_quantum_2018]. Indeed, these schemes can be referred to as measurement-based topological quantum computing and good readout fidelities are absolutely central to their performance. This makes it essential to develop a detailed theoretical understanding of the proposed readout schemes for Majorana-based topological qubits.
While a blessing for their characteristics as a quantum memory, the nonlocal nature of topological qubits complicates the readout of the encoded quantum information. Readout requires one to make the nonlocally encoded quantum information available locally. This can be achieved by exploiting interference effects which are sensitive to the Majorana parity operator of interest [@fu_electron_2010; @plugge_majorana_2017; @karzig_scalable_2017]. A schematic Majorana qubit is shown in Fig. \[fig:tetron\_quantum\_dot\_measurement\_device\] and involves four Majorana bound states located at the ends of two proximity-coupled semiconductor quantum wires. The Pauli operators associated with the Majorana qubit are parity operators involving products of two Majorana operators. Here, we focus on a readout procedure which measures these Majorana parity operators by tunnel coupling a quantum dot to the relevant pair of Majoranas as shown in Fig. \[fig:tetron\_quantum\_dot\_measurement\_device\]. Virtual tunneling processes between Majorana qubit and quantum dot shift the energy levels of the quantum dot in a manner that depends on the Majorana parity. As a result, the coupled time evolution of Majorana qubit and quantum dot entangles the two, and the charge state of the quantum dot becomes correlated with the Majorana parity. Measurements of the quantum dot charge, for instance by a nearby quantum point contact, can thus be used to read out the Majorana qubit.
![Setup for readout of Majorana qubit. A quantum dot (QD) is tunnel coupled to a Majorana qubit consisting of two topological superconducting wires (dark blue) with four Majorana bound states $\hat{\gamma}_1,\ldots,\hat{\gamma}_4$. The wires are connected by a conventional superconducting bridge (SC) allowing charge to move freely between the wires, so that only the overall charge of the Majorana qubit is fixed by the charging energy. The Majorana parity $\hat Z = - i\hat{\gamma}_1\hat{\gamma}_2$ defines the Pauli-$Z$ operator of the Majorana qubit, and can be read out by tunnel coupling the two Majoranas $\hat{\gamma}_1$ and $\hat{\gamma}_2$ to the quantum dot. The quantum dot charge is measured by capacitively coupling (with strength $k\propto |\chi|^2$) the dot to a quantum point contact.[]{data-label="fig:tetron_quantum_dot_measurement_device"}](tetron_quantum_dot_qpc_detailed.pdf){width="0.9\columnwidth"}
In principle, it is possible to design parity-to-charge conversion procedures which allow for a projective measurement of the Majorana parity based on a single-shot projective measurement of the charge $\hat{n}$ of the quantum dot [@plugge_majorana_2017]. However, these schemes are not robust and require fine tuning and rapid manipulation of system parameters. More generically, the charge state of the quantum dot becomes only weakly correlated with the state of the Majorana qubit, and qubit readout requires multiple measurements. This can be achieved by repeatedly coupling and decoupling qubit and quantum dot, with intervening projective measurements of $\hat{n}$ and resets of the qubit charge state.
In practice this coupling, decoupling, and resetting is challenging and prone to errors. It would be preferable and more natural to keep Majorana qubit and quantum dot coupled during the entire readout procedure and to *monitor* the charge of the quantum dot continuously. Here, we show that a projective readout of the Majorana qubit can indeed be robustly implemented in this manner. In particular, our strategy significantly relaxes the requirements on dynamical control over system parameters, and obviates the need for resets of the quantum dot charge state.
To describe the dynamics of the quantum measurement, we include the measurement device in the theoretical description. The continuous measurement decoheres the system in the basis of the quantum dot charge and outputs the noisy measurement signal $j(t)$ of the quantum point contact [@wiseman_quantum_2009]. The task is then twofold. First, one needs to show that the system of Majorana qubit and quantum dot decoheres in the actual basis of interest associated with the parity operator of the Majorana qubit. Second, one needs to ascertain that the measurement outcome can be extracted from the signal $j(t)$. Both criteria must be satisfied to effectively implement a projective (Born rule) readout of the Majorana qubit which can be employed for measurement-based topological quantum computation.
In section \[sec:basics\_majorana\_qubits\], we introduce the system under consideration, a Majorana qubit with two of the four Majorana bound states tunnel coupled to a quantum dot as illustrated in Fig. \[fig:tetron\_quantum\_dot\_measurement\_device\], and discuss how under idealized assumptions, quantum dot charge measurements can be used for single-shot readout of the Majorana qubit. We then turn to more realistic readout protocols which rely on continuous monitoring of the quantum dot charge, collecting our central results in Sec. \[sec:qpc\_readout\]. The basic master-equation formalism describing weak measurements of the quantum dot charge by a quantum point contact is described in Sec. \[sec:contcharmeas\]. As a backdrop, we first illustrate the formalism in Sec. \[sec:simple\_charge\_readout\] by reviewing charge monitoring of a quantum dot in the absence of coupling to a Majorana qubit. We then include the coupling to the Majorana qubit in Sec. \[sec:majorana-qubit\_readout\] and show how a two-Majorana parity (Pauli operator) of the Majorana qubit can be read out. While it suffices to monitor the average quantum-point-contact current for charge readout of an uncoupled quantum dot, we find that in general, readout of the Majorana qubit requires one to measure noise correlations of the current. Readout based on noise correlations can be avoided by tuning to a sweet spot in parameter space or, as shown in Sec. \[sec:with\_relaxation\], by including additional processes which cause relaxation of the coupled Majorana qubit-quantum dot system to its ground state. In both of these cases, it suffices in principle to monitor the average quantum-point-contact current. In Sec. \[sec:imperfect\], we discuss various processes which are detrimental to the readout protocol. Most importantly, the previous sections assume that the residual Majorana hybridizations of the qubit are negligible, and we show here how these hybridizations affect the measurement protocol. Finally, Sec. \[sec:qpc\_readout\] closes with a discussion of alternative readout schemes which rely on coupling the Majorana qubit to double quantum dots, see Sec. \[sec:DQDreadout\]. We find that this readout scheme adds flexibility in designing the coupling between Majorana qubit and quantum dots. We also discuss readout of Majorana parity operators involving more than two Majoranas, which represent two-qubit parities or stabilizer operators of topological quantum error correcting codes. While our results are mostly analytical in nature, we illustrate the various measurement protocols by simulations of the stochastic master equation. In addition to our analytical estimates throughout Sec. \[sec:qpc\_readout\], these simulations also illustrate the required measurement times. Sec. \[sec:general\_readout\_analysis\] discusses the measurement protocols from a more general point of view, not restricted to the readout of Majorana qubits. We finally summarize and conclude in Sec. \[sec:discussion\]. Throughout the paper, we focus on the principal arguments and results. Explicit calculations and background material are relegated to a series of appendices.
Majorana Qubits and Quantum Dot Readout {#sec:basics_majorana_qubits}
=======================================
\[sec:majorana\_qubits\_and\_quantum\_dots\]
Majorana qubit coupled to quantum dot
-------------------------------------
Majorana qubits are Coulomb-blockaded islands hosting $2m$ Majorana bound states $\hat\gamma_j$ as described by the Hamiltonian $$\label{eq:charging_ham}
\hat{H}_{\textrm{M}} =
E_C\pqty{\hat{N}-N_g}^2 +\ i\sum_{i < j}^{2m} \varepsilon_{ij}\hat{\gamma}_i\hat{\gamma}_j.$$ The first term reflects the charging energy $E_C$ of the device, which depends on the total charge $\hat{N}$ as well as a gate-controlled offset $N_g$. For a fixed charge and well-separated Majorana bound states, the ground state of the system is $2^{m-1}$-fold degenerate. Residual splittings are included in $H_M$ through the $\varepsilon_{ij}$. Above-gap excitations of the Majorana wires are ignored by virtue of a sufficiently large gap.
The minimal number of Majoranas required for a single qubit is four, in which case the Majorana island realizes a Majorana box qubit or tetron. Figure \[fig:tetron\_quantum\_dot\_measurement\_device\] shows such a Majorana qubit assembled from a pair of topological superconducting quantum wires hosting two Majorana bound states each. The superconducting bridge between the quantum wires provides a sufficiently large mutual capacitance so that the charging energy depends on the charges of the individual wires only via the total charge of the device [@karzig_scalable_2017]. For definiteness, we choose $N_g = 0$, so that the ground-state manifold has even fermion parity, $\hat P= (i\hat\gamma_1\hat\gamma_4) (i\hat\gamma_2\hat\gamma_3)=1$. We can then define the Pauli operators
$$\begin{aligned}
\hat{Z} &=& -i\hat{\gamma}_1\hat{\gamma}_2 \\
\hat{X} &=& -i\hat{\gamma}_2\hat{\gamma}_3 \\
\hat{Y} &=& -i\hat{\gamma}_3 \hat{\gamma}_1\end{aligned}$$
of the qubit. Fermion parity conservation implies that one can alternatively use the operators $\hat{Z}' = \hat{P} \hat{Z}$ (with $\hat{X}'$ and $\hat{Y}'$ defined analogously).
Readout of the qubit operators (say, $\hat Z$ for definiteness) can be effected by connecting the Majorana island to a quantum dot via tunnel junctions as depicted in Fig. \[fig:tetron\_quantum\_dot\_measurement\_device\] [@plugge_majorana_2017; @karzig_scalable_2017]. We assume that the quantum dot has a single nondegenerate level $\epsilon$, which is spin resolved due to the magnetic field required for realizing topological superconductivity. Then, the quantum dot is described by the Hamiltonian $$\label{eq:ham_qd}
\hat{H}_{\textrm{QD}} = \epsilon \hat{n},$$ where the quantum dot occupation $\hat{n} =\hat{d}^{\dagger}\hat{d}$ involves the annihilation operator $\hat d$ of the gate-tunable dot level. We assume that the quantum dot is unoccupied in the ground state, $n=0$.
Tunneling between quantum dot and Majorana qubit is described by $$\label{eq:tetron_qd}
\hat{H}_{\textrm{T}} = \pqty{t_1 \hat{\gamma}_1+t_2\hat{\gamma}_2}e^{i\hat{\phi}/2} \hat{d} + \textrm{h.c.}$$ Electrons tunneling into or out of the Majorana island affect the state of the system within the ground-state manifold and change its charge. We describe the first effect through the Majorana operators $\hat{\gamma}_j$ (which leave the charge state of the system unchanged, $[\hat{\gamma}_i,\hat{N}] = 0$), and the second through a charge shift operator $e^{i\hat{\phi}/2}$ (with $[\hat{\phi},\hat{N}] = 2i$ and $[\hat{\gamma}_i,\hat{\phi}] = 0$). In this formulation, physical states must satisfy the total parity constraint $(-i)^m\prod_{i=1}^{2m}\hat{\gamma}_{i} = (-1)^{\hat{N}}$.
For a topological qubit, we assume that the hybridizations $\varepsilon_{ij}$ are negligible, so that the Majorana bound states are true zero-energy modes of the Majorana island. This implies that in addition to the fermion parity $\hat P$ of the qubit, also all two-Majorana parities $i\hat\gamma_i\hat\gamma_j$ are good quantum numbers. In particular, this is the case for the Pauli-$\hat Z$ operator of the Majorana qubit. The resulting degeneracy is partially lifted when tunnel coupling the Majorana qubit to the quantum dot. It is important to notice, however, that unlike the Pauli-$\hat Z$ operator, the combined fermion parity operator $$\label{eq:combined_parity}
\hat{\pi} = \hat{Z}\pqty{-1}^{\hat{n}}$$ remains a conserved quantity [@akhmerov_topological_2010]. Restricting ourselves to states which have total fermion parity $P(-1)^n=1$ and can thus be reached from the ground state with $n=N=0$ by tunneling, the Hamiltonian $\hat{H} = \hat{H}_{\textrm{M}} + \hat{H}_{\textrm{QD}} + \hat{H}_{\textrm{T}} $ becomes block diagonal in the subspaces of the combined fermion parity $\hat\pi$, $$\label{eq:hamiltonian_parities}
\hat{H} = \left(\begin{array}{cc} h_+ & 0 \\ 0 & h_- \end{array}\right).$$ Here, we choose the basis $\{\ket{\uparrow,0},\ket{ \downarrow,1},\ket{\downarrow,0},\ket{\uparrow,1}\}$ with the first entry distinguishing the eigenstates of $\hat Z$ and the second entry denoting the occupation of the quantum dot level, see App. \[app:definitions\] for details. We use lower-case letters without hats to denote operators within subspaces of fixed combined parity $\hat \pi$. The corresponding $2\times2$ blocks take the explicit form $$\label{eq:sector_hams}
h_{\pi} =
\begin{pmatrix}
0 & t_1 - i\pi t_2 \\ t^*_1 + i\pi t^*_2 & \varepsilon
\end{pmatrix}
=
\frac{\varepsilon}{2} + \Omega_{\pi} \bm{h}_{\pi}\cdot \bm{\sigma},$$ where $\bm{\sigma}$ denotes a vector of Pauli matrices and $\varepsilon = \epsilon + E_C$ the detuning (which depends on both, the level energy $\epsilon$ of the quantum dot and the charging energy $E_C$ of the Majorana island). We also defined a Bloch vector $$\label{eq:bloch_vec}
\bm{h}_\pi = (\sin\theta_\pi \cos\phi_\pi, \sin\theta_\pi \sin\phi_\pi,\cos\theta_\pi)$$ with $\cos\theta_\pi = -\varepsilon/2 \Omega_\pi$ and $\sin\theta_\pi e^{i\phi_\pi} = (t^*_1+ i\pi t^*_2)/\Omega_\pi$, as well as the Rabi frequency $\Omega_{\pi}^2 = \varepsilon^2/4 + \abs{t^*_1 +i\pi t^*_2}^2$. The eigenenergies and Rabi frequencies of the two subspaces are different provided that $\Im\pqty{t_1 t_2^*} \neq 0$.
Basic measurement protocols {#sec:basic_protocols}
---------------------------
The tunnel couplings entangle the Pauli-$\hat Z$ operator of the Majorana qubit with the charge of the quantum dot. The entanglement emerges from processes in which an electron virtually occupies the Majorana island through one Majorana involved in $\hat Z$ and leaves it through the other. This makes the quantum information stored nonlocally in the Majorana qubit accessible locally in the quantum dot. We first discuss to which degree a single projective measurement of the charge of the quantum dot realizes a measurement of $\hat Z$.
We begin with a protocol in which the tunnel coupling between dot and Majorana qubit is turned on adiabatically on the scale of the system dynamics. We assume that prior to turning on the tunnel couplings, the Majorana qubit is in an arbitrary qubit state and the quantum dot is initialized in the $n=0$ state, $$\ket{\psi}=\pqty{\alpha\ket{\uparrow}+\beta\ket{\downarrow}} \ket{0}.
\label{initialstate}$$ Adiabatically turning on the tunnel couplings $t_1$ and $t_2$, this state evolves into the corresponding eigenstate of the coupled system, $$\label{eq:adiabatic_turn_on}
\ket{\psi'}=\ \alpha\ket{g_+}+\beta e^{i\chi}\ket{ g_- }.$$ Here, the relative phase $\chi$ depends on details of the protocol, and we defined the exact ground states
\[eq:ground\_states\] $$\begin{aligned}
\ket{g_+} &=& \sin\frac{\theta_+}{2} \ket{\uparrow,0} -\cos\frac{\theta_+}{2}e^{i\phi_+}
\ket{\downarrow,1}
\\
\ket{g_-} &=& \sin\frac{\theta_-}{2} \ket{\downarrow,0} -\cos\frac{\theta_-}{2}e^{i\phi_-}
\ket{\uparrow,1}\end{aligned}$$
of $h_\pm$. Notice that $\theta_\pm=\pi$ in the absence of the tunnel couplings. Then, a subsequent projective measurement of the charge $n$ of the quantum dot either yields $n=0$ and the state $$\label{eq:adiabatic_post_measurement_state_0}
\ket{\psi'_0}= \frac{1}{\sqrt{p'_0}}\pqty{ \alpha \sin\frac{\theta_+}{2} \ket{\uparrow} + \beta e^{i\chi} \sin\frac{\theta_-}{2} \ket{\downarrow} } \ket{0},$$ with probability $$p'_0=|\alpha|^2 \sin^2\frac{\theta_+}{2}
+|\beta|^2 \sin^2\frac{\theta_-}{2},$$ or $n=1$ and $$\label{eq:adiabatic_post_measurement_state_1}
\ket{\psi'_1}= \frac{1}{\sqrt{p'_1}}\pqty{ \alpha \cos\frac{\theta_+}{2}e^{i\phi_+} \ket{\downarrow} + \beta e^{i\chi} \cos\frac{\theta_-}{2}e^{i\phi_-} \ket{\uparrow} } \ket{1},$$ with probability $p'_1=1-p'_0$.
Unfortunately, such charge measurements provide only partial information on the qubit state. One can arrange perfect correlation between the measurement outcome $n=0$ and, say, the $\ket{\uparrow}$ state of the qubit by fine tuning $\sin\frac{\theta_-}{2}=0$. However, one readily proves that $$\abs{\sin^2\frac{\theta_+}{2}-\sin^2\frac{\theta_-}{2}}\leq \frac{1}{2}.$$ This implies that a measurement outcome of $n=1$ remains compatible with both qubit states even if $n=0$ is perfectly correlated with the $\ket{\uparrow}$ state.
In principle, a projective measurement can be implemented when turning on the quantum dot-Majorana qubit tunneling instantaneously. In this case, the initial state $\ket{\psi}$ is no longer an eigenstate, and its unitary evolution under the Hamiltonian $\hat H$ entangles qubit and quantum dot. Depending on the measurement outcome, a projective charge measurement after a waiting time $T$ yields the states
$$\begin{aligned}
\ket{\psi''_0}=&\ \frac{1}{\sqrt{p''_0}}\left[ \alpha A_+(T) \ket{\uparrow} + \beta A_-(T) \ket{\downarrow} \right] \ket{0}, \label{eq:sudden_post_measurement_state_0}\\
\ket{\psi''_1}=&\ \frac{1}{\sqrt{p''_1}}\left[ \alpha B_+(T) \ket{\downarrow} + \beta B_-(T) \ket{\uparrow} \right] \ket{1}. \label{eq:sudden_post_measurement_state_1}\end{aligned}$$
with probabilities $p''_0=|\alpha A_+|^2 + |\beta A_-|^2$ and $p''_1=1-p''_0$, respectively. Here, we defined the amplitudes $A_{\pm}(T) = \cos\Omega_{\pm}T + i\cos\theta_\pm \sin\Omega_{\pm}T$ and $B_{\pm}(T) = -i\sin\theta_\pm e^{i\phi_\pm} \sin\Omega_{\pm}T$, where $|A_\pm|^2+|B_\pm|^2=1$. This scheme implements a projective measurement of $\hat{Z}$, when these coefficients satisfy, say, $\vert A_- \vert^2 =\vert B_+ \vert^2=0$, with the remaining two coefficients being equal to unity. This can be realized for $t_1 = i t_2$, $\abs{t_{1,2}}^2 \gg \varepsilon^2/4$, and $T=\pi/2\Omega_-$. As such, the qubit is left in the $\ket{\uparrow}$ state for both measurement outcomes. Thus, for the outcome $n=1$, this protocol would need to be followed by another waiting period to rotate the state back to $\ket{\downarrow}$, before turning off the tunnel couplings.
While in principle this allows for single-shot projective measurements of $\hat{Z}$, the protocol relies on fine-tuned waiting periods and a hierarchy of time scales which would be challenging to fulfill in experiment. Most importantly, we need to assume that the charge measurement is fast compared to internal time scales of the coupled Majorana qubit-quantum dot system. This requirement derives from the fact that the measurement operator $\hat n$ does not commute with the Hamiltonian of the system. Alternatively, we can abruptly turn off the tunnel couplings after entangling Majorana qubit and quantum dot state, and measure the quantum dot charge only subsequently. Then, the charge measurement is quantum nondemolition. Under realistic circumstances, a single projective charge measurement would presumably provide only partial information on $\hat{Z}$. This can be remedied by repeating the above protocol sufficiently many times until the measurement outcome is certain (see App. \[app:repeated\_measurements\]).
Implementing this protocol is clearly challenging and requires detailed and fast control. One may also worry that the fast switching excites the qubit in unwanted ways. It would be preferable to implement a projective measurement of $\hat{Z}$ by monitoring the quantum dot charge [*while*]{} the quantum dot is coupled to the Majorana qubit. This obviates the need for repeated switching, ideally without compromising on the achievable measurement times. We now turn to describe such continuous measurement procedures.
Majorana qubit readout via quantum dot charge monitoring {#sec:qpc_readout}
========================================================
Continuous charge measurements {#sec:contcharmeas}
------------------------------
To describe the readout dynamics, we need to include a measurement device for the quantum dot charge in the microscopic description. We assume that the quantum dot is capacitively coupled to a voltage-biased quantum point contact in such a way that the transmission amplitude $\mathcal{T}$ of the quantum point contact depends on the charge state of the quantum dot [@korotkov_continuous_1999; @goan_dynamics_2001], $$\hat{\mathcal{T}} = \tau + \chi \hat{n}.$$ The current through the quantum point contact will then depend on the quantum dot charge, taking the values $I_0 \propto \abs{\tau}^2$ when the quantum dot is empty and $I_1 \propto \abs{\tau+\chi}^2$ when the quantum dot is occupied. (For simplicity, we assume that $\tau$ and $\chi$ have relative phase $\pi$.) Provided that the voltage applied to the quantum point contact is sufficiently large, $eV\gg \Omega_\pi$, this setup decoheres the system in the quantum dot charge basis (see App. \[app:derivation\_unconditional\]).
When $I_0 \gg \abs{\delta I} = \abs{I_1 - I_0}$, the fluctuating current through the quantum point contact becomes a Gaussian random process (see App. \[app:derivation\_conditional\_diffusive\]), $$I(t) = I_0 + \delta I \ev{\hat{n}(t)} + \sqrt{I_0} \xi(t),$$ involving the Langevin current $\xi(t)$ with $\mathbb{E}\bqty{\xi(t)} = 0$ and $\mathbb{E}\bqty{\xi(t)\xi(t+\tau)} = \delta(\tau)$. Here, $\mathbb{E}[.]$ denotes an ensemble average over many realizations of the measurement procedure. It will prove useful to work with a dimensionless measurement signal $$\label{eq:meas_signal}
j(t) = \frac{{I}(t) - I_0}{\delta I} = \ev{\hat{n}(t)} + \frac{1}{\sqrt{4k}}\xi(t)$$ relative to the measured background current $I_0$, with $k\propto \abs{\chi}^2$. The first term contains information on the quantum dot charge and the second is the noise added by the quantum point contact. Within the Born-Markov approximation, the state of the system evolves according to the stochastic master equation [@korotkov_continuous_1999; @goan_dynamics_2001] (see App. \[app:derivation\] for a derivation) $$\begin{gathered}
\label{eq:sme_mbq-qd-qpc}
\frac{\textrm{d}}{\textrm{d}t}\hat{\rho}_c(t) = \underbrace{-i\bqty{\hat{H},\hat{\rho}_c(t)} + k \mathcal{D}\bqty{\hat{n}}\hat{\rho}_c(t)}_{\equiv \mathcal{L} \hat{\rho}_c(t)}
\\
+ \sqrt{k} \xi(t) \mathcal{H}\bqty{\hat{n}}\hat{\rho}_c(t).\end{gathered}$$ The term involving the superoperator $\mathcal{D}[\hat{L}]\hat{\rho}=\hat{L}\hat{\rho}\hat{L}^{\dagger}-(\hat{L}^{\dagger}\hat{L}\hat{\rho}+\hat{\rho}\hat{L}^{\dagger}\hat{L})/2$ causes decoherence in the eigenbasis of $\hat{n}$, and the term with the superoperator $\mathcal{H}[\hat{L}]\hat{\rho} = \hat{L}\hat{\rho} + \hat{\rho} \hat{L}^{\dagger}- \langle \hat{L} + \hat{L}^{\dagger}\rangle \hat{\rho}$ describes the information gain by the measurement. The latter is a Langevin term due to the stochastic nature of the measurement. The stochastic master equation (\[eq:sme\_mbq-qd-qpc\]) describes the state of the system *conditioned* on the measurement signal $j(t)$, as denoted by the subscript $c$. The ensemble-averaged – and thus *unconditioned* – evolution of the system simply follows by dropping the stochastic term, $ \textrm{d}\hat{\rho}(t)/ \textrm{dt} = \mathcal{L} \hat{\rho}(t)$.
Quantum dot charge measurement {#sec:simple_charge_readout}
------------------------------
To recall how this formalism describes standard projective measurements, consider first a simple quantum dot charge readout with $\hat{H}_T = 0$. The deterministic terms in Eq. lead to a decay of the off-diagonal components of the density matrix $\hat{\rho}$ in the charge basis, but preserve the diagonal components. The action of the stochastic terms can be understood from the equation of motion for $n(t) \equiv \ev{\hat{n}(t)} = \textrm{tr}[\hat{n} \hat{\rho}_c(t)]$, $$\label{eq:charge_stochastic_equation}
\frac{\textrm{d}}{\textrm{d}t} n(t) = \sqrt{4k}[n(t) - n^2(t)]\xi(t).$$ Its two fixed points $n=0$ and $n=1$ correspond to the two possible measurement outcomes. Conservation of the diagonal components of the ensemble-averaged density matrix ensures that these outcomes occur with the correct probabilities. At the fixed point $n$, the quantum-point-contact current $j(t)$ becomes stationary, $j_{n}(t) = n + \xi(t)/\sqrt{4k}$, and directly reveals the measurement outcome $n$ after an integration time $\tau_m$. The latter is determined by the requirement that the integrated signal dominate over the integrated noise, which happens for $\tau_m \gg \pqty{4k}^{-1}$. We note that we use the steady-state measurement signal when estimating measurement times. While transients may also provide information in principle, we assume that in practice, typical measurement times will exceed the time scale on which the system decoheres.
![ Continuous measurement of the quantum dot charge for $H_T = 0$, with initial state $(\ket{\uparrow,0} + \ket{\downarrow,1})/\sqrt{2}$ corresponding to $n(0) = 1/2$. Top panel: Two sample trajectories $n_1(t)$ (blue) and $n_2(t)$ (orange) corresponding to measurement results $n=0$ and $n=1$, respectively. The ensemble averaged evolution of the quantum dot charge (green, obtained from $1000$ trajectories) stays near 1/2. The ensemble average of $4\pqty{n(t)-n^2(t)}$ (red) equals unity for uncertain charge and zero when a fixed point has been reached, and therefore quantifies the advance of the measurement process. Bottom panel: Instantaneous (green, right y-axis labels) and time-averaged (dark red, left y-axis labels) measurement current for sample trajectory $n_1(t)$ (blue, same as in top panel).[]{data-label="fig:three graphs"}](simple_qd_readout_combined.pdf){width="45.00000%"}
This time-resolved description of a projective measurement of $\hat{n}$ is illustrated in Fig. \[fig:three graphs\] based on a numerical solution of Eq. (\[eq:sme\_mbq-qd-qpc\]). We show sample trajectories of the expectation value of the quantum dot charge (experimentally inaccessible) as well as the corresponding measurement currents through the quantum point contact. While the instantaneous measurement current fluctuates strongly, its time average reveals the quantum dot charge.
Majorana-qubit readout {#sec:majorana-qubit_readout}
----------------------
Including the tunnel coupling $\hat{H}_T$ between quantum dot and Majorana qubit, the dot occupation is no longer a good quantum number. While the measurement tries to project the state of the system into charge eigenstates, tunneling continuously rotates it out of this basis. This can be seen explicitly from the equation of motion for the charge expectation value, $$\label{eq:charge_stochastic_equation_2}
\frac{\textrm{d}n}{\textrm{d}t} = i \ev{\bqty{\hat{H},\hat{n}}} + \sqrt{4k}[n - n^2]\xi.$$ As a result of the nonzero commutator on the right hand side, the evolution of $n(t)$ no longer tends towards fixed points. Similarly, $\langle\hat{Z}(t)\rangle$ also does not evolve towards fixed points as $\hat{Z}$ does not commute with $\hat{H}$, as well. At first sight, this seems to imply that quantum dot charge measurements will not suffice to read out the state of the qubit. Remarkably, we find that one may still read out $\hat{Z}$ from a measurement of the quantum dot charge, but the procedure is more subtle.
The key observation is that the evolution governed by Eq. implements a quantum nondemolition measurement of the combined local parity $\hat{\pi}$, which is in one-to-one correspondence with $\hat Z$ for the initial state in Eq. (\[initialstate\]). Under quantum dot charge measurements, the evolution of $\pi(t) = \ev{\hat{\pi}(t)}$ is a bistable process with fixed points $\pi = \pm 1$, which are reached with the correct probabilities $|\alpha|^2$ and $|\beta|^2$, respectively. This does not yet guarantee a projective measurement of $\hat Z$. First, as a consequence of the tunneling Hamiltonian, the measurement does not properly project the state of the system, but leaves it in an equal mixture of the two eigenstates with combined local parity $\pi$. However, once the measurement outcome $\pi$ is determined, the readout device can be decoupled and the state of the quantum dot-Majorana qubit system appropriately reset (see App. \[app:qd\_reset\]). Second, one needs to specify how to read out $\pi$ and thus $ Z$ from the measurement current. Unlike for pure quantum dot charge measurements, the average measurement current in general no longer distinguishes between the two measurement outcomes. Instead, a measurement readout generally requires one to analyze the frequency-dependent noise of the measurement current.
We now discuss these claims in more detail. We first show that under quantum dot charge measurements, the unconditioned evolution of the density matrix $\hat \rho(t)$ generically tends towards $$\hat{\rho}^{\infty} = \frac{1}{2} \textrm{diag}(\abs{\alpha}^2,\abs{\alpha}^2,\abs{\beta}^2,\abs{\beta}^2)$$ for the initial state in Eq. (\[initialstate\]). This follows because the evolution preserves the weight of the two $\pi$ subspaces and the set of steady states of $\mathcal{L}$ is spanned by $\hat{\rho}^\infty_{+} = \textrm{diag}(1,1,0,0)/2$ and $\hat{\rho}^\infty_{-} = \textrm{diag}(0,0,1,1)/2$. To see this, we decompose $\hat\rho$ into $2\times 2$ blocks ${\rho}_{\pi,\pi'}$ according to the combined parity eigenvalues. Since $\hat\pi$ is a good quantum number and commutes with the quantum dot charge operator, the evolution equation for $\hat\rho$ decouples into independent equations
$$\begin{aligned}
\dot{\rho}_{\pi,\pi} =&\ -i\bqty{h_\pi,\rho_{\pi,\pi}} + k\mathcal{D}
\bqty{ n }\rho_{\pi,\pi} \nonumber \\
=&\ \mathcal{L}_{\pi,\pi}\rho_{\pi,\pi}, \label{eq:++_block_meq}\\
\dot{\rho}_{+-} =&\ -i\pqty{h_{+}\rho_{+-} -\rho_{+-} h_{-}} + k\mathcal{D}\bqty{n} \rho_{+-} \nonumber \\
=&\ \mathcal{L}_{+-}\rho_{+-}. \label{eq:+-_block_meq}\end{aligned}$$
for the diagonal and off-diagonal blocks of $\hat\rho$.
![ Continuous readout of Majorana qubit, with initial state $(\ket{\uparrow,0} + \ket{\downarrow,0})/\sqrt{2}$ corresponding to $n(0) = 0$. Two sample trajectories $\pi_1(t)$ (green) and $\pi_2(t)$ (red) show different measurement outcomes $\pi=1$ and $\pi=-1$, respectively. The ensemble average of $\pi(t)$ (blue, computed for 100 trajectories) remains close to zero for all times. The ensemble average of $1-\pi^2(t)$ (orange) quantifies the distance from the fixed points $\pi=\pm1$. Parameters: $\varepsilon = 20k, t_1 = e^{-i\varphi} t_2 = 2k$ with $\varphi = \pi/4$. []{data-label="fig:majorana_measurement_generic"}](majorana_readout_generic_ensamble_averages_and_sample_trajectories.pdf){width="0.9\columnwidth"}
The equations for the diagonal blocks have themselves Lindblad form and preserve the trace. As $h_\pi$ does not commute with the quantum dot charge $n$ (unless $t_1 = \pm i t_2$ in which case the tunneling Hamiltonian vanishes for one of the blocks; we will comment on this case below) and as $n$ is hermitian, their only zero mode is the completely mixed state. Then, preservation of the trace implies that the diagonal blocks of the density matrix do indeed tend towards the fixed points $$\rho_{++}^{\infty} = \frac{\abs{\alpha}^2}{2}\mathds{1} \,\,\,\,\,\,\,\, {\rm and} \,\,\,\,\,\,\,\, \rho_{--}^{\infty} = \frac{\abs{\beta}^2}{2}\mathds{1},
\label{asymprho}$$ respectively. (We analyze the complete set of eigenvalues $\lambda_{\pi,n}$ and eigenmodes of $\mathcal{L}_{\pi,\pi}$ in App. \[app:liouvillian\_evs\_diag\].) Anticipating that the off-diagonal blocks generically decay to zero, we obtain the correct Born-rule probabilities for $\hat{Z}$. The final state has weight $\abs{\alpha}^2$ in the $\pi = +1$ subspace and $\abs{\beta}^2$ in the $\pi = -1$ subspace, which just corresponds to the probabilities of finding $Z = +1$ or $Z=-1$, as required.
{width="100.00000%"}
The equation for the off-diagonal block deviates from Lindblad form since the first term on the right hand side involves both Hamiltonians $h_+$ and $h_-$. We analyze the eigenvalues $\tilde{\lambda}_n$ of $\mathcal{L}_{+-}$ in App. \[app:liouvillian\_evs\_offdiag\] and find that they generically correspond to decaying modes. Then, the off-diagonal blocks decay to zero and the two $\pi$ subspaces decohere, $\rho_{+-}^{\infty} = 0$. The only exception occurs when $\Im{t_1t_2^*} = 0$. In this case, the off-diagonal block supports a nondecaying mode since the characteristic frequencies coincide for the $\pi=+1$ and $\pi = -1$ eigenspaces and we find $\rho_{+-}^{\infty} = {\alpha^*\beta}\mathds{1} / {2}$.
Consistent with these results, the conditional evolution of $\hat\rho$ is a bistable process for $\pi(t)$ and a projective measurement of $\hat{\pi}$. We can readily derive the stochastic evolution equation for $\pi(t)$, $$\dot{\pi} = \sqrt{4k}\xi \pqty{ \ev{\hat{n} \hat{\pi}} - n\pi }.$$ Clearly, $\pi = \pm 1$ are fixed points of this equation. This is illustrated in Fig. \[fig:majorana\_measurement\_generic\], where we show two representative sample trajectories of $\pi(t)$ for measurement outcomes $\pi = +1$ and $\pi = -1$. We find numerically that provided $\Im{t_1 t_2^*} \neq 0$, these are the only fixed points, cp. $\mathbb{E}[1-\pi^2(t)] \to 0$ in Fig. \[fig:majorana\_measurement\_generic\]. We analyze the stochastic evolution in more detail in App. \[app:stochastic\_evolution\].
The decay of the off-diagonal block to zero implies that $\pi$ can be extracted from the measurement current $j(t)$. Once the measurement current becomes stationary, $j(t) = j_\pi (t)$, the system explores the full subspace with fixed $\pi$ in an ergodic manner. This follows from the fact that the ensemble average over all trajectories with outcome $\pi$ yields a completely mixed state on the subspace. Importantly, this implies that the ensemble (or time) average of $j_\pi(t)$ does not carry information on $\pi$, $$\begin{aligned}
\label{eq:enavj}
\mathbb{E}[j_\pi (t)] =&\ \textrm{tr}\bqty{\hat{n}\mathbb{E}[\hat{\rho}_c (t)\vert_{\pi}]} + \frac{\mathbb{E}[\xi(t)]}{\sqrt{4k}} = n^{\infty}_\pi = \frac{1}{2}.\end{aligned}$$ In agreement with Eq. (\[eq:enavj\]) and in contrast to the simple quantum dot charge readout (see Sec. \[sec:simple\_charge\_readout\]), the integrated measurement currents converge to $1/2$ irrespective of the measurement outcome. This is shown in Fig. \[fig:majorana\_readout\_compare\_generic\_and\_sweet\_spot\] (left panel).
Nonetheless, information on $\pi$ is generically encoded in the noise correlations of the measurement current, $$\label{eq:autocorrelation_main}
S_\pi(\tau) = \mathbb{E}\bqty{j_\pi(t)j_\pi(t+\tau)}.$$ In the stationary (long-time) limit, the corresponding power spectrum $S_\pi(\omega)$ can be readily computed for the two values of $\pi$ (see App. \[app:spectrum\]). In the limit of weak measurements, $k \ll \Omega_\pi$, we find that in addition to a white-noise background, which is just the shot noise power of the quantum point contact, the power spectrum exhibits Lorentzian peaks at $\omega = 0$ and $\omega=\pm 2\Omega_\pi$, which reflect the dynamics of the Majorana qubit-quantum dot system. Explicitly, we find $$\begin{gathered}
\label{eq:spectrum_main_text}
S_\pi(\omega) = \frac{1}{4k} + \frac{\cos^2\theta_\pi}{2} \frac{ \kappa_\pi }{\omega^2 + \kappa_\pi^2} \\ +
\frac{\sin^2\theta_\pi}{4} \sum_\pm
\frac{ \tilde{\kappa}_\pi }{(\omega \pm 2\Omega_\pi)^2 +\tilde{\kappa}_\pi^2 } ,\end{gathered}$$ where we introduced the widths
$$\begin{aligned}
\kappa_\pi =&\ \frac{\sin^2\theta_\pi}{2} k, \\
\tilde{\kappa}_\pi =&\ \frac{\pqty{1+\cos^2\theta_\pi}}{4} k\end{aligned}$$
of the Lorentzians. The measurement result for $\pi$ can be read off in particular from the location of the Lorentzians in frequency. This is illustrated in Fig. \[fig:spectrum\] which shows the power spectrum for a numerically generated measurement current and compares it to Eq. (\[eq:spectrum\_main\_text\]).
Physically, the zero-frequency peak in Eq. (\[eq:spectrum\_main\_text\]) is associated with the telegraph noise of the charge expectation value shown in Fig. \[fig:majorana\_readout\_compare\_generic\_and\_sweet\_spot\]. In principle, the width of this peak also encodes the measurement outcome. Indeed, Fig. \[fig:majorana\_readout\_compare\_generic\_and\_sweet\_spot\] illustrates that the dwell time near $n=0$ and $n=1$ depends on the $\pi$ subspace. For suitable parameters, it may also be possible to monitor the transition rate directly by means of an appropriately smoothened measurement current, or to extract information on the measurement outcome from associated transients in the time-averaged measurement current. The peak at finite frequency originates from Rabi oscillations, which are seen in Fig. \[fig:majorana\_readout\_compare\_generic\_and\_sweet\_spot\] as the small deviations of the quantum dot charge from its eigenvalues $n=0$ and $n=1$ (see inset of right panel). The measurement pushes the system into a charge eigenstate and thus a superposition of energy eigenstates, which then leads to oscillations as a result of the Hamiltonian dynamics.
![ Power spectra of the measurement signal corresponding to outcomes $\pi = +1$ and $\pi = -1$. Numerical simulations based on Eq. (\[eq:sme\_mbq-qd-qpc\]) (green and red traces) are in excellent agreement with the analytical expression in Eq. (\[eq:spectrum\_main\_text\]) (blue and orange traces). The numerical power spectra were obtained by generating measurement signals for long time intervals of $T \sim 10^7 k^{-1}$. The long integration time is necessitated by the small weight of the finite-frequency Lorentzians when $\sin\theta_\pi \ll 1$. Other parameters as in Fig. \[fig:majorana\_measurement\_generic\]. The power spectrum for $\pi=-1$ is scaled up by a factor of five for better visibility.[]{data-label="fig:spectrum"}](meas_curr_spectrum){width="0.9\columnwidth"}
It is interesting to consider two special parameter choices. First, for $\Im{t_1 t_2^*} = 0$, the off-diagonal block $\rho_{+-}$ of the density matrix does not decay. Indeed, in this case, not only the average measurement signal, but also its noise correlations are independent of $\pi$. More generally, the failure to decohere in the $\hat{\pi}$ basis reflects the fact that the measurement signals for the two subspaces are indistinguishable.
While the measurement fails for $\Im{t_1 t_2^*} = 0$, it can be simplified at the sweet spot $t_1 = - i t_2$ ($t_1 = i t_2$ is analogous). In this case, the Hamiltonian commutes with $\hat{n}$ in the $\pi = -1$ block and charge conservation in this block makes all density matrices which are diagonal in the charge basis into zero modes of $\mathcal{L}_{--}$. In this fine-tuned situation, it is not necessary to measure noise correlations. Instead, the measurement outcome for $\hat{\pi}$ and hence $\hat{Z}$ can be extracted from the ensemble-averaged charge alone, which yields $\mathbb{E}[n]=1/2$ for $\pi = +1$ and $\mathbb{E}[n]=0$ for $\pi = -1$. For small deviations from the fine-tuned point, $t_1 = - i t_2 + \delta$ with $\delta \ll \abs{t_1}$, charge is no longer conserved in both blocks and $\mathbb{E}[n]=1/2$ regardless of $\pi$. However, the relaxation rate to the stationary state will be smaller in the $\pi=-1$ block by a factor $\abs{\delta}^2/\abs{t_1}^2$. For a sufficiently high measurement efficiency, it might then be possible to resolve $\pi$ from transient differences in the average charge. Figure \[fig:majorana\_readout\_compare\_generic\_and\_sweet\_spot\] (right panel) shows corresponding simulations (with $\delta = 0$), which confirm that in principle, the integrated measurement signals suffice to identify the measurement outcome in a measurement time $\tau_m \sim k^{-1}$.
![Decoherence rate as characterized by the real part of the slowest decaying eigenvalue of $\mathcal{L}_{+-}$, $\Re\{\tilde{\lambda}_{\textrm{slow}}\}$ as a function of the measurement strength $k$ and $\varphi$ defined through $t_1 = e^{-i\varphi} t_2 = 0.1 \varepsilon$.[]{data-label="fig:slowest_decaying_ev"}](slowly_decaying_ev_offdiag.pdf){width="\columnwidth"}
An important characteristic of the measurement is the measurement-induced decoherence time (see Refs. [@Goldstein2011; @Rainis2012; @Schmidt2012; @Hu2015; @Knapp2018; @Li2018; @Bauer2020] for discussions of Majorana qubit decoherence unrelated to measurements). The decoherence time is closely related to (and in fact upper bounded by) the inverse of the real part of the slowest decaying eigenvalue $\tilde{\lambda}_{\textrm{slow}}$ of $\mathcal{L}_{+-}$. Figure \[fig:slowest\_decaying\_ev\] shows numerical results for $\Re\{\tilde{\lambda}_{\textrm{slow}}\}$. We focus on weak measurements, $k, |t_1|,|t_2| \ll \varepsilon$, and nonzero but possibly small $\varphi$ defined through $t_1 = e^{-i\varphi}t_2$. We observe that there is no decoherence along the line $\varphi=0$ where $\Im{t_1t_2^*} = 0$. At fixed $k$, $\Re\{\tilde{\lambda}_{\textrm{slow}}\}$ grows quadratically in $\varphi$ up until discontinuous lines, where $\tilde{\lambda}_{\textrm{slow}}$ and the corresponding eigenmatrix coalesce with another eigenvalue-eigenmatrix pair. Then, the eigenmatrices of $\mathcal{L}_{+-}$ fail to span the space of $2 \times 2$ complex matrices along these *exceptional* lines emanating from $\pqty{\varphi,k}=\pqty{0,0}$. At fixed $\varphi$ and to leading order in $k$, we observe a linear decrease as $k$ increases towards the discontinuity. The measurement is more efficient when tuning to the large-$|\varphi|$ side of the exceptional lines. In this region, $\Re\{\tilde{\lambda}_{\textrm{slow}}\}$ depends only weakly on $\varphi$, which is why we only display the region of small $\varphi$ in Fig. \[fig:slowest\_decaying\_ev\]. Note that the discontinuities are regularized by relaxation terms, such as those discussed in the following Sec. \[sec:with\_relaxation\].
Majorana qubit readout in the presence of relaxation {#sec:with_relaxation}
----------------------------------------------------
The results of the previous section may be surprising in that the quantum point contact does not detect the dependence of the ground-state expectation value of the quantum dot charge on the parity sector $\pi$. Instead, the information on $\pi$ can generically only be extracted from noise correlations of the measurement current $j(t)$. Experimentally, however, it would be preferable if $\pi$ could be extracted from the average measurement signal.
The underlying reason for the insensitivity to the quantum dot charge in the ground state is that under the continuous measurement, the density matrix generically becomes proportional to the unit matrix within the subspace with fixed $\pi$ \[see Eq. (\[asymprho\])\]. Then, the expectation value of the quantum dot charge is just equal to 1/2, independent of the parity $\pi$. The difference in quantum dot charge between the ground states is compensated by the opposite difference between the excited states.
It is then natural to expect that the average measurement current distinguishes between the two subspaces once one includes additional relaxation processes from the excited state $\ket{e_\pi}$ to the ground state $\ket{g_\pi}$. Unlike the measurement which leads to relaxation in the basis of the quantum dot charge, this additional relaxation should operate within the eigenbasis of $h_\pi$. We now show that this expectation is indeed correct.
There are various processes which induce relaxation within the eigenstate basis. One relevant example are effective measurements of the Majorana-qubit charge by the environment. For definiteness, we effect relaxation within the eigenstate basis by coupling to the electromagnetic environment. Within the Born-Markov approximation and focusing on $T=0$ for simplicity, this leads to an additional dissipation term in Eq. (\[eq:sme\_mbq-qd-qpc\]) (see App. \[app:dissipation\] for details), $$\label{eq:dissipator_main}
\frac{\textrm{d}}{\textrm{dt}}\hat{\rho}\vert_{\textrm{relax}}
= \Gamma_- \sum_\pi \mathcal{D}\bqty{\frac{\sin\theta_\pi}{2} \hat{\tau}^\pi_-}\hat{\rho} \equiv \mathcal{L}'\hat{\rho}.$$ Here, we defined the lowering operator in the energy basis for subspace $\pi$, $\hat{\tau}^\pi_- = \ketbra{g_\pi}{e_\pi}$ and the zero-temperature relaxation rate $\Gamma_- = 2\pi J(2\Omega_\pi)$ governed by the spectral density $J(\omega)$ of the electromagnetic environment. While $\Gamma_-$ may depend on $\pi$ in principle, we assume $J(2\Omega_+) \simeq J(2\Omega_-)$ for simplicity. Note that at finite temperatures, there is an additional dephasing term $\Gamma_0 \mathcal{D}\bqty{\cos\theta_\pi \hat{\tau}^\pi_z / 2}\hat{\rho}$ in the eigenstate basis, where $\Gamma_0 = 2\pi \lim_{\omega \to 0} J(\omega)b(\omega)$ with the Bose distribution $b(\omega)$. However, for $k, \Gamma_-, \Gamma_0 \ll \Omega_\pi$, this term does not affect the results, cp. App. \[app:liouvillian\_evs\_diag\].
![Continuous readout of Majorana qubit in the presence of relaxation with rate $\Gamma_- = 5k$ (other parameters as in Fig. \[fig:majorana\_measurement\_generic\]). Top panel: Ensemble-averaged quantum dot occupations $n(t)$, restricted to the subspaces $\pi=+1$ (green trace) and $\pi=-1$ (red trace), as obtained from the unconditioned master equation (starting in the ground state with $n=0$). $\mathbb{E}[n(t)]_\pi$ converges to distinct expectation values $n_\pi^\infty$ for the two subspaces, so that $\pi$ can be read out from the time-averaged measurement current of a quantum dot charge measurement. This is shown in the bottom panel: Time averaged measurement signals for trajectories in the $\pi = +1$ (cyan trace) and $\pi = -1$ (orange) sectors, converging to $n_\pi^\infty$. Readout requires integration times which are significantly longer than convergence times of the ensemble-averaged quantum dot occupations in the top panel. From Eq. , the measurement time can be estimated as $\tau_m \sim 10^4 /k$ for the given parameters. This estimate is based on the fluctuations in the $\pi = -1$ sector, where fluctuations are larger since $\cot \theta_- \gg \cot\theta_+$. We also show the time average of the charge expectation value (not accessible in experiment). The similarity of the curves indicates that fluctuations of the measurement current are dominated by fluctuations of the charge expectation value. []{data-label="fig:majorana_readout_with_dissipation"}](majorana_readout_dissipation.pdf){width="0.9\columnwidth"}
Importantly, Eq. also conserves $\hat{\pi}$ and the total unconditioned master equation \[obtained by incorporating Eq. (\[eq:dissipator\_main\]) into Eq. \] still decouples into blocks. The off-diagonal block obeys $$\begin{aligned}
\dot{\rho}_{+-} =&\ \pqty{ \mathcal{L}_{+-} + \mathcal{L}'_{+-} } \rho_{+-}. \end{aligned}$$ Here, $\mathcal{L}'_{+-} = \frac{1}{4}\Gamma_- \sin\theta_+\sin\theta_-\mathcal{D}\bqty{\tau_-}$ has Lindblad form and a negative semidefinite real part. Since $\mathcal{L}_{+-}$ generically has only decaying eigenvalues, $\mathcal{L}_{+-} + \mathcal{L}'_{+-}$ is also decaying. The only exception occurs when $\Im{t_1 t_2^*} = 0$, as in the absence of relaxation. In this special case, $\mathcal{L}_{+-}$ and thus also $\mathcal{L}_{+-} + \mathcal{L}'_{+-}$ as a whole have Lindblad form and preserve the trace. Thus, the condition for the measurement to work remains unchanged in the presence of relaxation.
The evolution of the diagonal blocks tends to $$\rho_{\pi,\pi}^{\infty} = \frac{1}{2}\pqty{\tau_0^\pi + R \tau^{\pi}_z }$$ in the energy basis to leading order in $\Omega_\pi \gg k, \Gamma_-$. Here we defined the ratio $$\label{eq:measurement_relaxation_ratio}
R = \frac{\Gamma_-}{\Gamma_- + 2k}$$ characterizing the strength of the additional relaxation. The associated ensemble average of the quantum dot charge becomes $$n_{\pi}^{\infty} = \frac{1}{2}\pqty{1 + R\cos\theta_\pi} .$$ When the measurement is stronger than dissipation, we have $R\ll 1$ and the quantum dot charge is close to $1/2$, independent of $\pi$. However, in the opposite limit, when dissipation is stronger than the measurement and $R$ approaches unity, the average charge is approximately given by the ground-state expectation value of the charge in the respective sector, $\bra{g_\pi} \hat{n} \ket{g_\pi} = (1+\cos \theta_\pi)/2$. In this limit, the time-averaged measurement signal depends on $\pi$, in agreement with the heuristic arguments given above.
We illustrate these considerations by the numerical simulations shown in Fig. \[fig:majorana\_readout\_with\_dissipation\]. Including the relaxation process in Eq. in the simulations of the unconditioned master equation, we compute the time-averaged measurement currents and find that indeed, they converge towards $n_{\pi}^{\infty}$, albeit slowly.
The corresponding measurement time is determined by the requirement to resolve the difference $$\abs{n_{+}^{\infty} - n_{-}^{\infty}} = 4R\abs{\frac{\Im{t_1 t^*_2}}{\varepsilon^2}}$$ in quantum dot occupations. The time-averaged measurement current $$j_{\textrm{int},\pi}(T) = \frac{1}{T}\int_0^T\textrm{d}t\ j_\pi(t)$$ fluctuates around $n_{\pi}^{\infty}$, with decreasing magnitude of the fluctuations as $T$ grows. The fluctuations can be estimated via the variance $$\label{eq:int_curr_variance_main}
\mathbb{V}[j_{\textrm{int},\pi}(T)] = \frac{1}{4kT} + \cot^2\theta_\pi R^2\ \frac{4k}{\Gamma_-^2 T}.$$ The first term reflects the white noise background, whereas the second originates from fluctuations of $n(t)$. The latter depends on the sector $\pi$. The measurement time can then be estimated by comparing the variance with the resolution necessary to distinguish the two possible measurement outcomes. This gives $$\label{eq:meas_time_relax}
\tau_{m} \sim \frac{\varepsilon^4}{16 \abs{\Im{t_1 t^*_2}}^2} \pqty{ \frac{1}{4kR^2} + C \frac{4k}{\Gamma_-^2}},$$ where we use the variance of the sector with larger fluctuations, defining $$C = \max_\pi{ \cot^2\theta_\pi}.$$ We observe that a large splitting resulting from a large $\abs{\Im{t_1 t^*_2}}$ and a small $\varepsilon$ are advantageous for a fast measurement. (Note, however, that there is a tradeoff since a small $\varepsilon$ enhances quasiparticle poisoning rates.) Moreover, the terms in the brackets have interesting structure. They diverge for both $k \to 0$ and $k\to\infty$. Thus, at a given $\Gamma_-$, there is an optimal measurement strength $k_{\textrm{opt}} = \Gamma_-/2\sqrt{1 + 4C}$ to identify the measurement outcome based on the time-averaged signal. The corresponding optimal measurement time becomes $$\tau_{m,\textrm{opt}} \sim \frac{\varepsilon^4\pqty{1 + \sqrt{1+4C }}}{16 \Gamma_- \abs{\Im{t_1 t^*_2}}^2}.$$
![Effect of Majorana hybridization $\epsilon_{23} = 0.2k$ on Majorana-qubit readout (in the absence of relaxation, $\Gamma_- =0$). Blue and orange traces: Sample trajectories of $\pi(t)$, clearly not reaching a fixed point. Red and purple traces: Ensemble-averaged evolution of $\pi(t)$ for states initialized in the $\pi = +1$ (red) and $\pi = -1$ (purple) sectors. Both curves converge towards $\mathbb{E}[\pi] = 0$, corresponding to an equal mixture of the two sectors, regardless of initial condition. Similarly, $\mathbb{E}[1-\pi^2(t)]$ (green) does not approach $0$, so that $\pi \neq \pm 1$. Other parameters as in Fig. \[fig:majorana\_measurement\_generic\]. []{data-label="fig:majorana_readout_with_hybridization_no_dissipation_fast"}](majorana_readout_with_hybridization_no_dissipation_fast.pdf){width="0.9\columnwidth"}
Charge nonconservation and Majorana hybridizations {#sec:imperfect}
--------------------------------------------------
An essential assumption underlying the readout of the Majorana qubit is that the combined parity $\hat\pi$ is a good quantum number. In practice, there can be processes which do not conserve $\hat\pi$. First, the combined parity does not commute with the residual Majorana hybridizations $\varepsilon_{ij}$ in Eq. (except for $\varepsilon_{12}$). Second, $\hat\pi$ is no longer conserved in the presence of leakage of the quantum dot charge, say into additional reservoirs.
![ Effect of a weak Majorana hybridization $\epsilon_{23} = 0.02k$ on Majorana-qubit readout. For this hybridization strength, the measurement evolution, which tries to project $\pi$ onto an eigenvalue of $\hat{\pi}$, is stronger than the evolution due to $\hat{H}_{23}$. Thus, in contrast to Fig. \[fig:majorana\_readout\_with\_hybridization\_no\_dissipation\_fast\], individual $\pi_i(t)$ traces (blue and orange) remain predominantly near the fixed points $\pi = \pm 1$. This is also reflected in the fact that $\mathbb{E}[1-\pi^2]$ (green trace) reaches a steady state value which is different from but still close to zero. At short times, individual trajectories reach the fixed points with a probability reflecting the initial weights $\abs{\alpha}^2$ and $\abs{\beta}^2$ associated with the $\hat{\pi}$ eigenspaces. Eventually, hybridization flips $\pi(t)$ between $1$ to $-1$, as illustrated by trajectory $\pi_2(t)$. These jumps cause a decay of the ensemble average of $\pi(t)$ over trajectories initialized within one fixed point, see red trace for $\mathbb{E}[\pi]_+$ (enlarged in inset). Correspondingly, the initial weights are lost and in the long-time limit, trajectories are close to either fixed point with equal probability (as quantified by the decay of $\mathbb{E}[\pi]_+$) . For good readout fidelity, the measurement outcome must be identifiable as long as $\mathbb{E}[\pi]_+ \simeq 1$. Other parameters as in Fig. \[fig:majorana\_measurement\_generic\].[]{data-label="fig:majorana_readout_with_hybridization_no_dissipation_slow"}](majorana_readout_with_hybridization_no_dissipation_slow.pdf){width="0.9\columnwidth"}
It is natural to expect that these processes spoil the measurement by allowing weight to move between the $\pi$ subspaces and thereby scrambling the probabilities associated with the measurement outcomes. We analyze this in more detail for the Majorana hybridizations. For definiteness, we focus on $\varepsilon_{23}$ with the corresponding contributions $$\begin{aligned}
\hat{H}_{23} =&\ -i\varepsilon_{23} \hat{\gamma}_2\hat{\gamma}_3 \\
=&\ \varepsilon_{23} \sum_n \pqty{ \ketbra{\uparrow,n}{\downarrow,n}+\ketbra{\downarrow,n}{\uparrow,n}}\end{aligned}$$ to the Hamiltonian and $\mathcal{L}_{23} \hat{\rho} = -i[\hat{H}_{23},\hat{\rho}]$ to the Liouvillian. In the absence of relaxation and for $\Im{t_1 t_2^*} \neq 0$, the new total Liouvillian $ \mathcal{L}+ \mathcal{L}_{23}$ has $\hat{\rho}^{\infty} = \textrm{diag}(1,1,1,1)/4$ as the only zero mode and, consequently, does not preserve information on the weights $\abs{\alpha}^2$ and $\abs{\beta}^2$ of the initial Majorana-qubit state in the long-time limit (see App. \[app:steady\_state\_w\_hybridizations\]).
This is illustrated in Fig. \[fig:majorana\_readout\_with\_hybridization\_no\_dissipation\_fast\] which shows $\mathbb{E}[\pi(t)]_\pm$, the ensemble-averaged evolution of $\pi(t)$ for initial states $\pi(0) = \pm 1$. For significant values of $\epsilon_{23}$, $\mathbb{E}[\pi(t)]_\pm$ relaxes to $0$ faster than the measurement can project $\hat{\pi}$, as indicated by the fact that $\mathbb{E}[1-\pi^2(t)]$ remains large for all times $t$. This implies that the system forgets the weights associated with the $\pi$ eigensectors too fast to perform a measurement. In contrast, Fig. \[fig:majorana\_readout\_with\_hybridization\_no\_dissipation\_slow\] shows data for a much smaller value of $\epsilon_{23}$. Here, $\varepsilon_{23} \ll \tau_m^{-1}$ and the information on the weights is retained transiently. Still, in the long-time limit, this information is lost and Majorana hybridizations set an upper limit for the time a measurement may take. Including relaxation does not change this qualitatively. In this case, the steady state will no longer be completely mixed, but importantly, there is only one steady state and information on the qubit state is lost in the long time limit.
Readout via double quantum dot {#sec:DQDreadout}
------------------------------
### Readout of two-Majorana parities
![Majorana qubit readout by means of a double quantum dot (with inter-dot tunneling $t_0$), with charge monitoring by a quantum point contact of one (as shown) or both quantum dots. Symbols as in Fig. \[fig:tetron\_quantum\_dot\_measurement\_device\].[]{data-label="fig:tetron_double_dot_with_readout"}](tetron_double_dot_with_qpc.pdf){width="0.7\columnwidth"}
It is interesting to compare the scheme discussed so far with a modified readout setup which couples the Majorana qubit to a double quantum dot, such that Majoranas $\hat\gamma_1$ and $\hat\gamma_2$ entering into $\hat Z$ are coupled to one quantum dot each, see Fig. \[fig:tetron\_double\_dot\_with\_readout\]. In this case, the effective hopping amplitude between the quantum dots equals $$t_{\hat Z} = t_0+\frac{it_1t_2^*}{E_c}\hat Z.$$ Here, $t_0$ denotes direct hopping, while the second term originates from indirect hopping via the Majorana qubit. We consider the subspace in which a single electron in the double quantum dot can reside in either of the two quantum dots, with basis states $\ket{1,0}$ and $\ket{0,1}$. The Hamiltonian of the system, written in the basis $\{\ket{1,0;\uparrow},\ket{0,1;\uparrow},\ket{1,0;\downarrow},\ket{0,1;\downarrow}\}$ becomes block-diagonal, $${\hat H}=\left(\begin{array}{cc} h_\uparrow & 0 \\ 0 & h_\downarrow
\end{array}\right),$$ where the $2\times 2$ blocks take the form $$\label{eq:hamDQD}
h_Z= \left(\begin{array}{cc} \epsilon/2 & t_Z \\
t_Z^* & -\epsilon/2 \end{array}\right).$$ Unlike the single-dot case, the block-diagonal structure is now directly related to the operator of interest, $\hat Z$. At first sight, this may seem to simplify readout based on monitoring the charge of one of the quantum dots.
However, this is not the case and our analysis of the readout via a single quantum dot carries over to the present case with only small changes. In particular, the time-averaged measurement signal of the quantum point contact does not distinguish between the two $Z$ values, unless there is relaxation in the energy eigenbasis. This is because the measurement attempts to project the quantum dot into a charge eigenstate of one of the quantum dots, which is not an eigenstate, thus causing Rabi oscillations of the charge between the quantum dots. In the stationary limit, the system explores both charge states, $\ket{1,0}$ and $\ket{0,1}$, with equal probability and the ensemble-averaged charge becomes equal to 1/2, independent of $Z$. The similarities with the single-dot setup are, of course, rooted in the fact that the Hamiltonians (\[eq:sector\_hams\]) and (\[eq:hamDQD\]) for the single and double-dot setups, respectively, are closely analogous.
Despite these similarities, the present setup may have some advantages which could compensate for the additional effort. First, the diagonal elements of the Hamiltonian (\[eq:hamDQD\]) can now be tuned by a gate, making a wider parameter range accessible. Second, the double quantum dot presumably couples efficiently to the electromagnetic environment, which induces relaxation in the energy basis and enables readout of the qubit via the average measurement current. Third, the setup obviates the need for resetting the qubit as electrons enter the Majorana qubit only virtually.
### Readout of four-Majorana parities {#sec:four_majorana_readout}
Universal quantum computing requires a gate which entangles qubits such as the controlled NOT. For Majorana qubits, the entangling gate can be implemented using measurements of two-qubit Pauli operators [@Zilberberg2008; @karzig_scalable_2017; @litinski_quantum_2018], say $\hat{Z}_1 \hat{Z}_2$, where $\hat{Z}_1 = -i \hat{\gamma}_1\hat{\gamma}_2$ and $\hat{Z}_2 = -i \hat{\gamma}_3\hat{\gamma}_4$, cf. Fig. \[fig:four\_majorana\_measurement\]. This requires measurements of products of four Majorana operators. Measurements of Majorana parities with even more operators are required to read out stabilizer operators of various topological error correcting codes [@Oreg2020].
Measurements of four-Majorana parities can be implemented using double quantum dots as in Sec. \[sec:DQDreadout\], replacing the tunneling path through a single Majorana qubit in Fig. \[fig:tetron\_double\_dot\_with\_readout\] by a tunneling path through a sequence of two Majorana qubits, as shown in Fig. \[fig:four\_majorana\_measurement\]. If the path involves all four Majoranas included in $\hat{Z}_1 \hat{Z}_2$, the corresponding tunneling amplitude becomes $$t_{\hat{Z}_1\hat{Z}_2} = t_0+\frac{t_1 t_{23} t^*_4}{E^2_c} \hat{Z}_1\hat{Z}_2.$$ By analogy with our discussion in Sec. \[sec:DQDreadout\], the quantum dot charge measurement leads to decoherence in the eigenbasis of $\hat{Z}_1\hat{Z}_2$. At the same time, the density matrix remains unaffected within the diagonal blocks of fixed two-qubit parity $\hat{Z}_1\hat{Z}_2$, so that no information is gained on ${\hat Z}_1$ or ${\hat Z}_2$. Clearly, this can, at least in principle, be extended to the measurement of larger products of Majorana operators.
![Four-Majorana readout by charge measurements on a double quantum dot. The Majorana bound states $\hat{\gamma}_2$ and $\hat{\gamma}_3$ are tunnel coupled directly via the tunneling link $t_{23}$. Symbols as in Fig. 1.[]{data-label="fig:four_majorana_measurement"}](four_majorana_double_dot_one_side.pdf){width="0.9\columnwidth"}
Symmetry protected readout {#sec:general_readout_analysis}
==========================
We found in Sec. \[sec:majorana-qubit\_readout\] that even though $\hat{Z}$ was not a conserved quantity and the measurement device was coupled to $\hat{n}$, we could read out $\hat Z$ by effectively extracting the combined local parity $\hat{\pi}$ which is a symmetry of both the system and the measurement Hamiltonian. This is a special case of a more general result (see, e.g., [@baumgartner_analysis_2008; @Albert2014]). If an operator $\hat\Pi$ commutes with both, the Hamiltonian, $[\hat{H},\hat{\Pi}] = 0$, and the full set of jump operators describing the measurement and decoherence channels, $[\hat{L}_\alpha,\hat{\Pi}] = 0$, the system generically decoheres in the $\hat{\Pi}$ basis. In particular, decoherence occurs as long as the measurement current distinguishes between the eigenspaces of $\hat{\Pi}$ [@molmer_hypothesis_2015]. Before justifying the validity of this statement, we further illustrate its usefulness by additional applications to Majorana qubits.
It was shown by Akhmerov [@akhmerov_topological_2010] that coupling Majorana zero modes $\hat\gamma_i$ to other fermionic quasiparticles $\hat\alpha_{i,k}$ localized in their vicinity is not detrimental to topological protection. Due to their localized nature, the quasiparticles do not couple distant Majoranas and the operators $$\hat{\gamma}'_i = \hat{\gamma}_i (-1)^{\hat{N}_i}$$ with $\hat{N}_i = \sum_{k} \hat{\alpha}^{\dagger}_{i,k} \hat{\alpha}_{i,k}$ are dressed but protected zero modes of the system which commute with the Hamiltonian. This was recently studied further for a specific model in Ref. [@munk_fidelity_2019].
For these dressed zero modes to be useful for topological quantum computation, we need to be able to use them in Majorana qubits and to perform projective measurements of corresponding qubit operators such as $\hat{Z}' = -i\hat{\gamma}'_1 \hat{\gamma}'_2$ [@litinski_quantum_2018]. The general statement mentioned above implies that this is indeed possible. Consider a measurement of $\hat{Z}$ by coupling $\hat{\gamma}_1$ and $\hat{\gamma}_2$ to a quantum dot as before. We can define a modified combined local parity $$\hat{\pi}' = \hat{Z}(-1)^{\hat{n}+ \hat{N}_1 + \hat{N}_2} = \hat{Z}' (-1)^{\hat{n}},$$ which includes the localized quasiparticles. Unlike $\hat\pi$, the modified combined parity $\hat{\pi}'$ is a symmetry of the system in the absence of processes coupling to other Majorana bound states or changing the charge $\hat{n}+ \hat{N}_1 + \hat{N}_2$. A measurement which distinguishes between the two eigenspaces of $\hat{\pi}'$ will then no longer decohere the system in the eigenbasis of $\hat{\pi}$, but in the eigenbasis of $\hat{\pi}'$, as required for a projective readout of a qubit based on the dressed zero modes. There may, however, be a reduction in the readout speed, as the coupling to other localized modes reduces the hybridization of the zero mode with the quantum dot.
Our description of the measurement process in terms of the stochastic master equation (\[eq:sme\_mbq-qd-qpc\]) assumes a large bias applied to the quantum point contact, which might cause unnecessary heating of the quantum dot-Majorana qubit system as a consequence of the measurement. The general statement above implies that this assumption, although technically convenient, is unnecessary. Inspecting the derivation of the stochastic master equation in App. \[app:derivation\], we see that relaxing this assumption will change the argument of the decoherence operator $\mathcal{D}[\hat{n}]$ in Eq. (\[eq:sme\_mbq-qd-qpc\]). Nevertheless, $\hat{\pi}$ is conserved by all interactions and thus necessarily by the argument of $\mathcal{D}$, as well. Then, the system still decoheres in the $\hat{\pi}$ eigenbasis. It is worthwhile noting, however, that for smaller bias voltages the argument of $\mathcal{D}$ will in general no longer be hermitian and the associated steady state will not be completely mixed within each $\pi$ subspace.
Now we turn to justifying the general statement. If, for simplicity, the symmetry squares to one, $\hat{\Pi}^2 = \mathds{1}$, the unconditional master equation decouples into blocks labeled by the eigenvalues of $\hat{\Pi}$ (cp. Sec. \[sec:majorana-qubit\_readout\]),
$$\begin{aligned}
\dot{\rho}_{\pi\pi} =&\ -i\bqty{h_\pi,\rho_{\pi\pi}} + \sum_{\alpha} k_{\alpha} \mathcal{D}
\bqty{ l^{\alpha}_\pi }\rho_{\pi\pi} \nonumber \\
=&\ \mathcal{L}_{\pi\pi}\rho_{\pi\pi}, \label{eq:++_block_meq_general}\\
\dot{\rho}_{+-} =&\ -i\pqty{h_{+}\rho_{+-} -\rho_{+-} h_{-}} + \sum_{\alpha} \tilde{\mathcal{D}}\bqty{l^{\alpha}_+,l^{\alpha}_-} \rho_{+-} \nonumber \\
=&\ \mathcal{L}_{+-}\rho_{+-}. \label{eq:+-_block_meq_general}\end{aligned}$$
Here, we use the notation $\tilde{\mathcal{D}}\bqty{A,B} \rho = A\rho B^{\dagger} - (A^{\dagger}A\rho + \rho B^{\dagger}B)/2$ and decompose $\hat{L}_{\alpha} = \textrm{diag}[ l^{\alpha}_+ , l^{\alpha}_- ]$ as well as $\hat{H} = \textrm{diag}[ h_+ , h_- ]$. Just as in Sec. \[sec:majorana-qubit\_readout\], the diagonal blocks have Lindblad form and the evolution preserves the weights in the respective blocks.
We then need to understand when $\mathcal{L}_{+-}$ leads to a decay of $\rho_{+-}$. Baumgartner and Narnhofer [@baumgartner_analysis_2008] show that nontrivial off-diagonal steady states exist if and only if there is a unitary $$\hat{U}= \begin{pmatrix}
0 & u^{\dagger} \\ u & 0
\end{pmatrix}$$ connecting the two subspaces, $\hat{U}\hat{P}_+ = \hat{P}_-\hat{U}^{\dagger}$, which commutes with the Hamiltonian and all the $\hat{L}^{\alpha}$. Here, $\hat{P}_\pm$ denotes the projectors onto the two eigenspaces of $\hat\Pi$ and $u$ is a unitary acting on the $\hat\Pi$ eigenspaces. Then, one has $$h_- = u^{\dagger} h_+ u\,\, , \,\, l^{\alpha}_- = u^{\dagger} l^{\alpha}_+ u,$$ so that both the spectra of the Hamiltonians and the algebras formed by $\Bqty{h_{\pm},l^{\alpha}_{\pm}}$ are identical. This implies that the two sectors are unitarily equivalent and the associated measurement currents are indistinguishable. The existence of such a unitary $U$ requires finetuning. Generically, the subspaces are not related in this manner and the measurement signals distinguish between the two sectors. Then, decoherence occurs in the eigenbasis of $\hat{\Pi}$.
This holds true regardless of the details of the measurement procedure. For instance, one could alternatively base the charge measurement on circuit-QED reflectometry, where the coupling to the quantum dot charge takes the form $$\hat{H}_{\textrm{cQED}} = g\ \hat{n} \pqty{\hat{a}^{\dagger}_0 + \hat{a}_0}.$$ Here, $\hat{a}_0$ annihilates a bosonic resonator mode and $g$ quantifies the coupling strength. Since the coupling respects the symmetry $[\hat{H}_{\textrm{cQED}},\hat{\pi}] = 0$, this generically decoheres the system in the eigenbasis of $\hat{\pi}$. This emphasizes that it is really the symmetry that counts, not the details of the measurement, and we refer to this mechanism as symmetry protected decoherence or symmetry protected readout.
While decoherence generically occurs in the eigenbasis of $\hat{\pi}$, the decoherence rates depend on the specifics of the measurement and can be linked to the rate at which it is possible to distinguish the measurement signals of the two sectors [@molmer_hypothesis_2015]. In particular, $\abs{\textrm{tr} \rho_{+-}(t)}^2$ is closely related to the probability to correctly identify the measurement outcome from the measurement signal up to time $t$. Since this assumes an ideal measurement, the decay of $\rho_{+-}$ generally provides only bounds on the measurement time.
Discussion {#sec:discussion}
==========
Readout of Majorana-based topological qubits is an important problem and has attracted much attention in the literature for qubit designs based on Coulomb-blockaded superconducting islands [@karzig_scalable_2017; @plugge_majorana_2017; @grimsmo_majorana_2019; @Qin2019] or alternative settings [@flensberg_non-abelian_2011; @yavilberg_fermion_2015; @ohm_microwave_2015; @aasen_milestones_2016; @gharavi_readout_2016; @zhou2020doubledot]. Its importance is rooted in the fact that promising schemes for Majorana-based quantum computation [@karzig_scalable_2017; @litinski_quantum_2018; @Oreg2020] rely on measurements as an integral part of quantum information processing. This implies that the measurements must not only provide the measurement outcome, but also reliably project the qubit into the corresponding eigenstate.
A variety of techniques have been proposed to read out Majorana qubits, including interferometry of transport currents passed through the Majorana qubit [@plugge_majorana_2017; @Qin2019; @zhou2020doubledot], techniques borrowed from circuit quantum electrodynamics [@yavilberg_fermion_2015; @ohm_microwave_2015; @plugge_majorana_2017; @karzig_scalable_2017; @grimsmo_majorana_2019], or measurements relying on charge sensing [@flensberg_non-abelian_2011; @aasen_milestones_2016; @gharavi_readout_2016; @plugge_majorana_2017; @karzig_scalable_2017; @szechenyi_parity_2019]. Techniques borrowed from circuit quantum electrodynamics can frequently be treated theoretically in close analogy to the description for (nontopological) superconducting qubits [@blais_cavity_2004]. At the same time, these schemes involve a substantial hardware overhead and may significantly increase the effective dimensions of each qubit. Therefore, we focused here on readout of Majorana qubits based on coupling to a quantum dot whose charge is measured by means of a quantum point contact. This approach combines suitability to the basic design of Majorana qubits with conceptual simplicity, and thus relevance for near-term devices with accessibility of a thorough theoretical analysis at an analytical level.
Despite its apparent simplicity, this scheme poses nontrivial questions. In particular, we discuss charge-based readout protocols of parity-protected Majorana qubits which are distinctly different from charge-based readout protocols of other types of qubits. Spin qubits (by spin-charge conversion) [@petta_coherent_2005] or Majorana qubits without parity protection [@aasen_milestones_2016] (by parity-to-charge conversion) can also be effected by charge measurements. In these cases, the computational basis of the qubit is robustly brought into one-to-one correspondence with the charge basis. In contrast, the charge-based readout of parity-protected Majorana qubits projects in the charge basis, while the qubit operator enters through a tunneling Hamiltonian which does not commute with the charge. Generically, this makes a single projective charge measurement insufficient to identify the qubit state. Moreover, readout by repeated charge measurements would necessitate very high levels of control. Instead, the readout process is a weak continuous measurement and its theoretical description requires a time-resolved description of the measurement.
A systematic measurement theory of this readout scheme for (parity-protected) Majorana qubits is the central contribution of this paper. Our theory reveals under which conditions a measurement of the quantum dot charge constitutes a projective measurement of the Majorana parity of the qubit, describes the time it takes to decohere the system in the measurement basis, and includes the noisy measurement signal which can be analyzed to estimate required measurement times.
Our central insight is that generically, one does not directly measure the Majorana parity but rather a combined parity which includes the quantum dot charge in addition to the Majorana parity of the qubit. We find that this is not detrimental to the readout as the combined parity can eventually be converted into the desired Majorana parity. Importantly, this observation generalizes and our theory also applies more generally. In particular, this implies that other local charges which the Majorana might couple to are not a hindrance to topological protection. Topological quantum computation, including qubit readout, can be based on dressed zero modes which include these additional local charges. In its general form, the underlying result states that decoherence generically occurs in the eigenbasis of operators which commute with the system and readout Hamiltonians, which we refer to as symmetry protected decoherence or readout.
The theory also describes how to extract the measurement outcome from the measurement current through the quantum point contact. We find that generically, the measurement outcome cannot be reconstructed from the average measurement current, but only from its noise correlations. This can only be avoided when including additional dissipative processes, or by exploiting transient signals in fine-tuned situations. This surprising result can be traced back to the fact that the quantity to be read out enters into a tunneling amplitude which does not commute with the measured quantity, namely the quantum dot charge. We emphasize that for readout based on a single quantum dot, the Majorana qubit is in an excited state for a significant fraction of the measurement time. This may make the procedure susceptible to qubit errors by uncontrolled electron tunneling. This can be avoided in a measurement setup using a double quantum dot, which may thus promise better readout fidelities.
Finally, the theory naturally provides estimates of the measurement time. We find that the measurement times for Majorana qubit readout based on setups with a single quantum dot are consistently considerably larger than those for a conventional quantum dot charge readout at the same measurement strength. The reason for this is twofold. First, the decoherence rates are no longer simply controlled by the measurement strength $\sim k$ but involve the small tunnel couplings between quantum dot and Majorana qubit, $\sim \abs{t}^2 k / \varepsilon^2 \ll k$. Second, one generically cannot access the entire information contained in the measurement signal. Instead, typical experiments will only have access to its mean and two-point correlations. Thus, actual measurement times can be large compared to the decoherence time. However, slow readout should not be a generic feature of Majorana qubit readout. In fact, readout setups with two quantum dots can access an increased parameter range and should be less restricted. Investigating the double-dot setup in greater detail represents an interesting direction for future work.
[*Note added:*]{} Recently, we became aware of related unpublished work [@munk_parity_2020], which contains a partly complementary analysis and reaches similar conclusions where overlapping.
Majorana qubits and quantum dot – notation and definitions {#app:definitions}
==========================================================
In this appendix, we summarize details of our definitions and conventions for the quantum states of the coupled system of Majorana qubit and quantum dot.
Choice of basis states
----------------------
A set of $2m$ Majorana bound states labeled by $i=1,\ldots,2m$ is described by hermitian fermionic operators $\hat{\gamma}_i = \hat{\gamma}_i^{\dagger}$ which satisfy $\{\hat{\gamma}_i,\hat\gamma_j\} = 2\delta_{ij}$. The associated $2^m$-dimensional Hilbert space is spanned by the Fock occupations $\hat{n}_{ij} = \hat{f}_{ij}^{\dagger} \hat{f}_{ij}$ of complex fermions $\hat{f}_{ij}= \pqty{\hat{\gamma}_i + i\hat{\gamma}_j}/2$.
The total fermion parity of these basis states is given by the operator $$\label{eq:parity_op}
\hat{P} = \pqty{-i}^m\prod_{j=1}^{m}\hat{\gamma}_{2j-1}\hat{\gamma}_{2j}$$ As quantum superpositions exist only for states of the same fermion parity, a qubit requires at least four Majorana bound states. States with even and odd fermion parity can be split energetically by a charging energy, Eq. , as they have different charge $N=2N_C + N_{M}$, where $N_C$ is the number of Cooper pairs and $N_M$ the charge in the Majorana sector. For definiteness, we choose the ground (excited) states of the Majorana qubit to have $N=0$ ($N=-1$). The Hilbert space is spanned by $\Bqty{\ket{N,n_{12},n_{34}}}$. Due to the parity constraint $$\label{eq:parity_constraint}
P = \pqty{-1}^{N} = \pqty{-1}^{n_{12}+n_{34}},$$ it is sufficient to specify the state as $\ket{N,n_{12}}$.
In the main text, we choose a slightly different labeling of the basis states. First, instead of using the label $n_{12}$, we specify $n_{12}$ via the eigenvalue $z \in \Bqty{1,-1} \equiv \Bqty{\uparrow,\downarrow}$ of the Pauli-$Z$ operator $$\hat{Z} = -i\hat{\gamma}_1\hat{\gamma}_2 = 1-2\hat{n}_{12} = (-1)^{\hat{n}_{12}}.$$ Second, the Majorana qubit only exchanges charge with the quantum dot. We always initialize the system to have even total parity with Majorana-qubit charge $N=0$ and quantum dot charge $n=0$, so that these charges are related as $n=-N$ in general. We then label the basis states by the quantum dot charge $n$ and the eigenvalue $z$ of the Pauli-$Z$ operator of the Majorana qubit, $$\ket{z,n} \equiv \ket{N =-n, n_{12} = \frac{1-z}{2} , n_{34} = \frac{1-z(-1)^n}{2} }.$$ With this definition, the constraint is automatically satisfied. Furthermore, with our choices, all participating states have even total fermion parity $P_{\textrm{tot}} = P(-1)^n$.
The four basis states $\ket{z,n}$ differ in the local parity $\hat{\pi} = \hat{Z} (-1)^{ \hat{n}}$. We choose a basis $\{\ket{\uparrow,0},\ket{ \downarrow,1},\ket{\downarrow,0},\ket{\uparrow,1}\}$, where the first (last) two states have local parity $\pi=+1$ ($\pi=-1$). We can then define Pauli matrices acting in a subspace of fixed local parity,
$$\begin{aligned}
\sigma^\pi_z &=& \ketbra{\pi,0}{\pi,0} - \ketbra{-\pi,1}{-\pi,1}, \\
\sigma^\pi_x &=& \ketbra{\pi,0}{-\pi,1} + \ketbra{-\pi,1}{\pi,0}, \\
\sigma^\pi_y &=& i\sigma^\pi_x \sigma^\pi_z. \end{aligned}$$
In the main text, we drop the $\pi$ label whenever this does not lead to confusion. In terms of these Pauli matrices, the quantum dot charge is $$n = \frac{1}{2}\pqty{\sigma_0 - \sigma_z}.$$
Coupled Majorana qubit-quantum dot system
-----------------------------------------
In addition, we define a set of Pauli matrices using the eigenbasis of the Hamiltonian in Eq. which also act within the subspaces with fixed $\pi$,
$$\begin{aligned}
\tau^\pi_z &=& \ketbra{g_\pi}{g_\pi} - \ketbra{e_\pi}{e_\pi}, \\
\tau^\pi_x &=& \ketbra{e_\pi}{g_\pi} + \ketbra{g_\pi}{e_\pi} = \tau^{\pi}_+ + \tau^{\pi}_-, \\
\tau^\pi_y &=& i\tau^\pi_x\tau^\pi_z. \end{aligned}$$
Here, the ground states are given by Eq. (\[eq:ground\_states\]), while the excited states are
$$\begin{aligned}
\ket{e_+} &=& \cos\frac{\theta_+}{2} \ket{\uparrow,0} +\sin\frac{\theta_+}{2}e^{i\phi_+}
\ket{\downarrow,1},
\\
\ket{e_-} &=& \cos\frac{\theta_-}{2} \ket{\downarrow,0} +\sin\frac{\theta_-}{2}e^{i\phi_-}
\ket{\uparrow,1}.\end{aligned}$$
In terms of these Pauli matrices, $h_{\pi} = \Omega_\pi \tau^{\pi}_z$ and $$\begin{aligned}
\hat{n} =&\ \sum_{z=\pm 1} \ketbra{z,1}{z,1} \\
=&\ \sum_{\pi=\pm 1}\Big[ \cos^2\frac{\theta_\pi}{2}\ketbra{g_\pi}{g_\pi} + \sin^2\frac{\theta_\pi}{2}\ketbra{e_\pi}{e_\pi} \nonumber \\
&\ \ \ \ \ \ \ - 2\cos\frac{\theta_\pi}{2}\sin\frac{\theta_\pi}{2}\pqty{ \ketbra{e_\pi}{g_\pi} + \ketbra{g_\pi}{e_\pi} } \Big] \nonumber \\
=&\ \frac{1}{2} \sum_{\pi=\pm 1} \bqty{\tau^\pi_0 + \cos\theta_\pi \tau^\pi_z - \sin\theta_\pi \tau^\pi_x}.\end{aligned}$$ In the Heisenberg picture, the quantum dot charge $\hat{n}(t)=\textrm{exp} (i\hat{H}t)\ \hat{n}\ \textrm{exp}(-i\hat{H} t)$ becomes $$\begin{gathered}
\label{eq:n_heisenberg}
\hat{n}(t) = \sum_{\pi=\pm 1}\Big[ \frac{\tau^\pi_0 + \cos\theta_\pi \tau^\pi_z}{2} \\
- \frac{\sin\theta_\pi}{2} \pqty{ e^{2i\Omega_{\pi}t} \tau^{\pi}_+ + e^{-2i\Omega_{\pi}t} \tau^{\pi}_- } \Big].\end{gathered}$$ We find it convenient to define the time-independent part $$\label{eq:n_time_independent_part}
\hat{c} = \sum_{\pi=\pm 1} \frac{\tau^\pi_0 + \cos\theta_\pi \tau^\pi_z}{2}.$$ Finally, since we are performing our simulations in the charge basis, we need $\tau_{\pm}^{\pi}$ in the charge basis. Within a given $\pi$-block, it takes the form $$\tau_-^\pi = \frac{1}{2} \begin{pmatrix}
\sin\theta_\pi & \pqty{1-\cos\theta_\pi}e^{-i\phi_\pi} \\
\pqty{-1-\cos\theta_\pi}e^{i\phi_\pi} & -\sin\theta_\pi,
\end{pmatrix}$$ with $\tau_+^\pi$ given by hermitian conjugation.
Stroboscopic protocol for $\hat{Z}$ readout {#app:repeated_measurements}
===========================================
Measurement protocol
--------------------
Section \[sec:majorana\_qubits\_and\_quantum\_dots\] discusses projective measurements of $\hat{Z}$ based on a single projective measurement of $\hat{n}$. In addition to assuming instantaneous charge readout, this requires fine-tuned parameters. In subsequent sections of the main text, we relax both of these conditions. Here, we briefly discuss schemes which assume instantaneous charge readout, but allow for general system parameters $t_i$ and $\varepsilon$. For general parameters, the measurement outcome for the charge of the quantum dot is no longer perfectly correlated with eigenstates of $\hat{Z}$. However, unless $\Im{t_1t_2^*} = 0$, the charge measurement still provides partial information on the qubit state.
Once coupled, quantum dot and Majorana qubit evolve unitarily from initial state $\ket{\psi} = \pqty{\alpha\ket{\uparrow} + \beta \ket{\downarrow}}\ket{0}$ into $$\begin{aligned}
\hat{\mathcal{U}} \ket{\psi} = &\
\alpha\pqty{ c_{\uparrow ,0} \ket{\uparrow, 0} + c_{\downarrow, 1} \ket{\downarrow, 1} } \nonumber \\
&+ \beta\pqty{ c_{\downarrow, 0} \ket{\downarrow, 0} + c_{\uparrow, 1} \ket{\uparrow ,1} }.\end{aligned}$$ The unitary time evolution $\hat{\mathcal{U}}$, entangling Majorana qubit and quantum dot, satisfies $[\hat{\mathcal{U}},\hat{\pi}] = 0$, but otherwise depends on details of the protocol. If the charge readout can be effected instantaneously (on the time scale of the hybridization between dot and qubit), or more realistically, qubit and quantum dot are rapidly decoupled following the unitary evolution, the quantum dot charge becomes a good quantum number during charge readout. Then, the measurement leaves the system in the state $$\begin{aligned}
\ket{\psi'_0}=& \frac{1}{\sqrt{p_0}}\pqty{ \alpha c_{\uparrow, 0} \ket{\uparrow} + \beta c_{\downarrow, 0} \ket{\downarrow} } \ket{0},\end{aligned}$$ or $$\begin{aligned}
\ket{\psi'_1}=& \frac{1}{\sqrt{p_1}}\pqty{ \alpha c_{\downarrow, 1} \ket{\downarrow} + \beta c_{\uparrow, 1} \ket{\uparrow} } \ket{1},\end{aligned}$$ with probabilities $p_0 = \vert \alpha c_{\uparrow 0} \vert^2 + \vert \beta c_{\downarrow 0} \vert^2$ and $p_1 = 1- p_0$, respectively. The partial information on $\hat{Z}$ obtained from the measurement transfers weight between qubit states. This can be interpreted in terms of Bayesian inference [@korotkov_continuous_1999], $$p(\uparrow\vert n) = \frac{p (n \vert \uparrow) }{p_n} p(\uparrow),\ p(\downarrow \vert n) = \frac{p(n \vert \downarrow)}{p_n} p(\downarrow),$$ where we identify the prior probabilities of the qubit states with $p(\uparrow) = \abs{\alpha}^2$ and $p(\downarrow) = \abs{\beta}^2$ and $p( n \vert Z) = \vert c_{Z n}\vert^2$ with the conditional probabilities to observe measurement outcome $n$.
Repeating this protocol results in a random walk in the space of qubit and quantum dot states. We find that the random walk has two distinct steady states corresponding to the $\hat{\pi}$ eigensectors. This becomes equivalent to the eigenstates of $\hat{Z}$, if for simplicity, we reset the qubit-quantum dot system in between steps, $\ket{\downarrow, 1} \to \ket{\uparrow, 0}$ and $\ket{\uparrow, 1} \to \ket{\downarrow, 0}$ after every charge readout that gave $n=1$ (for a scheme implementing this reset, see App. \[app:qd\_reset\] below). With this reset, we effectively obtain a random walk in the space of qubit states, which can be described by the Kraus operators $$\begin{aligned}
\label{eq:kraus}
\hat{M}_0=&
\begin{pmatrix}
c_{\uparrow 0} & 0 \\ 0 & c_{\downarrow 0}
\end{pmatrix},\
\hat{M}_1= \hat{X}
\begin{pmatrix}
0 & c_{\uparrow 1} \\ c_{\downarrow 1} & 0
\end{pmatrix}\end{aligned}$$ acting on a qubit state $\ket{\phi}$, so that the qubit state $\ket{\phi'_n}$ conditioned on the measurement outcome $n$ is given by $$\ket{\phi'_n} = \frac{\hat{M}_n \ket{\phi}}{\sqrt{p_n}},$$ with $p_n = \bra{\phi} \hat{M}_n^{\dagger} \hat{M}_n \ket{\phi}$. The Pauli $\hat{X}$ in the definition of $\hat{M}_1$ makes the reset of the qubit-quantum dot system explicit.
![Evolution under the stroboscopic measurement protocol for adiabatic coupling with initial state $\ket{\psi} = (\ket{\uparrow}+\ket{\downarrow})\ket{0}/\sqrt{2}$, as function of time (number of iterations). Sample trajectories of $Z = \langle \hat{Z} \rangle$ (green and red traces) demonstrate the evolution towards the fixed points. The evolution tends slowly towards the $\downarrow$ state (corresponding to likely outcomes), interrupted by jumps (unlikely outcomes) that provide much more information, favoring the $\uparrow$ state and causing greater backaction. The measurement signal for two trajectories is shown by the blue and orange traces. Jumps in the trajectory are correlated with $n=1$ outcomes. The correct statistics of the measurement is illustrated by the constant ensemble average of $Z(N)$ (brown trace), which we computed for the initial state $\ket{\psi_2} = (\ket{\uparrow}+\sqrt{2}\ket{\downarrow})\ket{0}/\sqrt{3}$. Projection onto the fixed points is indicated by the decay of $\mathbb{E}[1-Z^2(N)]$ (purple trace) to zero. Ensemble averages are over $10^4$ trajectories. Parameters: $t_2 = t_1 e^{i\pi/4},\varepsilon = 10 t_1$.[]{data-label="fig:discrete_majorana_readout_adiabatic"}](discrete_majorana_readout_adiabatic.pdf){width="0.9\columnwidth"}
Figure \[fig:discrete\_majorana\_readout\_adiabatic\] shows numerical simulations of this protocol for the adiabatic coupling scheme discussed in Sec. \[sec:basic\_protocols\], so that $\mathcal{\hat{U}}$ involves the coefficients $c_{Z,n}$ of the state given in Eq. . The trajectories reach the fixed points $Z=\pm1$ with the correct probabilities. Moreover, the measurement outcome can be extracted from the sequence of $n$ outcomes of a single trajectory by noting that $n$ averages to the ground-state charge corresponding to the respective fixed point. Thus, this protocol implements a projective measurement of $\hat{Z}$.
We conclude this section with a number of comments. First, resetting the quantum dot charge is not essential. Without intermediate resetting, the protocol projects $\hat{\pi} = \hat{Z}(-1)^{\hat{n}}$ (see discussion in the main text) and one may reset once the outcome of the measurement is determined. However, in this case, the average charge no longer equals the ground state charge and a more involved signal analysis (for instance using the Bayes theorem) is required. Second, qubit and quantum dot can also be entangled through evolution with the Hamiltonian . In this case, decoupling qubit and quantum dot in between steps is necessary only if charge readout is slower than tunneling. The resulting evolution is closely related to the continuous evolution discussed in the main text, which arises naturally from sequences of repeated measurements when relaxing more and more assumptions on the strengths of various couplings.
![Spectrum of the quantum dot-Majorana qubit system as a function of the qubit gate charge $N_g$. The dashed lines for $N=-1$ (orange) and $N=0$ (blue) correspond to the energies of the system without coupling to the quantum dot. (The vertical shift of the orange curve reflects the energy of the quantum dot.) The full lines (red, green) refer to the coupled system and exhibit an avoided crossing. Initially, the Majorana qubit gate is tuned to $N_g = 0$ and the system is in the charge ground state $\ket{\psi} \propto \ket{N=0,n=0}$. (i) A measurement outcome of $n=1$ transfers the system into the excited charge state $\ket{\psi'} \propto \ket{N=-1,n=1}$. (ii) The charge state is reset by adiabatically changing $N_g \to -1$. The system state is now $\propto \ket{N=0,n=0}$ again. (iii) Suddenly set $N_g \to 0$, so that the system state does not change and the initial situation is restored. Alternatively, one may also decouple the quantum dot and slide down the dashed green curve adiabatically.[]{data-label="fig:qd_reset"}](reset_with_protocol.pdf){width="\columnwidth"}
Resetting the qubit-quantum dot system {#app:qd_reset}
--------------------------------------
When the charge measurement yields $n=1$, the Majorana qubit is in the excited charge state $N= - 1$. To avoid uncontrolled charging events, the electron should be swiftly returned from quantum dot to Majorana qubit.
This can be achieved by adiabatic variations of $\varepsilon$ and $t_i$, transforming $$\ket{\psi} = \pqty{\alpha \ket{\downarrow} + \beta \ket{\uparrow} }\ket{1}$$ into $$\ket{\psi_{\textrm{reset} }} = \pqty{\alpha \ket{\uparrow} + \beta \ket{\downarrow} }\ket{0}$$ for general $\alpha$ and $\beta$. We focus on the subspace $\pi = +1$ for definiteness.
Without fine tuning of the dynamical phase, $\ket{\downarrow, 1}$ can only be transformed into $\ket{\uparrow, 0}$ if it is an eigenstate of the initial Hamiltonian. Consider an initial Hamiltonian $h_+$ given by $\theta_+ = \theta_0$ and $\phi_+ = \phi_0$, and expand the initial state in eigenstates of $h_+$, $$\ket{\downarrow, 1} = e^{-i\phi_0}\pqty{ \sin\frac{\theta_0}{2} \ket{e_+}_0 - \cos\frac{\theta_0}{2} \ket{g_+}_0 }.$$ Adiabatically changing $\theta_0,\phi_0 \to \theta_1,\phi_1$, the eigenstates evolve as $\ket{e_+}_0 \to e^{i\chi_e} \ket{e_+}_1$ and $\ket{g_+}_0 \to e^{i\chi_g} \ket{g_+}_1$, where the subscripts distinguish eigenstates of the initial ($\theta_0,\phi_0$) and final ($\theta_1,\phi_1$) Hamiltonians. Then $$\ket{\downarrow , 1} \to e^{-i\phi_0}\pqty{ e^{i\chi_e}\sin\frac{\theta_0}{2} \ket{e_+}_1 - e^{i\chi_g} \cos\frac{\theta_0}{2} \ket{g_+}_1 }.$$ Writing $\ket{e_+}_1$ and $\ket{g_+}_1$ in the basis $\ket{\uparrow 0}, \ket{\downarrow 1}$ and setting the coefficient of $\ket{\downarrow 1}$ to zero yields the condition $$e^{i\chi_g} \sin\frac{\theta_0}{2}\sin\frac{\theta_1}{2} + e^{i\chi_e} \cos\frac{\theta_0}{2}\cos\frac{\theta_1}{2} = 0.$$ Without fine tuning, $\chi_g$ and $\chi_e$ are arbitrary phases and the two terms need to vanish separately. This implies $\theta_0 = 0$ and $\theta_1 = \pi$, or vice versa, so that the state $\ket{\downarrow, 1}$ was an eigenstate of $h_+(\theta_0,\phi_0)$ to begin with.
The charge state of the quantum dot-Majorana qubit system may then be reset from the initial state $\ket{\psi}$ as follows:
1. Suddenly decouple quantum dot and Majorana qubit ($t_i = 0$ and $\varepsilon>0$ or $(\theta_{\pm})_0 = \pi$). Then, $\ket{\downarrow, 1} = \ket{e_+}_0$ and $\ket{\uparrow ,1} = \ket{e_-}_0$ are energy eigenstates. The system state $\ket{\psi}$ is now a superposition $$\ket{\psi} = \alpha \ket{e_+}_0 + \beta\ket{e_-}_0.$$
2. Adiabatically swap $\ket{e_{\pm}}$ and $\ket{g_{\pm}}$, rotating the Hamiltonian $h_{\pm}$ from the south $(\theta_{\pm})_0 = \pi$ to the north pole $(\theta_{\pm})_1 = 0$ of the Bloch sphere, $$\begin{aligned}
\ket{\psi} \to&\ \alpha \ket{e_+}_1 + \beta e^{i\chi} \ket{e_-}_1
\\ &= \alpha \ket{g_+}_0 + \beta e^{i\chi} \ket{g_-}_0,
\\ &= \pqty{\alpha \ket{\uparrow} + \beta e^{i\chi} \ket{\downarrow}}\ket{0},
\end{aligned}$$ where $\chi$ is the relative dynamical phase between the $\hat{\pi}$ eigensectors introduced in the adiabatic evolution. Although uncontrolled, the relative phase does not affect the readout evolution as it preserves the weights in the $\hat{Z}$ eigenbasis. The step requires $t_i \neq 0$ at some point during the evolution to avoid gap closing, but eventually quantum dot and Majorana qubit are again decoupled.
3. Finally, suddenly reset the gates to their initial values, so that the quantum dot-Majorana qubit system again resides in a superposition of charge ground states.
This protocol requires a sign flip of $\varepsilon = E_C + \epsilon$, which can be realized by varying the quantum dot energy $\epsilon$ to compensate for the charging energy $E_C$ of the Majorana qubit, or by also varying the spectrum of Majorana-qubit charge states by the gate offset $N_g$, see Fig. \[fig:qd\_reset\].
The same procedure can be applied at the end of the continuous readout discussed in the main text which may also require a charge reset once the measurement outcome is certain.
Derivation of the stochastic master equation (\[eq:sme\_mbq-qd-qpc\]) {#app:derivation}
=====================================================================
For completeness, we include a derivation of the stochastic master equation which describes the evolution of the Majorana qubit-quantum dot system under continuous monitoring of the quantum dot charge by a quantum point contact, see, e.g., Ref. [@goan_continuous_2001].
The Hamiltonian $$\label{eq:qd_ham}
\hat{H}_{\textrm{readout}} = \hat{H}_{\textrm{leads}} + \hat{H}_{\textrm{jct}} + \hat{n}\ \delta\hat{H}_{\textrm{jct}}.$$ of the quantum point contact describes two (left and right) free-fermion leads, $$\hat{H}_{\textrm{leads}} = \sum_{\alpha} \pqty{ \xi_{L\alpha} \hat{c}^{\dagger}_{L\alpha}\hat{c}_{L\alpha} + \xi_{R\alpha} \hat{c}^{\dagger}_{R\alpha}\hat{c}_{R\alpha} },$$ and a tunneling Hamiltonian $$\hat{\mathcal{V}} = \hat{H}_{\textrm{jct}} + \hat{n}\ \delta\hat{H}_{\textrm{jct}}
= \pqty{ \tau + \chi \hat{n} } \hat{\psi}_L^{\dagger} \hat{\psi}_R e^{ieVt}+ \textrm{h.c.},
\label{Vcoup}$$ which includes the coupling to the quantum dot charge $\hat n$. The chemical potentials of the two leads differ by the bias voltage $V$ applied across the quantum point contact, $$eV = \mu_L-\mu_R$$ The factor $e^{ieVt}$ in the tunneling Hamiltonian (\[Vcoup\]) already accounts for a time-dependent unitary transformation such that the single-particle energies of the left and right leads are measured from the respective chemical potentials, $\xi_{L\alpha}= \epsilon_{L\alpha} - \mu - eV$ and $\xi_{R\alpha} = \epsilon_{R\alpha} - \mu$, with $\alpha$ labeling the single-particle eigenstates with corresponding electron operators $c_{L\alpha}$ and $c_{R\alpha}$. The electron operators evaluated at the junction position are denoted by $\hat{\psi}_L$ and $\hat{\psi}_R$. Importantly, there is only capacitive coupling, but no charge transfer between quantum dot and quantum point contact.
At zero temperature, the quantum point contact carries an average current $$\label{eq:current_unocc}
I_0 = 2\pi\nu_L\nu_R \abs{\tau}^2 eV,$$ when the quantum dot is unoccupied, and $$\label{eq:current_occ}
I_1 = 2\pi\nu_L\nu_R \abs{\tau+\chi}^2 eV,$$ when the quantum dot is occupied. Here, $\nu_{L/R}$ denotes the lead density of states. The sensitivity of the detector depends on $\delta I = I_1 - I_0$. We assume that the quantum dot affects the current weakly, $\delta I \ll I_0$.
Unconditional Lindblad master equation {#app:derivation_unconditional}
--------------------------------------
We first derive the unconditional master equation for the quantum dot state following the standard procedure of tracing over the leads, assuming factorization of the system-lead density matrix at all times (Born approximation), and finally assuming fast decay of the lead correlation functions to obtain a Markovian equation of motion. We will subsequently use the Lindblad equation to identify the Kraus operators, which allows one to derive the conditional master equation which accounts for the monitoring of the quantum-point-contact current.
We start in the interaction picture, with operators and states evolving according to $\hat{H}_0 = \hat{H} + \hat{H}_{\textrm{leads}}$ and $\hat{\mathcal{V}}=\hat{H}_{\textrm{jct}} + \hat{n}\ \delta\hat{H}_{\textrm{jct}}$, respectively. The corresponding density matrix for system and leads, $\hat{\chi}$, satisfies the equation of motion $$\begin{gathered}
\label{eq:IA_pic_eom}
\frac{\textrm{d}}{\textrm{d}t}\hat{\chi}(t)
= -i\ \bqty{ \hat{\mathcal{V}}(t) , \hat{\chi}(t_0) } \\
-\int_{t_0}^{t}\textrm{d}t'\ \bqty{ \hat{\mathcal{V}}(t) , \bqty{ \hat{\mathcal{V}}(t') , \hat{\chi}(t') } }.\end{gathered}$$ Weak coupling between system and quantum point contact allows for the Born approximation $\hat{\chi}(t) = \hat{\rho}(t) \otimes e^{-\beta H_{\textrm{leads}}}/Z$ since the effect on the density matrices of the leads remains small at all times. We can then trace out the leads, which enter the resulting equation for the (interaction-picture) density matrix $\hat{\rho}(t)$ of the system only through correlation functions. The first term in vanishes since cross-lead correlation functions are assumed zero. The second term gives
$$\begin{aligned}
\frac{\textrm{d}}{\textrm{d}t}\hat{\rho}(t) =
-\int_{t_0}^{t}\textrm{d}t'\
\Big\{ &
G^{<}_{L}(t'-t) G^{>}_{R}(t-t')
e^{ieV(t-t')}
\bqty{ \hat{m}(t) \hat{m}^{\dagger}(t') \hat{\rho}(t')
- \hat{m}^{\dagger}(t') \hat{\rho}(t') \hat{m}(t) } \nonumber \\
&
- G^{>}_L(t'-t) G^{<}_{R}(t-t')
e^{ieV(t-t')}
\bqty{ \hat{m}(t) \hat{\rho}(t') \hat{m}^{\dagger}(t')
- \hat{\rho}(t') \hat{m}^{\dagger}(t') \hat{m}(t) }\nonumber \\
&
+ G^{>}_{L}(t-t') G^{<}_{R}(t'-t)
e^{-ieV(t-t')}
\bqty{ \hat{m}^{\dagger}(t) \hat{m}(t') \hat{\rho}(t')
- \hat{m}(t') \hat{\rho}(t') \hat{m}^{\dagger}(t) } \phantom{\Big\{}\nonumber \\
&
- G^{<}_{L}(t-t') G^{>}_{R}(t'-t)
e^{-ieV(t-t')}
\bqty{ \hat{m}^{\dagger}(t) \hat{\rho}(t') \hat{m}(t')
- \hat{\rho}(t') \hat{m}(t') \hat{m}^{\dagger}(t) }
\Big\},
\label{eomrhoia}\end{aligned}$$
where we defined the shorthand $\hat{m}(t) = \tau + \chi\hat{n}(t)$ as well as the greater and lesser lead Green functions $$\begin{aligned}
G^{>}_{j}(t-t') =&\ -i \ev{ \hat{\psi}_{j}(t)\hat{\psi}^{\dagger}_{j}(t') }
= -\nu_j \frac{\pi/\beta}{\sinh \frac{\pi (t-t'-i\eta)}{\beta}} \\
G^{<}_{j}(t-t') =&\ i \ev{ \hat{\psi}^{\dagger}_{j}(t') \hat{\psi}_{j}(t)}
= -\nu_j \frac{\pi/\beta}{\sinh \frac{\pi (t-t'+i\eta)}{\beta}}. \end{aligned}$$ Here, $\beta$ denotes the inverse temperature and $\eta$ is a positive infinitesimal.
In the limit of a large bias voltage, the prefactors of the square brackets in the integrand on the right-hand side of Eq. (\[eomrhoia\]) effectively become sharply peaked functions in $t-t'$. Using the Fourier transform of the function $$g(E)= Eb(E) = -\int \frac{\mathrm{d}t}{2\pi} e^{iEt}\left(\frac{\pi/\beta}{\sinh \frac{\pi (t+i\eta)}{\beta}} \right)^2$$ with the Bose distribution $b(E)$, we can thus approximate $$\begin{aligned}
G^{<}_{L}(t'-t) & G^{>}_{R}(t-t') e^{ieV(t-t')}\nonumber\\
&\simeq 2\pi\nu_L\nu_Rg(-eV)\delta(t-t') + \textrm{imaginary} \nonumber\\
G^{>}_{L}(t'-t) & G^{<}_{R}(t-t') e^{ieV(t-t')}\nonumber\\
&\simeq 2\pi\nu_L\nu_Rg(eV)\delta(t-t') + \textrm{imaginary} \nonumber\\
G^{>}_{L}(t-t') & G^{<}_{R}(t'-t) e^{-ieV(t-t')}\nonumber\\
&\simeq 2\pi\nu_L\nu_Rg(eV)\delta(t-t') + \textrm{imaginary} \nonumber\\
G^{<}_{L}(t-t') & G^{>}_{R}(t'-t) e^{-ieV(t-t')}\nonumber\\
&\simeq 2\pi\nu_L\nu_Rg(-eV)\delta(t-t') + \textrm{imaginary}. \end{aligned}$$ Here, we do not specify the imaginary terms as they correspond to perturbative renormalizations of the system Hamiltonian. The approximate $\delta$-functions in time have a width of order $1/eV$. We assume that both the density matrix $\rho(t)$ and the quantum dot occupation $\hat n(t)$ vary slowly within times of order $1/eV$. In particular, this requires that the applied bias is large compared to characteristic system frequencies, $eV\gg \Omega_\pm$. In this limit, the equation of motion for $\hat\rho(t)$ becomes Markovian, and we obtain $$\begin{gathered}
\label{eq:master_eq_system_app_1}
\frac{\textrm{d}}{\textrm{d}t}\hat{\rho}(t)
= -i\bqty{\hat{H}, \hat{\rho}} \\
+ \frac{k}{eV\abs{\chi}^2}\Big\{ g(-eV) \mathcal{D}\bqty{\tau^*+\chi^* \hat{n}} \\
+ g(eV) \mathcal{D}\bqty{\tau + \chi \hat{n}} \Big\}\hat{\rho}(t).\end{gathered}$$ Here, we reverted from the interaction to the Schrödinger picture and defined the measurement strength $k=2\pi\nu_L\nu_R \abs{\chi}^2 eV$. The $(\tau^*+\chi^* \hat{n})$-term describes a process in which electrons tunnel from the left to the right lead. For positive $eV$, this happens even at $T=0$. The $(\tau+\chi \hat{n})$-term describes a process in which electrons tunnel from the right to the left lead which cannot occur at $T=0$. Absorbing a shift $2\pi \nu_L\nu_R eV \Im\bqty{\chi\tau^*}$ into the quantum dot energy $\epsilon$ in the Hamiltonian and taking the limit of zero temperature with $g(-eV) = eV$ and $g(eV) = 0$, we obtain the unconditional part of Eq.
Stochastic master equation {#app:derivation_conditional_jump}
--------------------------
Since the tunneling amplitude depends only weakly on the quantum dot occupation, the current measurement constitutes a weak measurement of the quantum-dot-qubit system. The change of the system state $$\ket{\psi} \to \frac{1}{\sqrt{p_i}}M_i\ket{\psi},$$ due to these weak measurements is described by Kraus operators $M_1$ and $M_0$ which can be respectively associated with transmission or absence of transmission of electrons by the quantum point contact (still assuming $T=0$ so that tunneling is unidirectional). The change in the density matrix $\hat{\rho}_c$ takes the form $$\hat{\rho}_c \to \frac{1}{p_i}M_i\hat{\rho}_c M_i^{\dagger}.$$ In these expressions, $$p_i = \bra{\psi}M_i^{\dagger} M_i \ket{\psi} = \textrm{tr}\bqty{M_i^{\dagger} M_i \hat{\rho}_c}$$ denotes the probability for outcome $i$.
The tunneling current through the quantum point contact can be described as a point process $$I_c(t) = e\frac{\textrm{d}N_c(t)}{\textrm{d}t},$$ where $\textrm{d}N_c(t) \in \Bqty{0,1}$ is a Poisson element which is not infinitesimal but has an infinitesimal ensemble average $$\begin{aligned}
\mathbb{E}\bqty{\textrm{d}N_c(t)}
= \textrm{tr}\bqty{M_1^{\dagger} M_1 \hat{\rho}_c(t)}\end{aligned}$$ equal to the probability that an electron is transmitted in time $\mathrm{d}t$. To find the Kraus operator $M_1$, we note that the ensemble average of the current $$\mathbb{E}\bqty{I_c(t)} = \frac{\textrm{tr}\bqty{M_1^{\dagger} M_1 \hat{\rho}_c(t)}} {\textrm{d}t}$$ has to equal $I_0$ for $n=0$ and $I_1$ for $n=1$. This is satisfied for $$M_1 = \pqty{\tau^*+\chi^*\hat{n}}\sqrt{\textrm{d}t},$$ where we rescaled $\sqrt{2\pi\nu_L\nu_R eV}\tau \to \tau$ and $\sqrt{2\pi\nu_L\nu_R eV}\chi \to \chi$ for notational simplicity (so that $I_0 = \abs{\tau}^2$, $I_1 = \abs{\tau+\chi}^2$, and $k = \abs{\chi}^2$, setting $e=1$).
To find the Kraus operator $M_0= \mathds{1} + \hat{A} \textrm{d}t$, we equate $\mathbb{E}\bqty{\hat{\rho}_c(t+\textrm{d}t)} = M_0 \hat{\rho}(t) M_0^{\dagger} + M_1 \hat{\rho}(t) M_1^{\dagger}$ and $\mathbb{E}\bqty{\hat{\rho}_c(t+\textrm{d}t)} =
\hat{\rho}(t+\textrm{d}t) = \pqty{1+\mathcal{L}\textrm{d}t}\hat{\rho}(t)$. Reading off the Liouvillian from Eq. (\[eq:master\_eq\_system\_app\_1\]), this yields $$M_0 = \mathds{1} - i \textrm{d}t \hat{H} - \frac{1}{2} M_1^{\dagger} M_1$$ and thus $\hat{A} = -i\hat{H} - (I_0 + \delta I \hat{n})/2$. As mentioned below Eq. (\[eq:master\_eq\_system\_app\_1\]) above, we absorb a shift of the quantum dot energy into the Hamiltonian, $$\begin{aligned}
\label{eq:a_operator}
\hat{A} = -i\pqty{\hat{H} + \Im\bqty{\tau^*\chi}\hat{n}} - \frac{1}{2}\bqty{I_0 + \pqty{2 \tau\chi^* + k} \hat{n}} .\end{aligned}$$ Below, we will no longer display this shift of $\hat{H}$ explicitly.
The conditional evolution of $\hat{\rho}_c(t)$ takes the form $$\begin{aligned}
\hat{\rho}_c(t+\textrm{d}t) =&\ \pqty{1-\textrm{d}N_c(t)} \frac{ M_0 \hat{\rho}_c(t) M_0^{\dagger} }{ \textrm{tr}\bqty{ M_0 \hat{\rho}_c(t) M_0^{\dagger}}} \nonumber \\
&+ \textrm{d}N_c(t) \frac{ M_1 \hat{\rho}_c(t) M_1^{\dagger}}{ \textrm{tr}\bqty{ M_1 \hat{\rho}_c(t) M_1^{\dagger}}},\end{aligned}$$ where the denominators ensure normalization. Expanding to linear order in $\textrm{d}t$ and neglecting higher-order terms of the form $\textrm{d}t\ \textrm{d}N_c(t)$, we obtain
$$\begin{gathered}
\label{eq:sme_jump}
\textrm{d}\hat{\rho}_c = \textrm{d}t
\Bqty{ -i \bqty{\hat{H},\hat{\rho}_c}
- \frac{1}{2}\pqty{
\Bqty{ I_0 + k \hat{n},\hat{\rho}_c}
+ 2 \chi^*\tau \hat{n}\hat{\rho}
+ 2\tau^*\chi \hat{\rho} \hat{n}
}
+ \pqty{ I_0 + \delta I \ev{ \hat{n} } } \hat{\rho}_c } \\
+ \textrm{d}N_c
\Bqty{
\frac{ \pqty{\tau^*+\chi^*\hat{n}} \hat{\rho}_c \pqty{\tau+\chi\hat{n}} }{ I_0 + \delta I \ev{ \hat{n}} }
- \hat{\rho}_c }.\end{gathered}$$
This describes the stochastic evolution of $\hat{\rho}_c$ as the quantum dot is monitored by the quantum point contact. The evolution of $\hat{\rho}_c$ is conditioned on the stochastic measurement current $I_c(t)$ of the quantum point contact.
We can alternatively describe the evolution in terms of a stochastic Schrödinger equation which takes the form
$$\textrm{d}\ket{\psi_c} =
\textrm{d}t \Bqty{
-i \hat{H}
- \frac{1}{2} \bqty{I_0 + \pqty{2\chi^*\tau + k} \hat{n}}
+ \frac{1}{2}\bqty{ I_0 + \delta I \ev{\hat{n}} }
} \ket{\psi_c}
+ \textrm{d}N_c \Bqty{
\frac{ \tau^*+\chi^*\hat{n} }{ \sqrt{ I_0 + \delta I\ \ev{ \hat{n}} } } - 1
} \ket{\psi_c}.$$
Diffusive approximation {#app:derivation_conditional_diffusive}
-----------------------
The assumption that the average current $I_0$ is much larger than the shift $\delta I$ induced by changes in the quantum-dot occupation allows one to approximate the point process by a Wiener process.
We consider a time interval $\delta t$ which is short enough that the changes in the density matrix $\hat{\rho}_c$ remain small and the quantum expectation value of $\hat n$ remains approximately constant. Then, the probability distribution of the number $N$ of tunneling events within the time interval $\delta t$ is given by the Poisson distribution $$\begin{aligned}
\label{eq:p_n}
P(N) = \frac{[(I_0+\delta I \langle\hat n\rangle)\delta t]^{N} }{ N ! } e^{-(I_0+\delta I \langle\hat n\rangle)\delta t}\end{aligned}$$ with ensemble average $$\begin{aligned}
\mathbb{E}\bqty{N} = [I_0 + {\delta I} \ev{\hat{n}}]\delta t, \end{aligned}$$ and variance $$\begin{aligned}
\mathbb{V}\bqty{N} \simeq I_0\delta t.\end{aligned}$$ The expression for the variance uses that $\delta I\ll I_0$. Assuming that $I_0\delta t$ is large, one can approximate the Poisson by a Gauss distribution with the same average and variance.
Assuming that $\delta I$ is sufficiently small and the Hamiltonian dynamics sufficiently slow, we can approximate both the unitary dynamics and the changes of the density matrix $\hat{\rho}_c$ induced by the weak measurements of the quantum dot charge to linear order in the time interval $\delta t$,
$$\begin{gathered}
\delta\hat{\rho}_c = {\delta}t
\Bqty{ -i \bqty{\hat{H},\hat{\rho}_c}
- \frac{1}{2}\pqty{
\Bqty{ I_0 + k \hat{n},\hat{\rho}_c}
+ 2 \chi^*\tau \hat{n}\hat{\rho}
+ 2\tau^*\chi \hat{\rho} \hat{n} }
+ \pqty{ I_0 + \delta I \ev{ \hat{n} } } \hat{\rho}_c } \\
+ {\delta}N_c
\Bqty{
\frac{ \pqty{\tau^*+\chi^*\hat{n}} \hat{\rho}_c \pqty{\tau+\chi\hat{n}} }{ I_0 + \delta I \ev{ \hat{n}} }
- \hat{\rho}_c }.
\label{eq:c23}\end{gathered}$$
Here, $\delta N_c$ describes the Wiener process $$\delta N_c(t) = [I_0 + {\delta I} \ev{\hat{n}}]\delta t + \sqrt{I_0} \xi(t) \delta t$$ with a Gaussian random process $\xi(t)$ with variance $1/\delta t$. Writing this in the continuum limit, we find $$\begin{gathered}
\label{eq:sme_diffusive_appendix}
\frac{\textrm{d}}{\textrm{d}t}\hat{\rho}_c = -i \bqty{\hat{H}_{\textrm{eff}} ,\hat{\rho}_c}
+ k\mathcal{D}\bqty{\hat{n}}\hat{\rho}_c \\
+ \sqrt{k}\xi(t) \mathcal{H}\bqty{\hat{n} e^{i \phi}}\hat{\rho}_c\end{gathered}$$ with $\tau^*\chi=\abs{\tau\chi}e^{i\phi}$ and the $\delta$-function correlator $\mathbb{E}\bqty{\xi(t)\xi(t')}=\delta(t-t')$. This simplifies to the evolution equation in the main text if one fixes $\phi = \pi$ (corresponding to a decrease in current through the quantum point contact due to the presence of an electron on the quantum dot).
The measurement current is obtained by subtracting the background current $I_0$ and normalizing. For $\phi = \pi$, this yields $$\begin{aligned}
\label{eq:measurement_current_app}
j(t) = \frac{1}{\delta I} \pqty{ \frac{{\delta} N_c(t)}{{\delta}t} - I_0 }
= \ev{\hat{n}(t) } + \frac{1}{\sqrt{4 k}} \xi(t)\end{aligned}$$ in agreement with Eq. (\[eq:meas\_signal\]) of the main text.
Relaxation by the electromagnetic environment {#app:dissipation}
---------------------------------------------
Coupling to the electromagnetic environment leads to relaxation in the eigenbasis of the Majorana qubit-quantum dot system, as described by Eq. (\[eq:dissipator\_main\]) in the main text. Here, we sketch its derivation.
The electrostatic potential of the electromagnetic environment is described as a free bosonic field $$\hat v(\mathbf{r}) = \sum_{\bf q} \left[ \tilde{m}_{\bm{q}}^* \hat{a}_{\bm{q}}^{\dagger} e^{i\mathbf{qr}}+ \tilde{m}_{-\bm{q}}\hat{a}_{\bm{-q}}e^{-i\mathbf{qr}} \right].$$ with Hamiltonian $$\hat{H}_{v} = \sum_{\bm{q}} \omega_{\bm{q}}\hat{a}_{\bm{q}}^{\dagger}\hat{a}_{\bm{q}}$$ and assumed to be in a thermal state $\hat{\rho}_v \propto \textrm{exp}(-\hat{H}_v/T)$. The potential $\hat v$ is an additional contribution to the gate voltages of Majorana qubit and quantum dot and varies slowly in space compared to the spatial extent of the system. Majorana qubit and quantum dot are then subject to the same potential $\hat v$, and we obtain the interaction $$\begin{aligned}
\hat{V} =& -\frac{2E_C C_g}{e} \hat{N} \hat{v} - \frac{2\epsilon_C c_g}{e} \hat{n}\hat{v} \\
=& \lambda \hat n \hat v \\
=& \hat n \sum_{\bf q} \left[ {m}_{\bm{q}}^* \hat{a}_{\bm{q}}^{\dagger} + {m}_{-\bm{q}}\hat{a}_{\bm{-q}} \right],\end{aligned}$$ where we used $\hat{N} = -\hat{n}$ due to charge conservation and absorbed $$\begin{aligned}
\lambda = \frac{2E_C C_g}{e} - \frac{2\epsilon_C c_g }{e} \end{aligned}$$ into the coefficients $m_{\mathbf{q}}$. (Here, $\epsilon_C$ and $c_g$ are charging energy and gate capacitance of the quantum dot, and $C_g$ is the gate capacitance of the Majorana qubit.)
We write the charge operator in the interaction picture with respect to the system Hamiltonian as given in Eq. . Then, the coupling Hamiltonian in rotating wave approximation becomes $$\begin{gathered}
\hat{V}(t) \simeq \hat{c} [\hat{B}(t) + \hat{B}^{\dagger}(t)] \\
- \frac{\sin\theta}{2} \pqty{ e^{2i\Omega t} \hat{\tau}_+ \hat{B}(t) + e^{-2i\Omega t} \hat{\tau}_- \hat{B}^{\dagger}(t) }, \end{gathered}$$ where we defined the shorthand $$\begin{aligned}
\label{eq:bath_ops_environment}
\hat{B}(t) =&\ \sum_{\bm{q}} m_{\bm{q}} e^{-i\omega_{\bm{q}}t}\hat{a}_{\bm{q}}.\end{aligned}$$ Notice that we suppressed $\pi$ indices as $\hat{V}(t)$ conserves $\hat{\pi}$ and does not mix the two subspaces.
Following the same steps as above, the master equation for a general system-bath interaction $$\label{eq:sb_schmidt_decomp}
\hat{H}_{SB} = \sum_{i} \hat{S}_{i} \otimes \hat{B}_{i}$$ with system operators $\hat S_i$, here associated with the Majorana qubit-quantum dot system, and bath operators $\hat B_i$, here associated with the electromagnetic environment, can be written as $$\begin{gathered}
\label{eq:reduced_evolution_equation}
\frac{\textrm{d}}{\textrm{d}t}\hat{\rho}(t)
= -i\bqty{\hat{H}, \hat{\rho}} \\
- \sum_{i} \Big\{ \hat{S}_{i}\hat{\mathcal{S}}^+_{i} \hat{\rho}(t)
- \hat{\mathcal{S}}^+_{i}\hat{\rho}(t) \hat{S}_{i} \\
+ \hat{\rho}(t) \hat{\mathcal{S}}^-_{i} \hat{S}_{i}
- \hat{S}_{i} \hat{\rho}(t) \hat{\mathcal{S}}^-_{i} \Big\}.\end{gathered}$$ Here, we defined the (calligraphic) operators
$$\begin{aligned}
\hat{\mathcal{S}}^+_{i} =&\ \sum_j \hat{\mathcal{S}}^+_{ij} = \sum_j\int_{0}^{\infty}\textrm{d}\tau\ C_{ij}(\tau) \hat{S}_{j}(-\tau), \\
\hat{\mathcal{S}}^-_{i} =&\ \sum_j \hat{\mathcal{S}}^-_{ij} =\sum_j\int_{0}^{\infty}\textrm{d}\tau\ C_{ji}(-\tau) \hat{S}_{j}(-\tau)\end{aligned}$$
including the bath correlation functions $$\label{eq:bath_correlators_general}
C_{ij}(t-t') = \ev{\hat{B}_{i}(t)\hat{B}_{j}(t')}.$$ We note that the $\hat{S}_{i}$ and $\hat{B}_{i}$ are not necessarily hermitian. If they are, $C^*_{ij}(\tau) = C_{ji}(-\tau)$ and thus $(\hat{\mathcal{S}}^-_{i})^{\dagger} = \hat{\mathcal{S}}^+_{i} \equiv \hat{\mathcal{S}}_{i}$.
Applying this to the problem at hand, we identify
$$\begin{aligned}
\hat S_1 =&\ \hat c \\
\hat S_2 =&\ -\sin\theta\ \textrm{exp}(2i\Omega t) \hat{\tau}_+ /2 \\
\hat S_3 =&\ -\sin\theta\ \textrm{exp}(-2i\Omega t) \hat{\tau}_- /2 \end{aligned}$$
and
$$\begin{aligned}
\hat B_1 =&\ \hat{B} + \hat{B}^{\dagger} \\
\hat B_2 =&\ \hat{B} \\
\hat B_3 =&\ \hat{B}^{\dagger}. \end{aligned}$$
One readily evaluates the basic bath correlation functions (with $\tau=t-t'$)
$$\begin{aligned}
C_{BB^{\dagger}}(\tau) =&\ \langle \hat B(t) \hat B^\dagger(t') \rangle
= \int_0^{\infty}\textrm{d}\omega\ J(\omega) e^{-i\omega\tau}(1+ b(\omega) ), \\
C_{B^{\dagger} B}(\tau) =&\ \langle \hat B^\dagger(t) \hat B(t') \rangle
= \int_0^{\infty}\textrm{d}\omega\ J(\omega) e^{i\omega\tau} b(\omega).\end{aligned}$$
Within the rotating wave approximation, we retain only terms which are slowly varying on the scale of the system dynamics. Moreover, we retain only dissipative terms and drop renormalizations of the system Hamiltonian. This yields the result $$\label{eq:relaxation_appendix}
\frac{\textrm{d}\hat{\rho}}{\textrm{d}t} =
\underbrace{\Bqty{\frac{\cos^2\theta}{4}\Gamma_0 \mathcal{D}\bqty{\hat{\tau}_z} + \frac{\sin^2\theta}{4}\pqty{\Gamma_+ \mathcal{D}\bqty{\hat{\tau}_+} + \Gamma_- \mathcal{D}\bqty{\hat{\tau}_-}} }}_{\equiv \mathcal{L}'}\hat{\rho},$$ where we defined
$$\begin{aligned}
\Gamma_0\, =&\ \pi \lim_{\omega \to 0} J(\omega)( 1 + 2 b(\omega)), \\
\Gamma_{+} =&\ 2\pi J(2\Omega) b(2\Omega) \\
\Gamma_{-} =&\ 2\pi J(2\Omega)\pqty{ 1 + b(2\Omega)}.\end{aligned}$$
The $\Gamma_0$ term causes decoherence in the energy basis, whereas the $\Gamma_\pm$ terms cause transitions between ground and excited states. At low temperatures, $\Gamma_- \gg \Gamma_+$ with $\Gamma_+$ vanishing at $T=0$ and $\Gamma_-$ remaining finite. For this reason, we neglect $\Gamma_+$ relative to $\Gamma_-$ in the main text.
Spectra and eigenmodes of Liouvillians {#app:liouvillian_evs}
======================================
Eigenvalues and eigenmatrices of diagonal block Liouvillian $\mathcal{L}_{\pi,\pi} + \mathcal{L}'_{\pi,\pi}$ {#app:liouvillian_evs_diag}
------------------------------------------------------------------------------------------------------------
This appendix gives the eigenvalues and eigenvectors of the diagonal blocks of the full Liouvillian including both measurement and relaxation dynamics, $\mathcal{L}_{\pi,\pi} + \mathcal{L}'_{\pi,\pi}$. To this end, we vectorize by columns, $\rho \to \ket{\rho} = (\rho_{11}, \rho_{21} , \rho_{12} , \rho_{22})^T$. In this notation, $\textrm{tr}[AB] = \braket{A^{\dagger}}{B}$ for square matrices $A$ and $B$. The Liouvillian matrix is then given by $$\begin{aligned}
\mathcal{L}_{\pi,\pi} + \mathcal{L}_{\pi,\pi}' =&\ -i\pqty{\mathds{1}\otimes h_\pi - h_\pi^T \otimes \mathds{1} } + k \mathrm{D}[n] \nonumber \\
&\ + \frac{\cos^2\theta_\pi}{4} \Gamma_0 \mathrm{D}\bqty{\tau_z} + \frac{\sin^2\theta_\pi}{4} \Gamma_- \mathrm{D}\bqty{ \tau_-} \nonumber \\
=&\ \sum_n \lambda_{\pi,n} \ketbra{\psi_{\pi,n}}{\phi_{\pi,n}}, \end{aligned}$$ where we defined the vectorized decoherence superoperator $$\mathrm{D}[L] = L^* \otimes L - \frac{1}{2}\bqty{ \mathds{1}\otimes L^{\dagger} L + \pqty{L^{\dagger} L}^T \otimes \mathds{1} },$$ and expanded in terms of left and right eigenvectors $\ket{\psi_{\pi,n}}$ and $\ket{\phi_{\pi,n}}$ to eigenvalue $\lambda_{\pi,n}$. In the following, we work to leading order in $t_i, k, \Gamma_-, \Gamma_0 \ll \Omega_\pi$ and will suppress $\pi$ subscripts of $\theta$ and $\Omega$. The eigenvalues are
$$\begin{aligned}
\lambda_0 =&\ 0, \\
\lambda_{\textrm{slow}} =&\ -\frac{\sin^2\theta}{4}\pqty{\Gamma_- + 2k}, \\
\lambda_{\textrm{fast} , \pm} =&\ \pm 2i\Omega - \frac{1+\cos^2\theta}{4}k \nonumber \\ & -\frac{\sin^2\theta}{8}\Gamma_- - \frac{\cos^2\theta}{2} \Gamma_0 .\end{aligned}$$
The corresponding right eigenvectors, written in the energy basis, are
$$\begin{aligned}
&\ket{\psi_0} \simeq \frac{1}{2}\pqty{ 1 + R , 0 , 0 , 1-R }^T , \\
&\ket{\psi_{\textrm{slow}}} \simeq \pqty{ 1 , 0 , 0, -1}^T, \\
&\ket{\psi_{\textrm{fast}, +}} \simeq \pqty{ 0 , 0, -1 , 0 }^T, \\
&\ket{\psi_{\textrm{fast}, -}} \simeq \pqty{ 0 , 1 , 0 , 0 }^T,\end{aligned}$$
where $R$ was defined in Eq. , and the left eigenvectors are
$$\begin{aligned}
&\bra{\phi_0} = \pqty{ 1 , 0 , 0 , 1 } ,\\
&\bra{\phi_{\textrm{slow}}} \simeq \pqty{ \frac{k}{\Gamma_- + 2k} , 0 , 0 , -\frac{\Gamma_-+k}{\Gamma_-+2k}}, \\
&\bra{\phi_{\textrm{fast}, +}} \simeq \pqty{ 0 , 0, -1 , 0 }, \\
&\bra{\phi_{\textrm{fast}, -}} \simeq \pqty{ 0 , 1 , 0 , 0 }.\end{aligned}$$
Note that $\ket{\psi_0} = \ket{\rho^{\infty}}$ and $\bra{\phi_0} = \bra{\mathds{1}_2}$. We normalized $\braket{\phi_m}{\psi_n} = \delta_{mn}$ to leading order.
Off-diagonal block Liouvillian $\mathcal{L}_{+-}$ in Eq. (\[eq:+-\_block\_meq\]) {#app:liouvillian_evs_offdiag}
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In this appendix, we analyze the off-diagonal Liouvillian $\mathcal{L}_{+-}$ in Eq. with $h_{\pm}$ in Eq. and one decoherence channel given by the quantum-point-contact coupling (i.e., without relaxation).
### Steady state of $\rho_{+-}$
To understand the long-time behavior of the density matrix, we find the eigenvalues of $ \mathcal{L}_{+-}$ and show that generically, the real parts of all eigenvalues are strictly negative. Thus, the only steady state is $\rho_{+-}=0$.
We again write the superoperator $ \mathcal{L}_{+-}$ in vectorized form with $\rho_{+-}\to (\rho_{+-}^{11},\rho_{+-}^{21},\rho_{+-}^{12},\rho_{+-}^{22})^T$, such that (in the charge basis) $$\begin{aligned}
\mathcal{L}_{+-} =&\ -i\pqty{\mathds{1}\otimes h_+ - h_-^T \otimes \mathds{1} } \nonumber \\
&\ + k\bqty{ n\otimes n - \frac{1}{2}\pqty{\mathds{1}\otimes n + n \otimes \mathds{1}} } \nonumber
\\
=&\
\begin{pmatrix}
0 & -it_1 - t_2 & i t^*_1 + t^*_2 & 0 \\
-i t^*_1 + t^*_2 & -i\varepsilon -k/2 & 0 & i t^*_1 + t^*_2 \\
i t_1 - t_2 & 0 & i \varepsilon -k/2 & -it_1 - t_2 \\
0 & i t_1 - t_2 & -i t_1^* + t^*_2 & 0
\end{pmatrix}.\end{aligned}$$ The characteristic polynomial becomes $$\begin{gathered}
\label{eq:char_pol}
\chi(\tilde{\lambda}) = \tilde{\lambda}^4 + k\tilde{\lambda}^3 + ({\varepsilon}^2 + 4|t_1|^2+4|t_2|^2 +{k^2}/{4}) \tilde{\lambda}^2
\\+2k(|t_1|^2+|t_2|^2)\tilde{\lambda} + 16|t_1|^2|t_2|^2\sin^2\varphi,\end{gathered}$$ where $\varphi$ denotes the phase difference between the tunneling amplitudes $t_1$ and $t_2$.
Evidently, $\mathcal{L}_{+-}$ has a zero eigenvalue when $\sin\varphi=0$ or, equivalently, $\Im{t_1 t_2^*}=0$. In this case, $\rho_{+-}$ does not decay to zero and neither $\hat{\pi}$ not $\hat{Z}$ are projectively measured. Physically, the Rabi frequencies of the $\pi = +1$ and the $\pi = -1$ sectors are identical and the steady-state measurement currents of the two sectors are indistinguishable. Thus, the measurement reveals no information on the qubit state and does not decohere the system in the measurement basis.
Conversely, if $\sin\varphi\neq 0$, there is no zero eigenvalue. We now show that in this case, the eigenvalues have strictly negative real parts. They are non-positive since $\mathcal{L}_{+-}+\mathcal{L}_{+-}^{\dagger}$ is negative semi-definite. To show that the real parts of the eigenvalues are strictly negative, consider the characteristic polynomial. Taking $\sin\varphi \neq 0$, we assume that there is an imaginary eigenvalue $\tilde{\lambda}=iy$ with $y \in \mathds{R}$. This eigenvalue satisfies $$\begin{aligned}
\label{eq:char_pol_2}
&&y^4 -i ky^3 - ({\varepsilon}^2 + 4|t_1|^2+4|t_2|^2 +{k^2}/{4}) y^2
\nonumber\\
&&+i2k(|t_1|^2+|t_2|^2)y + 16|t_1|^2|t_2|^2\sin^2\varphi = 0.\end{aligned}$$ The imaginary part of this equation, $y^3=2(|t_1|^2+|t_2|^2)y$ has solutions $y=0$ and $y=\pm [2(|t_1|^2+|t_2|^2)]^{1/2}$. For $y=0$, the real part of Eq. implies $\sin^2\varphi=0$, contradicting our assumptions. Similarly, for $y=\pm [2k(|t_1|^2+|t_2|^2)]^{1/2}$, the real part implies $$\frac{4|t_1|^2|t_2|^2}{(|t_1|^2+|t_2|^2)^2}\sin^2\varphi -1 = \frac{\epsilon^2+k^2/4}{2(|t_1|^2+|t_2|^2)}.$$ While the left hand side is non-positive, the right-hand side is strictly positive, so that there are no solutions. We conclude that $\mathcal{L}_{+-}$ has only eigenvalues with a strictly negative real part.
### Decoherence rate
The decoherence rate is governed by the eigenvalue $\tilde{\lambda}_{\textrm{slow}}$ of $\mathcal{L}_{+-}$ with the largest real part (corresponding to the slowest decay). We perform perturbative analyses for small $k$ as well for small $\sin^2\varphi$. Beyond the perturbative regime, we investigate the behavior of the eigenvalues numerically, see Fig. \[fig:slowest\_decaying\_ev\] in the main text. For simplicity, we specify to $t_1 = t_2 e^{-i\varphi}$ with $t_1$ real. We also define the shorthand $\tilde{\varepsilon}^2 = \varepsilon^2+8t_1^2$. Then, the characteristic polynomial takes the form $$\chi(\tilde{\lambda}) = \tilde{\lambda}^4 + k\tilde{\lambda}^3 + (\tilde{\varepsilon}^2 + \frac{k^2}{4}) \tilde{\lambda}^2 + 4kt_1^2\tilde{\lambda} + 16t_1^4 \sin^2\varphi.$$
#### Small $k$: {#small-k .unnumbered}
For weak coupling between quantum dot and quantum point contact, we determine the roots of the characteristic polynomial to first order in $k$. Expanding $\tilde{\lambda}=\tilde{\lambda}_0 + k \tilde{\lambda}_1 + ...$, we obtain
$$\begin{aligned}
0 =& \tilde{\lambda}_0^4 + \tilde{\varepsilon}^2 \tilde{\lambda}_0^2 + 16t_1^4\sin^2\varphi, \\
0 =& 4\tilde{\lambda}_0^3\tilde{\lambda}_1 + \tilde{\lambda}_0^3 + 2\tilde{\varepsilon}^2 \tilde{\lambda}_0 \tilde{\lambda}_1 + 4 t_1^2 \tilde{\lambda}_0\end{aligned}$$
with the solutions
$$\begin{aligned}
(\tilde{\lambda}^\pm_0)^2 =& -\frac{\tilde{\varepsilon}^2}{2} \pm \sqrt{\pqty{\frac{\tilde{\varepsilon}^2}{2}}^2-16t_1^4\sin^2\varphi} \\
\tilde{\lambda}^\pm_1 =& - \frac{1}{4}\left[1- \frac{\varepsilon^2}{2(\tilde{\lambda}^\pm_0)^2+\tilde\varepsilon^2}\right] \nonumber\\
=& -\frac{1}{4}\bqty{1 \pm \frac{\varepsilon^2}{\sqrt{\varepsilon^4+16\varepsilon^2t_1^2+64t_1^4\cos^2\varphi }}}.\end{aligned}$$
While $(\tilde{\lambda}^\pm_0)^2 \leq 0$, so that the $\tilde{\lambda}_0$ are purely imaginary, the first-order correction $k\tilde{\lambda}_1$ is manifestly real and negative. For the slowly decaying eigenvalues, we choose the minus sign in the above expressions. In the limit $t_1 \ll \varepsilon$, the decay of $\rho_{+-}$ is then controlled by $$\Re{\tilde{\lambda}_{\textrm{slow}}} \simeq \Re{\tilde{\lambda}^-_0+ k \tilde{\lambda}^-_{1}} = -2k\frac{t_1^2}{\varepsilon^2}.$$ For $\sin^2\varphi=0$, $\tilde{\lambda}_0^-$ vanishes while $\tilde{\lambda}_1^-$ remains finite. Hence, perturbation theory breaks down close to $\sin^2\varphi=0$. Requiring $\vert\tilde{\lambda}^-_1\vert \ll k\vert\tilde{\lambda}^-_2\vert$ yields the condition $$\frac{k}{\varepsilon} \ll \abs{ \abs{\sin\varphi} - \frac{1}{4\abs{\sin\varphi}}}^{-1}.$$ The right side of this inequality vanishes for $\varphi = 0$, i.e. the expansion indeed breaks down.
#### Small $\sin^2\varphi$: {#small-sin2varphi .unnumbered}
The slowest decaying eigenvalue $\tilde{\lambda}_{\textrm{slow}}$ evolves from the zero eigenvalue for $\sin^2\varphi=0$. For small $\sin^2\varphi$, we readily find $$\tilde{\lambda}_{\textrm{slow}} = -\frac{4 t_1^2 \sin^2 \varphi}{k} + ...$$ Notice that the expansion breaks down for vanishing $k$.
The fact that a perturbative expansion is impossible for small $k$ and $\varphi$ is in accordance with the numerical observation that exceptional lines emanate from this region, see Fig. \[fig:slowest\_decaying\_ev\].
Stochastic evolution of $\pi(t)$ {#app:stochastic_evolution}
================================
It is instructive to analyze the stochastic evolution of the expectation value of the combined fermion parity $\hat\pi$. For $\hat{H}_T = 0$, Eq. (\[eq:charge\_stochastic\_equation\]) showed that the evolution of $n(t)$ ceases once $n = 0$ or $n = 1$. In the presence of tunneling, $\hat{H}_T \neq 0$, this is no longer the case due to the additional term $\langle i [\hat{H}_T , \hat{n}]\rangle$ in Eq. (\[eq:charge\_stochastic\_equation\_2\]).
To analyze $\pi$, we consider the set of coupled stochastic differential equations obtained from Eq. . As a result of the transfer of weight between the $\pi$ subspaces due to measurements, the stochastic term couples $\rho_{++}$ and $\rho_{--}$. At the same time, these diagonal blocks of $\hat\rho$ remain uncoupled to $\rho_{+-}$. (However, the time evolution of $\rho_{+-}$ depends on $\mathrm{tr}\rho_{\pi,\pi}$.) We write the coupled equations for the diagonal blocks using the Bloch-vector notation $$\begin{aligned}
\rho_{\pi,\pi}
= \frac{p_\pi}{2} + \frac{1}{2}\pqty{\sigma_x x_\pi + \sigma_y y_\pi + \sigma_z z_\pi },\end{aligned}$$ where $p_\pi = \textrm{tr} \rho_{\pi,\pi} = \langle\hat{P}_\pi\rangle$ (with the projector $\hat{P}_\pi$ onto the $\pi$ subspace). With this parametrization, Eq. yields the stochastic differential equations
$$\begin{aligned}
\dot{p}_\pi =& -\sqrt{k} \xi \pqty{z_\pi - \mathcal{Z} p_\pi} \label{eq:stochastic_projector_plus} \\
\dot{z}_\pi =& 2\Omega_\pi \pqty{h_\pi^x y_\pi - h^y_\pi x_\pi} - \sqrt{k} \xi \pqty{p_\pi - \mathcal{Z} z_\pi}, \\
\dot{x}_\pi =& 2\Omega_\pi \pqty{h_\pi^y z_\pi - h^z_\pi y_\pi} - \frac{k}{2} x_\pi + \sqrt{k} \xi \mathcal{Z} x_\pi ,\\
\dot{y}_\pi =& 2\Omega_\pi \pqty{h_\pi^z x_\pi - h^x_\pi z_\pi} - \frac{k}{2} y_\pi + \sqrt{k} \xi \mathcal{Z} y_\pi,\end{aligned}$$
where $\mathcal{Z} = z_+ + z_-$ introduces the coupling between the diagonal blocks.
Fixed points of $\pi(t)=p_+-p_-=2p_+-1$ require that the right hand side of Eq. vanish, $M_\pi = z_\pi - \mathcal{Z} p_\pi = 0$. $p_\pi = 0$ implies $x_\pi = y_\pi = z_\pi =0$. We then find by direct evaluation that $p_+ = 0$ (or similarly $p_-=0$) implies $M_+ = M_- = 0$. We conclude that $p_+=0$ and $p_+=1$ are fixed points of the evolution of $p_+(t)$. These correspond to fixed points $\pi= -1$ and $\pi=+1$ of $\pi(t)$, respectively. We checked numerically that these are the only fixed points provided $\Im{t_1 t_2^*}\neq 0$. (This is indicated, e.g., by the fact that $\mathbb{E}[1-\pi^2] \to 0$ as shown in Fig. \[fig:majorana\_measurement\_generic\]).
Now consider the case $\Im{t_1 t_2^*} =0$. For simplicity we make the stronger assumption $h_+ = h_-$. To understand this case, it is easiest to consider the evolution in terms of $$\begin{aligned}
l^\perp_\pi =&\ \pqty{h_\pi^x y_\pi - h^y_\pi x_\pi}, \\
l^\parallel_\pi =&\ \pqty{h_\pi^x x_\pi + h^y_\pi y_\pi}.\end{aligned}$$ Then, the evolution equations are
$$\begin{aligned}
\dot{p}_\pi =&\ -\sqrt{k} \xi \pqty{z_\pi - \mathcal{Z} p_\pi}\\
\dot{z}_\pi =&\ 2\Omega_\pi l^\perp_\pi - \sqrt{k} \xi \pqty{p_\pi - \mathcal{Z} z_\pi}, \\
\dot{l}^\perp_\pi =&\ 2\Omega_\pi \pqty{h^z_\pi l^\parallel_\pi - \sin^2\theta_\pi z_\pi } - \frac{k}{2} l^\perp_\pi + \sqrt{k} \xi \mathcal{Z} l^\perp_\pi ,\\
\dot{l}^\parallel_\pi =&\ - 2\Omega_\pi h^z_\pi l^\perp_\pi - \frac{k}{2} l^\parallel_\pi + \sqrt{k} \xi \mathcal{Z} l^\parallel_\pi.\end{aligned}$$
We also define $L^\perp = l^\perp_+ + l^\perp_-$ and $L^\parallel = l^\parallel_+ + l^\parallel_-$. For $h_+ = h_- = h = \Omega \bm{h}\cdot \bm{\sigma}$, they satisfy a decoupled set of equations
$$\begin{aligned}
\dot{\mathcal{Z}} =&\ 2\Omega L^\perp - \sqrt{k} \xi \pqty{1 - \mathcal{Z}^2 }, \\
\dot{L}^\perp =&\ 2\Omega \pqty{h^z L^\parallel - \sin^2\theta \mathcal{Z} } - \frac{k}{2} L^\perp + \sqrt{k} \xi \mathcal{Z} L^\perp ,\\
\dot{L}^\parallel =&\ - 2\Omega h^z L^\perp - \frac{k}{2} L^\parallel + \sqrt{k} \xi \mathcal{Z} L^\parallel.\end{aligned}$$
Importantly, the equations for the $l$ and $L$ have identical form. We now show that the evolution of $p_+$ becomes frozen if the dynamics of the lower-case variables is locked to that of the upper-case variables, i.e., if
\[eq:fixed\_point\_h=h\] $$\begin{aligned}
0 =&\ z_+ - p_+ \mathcal{Z}, \\
0 =&\ l^\perp_+ - p_+ L^\perp, \\
0 =&\ l^\parallel_+ - p_+ L^\parallel, \end{aligned}$$
independently of the value of $p_+$. If $h_+ = u h_- u^\dagger$ similar statements hold for linear combinations of these variables. For the above relations to indeed be fixed points, we need to show that also their Ito differentials vanish. Consider first the two lower lines, $$\begin{aligned}
\textrm{d}\pqty{l_+ - p_+ L} =&\ \textrm{d} l_+ - p_+ \textrm{d} L - \textrm{d} p_+ \pqty{ { \ldots }} \nonumber \\
=&\ \ \ \ \pqty{ { \ldots } } \bqty{z_+ - p_+ \mathcal{Z}} \nonumber \\
&+ \pqty{ { \ldots } } \bqty{l^\perp_+ - p_+ L^\perp} \nonumber \\
&+ \pqty{ { \ldots } } \bqty{l^\parallel_+ - p_+ L^\parallel}, \end{aligned}$$ where we used that the evolution equations for $l$ and $L$ have identical form. The terms in square brackets vanish at the fixed point specified by Eq. , so that the detailed form of the terms in the parentheses does not matter. Similarly, $$\begin{aligned}
\textrm{d}\pqty{ z_+ - p_+ \mathcal{Z} } =&\ \sqrt{k} \xi \mathcal{Z} \pqty{z_+ - p_+ \mathcal{Z}} \nonumber \\ &+ 2\Omega \pqty{l^\perp_+ - p_+ L^\perp}.\end{aligned}$$ Thus, Eqs. indeed describe fixed points with arbitrary $p_+$. If the system is initialized, the dynamics of the two sectors will then tend to lock. Once this has happened, $p_+$ remains constant. The ensemble averaged evolution of course has $p_+(t) = p_+(0)$.
Noise spectrum of the measurement current {#app:spectrum}
=========================================
Autocorrelation function of the steady-state measurement signal {#app:auto_correlation}
---------------------------------------------------------------
The measurement outcomes $\pi = +1$ and $\pi = -1$ are distinguished by the noise spectrum of the steady-state measurement current $j_\pi(t)$ for a given $\pi$, $$\begin{aligned}
\label{eq:autocorrelation_expanded}
S(\tau) =&\ \mathbb{E}\bqty{j(t)j(t+\tau)} - \mathbb{E}\bqty{j(t)}\bqty{j(t+\tau)}\nonumber \\
=&\ \frac{\delta(\tau)}{4k}
+ \frac{1}{\sqrt{4k}} \left\{\mathbb{E}\bqty{n(t)\xi(t+\tau)} + \mathbb{E}\bqty{n(t+\tau)\xi(t)}\right\} \nonumber \\
&+ \mathbb{E}\bqty{n(t)n(t+\tau)} - \mathbb{E}\bqty{n(t)}\bqty{n(t+\tau)},\end{aligned}$$ where we suppress all labels indicating the measurement outcome and used Eq. (\[eq:meas\_signal\]). Our evaluation of $S(\omega)$ follows Ref. [@wiseman_quantum_1993] (see App. B).
First consider $\mathbb{E}\bqty{n(t+\tau)\xi(t)}$, which requires one to compute $\mathbb{E}\bqty{\hat\rho_c(t+\tau)\xi(t)}$. From Eqs. (\[eq:c23\]) and (\[eq:measurement\_current\_app\]), it is evident that $\xi(t)$ is only correlated with the stochastic contribution to $\delta\rho_c$ for the next time step, from $t$ to $t+\delta t$. We then find $$\begin{aligned}
\mathbb{E}\bqty{ \hat{\rho}_c(t+{\delta}t) \xi(t)}
=& \sqrt{k} \mathbb{E}\bqty{ \mathcal{H}[\hat{n}] \hat{\rho}_c(t) } \nonumber \\
=& \sqrt{4k}\pqty{ \frac{\Bqty{\hat{n},\hat{\rho}^{\infty}}}{2} - \mathbb{E}\bqty{n(t) \hat{\rho}_c(t) } },\end{aligned}$$ where we write $\mathbb{E}[\hat\rho_c] = \hat{\rho}^{\infty}$. Using the formal solution $\hat{\rho}_c(t) = \mathcal{U}(t,0) \hat{\rho}_c(0)$ of the stochastic master equation (\[eq:sme\_mbq-qd-qpc\]) with $$\mathcal{U}(t,t') = \mathcal{T} \exp\Bqty{\mathcal{L}(t-t') + \sqrt{k}\int_{t'}^{t} \textrm{d}t_1 \xi(t_1)\mathcal{H}[\hat{n}] }$$ ($\mathcal{T}$ is the time ordering operator) and exploiting that $\xi(t)$ is uncorrelated with any of the later $\xi(t_1)$, we conclude that $$\begin{gathered}
\mathbb{E}\bqty{ \hat{\rho}_c(t+\tau) \xi(t)}
= \sqrt{4k}\, \theta(\tau) \Big( e^{\mathcal{L}\tau}\frac{\Bqty{\hat{n},\hat{\rho}^{\infty}}}{2}
\\
- \mathbb{E}\bqty{n(t) e^{\mathcal{L}\tau} \hat{\rho}_c(t) } \Big) ,\end{gathered}$$ and thus $$\begin{gathered}
\label{eq:autocorrelation_term_tau_greater_0}
\frac{1}{\sqrt{4k}}\mathbb{E}\bqty{n(t+\tau)\xi(t)} \\
= \theta(\tau) \Big(
\textrm{tr}\bqty{\hat{n} e^{\mathcal{L}\tau} \frac{\Bqty{\hat{n},\hat{\rho}^{\infty}}}{2} }
- \mathbb{E}\bqty{n(t) \textrm{tr}\bqty{\hat{n}e^{\mathcal{L}\tau} \hat{\rho}_c(t) }} \Big) \\
= \theta(\tau) \Big(
\textrm{tr}\bqty{\hat{n} e^{\mathcal{L}\tau} \frac{\Bqty{\hat{n},\hat{\rho}^{\infty}}}{2} }
- \mathbb{E}\bqty{n(t) n(t+\tau) } \Big) .\end{gathered}$$ In the last step, we used that $$\mathbb{E}\bqty{n(t) \textrm{tr}\bqty{\hat{n}e^{\mathcal{L}\tau} \hat{\rho}_c(t) }}
= \mathbb{E}\bqty{n(t) \textrm{tr}\bqty{\hat{n}\mathcal{U}(t+\tau,t) \hat{\rho}_c(t) }} ,$$ since all the additional stochastic terms introduced on the right-hand side average to zero.
Similarly, $\mathbb{E}\bqty{n(t)\xi(t+\tau)}$ is nonzero for $\tau<0$ only. Then, time translation invariance of the stationary state implies $$\begin{aligned}
\mathbb{E}\bqty{n(t)\xi(t+\tau)} &= \theta(-\tau)\mathbb{E}\bqty{n(t)\xi(t-|\tau|)} \\
&= \theta(-\tau) \mathbb{E}\bqty{n(t+|\tau|)\xi(t)} \end{aligned}$$ and we conclude $$\begin{gathered}
\frac{1}{\sqrt{4k}}\Big(\mathbb{E}\bqty{n(t+\tau)\xi(t)}+\mathbb{E}\bqty{n(t)\xi(t+\tau)}\Big)\\
=
\textrm{tr}\bqty{\hat{n} e^{\mathcal{L}|\tau|} \frac{\Bqty{\hat{n},\hat{\rho}^{\infty}}}{2} }
- \mathbb{E}\bqty{n(t) n(t+\tau) } .\end{gathered}$$ Inserting this into Eq. (\[eq:autocorrelation\_expanded\]) gives $$\label{eq:autocorrelation_almost_final}
S(\tau) = \frac{\delta(\tau)}{4k} + \textrm{tr}\bqty{\hat{n} e^{\mathcal{L}\abs{\tau}}\frac{\Bqty{\hat{n},\hat{\rho}^{\infty}}}{2} } - \pqty{n^{\infty}}^2$$ with $n^\infty = \textrm{tr}[\hat{n} \hat\rho^\infty]$.
Explicit evaluation
-------------------
We now evaluate Eq. explicitly for the steady states $\hat{\rho}^{\infty}_+ = \textrm{diag}(1+R,1-R,0,0)/2$ and $\hat{\rho}^{\infty}_- = \textrm{diag}(0,0,1+R,1-R)/2$ of $\mathcal{L} + \mathcal{L}'$ \[as given in Eq. and Eq. \] corresponding to the two measurement outcomes $\pi=+1$ and $\pi=-1$, respectively. Note that we wrote the $\hat{\rho}^{\infty}_\pi$ in the energy basis here. Since $\mathcal{L} + \mathcal{L}'$ conserves $\hat\pi$, the trace in Eq. (\[eq:autocorrelation\_almost\_final\]) reduces to a trace in one of the $\pi$ subspaces. We thus have to evaluate $$\label{eq:expression_we_have_to_eval}
\textrm{tr}\bqty{ n e^{(\mathcal{L}_{\pi,\pi} + \mathcal{L}_{\pi,\pi}')\abs{\tau} }\frac{\Bqty{n, \rho^{\infty} }}{2}}$$ with $\rho^{\infty} = (\tau_0 + R\tau_z)/2$. Clearly, the remaining calculation is identical for the two subspaces. Suppressing $\pi$ labels, we evaluate $$\begin{aligned}
\frac{\Bqty{n, \rho^{\infty} }}{2} =&\ \frac{n^{\infty}}{2} \pqty{ \tau_0 + \frac{R+ \cos\theta}{2n^{\infty}} \tau_z - \frac{\sin\theta}{2n^{\infty}}\tau_x } \nonumber \\
=&\ n^{\infty}\bqty{ \rho^\infty + \frac{R+ \cos\theta - 2Rn^{\infty}}{4n^{\infty}}\tau_z - \frac{\sin\theta}{4n^{\infty}}\tau_x }.\nonumber\end{aligned}$$ The first term cancels against the $-(n_\infty)^2$ term in $S(\tau)$. We expand the exponential of the Liouvillian in eigenmodes, $$e^{(\mathcal{L}_{\pi,\pi} + \mathcal{L}_{\pi,\pi}')\abs{\tau} } = \sum_n e^{\lambda_n \abs{\tau}} \ketbra{\psi_n}{\phi_n},$$ where $\ket{\psi_n}$ and $\ket{\phi_n}$ are the right and left eigenmodes of $\mathcal{L}_{\pi,\pi} + \mathcal{L}_{\pi,\pi}'$ to eigenvalue $\lambda_n$, respectively (see App. \[app:liouvillian\_evs\_diag\]). Note that we can write $\ket{\tau_z} \simeq \ket{\psi_{\textrm{slow}}} $ and $\ket{\tau_x} \simeq -\ket{\psi_{\textrm{fast},+}} + \ket{\psi_{\textrm{fast},-}} $. With this, we can evaluate expression to leading order, $$\begin{aligned}
\bra{n} e^{(\mathcal{L}_{\pi,\pi} + \mathcal{L}_{\pi,\pi}')\abs{\tau}} \ket{\tau_z} \simeq&\ \cos\theta e^{\lambda_{\textrm{slow}} \abs{\tau}} , \\
\bra{n} e^{(\mathcal{L}_{\pi,\pi} + \mathcal{L}_{\pi,\pi}')\abs{\tau} } \ket{\tau_x} \simeq&\ -\sin\theta \cos(2\Omega\tau) e^{ \Re{ \lambda_{\textrm{fast}} } \abs{\tau}} .\end{aligned}$$ Here, we expanded $\bra{n} = (\bra{\phi_0} + \cos\theta \bra{\tau_z} - \sin\theta \bra{\tau_x})/2$ and used the overlaps $\braket{\tau_z}{\psi_{\textrm{slow}}} = 2, \braket{\tau_x}{\psi_{\textrm{slow}}} =0, \braket{\tau_z}{\psi_{\textrm{fast}}} \simeq 0$ and $\braket{\tau_x}{\psi_{\textrm{fast},\pm}} \simeq \mp 1 $. This yields the autocorrelation function $$\begin{gathered}
\label{eq:autocorrelation_detailed}
S(\tau) \simeq \frac{\delta(\tau)}{4k} + \frac{1}{4}\Bigg[ \frac{2k}{\Gamma_- + 2k} \cos^2\theta e^{\lambda_{\textrm{slow}} \abs{\tau}} \\
+ \sin^2\theta \cos\pqty{2\Omega\tau} e^{ \Re{ \lambda_{\textrm{fast}} } \abs{\tau}} \Bigg].\end{gathered}$$
Finally, we compute the noise spectrum $$\begin{aligned}
S (\omega) = \int_{-\infty}^{\infty}\textrm{d}\tau e^{i\omega\tau} S (\tau), \end{aligned}$$ which becomes $$\begin{gathered}
S (\omega) \simeq \frac{1}{4k} + \frac{ \cos^2\theta k}{\Gamma_- + 2k} \frac{\abs{\lambda_{\textrm{slow}}}}{\omega^2 + \abs{\lambda_{\textrm{slow}}}^2} \\
+ \frac{\sin^2\theta }{4} \sum_\pm
\frac{ \abs{\Re{\lambda_{\textrm{fast}} } }}{(\omega \pm 2\Omega )^2 + \abs{\Re{\lambda_{\textrm{fast}}}^2}}.\end{gathered}$$ Thus, the noise spectrum consists of Lorentzians centered at $\omega=0$ due to $\lambda_{\textrm{slow}}$ and at $\pm 2\Omega$ due to $\lambda_{\textrm{fast},\pm}$. For $\abs{t_i} \ll \varepsilon$, we have $\theta \simeq \pi$ and the zero-frequency peak is higher than the finite-frequency peaks by a factor of order $(\varepsilon / \abs{t_i})^4$. Without relaxation, i.e., for $\Gamma_-= \Gamma_0 = 0$ this reduces to the expression in the main text.
Fluctuations of time-averaged measurement signal {#app:fluctuations_time_avg}
------------------------------------------------
In the presence of relaxation, readout can be based on the time-averaged measurement signal. To estimate readout times it is necessary to obtain the variance of the time-averaged measurement signal, see Sec. \[sec:with\_relaxation\]. The time-averaged measurement signal is $$\begin{aligned}
j_{\textrm{int},\pi}(T) =&\ \frac{1}{T} \int_0^T \textrm{d}t\ j_\pi(t) \nonumber \\
=&\ \frac{1}{T} \int_0^T \textrm{d}t\ \left\{n_\pi(t) + \frac{X_0}{\sqrt{4kT}}\right\},\end{aligned}$$ where $X_0$ is a Gaussian random variable with zero mean and unit variance. Readout relies on $$\mathbb{E}[j_{\textrm{int},\pi}(T)] = n^{\infty}_\pi.$$ Here, we evaluate the variance of this quantity (suppressing $\pi$ labels), $$\begin{aligned}
\mathbb{V}[j_{\textrm{int},\pi}(T)] =&\ \frac{1}{T^2} \int_0^T \textrm{d}t \int_0^T \textrm{d}t'\ \mathbb{E}[j(t) j(t')] - \pqty{n^\infty}^2 \nonumber \\
=&\ \frac{2}{T^2} \int_0^T \textrm{d}t \int_0^t \textrm{d}\tau\ S(\tau).\end{aligned}$$ As $\abs{t_i} \ll \varepsilon$, we can neglect the $\sin^2\theta$ term in Eq. . Using $$\frac{2}{T^2} \int_0^T \textrm{d}t \int_0^t \textrm{d}\tau\ e^{\lambda \tau} = 2\pqty{ \frac{e^{\lambda T} - 1}{\lambda^2 T^2} - \frac{1}{\lambda T}} \simeq - \frac{2}{\lambda T}$$ for $\lambda T \ll -1$ (checking a posteriori that the measurement times indeed allow for this simplification), we obtain the final result $$\mathbb{V}[j_{\textrm{int}}(T)] = \frac{1}{T} \bqty{ \frac{1}{4k} + \frac{k \cos^2\theta }{\Gamma_- + 2k} \frac{1}{\abs{\lambda_{\textrm{slow}}}}}$$ which may be rewritten as $$\mathbb{V}[j_{\textrm{int}}(T)] = \frac{1}{T} \bqty{ \frac{1}{4k} + \frac{1}{\tan^2\theta} \frac{4k}{\pqty{\Gamma_- +2k}^2} },$$ from which we find Eq. (\[eq:int\_curr\_variance\_main\]).
Steady state in the presence of Majorana hybridizations {#app:steady_state_w_hybridizations}
=======================================================
Here, we justify the statement in Sec. \[sec:imperfect\] that $\hat{\rho}^{\infty} = \textrm{diag}(1,1,1,1)/4$ is the only zero mode of $\mathcal{L}+ \mathcal{L}_{23}$ (in the absence of relaxation). For small $\varepsilon_{23}$, this may be obtained as follows. We decompose $\hat{\rho}$ into the steady states $\hat{\rho}^{\infty}_{\pm}$ of $\mathcal{L}$ plus deviations $\delta\hat{\rho}$, $$\hat{\rho} = \abs{\alpha}^2 \hat{\rho}^{\infty}_{+} + \abs{\beta}^2 \hat{\rho}^{\infty}_{-}
+ \delta\hat{\rho}.$$ Thus, $\delta\hat{\rho}$ contains only the traceless parts of the diagonal blocks. Note that
\[eq:hybrid\_liou\_maps\_y\] $$\begin{aligned}
\mathcal{L}_{23} \hat{\rho}^{\infty}_{\pm} =&\ \mp \frac{\varepsilon_{23}}{2}\begin{pmatrix}
0 & -i \mathds{1}_2 \\ i \mathds{1}_2 & 0
\end{pmatrix}
= \mp \frac{\varepsilon_{23}}{2} \ket{Y}, \label{eq:hybrid_liou_maps_y_1} \\
\mathcal{L}_{23} \ket{Y} =&\ 4 \varepsilon_{23} ( \hat{\rho}^{\infty}_{+} - \hat{\rho}^{\infty}_{-} ) . \label{eq:hybrid_liou_maps_y_2} \end{aligned}$$
We also define the projector $\hat{P}$ onto the non-decaying subspace $\textrm{span}(\hat{\rho}^{\infty}_+ , \hat{\rho}^{\infty}_-)$, as well as its complement $\hat{P}_\perp = 1 - \hat{P}$ which projects onto the fast decaying subspace. We then project the eigenvalue equation $$(\mathcal{L}+\mathcal{L}_{23}) \hat{\rho} = \lambda \hat{\rho}$$ onto the two subspaces, $$\begin{aligned}
& \lambda\pqty{\abs{\alpha}^2 \hat{\rho}^{\infty}_{+} + \abs{\beta}^2 \hat{\rho}^{\infty}_{-} } = \hat{P} \mathcal{L}_{23} \delta \hat{\rho}, \\
& \lambda \delta \hat{\rho} = \mathcal{L} \delta \hat{\rho} + \mathcal{L}_{23} \pqty{ \abs{\alpha}^2 \hat{\rho}^{\infty}_{+} + \abs{\beta}^2 \hat{\rho}^{\infty}_{-} } + \hat{P}_\perp \mathcal{L}_{23} \delta \hat{\rho}.\end{aligned}$$ We formally solve the second equation for $\delta \hat{\rho}$ and insert it into the first equation. Using Eqs. and the fact that only $\ket{Y}$ is mapped onto $\textrm{span}(\hat{\rho}^{\infty}_+ , \hat{\rho}^{\infty}_-)$, i.e., $\hat{P} \mathcal{L}_{23}\ ... = \mathcal{L}_{23} \ketbra{Y}{Y}\ ... /4$ (the factor of $1/4$ stems from the fact that $\braket{Y}{Y} = 4$), we find, after tracing $\tr (\hat{\rho}^\infty_+ ... )$ and using $\abs{\beta}^2 = 1- \abs{\alpha}^2$, $$\lambda \abs{\alpha}^2 = \frac{1}{2} \varepsilon^2_{23} \mathcal{G}_{Y} (\lambda) \pqty{ 2\abs{\alpha}^2 - 1 }.$$ Here we defined the “propagator” $$\mathcal{G}_{Y} (\lambda) = \bra{Y} \frac{1}{ \lambda - \mathcal{L} - \hat{P}_\perp \mathcal{L}_{23} } \ket{Y}.$$ We are interested in $\lambda = 0$. If $\mathcal{G}_{Y} (0) \neq 0$, it follows that a zero mode necessarily has $\abs{\alpha}^2 = 1/2$. Inserting this into the eigenvalue equations, we obtain the relations $\hat{P} \mathcal{L}_{23} \delta \hat{\rho} = 0$ and $(\mathcal{L}+ \hat{P}_\perp \mathcal{L}_{23}) \delta \hat{\rho} = 0$. The first equation gives $\braket{Y}{ \delta \hat{\rho}}=0$, i.e., the steady state has no weight in the span of $\ket{Y}$. Then, the second equation becomes $$(\mathcal{L} + \mathcal{L}_{23}) \delta \hat{\rho} = 0.$$ $\mathcal{L}$ has no zero modes acting on the traceless $ \delta \hat{\rho}$. Hence, for weak perturbations $\varepsilon_{23} \ll \vert\Re\{\tilde{\lambda}_{\textrm{slow}}\} \vert , \vert\lambda_{\textrm{slow}, \pi} \vert$, it follows that $\delta \hat{\rho} = 0$. Finally, it is straightforward to check numerically that $\mathcal{G}_{Y} (0)$ is indeed non-vanishing within the relevant parameter range by expanding in left and right eigenvectors of $\mathcal{L} + \hat{P}_\perp \mathcal{L}_{23}$. Thus, the completely mixed state is indeed the only steady state.
|
---
abstract: 'In this paper, we present internal surface brightness profiles, using images in the F606W and F814W filter bands observed with the Advanced Camera for Surveys on the [*Hubble Space Telescope*]{}, for ten globular clusters (GCs) in the outer halo of M31. Standard King models are fitted to the profiles to derive their structural and dynamical parameters. The results show that, in general, the properties of clusters in M31 and the Milky Way fall in the same regions of parameter spaces. The outer halo GCs of M31 have larger ellipticities than most of GCs in M31 and the Milky Way. Their large ellipticities may be due to galaxy tides coming from satellite dwarf galaxies of M31 or may be related to the apparently more vigorous accretion or merger history that M31 has experienced. The tight correlation of cluster binding energy $E_b$ with mass $M_{\rm mod}$ indicates that, the “fundamental plane” does exist for clusters, regardless of their host environments, which is consistent with previous studies.'
author:
- 'Song Wang, Jun Ma'
title: STRUCTURAL PARAMETERS FOR GLOBULAR CLUSTERS IN THE OUTER HALO OF M31
---
INTRODUCTION {#Introduction.sec}
============
The mechanisms involved in galaxy formation is still one of the major unsolved problems in astrophysics [e.g., @per02]. Globular clusters (GCs), which are considered to be debris of the galaxy formation, have a record about the information on both their formation condition and dynamical evolution within the environment of their host galaxies, which are reflected by their spatial structures and kinematics [@barmby07; @maclau08]. So, GCs are regarded as a laboratory of galaxy history [@bs06]. In addition, GCs can be used as one of excellent tracers of substructures in the outer regions of their parent galaxies. For example, @bfi identified the accretion signature of the Sagittarius dwarf galaxy among the GCs in the outer halo of the Milky Way (MW); @mackey07 found some of the GCs in the outer halo of M31 are rather unlike their MW counterparts as they are metal-poor, compact, and very luminous, which may well offer important clues to differences in the early formation and evolution of the two galaxies or in their subsequent accretion histories [see @mackey07 for details]. Thus, a detailed study of GCs in the outer halo of a galaxy is important.
M31, with a distance of $\sim780$ kpc from us [@sg98; @mac01], is the largest galaxy in the Local Group, and it is so close to us that the GCs in it can be well resolved with the cameras on the [*Hubble Space Telescope*]{} ([*HST*]{}). M31 contains more GCs than all other Local Group galaxies combined with 654 confirmed GCs and 606 GC candidates in the version V4.0 of the Revised Bologna Catalogue (RBC) of M31 GCs [@gall04; @gall06; @gall07; @gall09]. M31 contains so many GCs that a variety of clusters may be included such as classic globulars, extended and diffuse globulars [@huxor08], intermediate-age globulars [@puzia05; @fan06; @ma09; @wang10] and young massive clusters [@perina09; @perina10; @maetal11]. GCs in the outer halo of M31 have been discussed by many authors [@martin06; @mackey06; @mackey07; @mackey10; @huxor08], which may provide important clues for the accretion and interaction events between M31 and surrounding galaxies. Recently, @mackey10 found a genuine physical association between GCs and multiple tidal debris streams in the outer regions of M31, implying that the remote GC system of M31 was largely accreted from the satellite galaxies [also reported in @huxor11].
Structures and kinematics of GCs can be determined by fitting different models to the surface brightness profiles, combined with mass-to-light ratios ($M/L$ values) estimated from velocity dispersions or population-synthesis models. In general, three models are used in the fits: the simple model of single-mass, isotropic, modified isothermal sphere developed by @king66, an alternate modified isothermal sphere based on the ad hoc stellar distribution function of @wilson75, and the $R^{1/n}$ surface-density profile of [@sersic68]. With these models, many authors have achieved some success in determining structures and kinematics of clusters from different galaxies, using images from ground-based telescopes or [*HST*]{}: the MW [@tdk93; @tkd95; @mm05]; the Large and Small Magellanic Clouds, Fornax and Sagittarius dwarf spheroidal galaxies [@mg03a; @mg03b; @mg03c; @mm05]; M31 [@grill96; @bhh02; @barmby07; @barmby09; @ma11; @ma06; @ma07; @ma12; @federici07; @str09; @huxor11]; M33 [@larsen02]; NGC 5128 [@hch99; @harris02; @mh04; @maclau08]. A number of studies focusing on the correlations between cluster parameters have been performed, which showed that there exists a fundamental plane among most clusters, regardless of their different “growing environment” in different host galaxies.
@barmby07 derived structural parameters for 34 GCs in M31 based on [*HST*]{} Advanced Camera for Surveys (ACS) observations, and the derived structural parameters were combined with corrected versions of those measured in an earlier survey in order to construct a comprehensive catalog of structural and dynamical parameters for 93 M31 GCs. @barmby09 measured structural parameters for 23 bright young clusters in M31 based on the [*HST*]{} Wide Field Planetary Camera 2 (WFPC2) observations, and suggested that on average they are larger and more concentrated than typical old clusters. However, the sample clusters from @barmby07 and @barmby09 lie at projected radii $R_{p}<20$ kpc except for five clusters G001, G002, G339, G353 and B468, the projected radii of which are 34.55, 33.62, 28.68, 26.32 and 20.05 kpc, respectively; and most of the sample clusters lie at projected radii $R_{p}<10$ kpc. So, structural parameters for GCs in the outer halo of M31 are worthwhile to be determined. In addition, @huxor11 derived structural parameters for 13 extended clusters (ECs) in the halo regions of M31 by fitting the @king62 profiles to the photometry data taken with the Wide Field Camera on the Isaac Newton Telescope (INT) and MegaCam on the Canada-France-Hawaii Telescope, which may provide an interesting comparison with the structural parameters for GCs in the outer halo of M31.
In this paper, we determined spatial structures and kinematics for ten GCs in the outskirts of M31. In Section 2, we give observations of the sample GCs and the data-processing steps to derive their surface brightness profiles. In Section 3, we determine structures and kinematics of the sample clusters with the model fitting. In Section 4, we discuss correlations of the structural and kinematic parameters of the sample clusters here combining with those of the Galactic and M31 clusters studied by other authors. Finally, we give our summaries in Section 5.
DATA AND ANALYSIS METHOD {#data.sec}
========================
Globular Cluster Sample
-----------------------
As mentioned in the introduction, a detailed study of GCs in the outer halo of a galaxy is important, since they can serve as one of excellent tracers of substructures in the outer regions of their parent galaxy. Till now, for M31, most of the clusters whose structural parameters have been determined, lie at projected radii $R_{p}<10$ kpc. So, structural parameters for GCs in the outer halo of M31 are worthwhile to be measured. In this paper, detailed studies of the structures of a sample of ten GCs in the outer halo of M31 from @mackey07 will be presented. These sample halo GCs are interesting. For example, @mackey07 found some of them are rather unlike their MW counterparts as they are metal-poor, compact, and very luminous [see @mackey07 for details]. Eight of the ten halo GCs lie at projected radii $R_{p}>30$ kpc, of which two lie at very large distances from M31: $R_{p}\sim 78$ and 100 kpc, respectively. In addition, @mackey07 estimated their metallicities, distance moduli and reddening values by fitting the Galactic GC fiducials from @brown05 to their observed color-magnitude diagrams in the F606W and F814W filters of deep images observed with the the ACS Wide Field Camera (WFC) under the [*HST*]{} program GO-10394 (PI: Tanvir). This program was aimed to obtain deep high resolution photometry of outer halo GCs in M31 to study their stellar populations, line-of-sight distances and structural parameters. Targets were imaged in the F606W and F814W filters for $\sim$ 1800s and $\sim$ 3000s, respectively, with small dithers between various sub-exposures [@rich09].
Surface Brightness Profiles
---------------------------
We used the analogous procedure adopted by @barmby07 to produce surface brightness profiles with [ellipse]{} in IRAF. The center positions of these clusters were determined by centroiding. Elliptical isophotes were fitted to the observed data, with no sigma clipping. Two passes of [ellipse]{} task were run in the procedure. In the first pass, ellipticity and position angle (P.A.) were allowed to vary with the isophote semimajor axes; in the second pass, surface brightness profiles were derived on fixed, zero-ellipticity isophotes, meaning that we always had circularly symmetric intensity profiles, which would be fitted with circular structure models. The overall ellipticity and position angle were determined by averaging the [ellipse]{} output over the isophotal semimajor axes, and the uncertainty is $\sigma$ . Table 1 lists the average ellipticity, P.A. and some additional integrated data for the sample GCs. $VI$ magnitudes of 9 GCs and $I$ magnitude of GC6 are from @huxor08, while $V$ magnitude of GC6 is from @rhh92. The galactocentric distances, distance moduli, reddening values and metallicities are from @mackey07, while the uncertainties of \[Fe/H\] are assumed to be 0.6 as @bh00 suggested for the standard deviation of the metallicity distribution of M31 GC system.
Raw output from package [ellipse]{} is in terms of counts s$^{-1}$ pixel$^{-1}$, which needs to multiply by 400 to convert to counts s$^{-1}$ arcsec$^{-1}$, since the ACS/WFC spatial resolution is 0.05 arcsec pixel$^{-1}$. For drizzled ACS data, the units of counts are ELECTRONS (ACS Handbook). Two formulas were used to transform the ACS counts to surface brightness calibrated on the [vegamag]{} system (ACS Handbook),
$\mu_{\rm F606W}/{\rm mag~arcsec^{-2}=26.398-2.5 \log(counts~s^{-1}}$ $${\rm arcsec^{-1})},$$ $\mu_{\rm F814W}/{\rm mag~arcsec^{-2}=25.501-2.5 \log(counts~s^{-1}}$ $${\rm arcsec^{-1})}.$$
However, occasional oversubtraction of background during the multidrizzling in the automatic reduction pipeline leads to “negative” counts in some pixels, so we worked in terms of linear intensity instead of surface brightness in magnitudes. Given ${M_{\odot,{\rm F606W}}} = +4.64$, ${M_{\odot,{\rm F814W}}} = +4.14$[^1], equations for transforming counts to surface brightness in intensity were derived [also see @barmby07 for details], $$I_{\rm F606W}/L_{\odot}~{\rm pc^{-2}\simeq0.8427\times(counts~s^{-1}~arcsec^{-1})},$$ $$I_{\rm F814W}/L_{\odot}~{\rm pc^{-2}\simeq1.2147\times(counts~s^{-1}~arcsec^{-1})}.$$
Table 2 gives the final, calibrated intensity profiles for the ten clusters but with no extinction corrected. The reported F606W- and F814W-band intensities are calibrated on the [vegamag]{} scale. Column (7) gives a flag for each point, which has the same meaning as @barmby07 and @maclau08 defined.
Point-spread Function
---------------------
As noted by @barmby07 and @maclau08 that, though the sample GCs here are well resolved with ACS/WFC, the core structures are still influenced by the point-spread function (PSF). We convolved the structural models developed by @king66 (hereafter ‘King model’) with a simple analytic description of the PSF before doing the model fitting, as given in @barmby07, $$I_{\rm PSF, F606W}/I_{\rm 0} = [1 + (R/0.0686~{\rm arcsec})^3]^{-3.69/3.0},
\label{eq:psf1}$$ and $$I_{\rm PSF, F814W}/I_{\rm 0} = [1 + (R/0.0783~{\rm arcsec})^3]^{-3.56/3.0},
\label{eq:psf2}$$ with FWHMs of $0.125~{\rm arcsec}$ and $0.145~{\rm arcsec}$ in the F606W and F814W filters, respectively.
Extinction and Magnitude Transformation
---------------------------------------
When we fit models to the brightness profiles of the sample clusters, we will correct the inferred intensity profiles for extinction. The effective wavelengths of the ACS F606W and F814W filters are $\lambda_{\rm eff}\simeq$ 5918 and 8060 [Å]{} [@siri05]. With the extinction curve $A_{\lambda}$ taken from @car89 with $R_V=3.1$, two formulas for computing ${A_{\rm F606W}}$ and ${A_{\rm F814W}}$ are derived: ${A_{\rm F606W}}\simeq2.8~E_{B-V}$; ${A_{\rm F814W}}\simeq1.8~E_{B-V}$. In addition, for easy comparison with catalogs of the GCs in the MW (see Section 4 for details), we transform the ACS/WFC magnitudes in the F606W filter to the standard $V$. @siri05 has given transformations from WFC to standard $BVRI$ magnitudes both on observed and synthetic methods (see their Table 22). As synthetic transformations are based on larger color range and more safely employed, they should be considered the norm, unless some indicated cases [@siri05]. We used the synthetic transformation from F606W to $V$ magnitude both on the [vegamag]{} scale with a quadratic dependence on dereddened $(V-I)_0$. With the magnitudes in $V$ and $I$ bands and reddening values listed in Table 1, we found the $(V-I)_0$ values of all the sample clusters are larger than 0.4. So, the following transformation formula was applied here,
$(V-{\rm F606W})_0=-0.067+0.340(V-I)_0-0.038$ $$(V-I)_0^2,$$ for which we estimated a precision of about $\pm 0.05$ mag.
MODEL FITTING {#model.sec}
=============
\[fig1\]
There are a number of possible choices of structural models for fitting star cluster surface profiles, including King model, @wilson75, and [@sersic68], as mentioned in the introduction. King model is the most commonly used model in studies of star clusters. In addition, @barmby07 [@barmby09] found that M31 clusters are better fitted by King models. So, in this paper, the intensity profiles of the ten GCs in M31 will be fitted by King models defined by the phase-space distribution function, $$f(E) \propto\left\{
\begin{array}{lcl}
\exp[-E/{\sigma}_0^2]-1, & & E < 0, \\
0, & & E \geq 0,
\end{array}\right.$$ where $E$ is the stellar energy, ${\sigma}_0$ is a velocity scale.
We first convolved King model with the ACS/WFC PSF for the F606W and F814W filters. Given a value for the scale radius $r_0$, we computed a dimensionless model profile $\widetilde{I}_{\rm mod}\equiv I_{\rm mod}/I_0$, and then carried out the convolution,
$\widetilde{I}_{\rm mod}^{*} (R | r_0) = \int\!\!\!\int_{-\infty}^{\infty}
\widetilde{I}_{\rm mod}(R^\prime/r_0)
\widetilde{I}_{\rm PSF}
\left[(x-x^\prime),(y-y^\prime)\right]$ $$\ dx^\prime \, dy^\prime\ ,
\label{eq:convol}$$ where $R^2=x^2+y^2$, and $R^{\prime2}=x^{\prime2}+y^{\prime2}$; and $\widetilde{I}_{\rm PSF}$ was approximated using the equations (\[eq:psf1\]) and (\[eq:psf2\]) [see @maclau08 for details]. The observed surface brightness profiles were fitted by calculating and minimizing $\chi^2$ as the sum of squared differences between model and observed intensities, with uncertainties listed in Table 2 being weights, $$\chi^2=\sum_{i}{\frac{[I_{\rm obs}(R_i)-I_0\widetilde{I}_{\rm mod}^{*}(R_i|r_0)
-I_{\rm bkg}]^2}{\sigma_{i}^{2}}},$$ in which a background $I_{\rm bkg}$ was also fitted.
Figure 1 displays the observed intensity profiles as a function of logarithmic projected radius and the best-fitting King model (solid red line) for each cluster. The observed data have been extinction corrected, following by a fitted $I_{\rm bkg}$ subtracted. The dashed blue lines represent the shape of the PSF for the WFC F606W or F814W filters. Most profiles of the sample clusters were well fitted by King model, except for those at the intermediate radii of GC3, GC7 and GC9. We checked the images, and found that the three clusters are very loose and there are several bright stars at the intermediate radii.
In Figure 1, open squares are [ellipse]{} data points included in the least-squares model fitting, and the crosses are points flagged as ‘DEP’ or ‘BAD’, which are not used to constrain the fit. In this paper, the [ellipse]{} gives isophotal intensities for $15$ radii inside $R < 2$ pixels, however, all of them are derived from the same innermost $13$ pixels, meaning that the isophotal intensities are not statistically independent. So, to avoid excessive weighting of the central regions of clusters in the fits, we only used intensities at radii $R_{\rm min}$, $R_{\rm min}+(0.5,1.0,2.0)~{\rm pixels}$, or $R>2.5$ pixels as @barmby07 used. In addition, we deleted some individual isophotes which deviated strongly from their neighbours or showed irregular features by hand.
Basic Model Parameters
----------------------
Table 3 lists the basic parameters of 20 model fits to the sample clusters here. Column (1) gives the cluster name, column (2) the detector/filter from which the observed data were derived. Column (3) gives the color correction $(V-{\rm F606W})_0$ to transform native instrumental magnitudes to the standard $V$ scale. The fourth column shows the number of points in the intensity profile that are flagged as ‘OK’ in Table 2, which were used for constraining the model fits. Column (5) is the fitting model which is always King model here. Column (6) gives the minimum $\chi^2$ obtained in the fits. Column (7) gives the best fitted background intensity. Column (8) gives the dimensionless central potential $W_0$ of the best-fitting model, defined as $W_0 \equiv -\phi(0)/\sigma_0^2$. Column (9) gives the concentration $c \equiv \log(r_t/r_0)$. Column (10) gives the best-fit central surface brightness in the native bandpass of the data. Column (11) and column (12) show the best model-fit scale radius $r_0$ in arcseconds and parsecs, respectively, while the latter was obtained from the angular scale with the distance moduli given in Table 1.
Uncertainties for the fitted model parameters were estimated following $\triangle\chi^2 \le 1$ for 68% confidence intervals. However, as @barmby07 pointed out, because the formal error bars estimated by [ellipse]{} for the isophotal intensities are artificially small, the best-fit $\chi_{\rm min}^2$ can be exceedingly high ($\gg\!N_{\rm pts}$; the number of points used in the model fitting) even when a model fit is actually very good (see the values of $\chi_{\rm min}^2$ in Table 3), and this would result in unrealistically small estimates of parameter uncertainties. So, we also re-scale the $\chi^2$ for all fitted models by a common factor chosen to make the global minimum $\chi_{\rm min}^2 = (N_{\rm pts}-4)$ as @barmby07 did. Under this re-scaling, the global minimum reduced $\chi^2$ per degree of freedom is exactly one [see @barmby07 for details].
Derived Quantities
------------------
Tables 4 and 5 give various derived parameters for the best-fitting models for each cluster [the details of their calculation are given by @maclau08].
The contents of Table 4 are:\
Column (4): $\log r_t$, the model tidal radius in parsecs.\
Column (5): $\log R_c$, the projected core radius of the model fitting a cluster, which is defined as $I(R_c) = I_0/2$.\
Column (6): $\log R_h$, the projected half-light, or effective, radius of a model, containing half the total luminosity in projection.\
Column (7): $\log (R_h/R_c)$, a measure of cluster concentration and relatively more model-independent than $W_0$ or $c$.\
Column (8): $\log I_0 = 0.4(26.402 - \mu_{V,0}$), the best-fit central ($R = 0$) luminosity surface density in the $V$ band, in units of $L_{\odot}$ pc$^{-2}$. The surface-brightness zero point of $26.402$ corresponds to a solar absolute magnitude $M_{V,\odot} = +4.83$[^2]. $\mu_{V,0}$ is derived from applying the term $(V-{\rm F606W})_0$ to the fitted central surface brightness in column (10) of Table 3.\
Column (9): $\log j_0$, the central ($r = 0$) luminosity volume density in the $V$ band in units of $L_{\odot}$ pc$^{-3}$.\
Column (10): $\log L_V$, the $V$-band total integrated model luminosity, in units of $L_{\odot}$.\
Column (11): $V_{\rm tot} = 4.83 - 2.5 \log (L_V/L_{\odot}) + 5\log (D/10$ pc) is the total $V$-band magnitude of a model cluster.\
Column (12): $\log I_h \equiv \log (L_V/2{\pi}R_h^2$), the luminosity surface density averaged over the half-light/effective radius in the $V$ band, in units of $L_{\odot,V}$ pc$^{-2}$.
The uncertainties of these derived parameters were estimated (separately for each given model family) by calculating them in each model that yields $\chi^2$ within 1 of the global minimum for a cluster, and then taking the differences between the extreme and best-fit values of the parameters [see @mm05 for details].
The contents of Table 5 are:\
Column (3): $\Upsilon_V^{\rm pop}$, the $V$-band mass-to-light ratio. The values of $\Upsilon_V^{\rm pop}$ were derived by applying the population synthesis models of @bc03, assuming a @chab03 initial mass function (IMF) and age of 13 Gyr for all these clusters, with metallicities given in Table 1. Uncertainties of $\Upsilon_V^{\rm pop}$ include a $\pm2$ Gyr uncertainty in age, as well as a $\pm0.6$ uncertainty in \[Fe/H\].\
Column (5): $\log M_{\rm tot} = \log \Upsilon_V^{\rm pop} + \log L_V$, the integrated cluster mass in solar units, estimated from the total model luminosity $L_V$.\
Column (6): $\log E_b$, the integrated binding energy in ergs, which is defined as $E_b \equiv -(1/2)
\int_0^{r_t} 4{\rm \pi}r^2\rho\phi{\rm d}r$.\
Column (7): $\log \Sigma_0=\log \Upsilon_V^{\rm pop} + \log I_0$, the central surface mass density in units of $M_{\odot}$ pc$^{-2}$.\
Column (8): $\log \rho_0 = \log \Upsilon_V^{\rm pop} + \log j_0$, the central volume density in units of $M_{\odot}$ pc$^{-3}$.\
Column (9): $\log \Sigma_h = \log \Upsilon_V^{\rm pop} + \log I_h$, the surface mass density averaged over the half-light/effective radius $R_h$, in units of $M_{\odot}$ pc$^{-2}$.\
Column (10): $\log \sigma_{p,0}$, the predicted line-of-sight velocity dispersion at the cluster center in units of km s$^{-1}$.\
Column (11): $\log \nu_{\rm esc,0}$, the predicted central “escape” velocity in units of km s$^{-1}$, with which a star can move out from the center of a cluster, which is defined as $\nu_{\rm esc,0}^2/\sigma_0^2 = 2[W_0 + GM_{\rm tot}/r_t\sigma_0^2]$.\
Column (12): $\log t_{r,h}$, the two-body relaxation time at the model-projected half-mass radius in units of years, estimated as $t_{r,h} = {\frac{2.06\times10^6 yr}{\ln(0.4M_{\rm tot}/m_{\star})}}{\frac{M_{\rm tot}^{1/2}R_h^{3/2}}{m_{\star}}}$. Here, $m_{\star}$, the average stellar mass in a cluster is assumed to be 0.5$M_{\odot}$ [see @maclau08 for the details]\
Column (13): $\log f_0 \equiv \log [\rho_0/(2\pi\sigma_c^2)^{3/2}$\], the model’s central phase-space density, in units of $M_{\odot}$ pc$^{-3}$ (km s$^{-1}$)$^{-3}$.
The uncertainties of these derived dynamical quantities were estimated from their variations in each model that yields $\chi^2$ within 1 of the global minimum for a cluster, as above, and combined in quadrature with the uncertainties in $\Upsilon_V^{\rm pop}$.
Comparison of Results in the F606W and F814W Filters
----------------------------------------------------
Model fits for the same cluster observed in different filters were compared to check whether there were systematic errors or color dependencies in the fits. Figure 2 shows the comparison of some parameters derived from fits to the sample clusters in both F606W and F814W filters. The left panel shows the comparison of projected half-light radius, while the right panel shows the ratio of half-light to core radii, all of which are from Table 4. The uncertainties for the parameters were also given in Figure 2. It is evident that the results between the two ACS bands are in good agreement. In following analysis, the F606W model fits were used for all the sample GCs.
DISCUSSION {#discussion.sec}
==========
We combined the GC parameters derived here with those derived by King-model fits for clusters in the MW [@mm05] and M31 [@bhh02; @barmby07; @barmby09; @huxor11] to form a large sample to look into the correlations between the parameters. The ellipticities and galactocentric distances for the MW GCs are from @harris96 (2010 edition). For M31 GCs of @bhh02 and @barmby07 which were not observed in WFC F606W filter, the data of Space Telescope Imaging Spectrograph V-band or High Resolution Channel (HRC) F606W-band or HRC F555W-band or WFPC2 V-band are used, except for B082, of which the data of WFC F814W-band was used as @barmby07 reported, since it was unsuccessfully fitted by King model in F606W filter. For clusters of @barmby09 observed in WFPC2, the data of F439W- or F450W-band are used as @barmby09 used, since these young clusters are dominated by blue stars and the measurements from the bluer filters are more preferred [see @barmby09 for the details]. @huxor11 derived the structure parameters of 13 ECs based on $V$-band photometry (for INT) or $g$-band (for MegaCam) photometry. However, only metallicities of 4 ECs (HEC4, HEC5, HEC7 and HEC12) were determined by @mackey06. We derived integrated cluster mass for the four ECs, using the $V$-band absolute magnitude obtained by @huxor11 and a mass-to-light ratio determined with the same approach here (see §3.2 for details).
Ellipticity Distribution
------------------------
As noted by @barmby07, @larsen01 listed several possible factors for the elongation of GCs: internal rotation, galaxy tides, cluster mergers, and “remnant elongation” from some clusters’ former lives as dwarf galaxy nuclei. Cluster rotation is generally accepted to be a major factor for cluster flattening [@dp90]. However, @vanden84 and @vanden96 presented that the brightest GCs in both the MW and M31 are most flattened, which can be explained by the cluster mergers and “remnant elongation”. In addition, as @harris02 noted that dynamical models show that internal relaxation coupled to the external tides will in most situations drive a cluster toward a rounder shape over several relaxation times. @harris02 presented that, the distributions of ellipticities for the M31 and NGC 5128 clusters and the old clusters in the Large Megallanic Cloud, are very similar, but different from the MW. With a large cluster sample, @barmby07 also showed the distribution of ellipticities for clusters in the MW, M31 and NGC 5128, and found the distributions of ellipticities for M31 and NGC 5128 are not statistically different; both differ from the MW distribution in having few very round clusters. So, @barmby07 concluded that there is no evidence that the overall galaxy environment is a major factor. @bhh02 discussed about correlations of GC ellipticities with other properties in detail, and presented some explanations for these correlations combined with other authors’ results [see @bhh02 and references therein]. In this paper, we show the distribution of ellipticity with galactocentric position for clusters in the MW and M31 in Figure 3, including the outer halo GCs in M31 from this study. A conclusion can be given that these outer halo GCs of M31 have larger ellipticities than most of GCs in M31 and the MW. These outer halo GCs lie at large projected radii than most sample clusters in @bhh02 [@barmby07], their large ellipticities may be due to galaxy tides coming from satellite dwarf galaxies of M31 or may be related to the apparently more vigorous accretion or merger history that M31 has experienced [e.g. @Ibata05; @Ibata07; @McConnachie09; @bekki10; @hammer10; @mackey10; @huxor11].
In order to show whether cluster ellipticities are caused by internal processes such as rotation or velocity anisotropy, @barmby07 showed ellipticity as a function of luminosity and half-mass relaxation time for clusters in M31, NGC 5128 and the MW, since if it is true, relaxation through dynamical evolution should act to reduce any initial flattening [see @barmby07 and references therein]. These authors found a mild systematic decrease in ellipticity with increased luminosity, although considerable scatter, and no correlation of ellipticity with relaxation time is evident. So, @barmby07 concluded that the observed distribution of GC ellipticity appears to be due to a number of factors. Figure 4 displays ellipticity as a function of model luminosity and half-mass relaxation time for clusters in the MW and M31, including the outer halo GCs in M31 studied here. It is evident that, when we add the data for the outer halo GCs in M31 obtained here, the conclusion of @barmby07 will not evidently change, although the mild systematic decrease in ellipticity with increased luminosity nearly disappears. In addition, we think that the larger ellipticities of the outer halo GCs of M31 than most of GCs in M31 and the MW may be due to galaxy tides coming from satellite dwarf galaxies of M31 or may be related to the apparently more vigorous accretion or merger history that M31 has experienced.
Correlations with Position and Metallicity
------------------------------------------
As @barmby07 noted that, previous studies have shown that structures of the MW GCs are largely independent of galactocentric distances and metallicity, except for the correlation of half-light radius with galactocentric distances. @bhh02 presented structural parameters as a function of galactocentric distance for clusters in M31 and the MW, and found that, there is no significant trend of $c$ with $R_{\rm gc}$, both $R_h$ and $r_0$ are correlated with $R_{\rm gc}$, and there is no clear correlation of $\mu_V(0)$ with $R_{\rm gc}$. The results of @bhh02 are in agreement with ones obtained for GCs in the MW [e.g. @maclau00] and NGC 5128 [e.g. @harris02]. @bhh02 concluded that the correlations of $R_h$ and $r_0$ with $R_{\rm gc}$ for GCs in both the MW and M31 are due to physical conditions at the time of cluster formation as suggested by @vanden91 for MW GCs. @barmby07 showed structural parameters as a function of galactocentric distance for GCs in the MW, the Magellanic Clouds and Fornax dwarf spheroidal, NGC 5128, and M31, and found similar results. In addition, @mv05 showed that there is a clear trend of increasing $R_h$ with increasing $R_{\rm gc}$ for the Galactic GCs. We should notice that the galactocentric distances are true three-dimensional distances for Galactic GCs and projected radii for M31 clusters.
Figure 5 shows structural parameters as a function of galactocentric distance $R_{\rm gc}$ for M31 outer halo GCs studied here, M31 young massive clusters [@barmby09], MW globulars [@mm05], M31 globulars [@bhh02; @barmby07] and M31 ECs [@huxor11]. It is evident that, when we add the data for the outer halo GCs in M31 obtained here, the conclusion of @bhh02 will not change, with the exception that the galactocentric distances of M31 clusters can reach to 100 kpc which are as distant as the MW clusters. In addition, it is true that M31 young massive clusters have larger $c$ and $R_h$ than old GCs at the same galactocentric distances. For comparing with @huxor11 (their Figure 9), we include the ECs of @huxor11 in the upper-right panel of Figure 5, in which $R_h$ is versus $R_{\rm gc}$. It can be seen that, at large raddi (from 30 to 100 kpc) there are few GCs having $R_h$ in the range from 8 to 15 pc, which is in agreement with the finding of @huxor11.
@bhh02 showed structural parameters as a function of \[Fe/H\] for clusters in M31 and the MW, and found that there is no correlation of metallicity with concentration $c$ or central surface brightness $\mu_V(0)$, but there does appear to be a correlation with size, as measured by $r_0$ or $R_h$, i.e. $r_0$ or $R_h$ deceases with increased metallicity. @harris02 showed a different correlation of $r_h$ with \[Fe/H\] for GCs in NGC 5128, where $r_h$ is the half-mass radius, and reported that the correlation may be due to a selection effect because of a small sample. @barmby07 showed structural parameters as a function of \[Fe/H\] for GCs in the MW, the Magellanic Clouds and Fornax dwarf spheroidal, NGC 5128, and M31, and found that, no correlation of $c$ with \[Fe/H\] exists; a weak correlation of $R_h$ with \[Fe/H\] is present: $R_h$ deceases with increased metallicity, except for GCs in NGC 5128; there is a slight systematic increase of $\mu_{V,0}$ with \[Fe/H\].
Figure 6 plots structural parameters as a function of \[Fe/H\] for M31 outer halo GCs studied here, M31 young massive clusters [@barmby09], MW globulars [@mm05], and M31 globulars [@bhh02; @barmby07] and M31 ECs [@huxor11]. It is evident that, the outer halo GCs of M31 fall in the same regions of parameter spaces of clusters in M31 and the MW. In addition, the conclusions of @bhh02 and @barmby07 do not change when adding the young massive clusters of @barmby09 and the outer halo GCs here. We also include four ECs of @huxor11 in the upper-right panel of Figure 6. An evident feature is that these four ECs are all metal-poor and have large $R_h$.
Figure 7 plots structural parameters as a function of model mass $M_{\rm mod}$ for M31 outer halo GCs studied here, M31 young massive clusters [@barmby09], MW globulars [@mm05], M31 globulars [@bhh02; @barmby07] and M31 ECs [@huxor11]. The properties of clusters in M31 and the MW fall in the same regions of parameter spaces, with the exception that, on average, the young massive clusters have larger sizes and higher concentrations than older clusters of the same mass [see @barmby09 for discussions in detail]. We also include four ECs of @huxor11 in the upper-right panel of Figure 7. Three ECs have intermediate masses as well as the outer halo GCs of M31 studied here, while one EC (HEC12) has very low mass ($\sim2\times 10^4~{\rm M_\odot}$). @barmby07 showed structural parameters as a function of model luminosity for GCs in M31, the MW, NGC 5128, the Magellanic Clouds, and the Fornax dSph, and found the properties of clusters in all six galaxies fall in the same regions of parameter spaces. The lower-right panel of Figure 7 shows one view of the fundamental plane, as defined by @maclau00. @Djorgovski95 found a pair of bivariate correlations in MW GC parameters which imply the existence of a “globular cluster fundamental plane”, similar to that expected if the cores were virialized structures. @harris02 found that the NGC 5128 GCs describe a relation between binding energy and luminosity that even tighter than in the MW, which occupy the same extremely narrow region of the parametric “fundamental plane” as do their MW counterparts.
SUMMARY
=======
GCs in the outer halo of M31 have recently been discovered in many surveys. We selected ten GCs (15 kpc $\lesssim R_p \lesssim$ 100 kpc) which have been studied by @mackey07 based on the [*HST*]{} observations used in this paper. We measured surface brightness profiles for them using the [*HST*]{} images of @mackey07. Structural and dynamical parameters were derived by fitting the King model to the light profiles. We discussed the properties of the sample GCs here combined with GCs in the MW [@mm05] and clusters in M31 [@bhh02; @barmby07; @barmby09]. In general, the properties of the M31 and the Galactic clusters fall in the same regions of parameter spaces.
The outer halo GCs of M31, which lie at large projected radii, have larger ellipticities than most of GCs in M31 and the MW. Their large ellipticities may be due to galaxy tides coming from satellite dwarf galaxies of M31 or may be related to the apparently more vigorous accretion or merger history that M31 has experienced. However, this conclusion remains to be checked because of the sample limitation. RBC V4.0 provides 39 GCs and 87 GC candidates which lie at $R_p > 20$ kpc. With more and more [*HST*]{} observations, structural and dynamical parameters for these clusters can be measured, which will provide a larger sample for discussion on the halo GCs.
The strong correlation of $E_b$ with model mass $M_{\rm mod}$ indicates a tight fundamental plane both for M31 and Galactic clusters, and no offset is apparent in the correlation between old and young clusters, especially including GCs in the outer halo of M31 studied here. This implies that some near-universal structural properties are present for clusters, regardless of their host environments, which is consistent with previous studies of @barmby07 [@barmby09].
We would like to thank Dr. McLaughlin for his help in finishing this paper. He provide us a table including some parameters being model-dependent function of $W_0$ or $c$. An anonymous referee is thanked for useful suggestions deriving from a careful and thorough reading of the original manuscript. This work was supported by the Chinese National Natural Science Foundation grant Nos. 10873016, and 10633020, and by the National Basic Research Program of China (973 Program) No. 2007CB815403.
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=2.5pt
[ccccccccccc]{} Name & $\epsilon^{a}$(F606W) & $\epsilon^{a}$(F814W) & $\theta^{b}$(F606W) & $\theta^{b}$(F814W) & $V$ & $I$ & ${R_{\rm gc}}$ & $(m-M)_0$ & $E(B-V)$ & ${\rm [Fe/H]}$\
& & & (deg E of N) & (deg E of N) & ([vegamag]{}) & ([vegamag]{}) & (kpc) & & &\
GC1 & $0.16 \pm 0.09 $ & $0.23 \pm 0.14 $ & $-91 \pm 60 $ & $-86 \pm49 $ & 16.050 & 15.070 & 46.4 & 24.41 & 0.09 & $-2.14 $\
GC2 & $0.18 \pm 0.07 $ & $0.21 \pm 0.09 $ & $-134 \pm 36 $ & $-132 \pm38 $ & 16.980 & 16.040 & 33.4 & 24.32 & 0.08 & $-1.94 $\
GC3 & $0.36 \pm 0.25 $ & $... $ & $-126 \pm 49 $ & $... $ & 16.310 & 15.360 & 31.8 & 24.37 & 0.11 & $-2.14 $\
GC4 & $0.33 \pm 0.20 $ & $0.30 \pm 0.21 $ & $-139 \pm 61 $ & $-147 \pm57 $ & 15.760 & 14.680 & 55.2 & 24.35 & 0.09 & $-2.14 $\
GC5 & $0.24 \pm 0.16 $ & $0.15 \pm 0.08 $ & $-165 \pm 69 $ & $-165 \pm49 $ & 16.090 & 15.010 & 78.5 & 24.45 & 0.08 & $-1.84 $\
GC6[$^c$]{} & $0.16 \pm 0.12 $ & $0.32 \pm 0.26 $ & $-123 \pm 68 $ & $-116 \pm60 $ & 16.590 & 15.460 & 14.0 & 24.49 & 0.09 & $-2.14 $\
GC7 & $... $ & $... $ & $... $ & $... $ & 18.270 & 17.070 & 18.2 & 24.13 & 0.06 & $-0.70 $\
GC8 & $0.15 \pm 0.08 $ & $0.22 \pm 0.15 $ & $-183 \pm 43 $ & $-137 \pm55 $ & 16.720 & 15.680 & 37.1 & 24.43 & 0.09 & $-1.54 $\
GC9 & $... $ & $0.35 \pm 0.16 $ & $... $ & $-135 \pm43 $ & 17.780 & 16.710 & 38.9 & 24.22 & 0.15 & $-1.54 $\
GC10 & $0.18 \pm 0.13 $ & $0.13 \pm 0.05 $ & $-160 \pm 49 $ & $-152 \pm37 $ & 16.500 & 15.590 & 99.9 & 24.42 & 0.09 & $-2.14 $\
[ccccccc]{} Name & Detector & Filter & $R$ & $I$ & Uncertainty & Flag\
& & & (arcsec) & $L_{\odot}$ pc$^{-2}$ & $L_{\odot}$ pc$^{-2}$ &\
(1) & (2) & (3) & (4) & (5) & (6) & (7)\
GC1 & WFC & F606W & 0.0260 & 31738.412 & 202.312 & OK\
GC1 & WFC & F606W & 0.0287 & 31294.604 & 223.745 & DEP\
GC1 & WFC & F606W & 0.0315 & 30807.689 & 243.173 & DEP\
GC1 & WFC & F606W & 0.0347 & 30277.402 & 263.883 & DEP\
GC1 & WFC & F606W & 0.0381 & 29710.863 & 294.703 & DEP\
GC1 & WFC & F606W & 0.0420 & 29090.574 & 324.043 & DEP\
GC1 & WFC & F606W & 0.0461 & 28412.213 & 351.340 & DEP\
GC1 & WFC & F606W & 0.0508 & 27656.385 & 368.209 & DEP\
GC1 & WFC & F606W & 0.0558 & 26734.781 & 357.023 & OK\
GC1 & WFC & F606W & 0.0614 & 25634.490 & 362.977 & DEP\
GC1 & WFC & F606W & 0.0676 & 24403.732 & 376.725 & DEP\
[cccccccccccc]{} Name & Detector & $(V-{\rm F606W})_0$ & $N_{\rm pts}$ & Model & $\chi_{\rm min}^2$ & $I_{\rm bkg}$ & $W_0$ & [$c$]{} & $\mu_0$ & $\log r_0$ & $\log r_0$\
& & (mag) & & & & $L_{\odot}$ pc$^{-2}$ & & & ${\rm (mag~arcsec^{-2})}$ & (arcsec) & (pc)\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) &(12)\
GC1 & WFC/F606 & $0.189 \pm 0.050 $ & 57 & K66 & 3401.76 & $0.20 \pm 0.08 $ & $8.96 ^{+0.27 }_{-0.31 }$ & $2.11 ^{+0.07 }_{-0.09 }$ & $14.73 ^{+0.18 }_{-0.13 }$ & $-1.100 ^{+0.039 }_{-0.049 }$ & $-0.532 ^{+0.039 }_{-0.049 }$\
& WFC/F814 & & 56 & & 2083.76 & $0.00 \pm 0.13 $ & $8.72 ^{+0.27 }_{-0.32 }$ & $2.04 ^{+0.07 }_{-0.09 }$ & $14.35 ^{+0.20 }_{-0.17 }$ & $-1.000 ^{+0.032 }_{-0.048 }$ & $-0.432 ^{+0.032 }_{-0.048 }$\
GC2 & WFC/F606 & $0.183 \pm 0.050 $ & 53 & K66 & 686.35 & $0.20 \pm 0.12 $ & $7.49 ^{+0.20 }_{-0.22 }$ & $1.67 ^{+0.06 }_{-0.07 }$ & $17.05 ^{+0.11 }_{-0.06 }$ & $-0.550 ^{+0.022 }_{-0.030 }$ & $0.000 ^{+0.022 }_{-0.030 }$\
& WFC/F814 & & 53 & & 743.06 & $0.10 \pm 0.15 $ & $7.58 ^{+0.35 }_{-0.21 }$ & $1.70 ^{+0.11 }_{-0.06 }$ & $16.24 ^{+0.09 }_{-0.09 }$ & $-0.600 ^{+0.034 }_{-0.043 }$ & $-0.050 ^{+0.034 }_{-0.043 }$\
GC3 & WFC/F606 & $0.172 \pm 0.050 $ & 30 & K66 & 262.8 & $0.70 \pm 0.12 $ & $5.06 ^{+0.16 }_{-0.14 }$ & $1.04 ^{+0.03 }_{-0.03 }$ & $18.25 ^{+0.08 }_{-0.09 }$ & $0.100 ^{+0.007 }_{-0.023 }$ & $0.660 ^{+0.007 }_{-0.023 }$\
& WFC/F814 & & 29 & & 195.46 & $0.80 \pm 0.18 $ & $5.09 ^{+0.18 }_{-0.22 }$ & $1.05 ^{+0.04 }_{-0.05 }$ & $17.63 ^{+0.12 }_{-0.09 }$ & $0.100 ^{+0.012 }_{-0.012 }$ & $0.660 ^{+0.012 }_{-0.012 }$\
GC4 & WFC/F606 & $0.216 \pm 0.050 $ & 57 & K66 & 1378.26 & $0.60 \pm 0.16 $ & $7.31 ^{+0.21 }_{-0.24 }$ & $1.62 ^{+0.07 }_{-0.07 }$ & $16.24 ^{+0.14 }_{-0.12 }$ & $-0.450 ^{+0.031 }_{-0.027 }$ & $0.106 ^{+0.031 }_{-0.027 }$\
& WFC/F814 & & 57 & & 935.37 & $0.90 \pm 0.34 $ & $7.37 ^{+0.23 }_{-0.23 }$ & $1.64 ^{+0.07 }_{-0.07 }$ & $15.62 ^{+0.13 }_{-0.14 }$ & $-0.450 ^{+0.025 }_{-0.030 }$ & $0.106 ^{+0.025 }_{-0.030 }$\
GC5 & WFC/F606 & $0.222 \pm 0.050 $ & 49 & K66 & 687.47 & $1.00 \pm 0.31 $ & $6.62 ^{+0.28 }_{-0.20 }$ & $1.42 ^{+0.08 }_{-0.05 }$ & $16.65 ^{+0.16 }_{-0.11 }$ & $-0.350 ^{+0.020 }_{-0.022 }$ & $0.226 ^{+0.020 }_{-0.022 }$\
& WFC/F814 & & 54 & & 2066.58 & $0.40 \pm 1.03 $ & $7.43 ^{+0.27 }_{-0.33 }$ & $1.66 ^{+0.08 }_{-0.10 }$ & $15.87 ^{+0.16 }_{-0.09 }$ & $-0.450 ^{+0.029 }_{-0.060 }$ & $0.126 ^{+0.029 }_{-0.060 }$\
GC6 & WFC/F606 & $0.230 \pm 0.050 $ & 54 & K66 & 477.9 & $0.60 \pm 0.11 $ & $6.95 ^{+0.11 }_{-0.13 }$ & $1.51 ^{+0.03 }_{-0.04 }$ & $16.27 ^{+0.04 }_{-0.03 }$ & $-0.550 ^{+0.018 }_{-0.017 }$ & $0.034 ^{+0.018 }_{-0.017 }$\
& WFC/F814 & & 54 & & 404.46 & $0.90 \pm 0.25 $ & $6.98 ^{+0.22 }_{-0.26 }$ & $1.52 ^{+0.07 }_{-0.08 }$ & $15.67 ^{+0.09 }_{-0.12 }$ & $-0.550 ^{+0.015 }_{-0.026 }$ & $0.034 ^{+0.015 }_{-0.026 }$\
GC7 & WFC/F606 & $0.261 \pm 0.050 $ & 34 & K66 & 196.86 & $0.10 \pm 0.00 $ & $7.43 ^{+0.11 }_{-0.13 }$ & $1.66 ^{+0.04 }_{-0.04 }$ & $18.46 ^{+0.09 }_{-0.06 }$ & $-0.450 ^{+0.020 }_{-0.027 }$ & $0.062 ^{+0.020 }_{-0.027 }$\
& WFC/F814 & & 34 & & 229.16 & $0.20 \pm 0.00 $ & $7.22 ^{+0.14 }_{-0.11 }$ & $1.59 ^{+0.04 }_{-0.03 }$ & $17.18 ^{+0.08 }_{-0.05 }$ & $-0.500 ^{+0.021 }_{-0.049 }$ & $0.012 ^{+0.021 }_{-0.049 }$\
GC8 & WFC/F606 & $0.205 \pm 0.050 $ & 48 & K66 & 170.88 & $0.60 \pm 0.15 $ & $7.40 ^{+0.08 }_{-0.07 }$ & $1.65 ^{+0.03 }_{-0.02 }$ & $15.98 ^{+0.04 }_{-0.03 }$ & $-0.700 ^{+0.006 }_{-0.013 }$ & $-0.128 ^{+0.006 }_{-0.013 }$\
& WFC/F814 & & 47 & & 186.59 & $0.70 \pm 0.19 $ & $7.82 ^{+0.09 }_{-0.12 }$ & $1.78 ^{+0.03 }_{-0.04 }$ & $14.99 ^{+0.05 }_{-0.06 }$ & $-0.850 ^{+0.016 }_{-0.014 }$ & $-0.278 ^{+0.016 }_{-0.014 }$\
GC9 & WFC/F606 & $0.189 \pm 0.050 $ & 33 & K66 & 321.09 & $0.40 \pm 0.03 $ & $5.75 ^{+0.07 }_{-0.06 }$ & $1.19 ^{+0.02 }_{-0.01 }$ & $19.33 ^{+0.03 }_{-0.02 }$ & $0.000 ^{+0.022 }_{-0.024 }$ & $0.530 ^{+0.022 }_{-0.024 }$\
& WFC/F814 & & 32 & & 82.88 & $-0.30 \pm 0.09 $ & $6.02 ^{+0.20 }_{-0.16 }$ & $1.26 ^{+0.05 }_{-0.04 }$ & $18.63 ^{+0.13 }_{-0.07 }$ & $-0.050 ^{+0.017 }_{-0.015 }$ & $0.480 ^{+0.017 }_{-0.015 }$\
GC10 & WFC/F606 & $0.169 \pm 0.050 $ & 54 & K66 & 1433.16 & $1.20 \pm 0.14 $ & $8.45 ^{+0.25 }_{-0.24 }$ & $1.97 ^{+0.07 }_{-0.07 }$ & $15.42 ^{+0.10 }_{-0.09 }$ & $-0.950 ^{+0.015 }_{-0.033 }$ & $-0.380 ^{+0.015 }_{-0.033 }$\
& WFC/F814 & & 54 & & 1280.34 & $1.10 \pm 0.17 $ & $8.54 ^{+0.30 }_{-0.25 }$ & $1.99 ^{+0.08 }_{-0.07 }$ & $14.77 ^{+0.11 }_{-0.09 }$ & $-1.000 ^{+0.021 }_{-0.039 }$ & $-0.430 ^{+0.021 }_{-0.039 }$\
=2.5pt
[ccccccccccccc]{} Name & Detector & Model & $\log r_{\rm tid}$ & $\log R_c$ & $\log R_h$ & $\log R_h/R_c$ & $\log I_{\rm 0}$ & $\log j_{\rm 0}$ & $\log L_V$ & $V_{\rm tot}$ & $\log I_h$ & $<\mu_V>_h$\
& & & (pc) & (pc) & (pc) & & $L_{\odot,V}$ pc$^{-2}$ & $L_{\odot,V}$ pc$^{-3}$ & $L_{\odot,V}$ & (mag) & $L_{\odot,V}$ pc$^{-2}$ & ${\rm (mag~arcsec^{-2})}$\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) &(12) & (14)\
GC1 & WFC/F606 & K66 & $1.58 ^{+0.07 }_{-0.08 }$ & $-0.540 ^{+0.037 }_{-0.048 }$ & $0.512 ^{+0.106 }_{-0.117 }$ & $1.052 ^{+0.154 }_{-0.154 }$ & $4.59 ^{+0.06 }_{-0.07 }$ & $4.83 ^{+0.10 }_{-0.11 }$ & $5.38 ^{+0.02 }_{-0.03 }$ & $15.79 ^{+0.07 }_{-0.06 }$ & $3.56 ^{+0.21 }_{-0.19 }$ & $17.50 ^{+0.47 }_{-0.52 }$\
& WFC/F814 & & $1.61 ^{+0.07 }_{-0.09 }$ & $-0.441 ^{+0.030 }_{-0.047 }$ & $0.532 ^{+0.109 }_{-0.117 }$ & $0.974 ^{+0.156 }_{-0.147 }$ & & $4.73 ^{+0.10 }_{-0.10 }$ & & & $3.52 ^{+0.21 }_{-0.19 }$ & $17.61 ^{+0.48 }_{-0.52 }$\
GC2 & WFC/F606 & K66 & $1.67 ^{+0.06 }_{-0.07 }$ & $-0.017 ^{+0.020 }_{-0.028 }$ & $0.584 ^{+0.062 }_{-0.069 }$ & $0.601 ^{+0.091 }_{-0.089 }$ & $3.67 ^{+0.03 }_{-0.05 }$ & $3.38 ^{+0.06 }_{-0.07 }$ & $4.96 ^{+0.02 }_{-0.02 }$ & $16.74 ^{+0.05 }_{-0.05 }$ & $3.00 ^{+0.12 }_{-0.10 }$ & $18.91 ^{+0.26 }_{-0.29 }$\
& WFC/F814 & & $1.65 ^{+0.11 }_{-0.06 }$ & $-0.066 ^{+0.032 }_{-0.041 }$ & $0.559 ^{+0.099 }_{-0.051 }$ & $0.625 ^{+0.139 }_{-0.083 }$ & & $3.43 ^{+0.07 }_{-0.08 }$ & & & $3.05 ^{+0.08 }_{-0.18 }$ & $18.78 ^{+0.44 }_{-0.20 }$\
GC3 & WFC/F606 & K66 & $1.70 ^{+0.03 }_{-0.03 }$ & $0.607 ^{+0.004 }_{-0.019 }$ & $0.850 ^{+0.023 }_{-0.015 }$ & $0.244 ^{+0.043 }_{-0.019 }$ & $3.19 ^{+0.04 }_{-0.04 }$ & $2.28 ^{+0.06 }_{-0.04 }$ & $5.37 ^{+0.03 }_{-0.03 }$ & $15.77 ^{+0.08 }_{-0.08 }$ & $2.87 ^{+0.00 }_{-0.01 }$ & $19.21 ^{+0.03 }_{0.00 }$\
& WFC/F814 & & $1.71 ^{+0.04 }_{-0.04 }$ & $0.608 ^{+0.007 }_{-0.008 }$ & $0.854 ^{+0.020 }_{-0.028 }$ & $0.246 ^{+0.028 }_{-0.035 }$ & & $2.28 ^{+0.05 }_{-0.05 }$ & & & $2.87 ^{+0.02 }_{-0.01 }$ & $19.23 ^{+0.02 }_{-0.06 }$\
GC4 & WFC/F606 & K66 & $1.73 ^{+0.07 }_{-0.07 }$ & $0.087 ^{+0.028 }_{-0.025 }$ & $0.645 ^{+0.066 }_{-0.058 }$ & $0.558 ^{+0.091 }_{-0.086 }$ & $3.98 ^{+0.05 }_{-0.06 }$ & $3.59 ^{+0.08 }_{-0.09 }$ & $5.43 ^{+0.02 }_{-0.02 }$ & $15.60 ^{+0.06 }_{-0.05 }$ & $3.34 ^{+0.09 }_{-0.11 }$ & $18.04 ^{+0.28 }_{-0.23 }$\
& WFC/F814 & & $1.74 ^{+0.07 }_{-0.07 }$ & $0.088 ^{+0.023 }_{-0.028 }$ & $0.660 ^{+0.060 }_{-0.054 }$ & $0.572 ^{+0.087 }_{-0.077 }$ & & $3.59 ^{+0.08 }_{-0.08 }$ & & & $3.31 ^{+0.09 }_{-0.10 }$ & $18.12 ^{+0.25 }_{-0.21 }$\
GC5 & WFC/F606 & K66 & $1.64 ^{+0.08 }_{-0.05 }$ & $0.200 ^{+0.018 }_{-0.018 }$ & $0.627 ^{+0.054 }_{-0.038 }$ & $0.427 ^{+0.073 }_{-0.056 }$ & $3.81 ^{+0.05 }_{-0.07 }$ & $3.31 ^{+0.07 }_{-0.09 }$ & $5.38 ^{+0.02 }_{-0.02 }$ & $15.84 ^{+0.05 }_{-0.05 }$ & $3.32 ^{+0.06 }_{-0.09 }$ & $18.09 ^{+0.22 }_{-0.14 }$\
& WFC/F814 & & $1.78 ^{+0.09 }_{-0.10 }$ & $0.108 ^{+0.025 }_{-0.057 }$ & $0.694 ^{+0.077 }_{-0.074 }$ & $0.586 ^{+0.134 }_{-0.100 }$ & & $3.40 ^{+0.11 }_{-0.09 }$ & & & $3.19 ^{+0.13 }_{-0.13 }$ & $18.43 ^{+0.33 }_{-0.32 }$\
GC6 & WFC/F606 & K66 & $1.55 ^{+0.03 }_{-0.04 }$ & $0.012 ^{+0.017 }_{-0.016 }$ & $0.500 ^{+0.030 }_{-0.033 }$ & $0.488 ^{+0.045 }_{-0.050 }$ & $3.96 ^{+0.02 }_{-0.02 }$ & $3.64 ^{+0.04 }_{-0.04 }$ & $5.23 ^{+0.02 }_{-0.02 }$ & $16.26 ^{+0.05 }_{-0.05 }$ & $3.43 ^{+0.05 }_{-0.04 }$ & $17.83 ^{+0.10 }_{-0.12 }$\
& WFC/F814 & & $1.55 ^{+0.07 }_{-0.07 }$ & $0.012 ^{+0.012 }_{-0.024 }$ & $0.508 ^{+0.056 }_{-0.054 }$ & $0.495 ^{+0.080 }_{-0.066 }$ & & $3.64 ^{+0.05 }_{-0.04 }$ & & & $3.41 ^{+0.09 }_{-0.09 }$ & $17.87 ^{+0.23 }_{-0.22 }$\
GC7 & WFC/F606 & K66 & $1.72 ^{+0.04 }_{-0.04 }$ & $0.044 ^{+0.019 }_{-0.026 }$ & $0.630 ^{+0.029 }_{-0.035 }$ & $0.585 ^{+0.056 }_{-0.054 }$ & $3.07 ^{+0.03 }_{-0.04 }$ & $2.72 ^{+0.06 }_{-0.06 }$ & $4.46 ^{+0.03 }_{-0.03 }$ & $17.80 ^{+0.06 }_{-0.06 }$ & $2.40 ^{+0.04 }_{-0.03 }$ & $20.39 ^{+0.08 }_{-0.11 }$\
& WFC/F814 & & $1.60 ^{+0.04 }_{-0.03 }$ & $-0.008 ^{+0.020 }_{-0.048 }$ & $0.533 ^{+0.033 }_{-0.026 }$ & $0.541 ^{+0.082 }_{-0.046 }$ & & $2.78 ^{+0.08 }_{-0.06 }$ & & & $2.60 ^{+0.03 }_{-0.04 }$ & $19.91 ^{+0.10 }_{-0.07 }$\
GC8 & WFC/F606 & K66 & $1.52 ^{+0.03 }_{-0.02 }$ & $-0.146 ^{+0.005 }_{-0.013 }$ & $0.438 ^{+0.028 }_{-0.021 }$ & $0.584 ^{+0.041 }_{-0.026 }$ & $4.09 ^{+0.02 }_{-0.03 }$ & $3.93 ^{+0.04 }_{-0.03 }$ & $5.14 ^{+0.02 }_{-0.02 }$ & $16.40 ^{+0.05 }_{-0.05 }$ & $3.47 ^{+0.02 }_{-0.04 }$ & $17.73 ^{+0.09 }_{-0.05 }$\
& WFC/F814 & & $1.50 ^{+0.03 }_{-0.04 }$ & $-0.293 ^{+0.015 }_{-0.013 }$ & $0.404 ^{+0.033 }_{-0.039 }$ & $0.696 ^{+0.046 }_{-0.055 }$ & & $4.07 ^{+0.04 }_{-0.04 }$ & & & $3.54 ^{+0.06 }_{-0.05 }$ & $17.55 ^{+0.11 }_{-0.15 }$\
GC9 & WFC/F606 & K66 & $1.72 ^{+0.02 }_{-0.01 }$ & $0.491 ^{+0.021 }_{-0.023 }$ & $0.801 ^{+0.015 }_{-0.008 }$ & $0.310 ^{+0.038 }_{-0.028 }$ & $2.75 ^{+0.02 }_{-0.02 }$ & $1.96 ^{+0.05 }_{-0.04 }$ & $4.77 ^{+0.02 }_{-0.02 }$ & $17.12 ^{+0.05 }_{-0.06 }$ & $2.37 ^{+-0.01 }_{-0.01 }$ & $20.47 ^{+0.02 }_{0.01 }$\
& WFC/F814 & & $1.74 ^{+0.05 }_{-0.04 }$ & $0.446 ^{+0.015 }_{-0.012 }$ & $0.787 ^{+0.030 }_{-0.028 }$ & $0.342 ^{+0.042 }_{-0.042 }$ & & $2.00 ^{+0.03 }_{-0.04 }$ & & & $2.40 ^{+0.03 }_{-0.04 }$ & $20.40 ^{+0.09 }_{-0.09 }$\
GC10 & WFC/F606 & K66 & $1.59 ^{+0.07 }_{-0.07 }$ & $-0.391 ^{+0.014 }_{-0.032 }$ & $0.493 ^{+0.089 }_{-0.077 }$ & $0.883 ^{+0.121 }_{-0.090 }$ & $4.33 ^{+0.04 }_{-0.04 }$ & $4.41 ^{+0.07 }_{-0.06 }$ & $5.20 ^{+0.03 }_{-0.02 }$ & $16.24 ^{+0.06 }_{-0.07 }$ & $3.42 ^{+0.13 }_{-0.15 }$ & $17.85 ^{+0.38 }_{-0.33 }$\
& WFC/F814 & & $1.56 ^{+0.09 }_{-0.07 }$ & $-0.440 ^{+0.020 }_{-0.037 }$ & $0.475 ^{+0.119 }_{-0.094 }$ & $0.915 ^{+0.156 }_{-0.114 }$ & & $4.46 ^{+0.08 }_{-0.06 }$ & & & $3.46 ^{+0.17 }_{-0.21 }$ & $17.76 ^{+0.53 }_{-0.41 }$\
=2.5pt
[ccccccccccccc]{} Name & Detector & $\Upsilon_V^{\rm pop}$ & Model & $\log M_{\rm tot}$ & $\log E_b$ & $\log \Sigma_{\rm 0}$ & $\log \rho_{\rm 0}$ & $\log \Sigma_h$ & $\log \sigma_{p,0}$ & $\log \nu_{\rm esc,0}$ & $\log t_{r,h}$ & $\log f_{\rm 0}$\
& & $M_{\odot}~L_{\odot,V}^{-1}$ & & $M_{\odot}$ & (erg) & $M_{\odot}$ pc$^{-2}$ & $M_{\odot}$ pc$^{-3}$ & $M_{\odot}$ pc$^{-2}$ & (km s$^{-1}$) & (km s$^{-1}$) & yr & $M_{\odot}$ (pc km s$^{-1})^{-3})$\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) &(12) & (13)\
GC1 & WFC/F606 & $1.918 ^{+0.244 }_{-0.237 }$ & K66 & $5.66 ^{+0.06 }_{-0.06 }$ & $50.68 ^{+0.21 }_{-0.25 }$ & $4.88 ^{+0.08 }_{-0.09 }$ & $5.11 ^{+0.12 }_{-0.13 }$ & $3.84 ^{+0.20 }_{-0.22 }$ & $0.906 ^{+0.026 }_{-0.034 }$ & $1.539 ^{+0.028 }_{-0.038 }$ & $9.11 ^{+0.18 }_{-0.20 }$ & $1.183 ^{+0.098 }_{-0.065 }$\
& WFC/F814 & $1.918 ^{+0.244 }_{-0.237 }$ & & & $50.93 ^{+0.21 }_{-0.26 }$ & $4.88 ^{+0.08 }_{-0.09 }$ & $5.01 ^{+0.12 }_{-0.12 }$ & $3.80 ^{+0.20 }_{-0.21 }$ & $0.956 ^{+0.026 }_{-0.036 }$ & $1.584 ^{+0.028 }_{-0.041 }$ & $9.14 ^{+0.18 }_{-0.19 }$ & $0.933 ^{+0.096 }_{-0.050 }$\
GC2 & WFC/F606 & $1.889 ^{+0.257 }_{-0.226 }$ & K66 & $5.24 ^{+0.06 }_{-0.06 }$ & $50.14 ^{+0.22 }_{-0.23 }$ & $3.94 ^{+0.06 }_{-0.07 }$ & $3.66 ^{+0.08 }_{-0.09 }$ & $3.27 ^{+0.12 }_{-0.13 }$ & $0.704 ^{+0.028 }_{-0.031 }$ & $1.305 ^{+0.028 }_{-0.034 }$ & $9.04 ^{+0.12 }_{-0.13 }$ & $0.325 ^{+0.064 }_{-0.040 }$\
& WFC/F814 & $1.889 ^{+0.257 }_{-0.226 }$ & & & $50.00 ^{+0.22 }_{-0.22 }$ & $3.94 ^{+0.06 }_{-0.07 }$ & $3.71 ^{+0.09 }_{-0.10 }$ & $3.32 ^{+0.18 }_{-0.10 }$ & $0.679 ^{+0.028 }_{-0.029 }$ & $1.282 ^{+0.028 }_{-0.031 }$ & $9.00 ^{+0.17 }_{-0.10 }$ & $0.449 ^{+0.094 }_{-0.064 }$\
GC3 & WFC/F606 & $1.918 ^{+0.244 }_{-0.237 }$ & K66 & $5.66 ^{+0.06 }_{-0.07 }$ & $50.81 ^{+0.21 }_{-0.24 }$ & $3.47 ^{+0.07 }_{-0.07 }$ & $2.56 ^{+0.08 }_{-0.07 }$ & $3.16 ^{+0.05 }_{-0.06 }$ & $0.784 ^{+0.028 }_{-0.034 }$ & $1.331 ^{+0.030 }_{-0.035 }$ & $9.61 ^{+0.07 }_{-0.07 }$ & $-1.041 ^{+0.040 }_{-0.030 }$\
& WFC/F814 & $1.918 ^{+0.244 }_{-0.237 }$ & & & $50.82 ^{+0.22 }_{-0.24 }$ & $3.47 ^{+0.07 }_{-0.07 }$ & $2.56 ^{+0.07 }_{-0.07 }$ & $3.15 ^{+0.05 }_{-0.06 }$ & $0.784 ^{+0.031 }_{-0.033 }$ & $1.332 ^{+0.033 }_{-0.035 }$ & $9.62 ^{+0.07 }_{-0.08 }$ & $-1.042 ^{+0.026 }_{-0.029 }$\
GC4 & WFC/F606 & $1.918 ^{+0.244 }_{-0.237 }$ & K66 & $5.72 ^{+0.06 }_{-0.06 }$ & $51.06 ^{+0.22 }_{-0.24 }$ & $4.26 ^{+0.07 }_{-0.08 }$ & $3.87 ^{+0.09 }_{-0.10 }$ & $3.63 ^{+0.12 }_{-0.11 }$ & $0.914 ^{+0.029 }_{-0.032 }$ & $1.511 ^{+0.031 }_{-0.035 }$ & $9.33 ^{+0.12 }_{-0.11 }$ & $-0.096 ^{+0.048 }_{-0.054 }$\
& WFC/F814 & $1.918 ^{+0.244 }_{-0.237 }$ & & & $51.07 ^{+0.21 }_{-0.24 }$ & $4.26 ^{+0.07 }_{-0.08 }$ & $3.87 ^{+0.09 }_{-0.10 }$ & $3.60 ^{+0.11 }_{-0.10 }$ & $0.914 ^{+0.028 }_{-0.033 }$ & $1.513 ^{+0.031 }_{-0.036 }$ & $9.35 ^{+0.11 }_{-0.11 }$ & $-0.097 ^{+0.054 }_{-0.043 }$\
GC5 & WFC/F606 & $1.881 ^{+0.275 }_{-0.223 }$ & K66 & $5.65 ^{+0.06 }_{-0.06 }$ & $50.97 ^{+0.25 }_{-0.24 }$ & $4.09 ^{+0.08 }_{-0.09 }$ & $3.58 ^{+0.09 }_{-0.10 }$ & $3.60 ^{+0.11 }_{-0.08 }$ & $0.885 ^{+0.033 }_{-0.037 }$ & $1.467 ^{+0.036 }_{-0.040 }$ & $9.27 ^{+0.11 }_{-0.09 }$ & $-0.302 ^{+0.039 }_{-0.030 }$\
& WFC/F814 & $1.881 ^{+0.275 }_{-0.223 }$ & & & $50.79 ^{+0.24 }_{-0.24 }$ & $4.09 ^{+0.08 }_{-0.09 }$ & $3.68 ^{+0.12 }_{-0.11 }$ & $3.46 ^{+0.15 }_{-0.14 }$ & $0.838 ^{+0.030 }_{-0.035 }$ & $1.438 ^{+0.030 }_{-0.040 }$ & $9.38 ^{+0.14 }_{-0.13 }$ & $-0.061 ^{+0.126 }_{-0.044 }$\
GC6 & WFC/F606 & $1.918 ^{+0.244 }_{-0.237 }$ & K66 & $5.51 ^{+0.06 }_{-0.06 }$ & $50.75 ^{+0.21 }_{-0.23 }$ & $4.24 ^{+0.06 }_{-0.06 }$ & $3.93 ^{+0.07 }_{-0.07 }$ & $3.71 ^{+0.07 }_{-0.07 }$ & $0.868 ^{+0.026 }_{-0.029 }$ & $1.458 ^{+0.027 }_{-0.029 }$ & $9.02 ^{+0.07 }_{-0.08 }$ & $0.096 ^{+0.039 }_{-0.042 }$\
& WFC/F814 & $1.918 ^{+0.244 }_{-0.237 }$ & & & $50.76 ^{+0.21 }_{-0.23 }$ & $4.24 ^{+0.06 }_{-0.06 }$ & $3.93 ^{+0.07 }_{-0.07 }$ & $3.70 ^{+0.11 }_{-0.11 }$ & $0.868 ^{+0.026 }_{-0.029 }$ & $1.458 ^{+0.026 }_{-0.031 }$ & $9.03 ^{+0.11 }_{-0.11 }$ & $0.096 ^{+0.057 }_{-0.036 }$\
GC7 & WFC/F606 & $2.441 ^{+1.121 }_{-0.573 }$ & K66 & $4.85 ^{+0.17 }_{-0.12 }$ & $49.35 ^{+0.66 }_{-0.47 }$ & $3.46 ^{+0.17 }_{-0.12 }$ & $3.11 ^{+0.17 }_{-0.13 }$ & $2.79 ^{+0.17 }_{-0.12 }$ & $0.492 ^{+0.082 }_{-0.059 }$ & $1.093 ^{+0.082 }_{-0.060 }$ & $8.94 ^{+0.17 }_{-0.13 }$ & $0.413 ^{+0.097 }_{-0.065 }$\
& WFC/F814 & $2.441 ^{+1.121 }_{-0.573 }$ & & & $49.16 ^{+0.66 }_{-0.47 }$ & $3.46 ^{+0.17 }_{-0.12 }$ & $3.16 ^{+0.18 }_{-0.13 }$ & $2.98 ^{+0.17 }_{-0.12 }$ & $0.467 ^{+0.083 }_{-0.059 }$ & $1.062 ^{+0.082 }_{-0.060 }$ & $8.80 ^{+0.17 }_{-0.13 }$ & $0.540 ^{+0.135 }_{-0.065 }$\
GC8 & WFC/F606 & $1.897 ^{+0.387 }_{-0.215 }$ & K66 & $5.42 ^{+0.08 }_{-0.06 }$ & $50.58 ^{+0.32 }_{-0.21 }$ & $4.36 ^{+0.08 }_{-0.06 }$ & $4.21 ^{+0.09 }_{-0.06 }$ & $3.75 ^{+0.09 }_{-0.06 }$ & $0.849 ^{+0.041 }_{-0.028 }$ & $1.449 ^{+0.041 }_{-0.029 }$ & $8.90 ^{+0.10 }_{-0.07 }$ & $0.436 ^{+0.046 }_{-0.026 }$\
& WFC/F814 & $1.897 ^{+0.387 }_{-0.215 }$ & & & $50.20 ^{+0.32 }_{-0.21 }$ & $4.36 ^{+0.08 }_{-0.06 }$ & $4.35 ^{+0.09 }_{-0.07 }$ & $3.82 ^{+0.09 }_{-0.08 }$ & $0.775 ^{+0.041 }_{-0.027 }$ & $1.384 ^{+0.041 }_{-0.027 }$ & $8.84 ^{+0.10 }_{-0.09 }$ & $0.808 ^{+0.046 }_{-0.037 }$\
GC9 & WFC/F606 & $1.897 ^{+0.387 }_{-0.215 }$ & K66 & $5.05 ^{+0.08 }_{-0.06 }$ & $49.64 ^{+0.32 }_{-0.21 }$ & $3.03 ^{+0.08 }_{-0.06 }$ & $2.23 ^{+0.09 }_{-0.07 }$ & $2.65 ^{+0.08 }_{-0.05 }$ & $0.504 ^{+0.040 }_{-0.026 }$ & $1.066 ^{+0.040 }_{-0.026 }$ & $9.28 ^{+0.09 }_{-0.06 }$ & $-0.516 ^{+0.063 }_{-0.049 }$\
& WFC/F814 & $1.897 ^{+0.387 }_{-0.215 }$ & & & $49.53 ^{+0.32 }_{-0.21 }$ & $3.03 ^{+0.08 }_{-0.06 }$ & $2.28 ^{+0.09 }_{-0.06 }$ & $2.68 ^{+0.09 }_{-0.06 }$ & $0.480 ^{+0.041 }_{-0.026 }$ & $1.049 ^{+0.041 }_{-0.027 }$ & $9.26 ^{+0.10 }_{-0.07 }$ & $-0.398 ^{+0.046 }_{-0.039 }$\
GC10 & WFC/F606 & $1.918 ^{+0.244 }_{-0.237 }$ & K66 & $5.49 ^{+0.06 }_{-0.06 }$ & $50.49 ^{+0.21 }_{-0.25 }$ & $4.61 ^{+0.07 }_{-0.07 }$ & $4.69 ^{+0.09 }_{-0.08 }$ & $3.70 ^{+0.16 }_{-0.14 }$ & $0.848 ^{+0.026 }_{-0.032 }$ & $1.470 ^{+0.028 }_{-0.035 }$ & $9.00 ^{+0.15 }_{-0.14 }$ & $0.938 ^{+0.067 }_{-0.032 }$\
& WFC/F814 & $1.918 ^{+0.244 }_{-0.237 }$ & & & $50.36 ^{+0.21 }_{-0.24 }$ & $4.61 ^{+0.07 }_{-0.07 }$ & $4.74 ^{+0.09 }_{-0.09 }$ & $3.74 ^{+0.22 }_{-0.18 }$ & $0.823 ^{+0.026 }_{-0.031 }$ & $1.447 ^{+0.027 }_{-0.033 }$ & $8.98 ^{+0.20 }_{-0.16 }$ & $1.063 ^{+0.080 }_{-0.042 }$\
[^1]: See http://www.ucolick.org/$\sim$cnaw/sun.html.
[^2]: See http://www.ucolick.org/$\sim$cnaw/sun.html.
|
---
abstract: 'Activity studies of solar-type stars, especially with reference to the status of our current Sun among them, have exposed the importance of (1) homogeneously selecting the sample stars and (2) reliably evaluating their activities down to a considerably low level. Motivated by these requirements, we conducted an extensive study on the activities of 118 solar-analog stars (of sufficiently similar properties to each other) by measuring the emission strength at the core of Ca [ii]{} 3933.663 line (K line) on the high-dispersion spectrogram obtained by Subaru/HDS, where special attention was paid to correctly detecting the chromospheric emission by removing the wing-fitted photospheric profile calculated from the classical solar model atmosphere. This enabled us to detect low-level activities down to $\log R'' \sim -5.4$ ($R''$ is the ratio of the chromospheric core emission flux to the total bolometric flux), by which we could detect subtle activity differences which were indiscernible in previous studies. Regarding the Sun, we found $\log R''_{\odot} = -5.33$ near to the low end of the distribution, which means that it belongs to the distinctly low activity group among solar analogs. This excludes the once-suggested possibility for the high frequency of Maunder-minimum stars showing appreciably lower activities than the minimum-Sun.'
author:
- |
Yoichi <span style="font-variant:small-caps;">Takeda</span>, Akito <span style="font-variant:small-caps;">Tajitsu</span>, Satoshi <span style="font-variant:small-caps;">Honda</span>, Satoshi <span style="font-variant:small-caps;">Kawanomoto</span>,\
Hiroyasu <span style="font-variant:small-caps;">Ando</span>, and Takashi <span style="font-variant:small-caps;">Sakurai</span>
title: |
Detection of Low-Level Activities in Solar-Analog Stars\
from the Emission Strengths of Ca II 3934 Line [^1]
---
Introduction
============
It is of great interest for solar as well as stellar astrophysicists to compare the activity of our Sun to those of a number of other similar solar-type stars, since it may provide us with an opportunity to infer the trend of solar activity on a very long astronomical time scale (i.e., investigating the long-time behavior of a star may be replaced by studying many similar stars at a given time).
Baliunas and Jastrow (1990) argued based on the results of Mt. Wilson Observatory’s HK survey project for 74 solar-type stars that the distribution of $S$-index (nearly equivalent to $\propto \int_{\rm line}{F_{\lambda}}d\lambda /\int_{\rm cont}{F_{\lambda}}d\lambda$; i.e., the ratio of integrated core-flux of Ca [ii]{} HK lines to the continuum flux; cf. Vaughan et al. 1978) is bimodal, with about 1/3 showing appreciably smaller activities than the Sun, which they interpreted as being in the “Maunder-minimum” state of activity. If this is true, it may mean that the spotless phase of considerably low-activity such as occurred in late 17th century in our Sun (Eddy 1976) may not necessarily be an unusual phenomenon in the long run.
However, this result could not be confirmed by a similar analysis done by Hall and Lockwood (2004), who reported based on many repeated observations of Ca [ii]{} H and K lines for 57 Sun-like stars along with the Sun at Lowell Observatory that such a bimodal distribution of $S$-index (as suggesting the existence of a considerable fraction of appreciably lower activity stars than the current Sun) is not observed; actually, even the $S$-values of 10 “flat-activity” stars turned out to be comparable with (or somewhat larger than) the typical solar-minimum value.
Furthermore, Wright (2004) pointed out an important problem in Baliunas and Jastrow’s (1990) sample selection. He concluded by examining the absolute magnitudes of their sample based on Hipparcos parallaxes that many of those “Maunder-minimum stars” with considerably low $S$ indices are old stars evolved-off the main sequence, which suggests that their apparently low activity is nothing but due to the aging effect without any relevance to the cyclic or irregular change of activity in solar-type dwarfs. Thus, fairly speaking, the original claim by Baliunas and Jastrow (1990) appears to be rather premature and difficult to be justified from the viewpoint of these recent studies.
Yet, the issue of clarifying the status of solar activity among Sun-like stars does not seem to have been fully settled and further investigations still remain to be done:\
— First, since understanding the activity of the Sun from a comprehensive perspective is in question, comparison samples should comprise stars as closer to the Sun as possible. Admittedly, those authors surely paid attention to this point: Baliunas and Jastrow’s (1990) HK project targets were in the $B-V$ range of 0.60–0.76, while Hall and Lockwood’s (2004) Sun-like stars sample were chosen from stars of $0.58 \le B-V \le 0.72$, both narrowly encompassing the solar $(B-V)_{\odot}$ of 0.65 (Cox 2000). However, the homogeneity of these samples are not yet satisfactory. Could they be made up of further more Sun-like stars or solar analogs?\
— Second, it appears that the precision of detecting low-level activity has been insufficient. Although Mt. Wilson $S$ index reflects the core-emission strength (equivalent width) of Ca [ii]{} HK lines, it would not be a sensitive activity indicator any more, when the emission becomes weak, as it stabilizes at a constant value determined by the photospheric absorption profile. While an indicator for the pure-emission strength, $R'_{\rm HK} (\equiv R_{\rm HK} - R_{\rm phot}$; where $R_{\rm HK} \equiv F_{\rm HK}/F_{\rm bol}$ and $R_{\rm phot} \equiv F_{\rm phot}/F_{\rm bol}$), has also been introduced to rectify this shortcoming and widely used, the photospheric component $R_{\rm phot}$ is in most cases only roughly evaluated as a simple function of $B-V$ (e.g., Noyes et al. 1984) and thus its accuracy is rather questionable. Wright (2004) also pointed out the importance of correctly subtracting this component, in view of its possible dependence on other stellar parameters (i.e., not only on $B-V$ or $T_{\rm eff}$, but also on $\log g$ and \[Fe/H\]).
These requirements motivated us to conduct a new investigation on this subject, since we have been engaged these years with the extensive study of 118 Sun-like stars selected by the criteria of $0.62 \ltsim B-V \ltsim 0.67$ and $4.5 \ltsim M_{V} \ltsim 5.1$ ($M_{V,\odot} = 4.82$; Cox 2000), a quantitatively as well as qualitatively ideal sample of solar-analog stars. This project was originally started for the purpose of clarifying the behavior of Li abundances ($A$(Li)), and revealed that the rotational velocity ($v_{\rm e}\sin i$) is the most influential key parameter (Takeda et al. 2007; hereinafter referred to as Paper I). In a successive study, we investigated the activities of these solar analogs by using the residual flux at the core of the Ca [ii]{} 8542 line ($r_{0}$(8542)), and confirmed a clear correlation between $A$(Li), $v_{\rm e}\sin i$, and $r_{0}$(8542), as expected (Takeda et al. 2010; hereinafter referred to as Paper II). However, it turned out hard to discriminate the differences in $r_{0}$(8542) when the activity is as low as that of the Sun, since it tends to get settled at $\sim 0.2$ and does not serve as a sensitive indicator any more. Actually, test calculations of non-LTE line formation suggested (cf. Appendix B in Paper II) that the core flux of the Ca [ii]{} 8498/8542/8662 triplet lines are rather inert to the chromospheric temperature rise in the upper atmosphere when the activity is low, but that for the Ca [ii]{} 3934/3968 doublet (H+K lines) is still sensitive and thus more advantageous for detecting the low-level activity. Then, we recently studied the Be abundances of these sample stars by using the Be [ii]{} 3131 line based on the near-UV spectra obtained with Subaru/HDS (Takeda et al. 2011; hereinafter referred to as Paper III). Since these HDS spectra fortunately cover the Ca [ii]{} H+K lines in the violet region, we decided to reinvestigate the activities of these 118 Sun-like stars by measuring the core-emission strength of the Ca [ii]{} 3934 (K) line, in order to clarify the activity status of our Sun in comparison with similar solar analogs, where special attention was given to correctly removing the background line profile computed from the solar photospheric model, while taking advantage of the fact that atmospheric parameters of all these targets are well established. The purpose of this paper is to report the outcome of this investigation.
Basic Observational Data
========================
Target Sample
-------------
We use the same targets (118 solar analogs) as used in Papers I–III, which were selected by the criteria of having $B-V$ and $M_{V}$ values sufficiently similar to those of the Sun ($|\Delta (B-V)| \ltsim$ 0.2–0.3 and $|\Delta M_{V}| \ltsim 0.3$). See section 2 in Paper I for a detailed description about the sample selection. We also determined the atmospheric parameters ($T_{\rm eff}$, $\log g$, $v_{\rm t}$, and \[Fe/H\]) from the equivalent widths of Fe lines, and the stellar parameters ($M$ and $age$) by comparing the positions on the HR diagram with the theoretical evolutionary tracks (cf. section 3 in Paper I). These parameters for most of the targets were actually confirmed to proximally distribute around the solar values (i.e., $|\Delta T_{\rm eff}| \ltsim 100$ K, $|\Delta \log g| \ltsim 0.1$ dex, $|\Delta v_{\rm t}| \ltsim 0.2$ km s$^{-1}$, $|\Delta{\rm [Fe/H]}|\ltsim 0.2$ dex, $|\Delta M|\ltsim 0.1\;M_{\odot}$, and $|\Delta \log age| \ltsim 0.5$ dex; cf. figures 4 and 5 in Paper I).
However, the following characteristics regarding the relations between these parameters are to be noted, which we had better bear in mind in discussing the behavior of stellar activities.\
— Since the effect of a decreased metallicity on $B-V$ is compensated by a lowering of $T_{\rm eff}$, several outlier stars with appreciably lower $T_{\rm eff}$ as well as \[Fe/H\] ($\Delta T_{\rm eff} \ltsim -200$ K and \[Fe/H\] $\ltsim -0.4$ dex) are included in our sample (cf. figure 4c in Paper I), such as HIP 26381, HIP 39506, HIP 40118, and HIP 113989; and they belong to the oldest group ($age \sim 10^{10}$ yr).\
— These parameters are not independent from each other and some correlations appear to exist between specific combinations; such as $age$ vs. $T_{\rm eff}$ (lower $T_{\rm eff}$ stars tend to be older), $age$ vs. \[Fe/H\] (lower \[Fe/H\] stars tend be older), $M$ vs. $age$ (lower-mass stars tend to be older), and $v_{\rm t}$ vs. $T_{\rm eff}$ ($v_{\rm t}$ tends to decrease with a lowered $T_{\rm eff}$), as recognized from figure 5 or figure 10 in Paper I, though the existence of outlier stars mentioned above partly plays a role in these tendencies.
It should also be remarked that the data for HIP 41484 given in Paper I were incorrect, as reported in appendix A of Paper II, where the correct results derived from a reanalysis of this star are presented.
Observational Material
----------------------
The spectroscopic observations of these 118 solar analogs and Vesta (substitute for the Sun) were carried out on 2009 August 6, 2009 November 27, 2010 February 4, and 2010 May 24 (Hawaii Standard Time), with the High Dispersion Spectrograph (HDS; Noguchi et al. 2002) placed at the Nasmyth platform of the 8.2-m Subaru Telescope atop Mauna Kea, by which we obtained high-dispersion spectra covering $\sim$ 3000–4600 $\rm\AA$ with a resolving power of $R \simeq 60000$. See section 2 of Paper III and electronic table E1 therein for details of the observations and the data reduction, as well as the basic data of the spectra (e.g., observing date, exposure time, S/N ratio at $\lambda \sim 3131 \rm\AA$).
The counts of raw echelle spectra (including the effect of blaze function) around $\lambda \sim 3950 \rm\AA$ (the broad maximum between the two large depressions of Ca [ii]{} 3934 (K) and 3968 (H) lines) are typically about $\sim 15$ times as large as those at the UV region of $\lambda \sim 3131 \rm\AA$, while the counts at core of the Ca [ii]{} K line at 3934 $\rm\AA$ is about $\sim 10\%$ of those at $\lambda \sim 3950 \rm\AA$. Therefore, the S/N ratio at the deep absorption core of the K line is not much different from the value at $\lambda \sim 3131 \rm\AA$ given in electronic table E1 of Paper III; that is, on the order of S/N $\sim 100$.
Core Emission Measurement
=========================
K line of ionized calcium
-------------------------
Most activity studies of solar-type stars so far based on the core emission strengths of Ca [ii]{} resonance lines appear to utilize both K (3934 $\rm\AA$) and H (3968 $\rm\AA$) lines, presumably due to the intention of reducing the systematic errors by averaging both two, since measurements tend to be done rather roughly by directly integrating raw spectra at the specified wavelength regions.
In this investigation, however, we focus only on the former K line at 3933.66 $\rm\AA$, since (1) it is by two times stronger than the H line and thus comparatively more suitable as a probe of the condition at the optically-thin chromospheric layer, and (2) the latter Ca [ii]{} H line at 3968.47 $\rm\AA$ is blended with the Balmer line (H$\epsilon$ at 3970.07 $\rm\AA$) which would make the situation more complicated in simulating the photospheric profile to be subtracted.
Our spectra in the selected wavelength region (3930–3937 $\rm\AA$) including the relevant Ca [ii]{} K line are shown in figures 1 (Vesta/Sun), 2 (all 118 stars including Vesta/Sun), and 3 (all stars overplotted), where the continuum levels of the observed spectra are so adjusted as to match the theoretical ones as explained below. We can see from these figures that the strengths of core emission considerably vary from star to star, while that for the Sun is apparently weak.
(80mm,60mm)[figure1.eps]{}
(80mm,60mm)[figure3.eps]{}
Photospheric profile matching
-----------------------------
Given that the very strong Ca [ii]{} K and H lines with considerably extended damping wings are predominant at $\sim$ 3900–4000 $\rm\AA$, it is hopeless to empirically establish the continuum position from the observed spectrum $D_{\lambda}^{\rm obs}$ (where an echelle order covers only $\sim 50$ $\rm\AA$). We thus “adjusted” the continuum position ($D_{\rm cont}^{\rm obs}$) of the spectrum (judged by eye-inspection) in such a way that $r_{\lambda}^{\rm obs} (\equiv D_{\lambda}^{\rm obs}/D_{\rm cont}^{\rm obs})$ satisfactorily matches the theoretically calculated residual flux $r_{\lambda}^{\rm th} (\equiv F_{\lambda}^{\rm th}/F_{\rm cont}^{\rm th})$[^2] in the inner wing of the line (within $|\Delta\lambda| \ltsim$ 2–3 $\rm\AA$ from the line center, excepting the core-emission region). An example of such an accomplished match is displayed in figure 1 for the case of the Sun (Vesta).
Regarding the computation of theoretical spectra, we used Kurucz’s (1993) WIDTH9 program, which was modified by Y. Takeda to enable spectrum synthesis by including many lines. As to the atomic line data, we invoked Kurucz and Bell’s (1995) compilation and included all available lines in the relevant region. In particular, the data for the Ca [ii]{} K line at 3933.663 $\rm\AA$ of our primary concern are as follows: $\chi_{\rm low}$ = 0.00 eV, $\log gf = +0.134$, $\log \Gamma_{\rm R} = 8.20$ \[radiation damping width (s$^{-1}$)\], $\log \Gamma_{e}/N_{\rm e} = -5.52$ \[Stark effect damping width (s$^{-1}$) per electron density (cm$^{-3}$) at $10^{4}$ K\], and $\log \Gamma_{w}/N_{\rm H} = -7.80$ \[van der Waals damping width (s$^{-1}$) per hydrogen density (cm$^{-3}$) at $10^{4}$ K\]. Since the atmospheric parameters ($T_{\rm eff}$, $\log g$, \[Fe/H\], and $v_{\rm t}$) are already established (as summarized in table 1), the model atmosphere for each star was generated by 3-dimensionally interpolating Kurucz’s (1993) ATLAS9 model grids (LTE, plane-parallel model) in terms of $T_{\rm eff}$, $\log g$, and \[Fe/H\]. Then, the synthetic spectrum was computed by using the relevant atmospheric model along with the metallicity-scaled abundances (for all elements including Ca; i.e., \[X/H\] = \[Fe/H\] for any X) as well as the microturbulence ($v_{\rm t}$), and further broadened according to the macrobroadening parameter determined in Paper III.
Evaluation of chromospheric emission
------------------------------------
Now that the theoretical photospheric background profile ($r_{\lambda}^{\rm th}$) used for subtraction has been successfully fitted with $r_{\lambda}^{\rm obs}$ by appropriately adjusting the continuum position, we can calculate the absolute emission flux ($F'_{\rm Kp}$) at the K-line core originating from the chromosphere (i.e., after subtraction of the photospheric component) as $$F'_{\rm Kp} \equiv
F_{\rm cont}^{\rm th}\int _{\lambda_{1}}^{\lambda_{2}}
(r_{\lambda}^{\rm obs} - r_{\lambda}^{\rm th})d \lambda$$ where $\lambda_{1}$ and $\lambda_{2}$ defining the integration range were chosen to be 3932.8 and 3934.6 $\rm\AA$, respectively (cf. figure 1). In order to demonstrate how this subtraction process works well, the photospheric profile ($r_{\lambda}^{\rm th}$) at \[$\lambda_{1}$, $\lambda_{2}$\] is displayed by red dashed line (along with the observed spectrum shown by symbols) for each star in figure 2. Finally, we can obtain $R'_{\rm Kp}$ (the ratio of the chromospheric emission flux at the K line to the total bolometric flux) as $$R'_{\rm Kp} \equiv F'_{\rm Kp}/ F_{\rm bol}
= \pi F'_{\rm Kp} / (\sigma T_{\rm eff}^{4}).$$ The resulting $\log R'_{\rm Kp}$ values for each of the 118 solar analogs (+Sun) are presented in table 1, where other activity-related quantities ($r_{0}$(8542) and $v_{\rm e}\sin i$) determined in Paper II are also given, along with the Li/Be abundances and stellar parameters established in Papers I and III.
Zero-Point Uncertainties in $R'_{Kp}$
-------------------------------------
We would like to remark here that the “absolute” values of $R'_{\rm Kp}$ are not very meaningful in the low-activity regime because of the uncertainties in its zero-point. That is, the theoretical profile we have computed for subtraction of the background photospheric component is by no means uniquely defined. Actually, apparently different results may be obtained depending on how it is calculated.
This situation is demonstrated in figure 1 for the solar case. The solar atmospheric model (which was obtained by interpolating the grids of Kurucz’s ATLAS9 model atmospheres) we adopted for calculating the LTE photospheric profile (red line) has its surface at $\log \tau_{5000}^{\rm surf} = -5$ ($T^{\rm surf} \sim 4000$ K). However, we note that the same but simply extrapolated model up to $\log \tau_{5000}^{\rm surf} = -7$ ($T^{\rm surf} \sim 3600$ K) yields an appreciably deeper core (upper blue line), reflecting the fact that the residual flux at the center is determined by $\sim B_{\lambda}(T^{\rm surf})/B_{\lambda}(T^{\rm ph})$ ($T^{\rm ph}$: photospheric temperature) and very sensitive to $T^{\rm surf}$ in this violet region where Wien’s approximation nearly holds. Moreover, the core of the non-LTE profile (simulated by using the departure coefficients computed in Paper II for the case of Model E; cf. Appendix B therein) gets even more deeper approaching a completely dark core (lower blue line). Besides, the core shape strongly depends on the turbulent velocity field in the upper atmosphere; e.g., compare the non-LTE profile in figure 1 (computed with a depth-independent microturbulence of 1 km s$^{-1}$) with that for Model E depicted in figure B.1(b) of Paper II (where a variable microturbulent velocity field increasing with height was adopted).
Accordingly, $R'_{\rm Kp}$ values \[equation (2)\] may be uncertain by an arbitrary constant, because $\int_{\lambda_{1}}^{\lambda_{2}}r_{\lambda}^{\rm th}d\lambda$ in equation (1) can have different values depending on how $r_{\lambda}^{\rm th}$ is computed, though “relative” differences between $R'_{\rm Kp}$ values of each star are surely meaningful as long as they are evaluated in the same system. Therefore, we should keep in mind that comparison of absolute values of our $R'_{\rm Kp}$ with those of similar $R'$ parameters derived by other groups, which we will try in subsection 4.1, is not much meaningful when the low-activity region (e.g., $\log R'_{\rm Kp} \ltsim -5$) is concerned. (On the other hand, such a zero-point problem should not be so serious for higher-activity cases, where $\log R'_{\rm Kp}$ tends to be dominated by the emission component and the role of $r_{\lambda}^{\rm th}$ subtraction is less significant.) In any event, we consider that our choice of shallower $r_{\lambda}^{\rm th}$ is adequate, since it leads to smaller values of $R'_{\rm Kp}$ (as a result of larger subtraction), which eventually realizes a larger contrast in the $\log R'_{\rm Kp}$ values of low-activity stars.
Discussion
==========
Comparison of $R'_{\rm Kp}$ with previous studies
-------------------------------------------------
The $\log R'_{\rm Kp}$ values determined in subsection 3.3 are compared with the equivalent activity indices[^3] derived by Strassmeier et al. (2000) \[$\log R'_{\rm K}$\], Wright et al. (2004) \[$\log R'_{\rm HK}$\], and Isaacson and Fischer (2010) \[$\log R'_{\rm HK}$\] in figures 4a, 4b, and 4c, respectively. These figures reveal almost the same tendency of correlation between our $\log R'$ and those of three studies: The agreement at $\log R' \gtsim -5$ is mostly good, though our $\log R'$ tends to be slightly larger by $\sim 0.1$ dex for high-activity stars of $\log R' \gtsim -4.5$. Meanwhile, a distinct difference is observed at the low-activity region where their $\log R'$ values tend to settle down at $\sim -5$, while ours are dispersed over $-5.5 \ltsim \log R' \ltsim -5$. Admittedly, there is not much meaning in comparing the absolute $R'$ values of different systems with each other, as remarked in subsection 3.4. The important point is, however, that we could detect the subtle difference in the low-level activity by measuring the weak core-emission at Ca [ii]{} K with a careful subtraction of the background photospheric profile, while such a precision for distinguishing the delicate difference in the weak emission strength could not be accomplished by comparatively rough measurements in those previous studies.
(70mm,140mm)[figure4.eps]{}
Connection with stellar parameters
----------------------------------
Figures 5a, 5b, and 5c display how the three activity-related parameters ($r_{0}$(8542), $v_{\rm e}\sin i$, and $\log age$; cf. Paper II) are correlated with $\log R'_{\rm Kp}$.
(70mm,140mm)[figure5.eps]{}
We can observe in figure 5a a marked sensitivity-difference between $\log R'_{\rm Kp}$ and $r_{0}$(8542) in the low-activity region; i.e., the latter is inert to a variation of low-level activity and stabilizes at $\sim 0.2$, despite that the former still shows an appreciable variability over $-5.5 \ltsim \log R'_{\rm Kp} \ltsim -5$. This is just what we have expected (cf. Appendix B in Paper II), and demonstrates the superiority of the Ca [ii]{} HK core emission (as long as correctly measured) to the line-center residual flux of Ca [ii]{} 8542 when it comes to investigating the activities of solar-type stars as low as the Sun.
Figure 5b shows a positive correlation between $R'_{\rm Kp}$ and $v_{\rm e}\sin i$, suggesting that stellar activity depends on the rotation rate, as we already confirmed in Paper II (cf. figure 5a therein). However, since $v_{\rm e}\sin i$ values cluster around $\sim 2$ km s$^{-1}$ at $-5.5 \ltsim \log R'_{\rm Kp} \ltsim -5$, we can not state much about whether this activity–rotation connection persists down to such a low-activity region (also, uncertainties in the projection factor prevent from a meaningful discussion).
When we compare the $age$ vs. $R'_{\rm Kp}$ relation depicted in figure 5c with the similar $r_{0}$(8542) vs. $age$ plot (cf. figure 5c of Paper II), the anti-correlation is more clearly (or less unambiguously, to say the least) recognized in the present case, thanks to the extended dynamic range of the activity indicator for low-activity stars, though the dispersion is still considerably large.
How the abundances of Li and Be depend on $R'_{\rm Kp}$ determined in this study is illustrated in figure 6, where their dependences upon $r_{0}$(8542) and $v_{\rm e}\sin i$ already discussed in Papers II and III are also shown for comparison. We can see from figures 6a that the near-linear relation between $A$(Li) and $\log R'_{\rm Kp}$ ($A$(Li) $\simeq 7 + \log R'_{\rm Kp}$) holds widely from highly active ($\log R'_{\rm Kp} \sim -4$) to less active ($\log R'_{\rm Kp} \sim -5.5$) stars, in contrast to the case of $A$(Li) vs. $r_{0}$(8542) where $A$(Li) shows a considerable dispersion at $r_{0}$(8542) $\sim 0.2$ (as if compressed) because of the less sensitivity of $r_{0}$(8542). This substantiates the observational conclusion in Paper II (or corroborates its validity even for less-active cases) that $A$(Li) closely depends upon the stellar activity, which further lends support for our previous argument that the key parameter for controlling the surface lithium in solar-analog stars is the stellar rotation.
Regarding $A$(Be), we suspected in Paper III that the peculiar 4 stars showing drastically depleted Be (by $\gtsim 2$ dex) in comparison with the solar abundance (while others have more or less near-solar Be) might be very slowly-rotating and and thus less active star than the Sun. However, since these 4 stars have appreciably different $\log R'_{\rm Kp}$ from each other ($-5.4 \ltsim \log R'_{\rm Kp} \ltsim -4.9$; cf. figure 6a$'$) and are not necessarily less active than the Sun, this speculation does not seem likely. Some other explanation would have to be sought for.
(80mm,160mm)[figure6.eps]{}
Activities of solar analogs and the Sun
---------------------------------------
We now discuss the activity trends of 118 solar analogs, with special attention paid to the status of our Sun among its close associates, as the main subject of this study. While our discussion is based on the $\log R'_{\rm Kp}$ indices determined by ourselves, we should keep in mind that only one snapshot data is available for each star. This means that uncertainties due to possible time-variations of stellar activities (whichever cyclic or irregular) are inevitably involved. For example, the Mt. Wilson $S$ value for the Sun varies over the range of $0.16 \ltsim S_{\odot} \ltsim 0.20$ (Baliunas et al. 1995), which may be translated into a variability in $\log R'_{\odot}$ by $\sim 0.2$ dex ($-5.0 \ltsim \log R'_{\odot} \ltsim -4.8$) according to the transformation formula given by Noyes et al. (1984). Since the observation time (2010 February 5) of our solar spectrum (Vesta) had better be regarded as corresponding to the near-minimum phase because of the appreciably retarded beginning of cycle 24 after the minimum in 2008, our $\log R'_{\rm Kp,\odot} (= -5.33)$ may as well be raised by $\ltsim 0.2$ dex at the solar maximum phase. It is thus reasonable to assume that our $\log R'_{\rm Kp,\odot}$ ranges from $-5.35$ (solar minimum) to $-5.15$ (solar maximum), as indicated by a short horizontal bar in figures 5, 6, and 7.
The distribution histogram of $\log R'_{\rm Kp}$ for all the 118 stars (and the Sun) and a similar histogram of $r_{0}$(8542) (for comparison; cf. Paper II) are shown in figures 7a and 7b, respectively. We immediately notice in figure 7a the bimodal distribution of $\log R'_{\rm Kp}$ having two peaks at $\sim -5.3$ and $\sim -4.3$, constituting a well-known Vaughan–Preston gap (Vaughan & Preston 1980), while this bimodal trend is not clear in the distribution of $r_{0}$(8542) (figure 7b) because of the densely peaked population around $r_{0}$(8542) $\sim 0.2$, reflecting its insensitivity when the activity is low (cf. figure 5a).
As we can see from figure 7a, the Sun with $\log R'_{\rm Kp,\odot}$ of $-5.33$ manifestly belongs to the low-activity group (ranging from $\sim -5.4$ to $\sim -5.0$). As a matter of fact, only 11 ($\sim 10\%$)[^4] out of 118 solar analogs have $\log R'_{\rm Kp}$ values smaller than $-5.33$. This distinctly low-activity nature of the Sun is also recognized by eye-inspection of figure 2, revealing that the Ca [ii]{} K line emission strength in our solar spectrum is near to the minimum level among other stars.
(70mm,120mm)[figure7.eps]{}
Thus, we can rule out the possibility for the existence of a significant fraction of Maunder-minimum stars (i.e., solar-type stars with appreciably lower activity than the Sun, even showing an another peak well below the current solar-minimum level), such as those once suggested by Baliunas and Jastrow (1990). This result corroborates the arguments raised by recent studies (e.g., Hall & Lockwood 2004; Wright 2004), which cast doubts about the reality of such a high frequency of Maunder-minimum stars. Thus, our Sun belongs to the group of manifestly low activity level among solar analogs, the fraction of stars below which is essentially insignificant.
Stars of Subsolar Activity
--------------------------
Although we have concluded that the Sun belongs to nearly the lowest activity group, some stars do exist showing activities still lower than that of the minimum-Sun, which are worth being examined more in detail. Since we defined that solar $\log R'_{\rm Kp, \odot}$ varies from $-5.35$ (minimum) to $-5.15$ (maximum) (cf. section 4.1), such stars may be sorted out by the criterion of $\log R'_{\rm Kp} < -5.35$, which resulted in the following 8 objects ($\log R'_{\rm Kp}$, $A$(Li), $\log age$, \[Fe/H\], remark): HIP 7918 ($-5.42$, 1.89, 9.56, +0.01), HIP 31965 ($-5.39$, 1.00, 9.87, +0.05), HIP 39506 ($-5.44$, $<0.9$, 10.36, $-0.62$) HIP 53721 ($-5.43$, 1.75, 9.89, $-0.02$, PHS), HIP 59610 ($-5.40$, 1.62, 9.63, $-0.06$, PHS), HIP 64150 ($-5.38$, $<1.0$, 9.63, +0.05, Be depleted), HIP 64747 ($-5.40$, $<1.1$, 9.80, $-0.18$), and HIP 96901 ($-5.38$, $<1.1$, 9.77, +0.08, PHS).
While only one (HIP 39506) of these is an outlier of lower \[Fe/H\] as well as lower $T_{\rm eff}$ (belonging to rather old population) as mentioned in subsection 2.1, the remaining 7 stars are Sun-like stars with sufficiently similar parameters, which excludes the possibility of such a low-level activity being due to stellar-evolution (i.e., evolved subgiants; cf. section 1).
We note here that (1) three (out of only five in our sample of 118 stars) planet-host stars are included, (2) all these stars have low-scale Li abundances ($A$(Li) $\ltsim 2$), and (3) their ages are similar to or older than that of the Sun. Combining these facts with the consequences in Papers I and II, we consider that these stars have actually low activities even compared with the solar-minimum level, and this is presumably attributed to their intrinsically slow rotation (which is closely related with the Li abundance as well as with the existence of giant planets).
It is, therefore, interesting to investigate the variabilities of these low-activity stars by long-term monitoring observations, in order to see how their activities behave with time (cyclic? flat? irregular?). Admittedly, activity observations for these stars have been reported in several published studies so far, for example: HIP 7918, 53721, 64150, and 96901 by Duncan et al. (1991); HIP 53721, 59610, 64150, and 96901 by Wright et al. (2004); HIP 7918, 53721, and 96901 by Hall et al. (2007); HIP 31965, 59610, 64150, and 96901 by Isaacson and Fischer (2010). However, as they are still quantitatively insufficient for establishing the long-term behavior of their activities[^5], much more observations are evidently needed.
Conclusion
==========
There have been several arguments regarding the status of solar activity among similar Sun-like stars. which began with the implication of Baliunas and Jastrow (1990) based on their Mt. Wilson HK survey project that a considerable portion ($\sim 1/3$) of solar-type stars have activities significantly lower than the present-day Sun, which they called “Maunder-minimum stars.” However, their conclusion could not be confirmed by Hall and Lockwood’s (2004) follow-up study, and Wright (2004) criticized the reality of such considerably low-active solar-type stars by pointing out that most of them are not so much dwarfs as evolved subgiants.
Given this controversial situation, we decided to contend with this problem by ourselves based on carefully selected sample of 118 solar-analogs sufficiently similar to each other (which we already investigated their stellar parameters as well as Li/Be abundances in a series of our previous papers), with a special attention being paid to reliably evaluating their activities down to a considerably low level.
Practically, we measured the emission strength at the core of Ca [ii]{} 3933.663 line (K line) on the high-dispersion spectrogram obtained by Subaru/HDS, where we gave effort to correctly evaluating the pure emission component by removing the wing-fitted photospheric profile calculated from the classical solar model atmosphere, which enabled us to detect low-level activities down to $\log R'_{\rm Kp} \sim -5.5$.
A comparison of our $\log R'_{\rm Kp}$ results with the corresponding $\log R'$ values of Strassmeier et al. (2000), Wright et al. (2004), and Isaacson and Fischer (2010) revealed that low-active stars (for which they derived $\log R' \sim -5.1$ at the minimum limit) actually have a dispersion of $\sim 0.4$ dex ($-5.5 \ltsim \log R'_{\rm Kp} \ltsim -5.0$) in our measurement, suggesting that our $\log R'_{\rm Kp}$ has a higher sensitivity and thus advantageous. A similar situation holds regarding the comparison with $r_{0}$(8542) we used in Paper II; i,e., this index stabilizes at $\sim 0.2$ and becomes insensitive for low-active stars in contrast to $\log R'_{\rm Kp}$.
As another merit of using $\log R'_{\rm Kp}$, we can state that the visibility of the $A$(Li)–activity relation as well as the age–activity relation becomes comparatively clearer, because this activity index turns out to have well diversified values for low-activity stars thanks to its high sensitivity, which can not be accomplished by using, e.g., $r_{0}$(8542).
From the distribution histogram of $\log R'_{\rm Kp}$, we could recognize a clear Vaughan–Preston gap between two peaks at $\sim -5.3$ and $\sim -4.3$. Our result of $\log R'_{\rm Kp,\odot} = -5.33$ manifestly suggests that the Sun belongs to the group of the former peak and has a distinctly low-active nature among solar analogs. Actually, a fraction of stars with $\log R'_{\rm Kp} \le \log R'_{\rm Kp,\odot}$ is only $\sim 10\%$. This consequence exclude the possibility for the existence of a considerable fraction (e.g., $\sim 1/3$) of “Maunder-minimum stars” such that having activities significantly lower than the current solar-minimum level as once suggested by Baliunas and Jastrow (1990).
Yet, some stars (only a minor fraction of the sample) do exist showing activities still lower than that of the solar-minimum level. Having examined such 8 low-activity stars, we found that they tend to include planet-host stars and have low Li abundances, from which we suspect that their activities are actually low as a result of intrinsically slow rotation. It would be an important task to clarify the behavior of their activity variations by long-term monitoring observations.
Baliunas, S. L., et al. 1995, ApJ, 438, 269 Baliunas, S., & Jastrow, R. 1990, Nature, 348, 520 Cox, A. N. 2000, Allen’s Astrophysical Quantities, 4th ed. (Berlin: Springer) Duncan, D. K., et al. 1991, ApJS, 76, 383 Eddy, J. A. 1976, Science, 192, 1189 Hall, J. C., & Lockwood, G. W. 2004, ApJ, 614, 942 Hall, J. C., Lockwood, G. W., & Skiff, B. A. 2007, AJ, 133, 862 Isaacson, H., & Fischer, D. 2010, ApJ, 725, 875 Kurucz, R. L. 1993, Kurucz CD-ROM, No. 13 (Harvard-Smithsonian Center for Astrophysics) \[also available at `http://kurucz.harvard.edu/PROGRAMS.html`\] Kurucz, R. L., & Bell, B. 1995, Kurucz CD-ROM, No. 23 (Harvard-Smithsonian Center for Astrophysics) \[also available at `http://kurucz.harvard.edu/LINELISTS.html`\] Noguchi, K., et al. 2002, PASJ, 54, 855 Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, A. H. 1984, ApJ, 279, 763 Strassmeier, K., Washuettl, A., Granzer, Th., Scheck, M., & Weber, M. 2000, A&AS, 142, 275 Takeda, Y., Honda, S., Kawanomoto, S., Ando, H., & Sakurai, T. 2010, A&A, 515, A93 (Paper II) Takeda, Y., Kawanomoto, S., Honda, S., Ando, H., & Sakurai, T. 2007, A&A, 468, 663 (Paper I) Takeda, Y., Tajitsu, A., Honda, S., Kawanomoto, S., Ando, H., & Sakurai, T. 2011, PASJ, 63, 697 (Paper III) Vaughan, A. H., & Preston, G. W. 1980, PASP, 92, 385 Vaughan, A. H., Preston, G. W., Wilson, O. C. 1978, PASP, 90, 267 Wright, J. T. 2004, AJ, 128, 1273 Wright, J. T., Marcy, G. W., Butler, R. P., & Vogt, S. S. 2004, ApJS, 152, 261
(160mm,240mm)[figure2.eps]{}
HIP $\log R'_{\rm Kp}$ $r_{0}$(8542) $v_{\rm e}\sin i$ $A$(Li) $A$(Be) $\log age$ $T_{\rm eff}$ $\log g$ $v_{\rm t}$ \[Fe/H\] Remark
----------- -------------------- --------------- ------------------- ---------- ----------- ------------ --------------- ---------- ------------- ---------- ----------------------------------------
001499 $-$5.32 0.190 2.01 ($<$1.1) 1.42 9.59 5724 4.45 0.95 $+0.20$
001598 $-$5.12 0.222 1.94 1.81 1.10 9.98 5693 4.33 0.96 $-0.27$
001803 $-$4.32 0.389 5.55 2.65 1.42 8.88 5817 4.41 1.17 $+0.24$
004290 $-$4.50 0.320 2.34 2.18 1.14 9.44 5719 4.40 1.10 $-0.12$
005176 $-$5.28 0.195 2.52 1.89 1.37 9.33 5855 4.39 1.03 $+0.19$
006405 $-$5.33 0.213 2.07 1.71 1.24 9.55 5728 4.38 0.96 $-0.14$
006455 $-$4.81 0.227 1.41 1.74 1.27 9.43 5716 4.57 0.99 $-0.09$
007244 $-$4.39 0.351 2.31 2.29 1.23 9.56 5755 4.52 1.12 $-0.04$
007585 $-$5.06 0.209 2.34 1.84 1.35 9.68 5784 4.50 1.04 $+0.07$
007902 $-$5.31 0.182 2.13 ($<$0.9) 1.28 9.39 5613 4.39 0.91 $-0.01$
007918 $-$5.42 0.191 2.57 1.89 1.25 9.56 5841 4.30 1.12 $+0.01$
008486 $-$4.34 0.311 2.41 2.55 1.33 9.69 5805 4.45 1.13 $-0.06$
009172 $-$4.25 0.282 4.29 2.50 1.37 9.47 5763 4.56 1.12 $+0.06$
009349 $-$4.54 0.245 2.26 2.06 1.14 9.73 5788 4.35 1.07 $+0.01$
009519 $-$4.27 0.465 6.94 2.97 1.50 9.29 5853 4.45 1.22 $+0.14$
009829 $-$5.31 0.211 1.77 ($<$0.9) 1.03 10.03 5579 4.25 0.94 $-0.31$
010321 $-$4.34 0.395 3.27 2.50 1.35 9.41 5707 4.60 1.04 $-0.01$
011728 $-$5.04 0.205 2.11 ($<$1.0) 1.27 9.48 5708 4.40 1.02 $+0.02$
012067 $-$4.71 0.240 1.97 ($<$1.3) 1.44 9.42 5709 4.41 0.96 $+0.20$
014614 $-$5.21 0.220 2.49 1.58 1.20 9.58 5726 4.26 1.00 $-0.12$
014623 $-$4.36 0.389 3.31 2.00 1.35 9.49 5742 4.52 1.09 $+0.12$
015062 $-$4.96 0.222 2.32 2.02 1.10 9.47 5735 4.49 0.94 $-0.29$
015442 $-$4.89 0.233 1.80 1.67 1.18 9.70 5682 4.50 0.87 $-0.19$
016405 $-$5.32 0.184 2.65 ($<$1.2) 1.38 9.48 5738 4.32 1.03 $+0.26$
017336 $-$4.87 0.164 2.07 ($<$0.8) ($<-0.9$) 9.59 5671 4.55 0.94 $-0.13$ Be depleted
018261 $-$4.94 0.242 2.71 2.27 1.23 9.40 5873 4.43 0.97 $+0.02$
019793 $-$4.32 0.385 5.17 2.53 1.35 9.23 5828 4.51 1.26 $+0.19$
019911 $-$4.54 0.351 3.73 2.26 1.33 9.76 5672 4.34 1.10 $-0.13$
019925 $-$4.75 0.262 2.39 1.61 1.34 9.47 5767 4.53 0.99 $+0.07$
020441 $-$4.42 0.376 2.84 2.28 1.16 9.41 5771 4.42 1.10 $+0.13$
020719 $-$4.30 0.462 5.40 2.60 1.29 9.27 5831 4.36 1.24 $+0.13$
020741 $-$4.35 0.391 3.33 2.41 1.25 9.05 5797 4.37 1.20 $+0.16$
020752 $-$4.38 0.375 4.63 2.72 1.40 9.28 5923 4.46 1.13 $+0.16$
021165 $-$5.20 0.220 2.53 1.61 1.24 9.66 5760 4.28 0.99 $-0.16$
021172 $-$5.23 0.196 1.85 ($<$1.0) 1.11 9.63 5625 4.27 0.90 $-0.10$
022203 $-$4.36 0.340 3.92 2.30 1.27 9.55 5740 4.33 1.07 $+0.13$
023530 $-$5.21 0.211 1.24 ($<$0.8) 0.84 9.94 5601 4.36 0.91 $-0.24$
025002 $-$4.32 0.369 3.40 2.38 1.31 9.56 5729 4.47 1.07 $-0.08$
025414 $-$5.13 0.220 1.85 ($<$1.1) 1.25 9.37 5635 4.49 0.89 $+0.10$
025670 $-$5.19 0.218 2.05 ($<$1.2) 1.34 9.52 5759 4.55 0.88 $+0.10$
026381 $-$4.98 0.240 0.96 ($<$0.7) 1.22 10.26 5518 4.47 0.87 $-0.45$ PHS, outlier in \[Fe/H\]/$T_{\rm eff}$
027435 $-$5.27 0.231 1.92 1.53 1.13 9.87 5697 4.45 0.93 $-0.22$
029432 $-$5.27 0.200 2.14 1.06 1.12 9.67 5712 4.32 1.00 $-0.12$
031965 $-$5.39 0.167 2.11 1.00 1.33 9.87 5770 4.31 0.99 $+0.05$
032673 $-$5.02 0.176 1.83 ($<$1.0) ($<-0.8$) 9.50 5724 4.57 0.95 $+0.06$ Be depleted
033932 $-$4.67 0.373 3.50 2.48 1.16 9.36 5891 4.38 1.10 $-0.12$
035185 $-$4.33 0.444 4.81 2.71 1.29 9.76 5793 4.19 1.35 $ 0.00$
035265 $-$4.92 0.191 1.77 2.01 1.15 9.79 5804 4.37 1.04 $-0.02$
036512 $-$5.31 0.218 1.42 1.25 1.24 9.52 5718 4.49 0.89 $-0.09$
038647 $-$4.47 0.315 2.55 2.13 1.30 9.43 5714 4.43 0.95 $+0.01$
038747 $-$4.34 0.424 5.53 2.75 1.38 9.42 5804 4.42 1.05 $+0.07$
038853 $-$5.31 0.222 2.53 2.44 1.16 9.60 5899 4.27 1.03 $-0.05$
039506 $-$5.44 0.227 2.14 ($<$0.9) 1.06 10.36 5600 4.24 0.83 $-0.62$ outlier in \[Fe/H\]/$T_{\rm eff}$
039822 $-$5.19 0.231 1.97 ($<$1.2) 1.13 9.63 5758 4.35 0.90 $-0.22$
040118 $-$5.12 0.202 1.36 ($<$0.9) 1.08 10.12 5541 4.45 0.84 $-0.42$ outlier in \[Fe/H\]/$T_{\rm eff}$
040133 $-$5.26 0.184 2.36 1.54 1.35 9.59 5698 4.33 0.97 $+0.12$
041184 $-$4.18 0.545 10.32 2.83 1.70 9.45 5705 4.43 1.51 $+0.11$
041484 $-$5.07 0.202 2.64 1.73 1.26 9.79 5864 4.33 0.92 $+0.05$
041526 $-$5.24 0.222 2.13 2.03 1.23 9.78 5801 4.27 0.98 $-0.02$
042333 $-$4.46 0.293 3.31 2.35 1.35 9.21 5816 4.44 1.08 $+0.14$
042575 $-$5.07 0.236 2.03 1.17 1.28 9.42 5675 4.40 0.96 $+0.06$
043297 $-$4.71 0.296 2.22 1.60 1.23 9.30 5691 4.46 1.05 $+0.08$
043557 $-$4.73 0.255 3.28 1.50 1.19 9.80 5805 4.42 1.05 $-0.06$
043726 $-$4.57 0.240 1.59 1.88 1.27 9.57 5769 4.48 1.01 $+0.11$
044324 $-$4.80 0.285 2.35 2.41 1.27 9.34 5888 4.46 1.09 $-0.01$
044997 $-$5.31 0.198 1.84 ($<$1.2) 1.32 9.49 5696 4.54 0.75 $+0.04$
045325 $-$4.79 0.311 3.09 2.35 1.28 9.24 5935 4.47 0.97 $+0.18$
046903 $-$4.89 0.238 2.91 2.02 1.22 9.53 5746 4.40 1.11 $-0.03$
049580 $-$4.89 0.215 2.48 1.98 1.26 9.58 5782 4.41 0.87 $+0.02$
049586 $-$4.98 0.222 2.21 ($<$1.3) 1.29 9.49 5786 4.42 1.06 $+0.20$
049728 $-$5.34 0.175 2.01 ($<$1.0) 1.30 9.60 5744 4.40 0.98 $-0.07$
049756 $-$5.29 0.185 2.14 1.34 1.21 9.58 5720 4.29 0.99 $+0.02$
050505 $-$5.12 0.218 1.49 ($<$0.9) 1.25 9.85 5590 4.44 0.84 $-0.17$
051178 $-$5.14 0.229 1.77 ($<$1.2) 0.80 9.66 5801 4.47 0.87 $-0.17$
053721 $-$5.43 0.193 2.69 1.75 1.23 9.89 5819 4.19 1.15 $-0.02$ PHS
054375 $-$4.38 0.353 4.06 2.44 1.31 9.36 5803 4.37 0.96 $+0.14$
055459 $-$5.34 0.218 2.39 1.58 1.37 9.58 5812 4.36 1.03 $+0.07$
055868 $-$4.75 0.256 1.92 2.14 1.19 9.55 5757 4.49 0.95 $-0.15$
059589 $-$5.13 0.205 1.81 ($<$1.2) 1.49 9.39 5654 4.52 0.70 $-0.01$
059610 $-$5.40 0.229 2.22 1.62 1.23 9.63 5829 4.34 1.04 $-0.06$ PHS
062175 $-$5.12 0.198 2.52 1.83 1.34 9.53 5683 4.19 0.90 $+0.13$
062816 $-$4.49 0.296 3.57 2.16 1.33 9.49 5804 4.44 0.97 $+0.06$
063048 $-$5.26 0.189 1.81 ($<$1.1) 1.34 9.50 5655 4.32 0.91 $-0.02$
063636 $-$4.45 0.351 2.93 2.26 1.23 9.62 5799 4.52 1.10 $-0.01$
064150 $-$5.38 0.187 2.21 ($<$1.0) ($<-0.9$) 9.63 5726 4.42 0.99 $+0.05$ Be depleted
064747 $-$5.40 0.204 1.77 ($<$1.1) 1.19 9.80 5710 4.42 0.93 $-0.18$
070319 $-$5.23 0.207 1.56 1.09 1.20 10.07 5678 4.42 0.96 $-0.33$
072604 $-$5.28 0.191 2.26 ($<$1.1) 1.31 10.03 5655 4.24 0.84 $-0.14$
075676 $-$5.19 0.200 2.18 ($<$1.1) ($<-1.0$) 9.62 5772 4.44 0.88 $-0.08$ Be depleted
076114 $-$5.29 0.195 1.76 0.91 1.22 9.59 5709 4.42 1.02 $-0.02$
077749 $-$4.27 0.416 4.77 2.70 1.56 9.22 5836 4.61 1.14 $+0.22$
078217 $-$5.22 0.213 1.17 1.89 1.15 9.54 5749 4.43 1.10 $-0.22$
079672 $-$5.12 0.193 2.34 1.63 1.30 9.66 5768 4.40 0.96 $+0.04$
085042 $-$5.33 0.187 2.01 ($<$0.8) 1.34 9.43 5676 4.48 0.99 $+0.03$
085810 $-$5.05 0.193 2.86 2.05 1.35 9.43 5856 4.46 1.08 $+0.15$
088194 $-$5.32 0.196 1.78 ($<$0.8) 1.25 9.67 5693 4.33 0.98 $-0.08$
088945 $-$4.20 0.558 6.51 ($<$1.1) $-$0.19 9.73 5800 4.38 1.44 $-0.01$ outlier in $A$(Li)
089282 $-$4.80 0.240 3.41 2.45 1.22 9.34 5833 4.22 1.00 $ 0.00$
089474 $-$5.30 0.189 2.50 ($<$0.9) 1.25 9.89 5755 4.20 1.04 $+0.01$
089912 $-$4.36 0.436 6.04 2.69 1.42 9.43 5846 4.38 1.24 $+0.04$
090004 $-$5.25 0.176 1.76 ($<$1.2) 1.33 9.83 5607 4.42 0.85 $-0.02$ PHS
091287 $-$5.07 0.185 2.05 1.74 1.34 9.21 5648 4.46 0.88 $-0.01$
096184 $-$5.34 0.165 2.03 1.65 1.40 9.25 5863 4.45 1.00 $+0.13$
096395 $-$5.02 0.218 2.30 2.18 1.25 9.76 5816 4.48 1.00 $-0.10$
096402 $-$5.32 0.182 1.99 ($<$1.1) 1.28 9.90 5661 4.20 1.00 $-0.03$
096901 $-$5.38 0.185 1.97 ($<$1.1) 1.37 9.77 5742 4.32 1.01 $+0.08$ PHS
096948 $-$5.21 0.175 1.85 1.52 1.30 9.62 5725 4.36 1.07 $+0.07$
097420 $-$4.72 0.209 2.57 2.22 1.41 9.65 5780 4.42 1.04 $+0.05$
098921 $-$4.38 0.329 3.66 2.47 1.39 9.13 5810 4.50 1.19 $+0.17$
100963 $-$4.97 0.200 2.39 1.72 1.24 9.71 5779 4.46 0.98 $ 0.00$
104075 $-$4.31 0.409 5.01 2.72 1.32 9.20 5881 4.37 1.08 $+0.05$
109110 $-$4.35 0.409 4.78 2.49 1.41 9.28 5835 4.51 1.11 $+0.07$
110205 $-$5.31 0.216 1.98 1.09 1.12 9.96 5708 4.28 1.08 $-0.23$
112504 $-$5.06 0.209 1.99 2.01 1.37 9.63 5741 4.34 1.00 $+0.01$
113579 $-$4.20 0.573 10.31 3.06 1.49 9.57 5759 4.21 1.44 $+0.05$
113989 $-$5.17 0.222 1.94 ($<$0.8) 0.99 10.13 5506 4.38 0.74 $-0.46$ outlier in \[Fe/H\]/$T_{\rm eff}$
115715 $-$5.25 0.215 2.34 1.72 1.22 10.08 5684 4.15 1.05 $-0.19$
116613 $-$4.32 0.349 2.96 2.12 1.22 8.95 5869 4.49 1.11 $+0.16$
Sun/Vesta $-$5.33 0.193 2.29 0.92 1.22 9.66 5761 4.44 1.00 $-0.01$
: Activity index, rotation, Li abundance, age, and the atmospheric parameters.
[^1]: Based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
[^2]: We here use the astrophysical flux ($F$) defined by $\pi F_{\lambda}^{\rm th}
\equiv 2\pi \int_{0}^{1} \mu I_{\lambda}^{\rm th}(0, \mu) d\mu$, with which the effective temperature $T_{\rm eff}$ is related as $\pi F_{\rm bol} = \pi \int_{0}^{\infty} F_{\lambda} d\lambda = \sigma T_{\rm eff}^{4}$ ($\sigma$: Stephan–Boltzmann constant).
[^3]: Since Strassmeier et al. (2000) treated H and K lines separately and presented each data of $F'_{\rm H}$ and $F'_{\rm K}$, we could convert their $R'_{\rm HK} [\equiv (F'_{\rm H} + F'_{\rm K})/F_{\rm bol}]$ (as clearly defined by them) into $R'_{\rm K} (\equiv F'_{\rm K}/F_{\rm bol})$ which is directly comparable with our $R'_{\rm Kp}$. Meanwhile, the $R'_{\rm HK}$ data derived by Wright et al.’s (2004) as well as Isaacson and Fischer (2010) appear to be essentially the [*average*]{} (not the sum) of $R'_{\rm H}$ and $R'_{\rm K}$ as judged by their extents. Therefore, we should be cautious about different definitions in the meaning of $R'_{\rm HK}$. This situation is manifestly displayed in figure 7 of Paper II, where we can see that $R'_{\rm HK}$(Strassmeier) is systematically larger than $R'_{\rm HK}$(Wright) by 0.3 dex. Therefore, we compared our $\log R'_{\rm Kp}$ with $R'_{\rm HK}$(Wright, Isaacson) without applying any correction, since the latter is practically equivalent to $R'_{\rm K}$ in any case.
[^4]: This ratio naturally depends on the reference solar activity, which may be subject to uncertainties due to cyclic variations as mentioned at the beginning of subsection 4.3. For example, if we assume a somewhat higher value of $-5.2$ for $\log R'_{\rm Kp,\odot}$ corresponding to the active phase of the Sun, this fraction becomes $\sim 30\%$.
[^5]: As a comparatively well studied case where a sufficient amount of data are available, we may presumably state that HIP 96901 = HD 186427 showed a Sun-like cyclic variation in the 1995–2007 period; cf. figure 7 of Hall et al. (2007).
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